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Robust a posteriori error estimation for discontinuous Galerkin methods for convection diffusion problems Zhu, Liang 2010

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Robust A Posteriori Error Estimation for Discontinuous Galerkin Methods for Convection Diffusion Problems by Liang Zhu  B.Sc., Tsinghua University, China, 2003 M.Sc., Tsinghua University, China, 2005  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Mathematics)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) April 2010 c Liang Zhu 2010  Abstract The present thesis is concerned with the development and practical implementation of robust a-posteriori error estimators for discontinuous Galerkin (DG) methods for convection-diffusion problems. It is well-known that solutions to convection-diffusion problems may have boundary and internal layers of small width where their gradients change rapidly. A powerful approach to numerically resolve these layers is based on using hp-adaptive finite element methods, which control and minimize the discretization errors by locally adapting the mesh sizes (h-refinement) and the approximation orders (p-refinement) to the features of the problems. In this work, we choose DG methods to realize adaptive algorithms. These methods yield stable and robust discretization schemes for convectiondominated problems, and are naturally suited to handle local variations in the mesh sizes and approximation degrees as required for hp-adaptivity. At the heart of adaptive finite element methods are a-posteriori error estimators. They provide information on the errors on each element and indicate where local refinement/derefinement should be applied. An efficient error estimator should always yield an upper and lower bound of the discretization error in a suitable norm. For convection-diffusion problems, it is desirable that the estimator is also robust, meaning that the upper and lower bounds differ by a factor that is independent of the mesh Péclet number of the problem. We develop a new approach to obtain robust a-posteriori error estimates for convection-diffusion problems for h-version and hp-version DG methods. The main technical tools in our analysis are new hp-version approximation results of an averaging operator, which are derived for irregular hexahedral meshes in three dimensions, as well as for irregular anisotropic rectangular meshes in two dimensions. We present a series of numerical examples based on C++ implementations of our methods. The numerical results indicate that the error estimator is effective in locating and resolving interior and boundary layers. For the hp-adaptive algorithms, once the local mesh size is of the same order as the width of boundary or interior layers, both the energy error and the error esii  Abstract timator are observed to converge exponentially fast in the number of degrees of freedom.  iii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iv  Abstract  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . Dedication  xi  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii  Statement of Co-Authorship . . . . . . . . . . . . . . . . . . . . . xiii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Motivation and objectives . . . . . . . . . . . . . . . . . . . . 1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Discontinuous Galerkin methods . . . . . . . . . . . . 1.2.2 A-posteriori error estimates for convection-diffusion problems . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Model problem . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Setting for error estimation . . . . . . . . . . . . . . . 1.3.3 Implementation . . . . . . . . . . . . . . . . . . . . . 1.3.4 Robust a-posteriori error estimation . . . . . . . . . . 1.3.5 Averaging operator . . . . . . . . . . . . . . . . . . . 1.3.6 Diffusion problems in three dimensions . . . . . . . . 1.3.7 Anisotropically refined meshes . . . . . . . . . . . . . 1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . .  1 1 5 5 6 7 7 7 8 9 10 13 14 15 17  2 An h-version a-posteriori error estimator . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Interior penalty discretization . . . . . . . . . . . . . . . . .  24 24 26  iv  Table of Contents  2.3  2.4  2.5  2.6 2.7  2.2.1 Model problem . . . . . . . . . . . . . . 2.2.2 Discretization . . . . . . . . . . . . . . Robust a-posteriori error estimation . . . . . . 2.3.1 Norms . . . . . . . . . . . . . . . . . . 2.3.2 A robust a-posteriori error estimator . 2.3.3 Reliability and efficiency . . . . . . . . 2.3.4 A robust estimator for reaction-diffusion Proofs . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Auxiliary forms and their properties . . 2.4.2 Approximation operators . . . . . . . . 2.4.3 Proof of Theorem 2.3.2 . . . . . . . . . 2.4.4 Proof of Theorem 2.3.3 . . . . . . . . . Numerical experiments . . . . . . . . . . . . . 2.5.1 Example 1 . . . . . . . . . . . . . . . . 2.5.2 Example 2 . . . . . . . . . . . . . . . . 2.5.3 Example 3 . . . . . . . . . . . . . . . . 2.5.4 Example 4 . . . . . . . . . . . . . . . . 2.5.5 Example 5 . . . . . . . . . . . . . . . . 2.5.6 Example 6 . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . .  3 An hp-version a-posteriori error estimator 3.1 Introduction . . . . . . . . . . . . . . . . . 3.2 Interior penalty discretization . . . . . . . 3.2.1 Model problem . . . . . . . . . . . . 3.2.2 Discretization . . . . . . . . . . . . 3.3 Robust a-posteriori error estimates . . . . . 3.3.1 Norms . . . . . . . . . . . . . . . . 3.3.2 A robust a-posteriori error estimate 3.4 Proofs . . . . . . . . . . . . . . . . . . . . . 3.4.1 Stability and auxiliary forms . . . . 3.4.2 Auxiliary meshes . . . . . . . . . . 3.4.3 Averaging operator . . . . . . . . . 3.4.4 Proof of Theorem 3.3.1 . . . . . . . 3.4.5 Proof of Theorem 3.3.3 . . . . . . . 3.5 Proof of Theorem 3.4.4 . . . . . . . . . . . 3.5.1 Polynomial basis functions . . . . . 3.5.2 Extension operators . . . . . . . . . 3.5.3 Decomposition of functions in Sp (T )  . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . .  26 27 28 29 30 31 32 34 34 37 37 41 46 46 47 49 50 52 53 56 59  . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . .  62 62 64 64 65 66 66 67 69 69 70 72 72 77 81 81 83 84  . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . .  v  Table of Contents  3.6  3.7 3.8  3.5.4 Proof of Theorem 3.4.4 3.5.5 Proof of Lemma 3.5.3 . 3.5.6 Proof of Lemma 3.5.4 . Numerical experiments . . . . 3.6.1 Example 1 . . . . . . . 3.6.2 Example 2 . . . . . . . 3.6.3 Example 3 . . . . . . . Conclusions . . . . . . . . . . . Bibliography . . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  4 Diffusion problems in three dimensions . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . 4.2 Discontinuous Galerkin discretization of a diffusion 4.2.1 Meshes and traces . . . . . . . . . . . . . . 4.2.2 Finite element spaces . . . . . . . . . . . . 4.2.3 Interior penalty discretization . . . . . . . 4.3 Energy norm a-posteriori error estimates . . . . . 4.3.1 Energy norm and residuals . . . . . . . . . 4.3.2 Reliability . . . . . . . . . . . . . . . . . . 4.3.3 Efficiency . . . . . . . . . . . . . . . . . . . 4.4 Proof of Theorem 4.3.1 . . . . . . . . . . . . . . . 4.4.1 Edges and nodes . . . . . . . . . . . . . . . 4.4.2 Auxiliary meshes . . . . . . . . . . . . . . 4.4.3 Averaging operator . . . . . . . . . . . . . 4.4.4 Proof of Theorem 4.3.1 . . . . . . . . . . . 4.5 Proof of Theorem 4.3.3 . . . . . . . . . . . . . . . 4.5.1 Inverse estimates . . . . . . . . . . . . . . 4.5.2 Polynomial extension over faces . . . . . . 4.5.3 Proof of Theorem 4.3.3 . . . . . . . . . . . 4.6 Proof of Theorem 4.4.1 . . . . . . . . . . . . . . . 4.6.1 Polynomial basis functions . . . . . . . . . 4.6.2 Edge extension operators . . . . . . . . . . 4.6.3 Face extension operators . . . . . . . . . . 4.6.4 Decomposition of functions in Sp (T ) . . . 4.6.5 Interior part . . . . . . . . . . . . . . . . . 4.6.6 Proof of Theorem 4.4.1 . . . . . . . . . . . 4.6.7 Proof of Proposition 4.6.4 . . . . . . . . . . 4.7 Numerical experiments . . . . . . . . . . . . . . . 4.7.1 Example 1 . . . . . . . . . . . . . . . . . . 4.7.2 Example 2 . . . . . . . . . . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . 86 . 88 . 92 . 93 . 94 . 96 . 99 . 100 . 102  . . . . . . . . . . problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . 106 . 106 108 . 108 . 109 . 110 . 111 . 111 . 112 . 113 . 114 . 114 . 114 . 116 . 116 . 121 . 121 . 122 . 124 . 127 . 128 . 131 . 132 . 134 . 138 . 138 . 140 . 148 . 149 . 151 vi  Table of Contents 4.8 4.9  Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . 154  5 Anisotropic meshes . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . 5.2 Interior penalty discretization . . . 5.2.1 Model problem . . . . . . . . 5.2.2 Discretization . . . . . . . . 5.2.3 Mesh sizes . . . . . . . . . . 5.2.4 Polynomial degrees . . . . . 5.2.5 Bilinear form . . . . . . . . . 5.3 A-posteriori error estimates . . . . . 5.3.1 Norms . . . . . . . . . . . . 5.3.2 An a-posteriori error estimate 5.4 Proofs . . . . . . . . . . . . . . . . . 5.4.1 Stability and auxiliary forms 5.4.2 Auxiliary meshes . . . . . . 5.4.3 Averaging operator . . . . . 5.4.4 Proof of Theorem 5.3.2 . . . 5.4.5 Proof of Theorem 5.3.4 . . . 5.5 Proof of Theorem 5.4.4 . . . . . . . 5.5.1 Polynomial basis functions . 5.5.2 Extension operators . . . . . 5.5.3 Decomposition of functions in 5.5.4 Proof of Theorem 5.4.4 . . . 5.5.5 Proof of Lemma 5.5.3 . . . . 5.5.6 Proof of Lemma 5.5.4 . . . . 5.6 Numerical experiments . . . . . . . 5.6.1 Example 1 . . . . . . . . . . 5.6.2 Example 2 . . . . . . . . . . 5.6.3 Example 3 . . . . . . . . . . 5.7 Conclusions . . . . . . . . . . . . . . 5.8 Bibliography . . . . . . . . . . . . . 6 Conclusions and future 6.1 Conclusions . . . . . 6.2 Future work . . . . 6.3 Bibliography . . . .  work . . . . . . . . . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sp (T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . .  . . . .  . . . .  . . . .  . . . .  . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  158 158 159 160 160 161 162 163 164 164 165 167 167 168 170 170 175 180 180 182 184 186 188 192 193 194 196 200 202 204  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  207 207 209 211  vii  List of Figures 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8  The piecewise linear solutions for the convection-diffusion equation (1.1.1) on a uniform mesh. . . . . . . . . . . . . . . . . . The piecewise linear DG approximation for the convectiondiffusion equation (1.1.1) with ε = 10−4 , h = 0.01 and Pe = 100. Adaptively generated h-version meshes after 7 refinement steps. Adaptively generated hp-version meshes after 7 refinement steps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of h-adaptive and hp-adaptive DG methods. . . . The construction of the auxiliary mesh Te from T . . . . . . . Adaptively generated hp-version mesh after 7 refinement steps. Anisotropically generated hp-meshes after 7 refinement steps for ε = 2 · 10−4 . . . . . . . . . . . . . . . . . . . . . . . . . . . Convergence behavior for ε = 2 · 10−4 . . . . . . . . . . . . . . Example 1: Convergence behavior for ε = 1, 10−2 , 10−4 and p = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 1: Convergence behavior for ε = 1, 10−2 , 10−4 and p = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 1: Adaptively generated meshes after 7 refinement steps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 2: Convergence behavior for ε = 10−2 , 10−3 and p = 1, 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 2: Adaptively generated meshes after 7 refinement steps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 3: Convergence behavior for ε = 10−2 , 10−4 and p = 1, 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 3: Adaptively generated meshes after 7 refinement steps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 4: Convergence behavior with ε = 10−2 , 10−4 and p = 1 (left), 2 (right). . . . . . . . . . . . . . . . . . . . . . .  2 3 9 11 12 12 13 15 15 48 49 50 51 52 53 54 54  viii  List of Figures 2.9 2.10 2.11 2.12 2.13  Example 4: The adaptively generated meshes after 7 refinement steps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 5: Convergence behavior for ε = 10−2 , 10−3 and p = 1 (left), 2 (right). . . . . . . . . . . . . . . . . . . . . . . Example 5: Adaptively generated meshes after 7 refinement steps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 6: Convergence behavior for ε = 10−2 , 10−4 and p = 1, 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 6: Adaptively generated meshes after 7 refinement steps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  55 55 56 57 58  The construction of the auxiliary mesh Te from T . . . . . . . 70 Reference element with variable edge polynomial degrees: p = 5, pEb1 = 2, pEb2 = 3, pEb3 = 4, pEb4 = 1. . . . . . . . . . . . . . 82 3.3 Left: Partition of K into K1 and K2 . Right: Element K and e ∈ R(K). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 K 3.4 Example 1: Convergence behavior for ε = 10−3 . . . . . . . . . 96 3.5 Example 1: Convergence behavior for ε = 2 · 10−4 . . . . . . . 97 3.6 Example 1: Adaptively generated meshes after 7 refinement steps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.7 Example 2: Convergence behavior for ε = 10−3 . . . . . . . . . 98 3.8 Example 2: Convergence behavior for ε = 5 · 10−6 . . . . . . . 99 3.9 Example 2: Adaptively generated meshes after 9 and 15 refinement steps. . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.10 Example 3: Convergence behavior for ε = 10−3 . . . . . . . . . 101 3.11 Example 3: Adaptively generated meshes after 7 refinement steps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101  3.1 3.2  4.1 4.2 4.3 4.4 4.5 4.6  b with the numbering of faces, edges and Reference element K vertices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Case 2: The elemental edge E ∈ E(K) has a hanging node located in its midpoint. . . . . . . . . . . . . . . . . . . . . . 132 Case 3: The mesh edges Ei belong to EF (K) for the elemental face F . The element K is then divided into four elements. . . 132 Case 2: Partition of K associated with the partition of face F . 133 Case 3: Partition of K associated with the partition of face F . 133 e ∈ R(K). . . . . . 135 The element K is refined into 8 elements K  ix  List of Figures 4.7  Example 1. (a) Comparison of the actual and estimated energy norm of the error with respect to the (third root of the) number of degrees of freedom with hp-adaptive mesh refinement; (b) Effectivity indices; (c) Comparison of the actual error with h- and hp-adaptive mesh refinement. . . . . . . . . 4.8 Example 1. Finite element mesh after 8 adaptive refinements, with 440 elements and 100578 degrees of freedom: (a) hpmesh; (b) Three-slice of the hp-mesh. . . . . . . . . . . . . . . 4.9 Example 2. (a) Comparison of the actual and estimated energy norm of the error with respect to the (fourth root of the) number of degrees of freedom with hp-adaptive mesh refinement; (b) Effectivity indices; (c) Comparison of the actual error with h- and hp-adaptive mesh refinement. . . . . . . . . 4.10 Example 2. Finite element mesh after 7 adaptive refinements, with 686 elements and 197670 degrees of freedom: (a) hpmesh; (b) Three-slice of the hp-mesh. . . . . . . . . . . . . . . 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17  Mapping of the element K. . . . . . . . . . . . . . . . . . . . The construction of the auxiliary mesh Te from T . . . . . . . Reference element with variable edge polynomial degrees: pKb = (5, 4), pEb1 = 2, pEb2 = 3, pEb3 = 4, pEb4 = 1. . . . . . . . . Partition of K into K1 , · · · , KN . . . . . . . . . . . . . . . . . e ∈ R(K) at the corner of K. . . . . . . . . . . . . . . . . . K e K ∈ R(K) locates inside K. . . . . . . . . . . . . . . . . . . . e ∈ R(K) shares an edge of K. . . . . . . . . . . . . . . . . . K Example 1: Convergence behavior for ε = 10−3 . . . . . . . . . Example 1: Convergence behavior for ε = 2 · 10−4 . . . . . . . Example 1: Adaptively generated meshes after 9 and 15 refinement steps for ε = 10−3 . . . . . . . . . . . . . . . . . . . . Example 1: Adaptively generated meshes after 9 and 15 refinement steps for ε = 2 · 10−4 . . . . . . . . . . . . . . . . . . Example 2: Convergence behavior for ε = 10−3 . . . . . . . . . Example 2: Convergence behavior for ε = 5 · 10−6 . . . . . . . Example 2: Adaptively generated meshes after 9 and 15 refinement steps for ε = 10−3 . . . . . . . . . . . . . . . . . . . . Example 2: Adaptively generated meshes after 9 and 15 refinement steps for ε = 5 · 10−6 . . . . . . . . . . . . . . . . . . Example 3: Convergence behavior for ε = 10−3 . . . . . . . . . Example 3: Adaptively generated meshes after 9 and 15 refinement steps for ε = 10−3 . . . . . . . . . . . . . . . . . . . .  149  150  151  153 160 169 181 183 185 185 185 195 196 197 198 199 199 200 201 202 203 x  Acknowledgements First and foremost I express my sincerest gratitude to my supervisor, Professor Dominik Schötzau, who has supported me thoughout my thesis with his patience and knowledge whilst allowing me the room to work in my own way. This thesis would not have been completed or written without him. One simply could not wish for a better or friendlier supervisor. I would like to sincerely thank Professor Paul Houston (University of Nottingham) for helpful discussions on the local smoothness estimation strategy in hp-adaptive algorithms. I would like also to show my gratitude to my supervisory committee members, Professor James Feng, Professor Chen Greif, Professor Michael Ward and Professor Brian Wetton, for their helpful comments and advices.  xi  Dedication To my father, Baoping Zhu, my mother, Yueying Zhang and my wife, Kelan Zhai, my light on earth.  xii  Statement of Co-Authorship Chapter 4 of this thesis is a joint work with Dr. Stefano Giani, Professor Paul Houston (University of Nottingham) and Professor Dominik Schötzau. The numerical tests in Section 4.7 have been implemented and carried out by Dr. Stefano Giani and Professor Paul Houston. My contribution in this chapter has been the construction and analysis of the hp-version averaging operator on three-dimensional meshes, the theoretical derivation of the a-posteriori error estimator in Theorems 4.3.1 and 4.3.3 for diffusion problems and the proof of its reliability and efficiency. The first draft of this chapter but Section 4.7 was written by me and revised by Professor Dominik Schötzau. The other chapters of this thesis (Chapters 2, 3 and 5) are a joint work with Professor Dominik Schötzau. In these three chapters, I have derived a-posteriori error estimators for h- and hp-adaptive DG methods for convection-diffusion problems and proved three key properties of these estimators: reliability, efficiency and robustness. In Chapters 3 and 5, I have constructed and analyzed the hp-version averaging operator on twodimensional isotropically and anisotropically refined meshes, respectively. The numerical examples in these three chapters were all implemented by myself. I wrote the first draft of each chapter and Professor Dominik Schötzau revised them.  xiii  Chapter 1  Introduction 1.1  Motivation and objectives  Convection-dominated flow problems play an important role in a wide range of applications, such as gas and fluid dynamics, meteorology, transport of contaminants in porous media, electro-magnetism, and many more [14, 48]. Devising robust, accurate and efficient methods for the numerical approximation of such problems is of significant importance in science and engineering. Over the last few decades, finite element methods (FEMs) have emerged as one of the methods of choice for several classes of partial differential equations (PDEs). However, the design of stable finite element methods for convection-dominated problems has been a longstanding problem. Indeed, it is well-known that standard FEMs break down for highly convectiondominated problems; see e.g., [52, 54, 56]. To illustrate this, we consider the simple one-dimensional convection-diffusion problem: −εu00 (x) + u0 (x) = 1, u(0) = u(1) = 0,  in (0, 1),  (1.1.1)  with 0 < ε  1 a small diffusion parameter. The analytical solution for this equation is smooth, but has a boundary layer at x = 1 of width O(ε). The critical parameter that determines the behavior of the finite element method for this problem is the ratio of the mesh size h and the diffusion parameter ε, Pe = h/ε, which is called the mesh Péclet number. The stability of standard FEMs deteriorates as the mesh Péclet number increases. If Pe is too large, oscillations occur in the approximations. In Figure 1.1(a), 1.1(b) and 1.1(c), we show the piecewise linear finite element approximations for problem (1.1.1) on a uniform mesh of size h = 0.01 for ε = 1, ε = 10−4 and ε = 10−6 , respectively. For ε = 1, the diffusive term is dominating, and the finite element discretization approximates the solution u(x) of (1.1.1) accurately; see 1  1.1. Motivation and objectives Figure 1.1(a). As the parameter ε becomes smaller, however, the numerical approximation becomes more and more oscillatory and eventually is useless; see Figure 1.1(b) and 1.1(c). This is because at the boundary layer near x = 1, the mesh size is not sufficiently small. Indeed, if ε = 10−6 and the mesh size is chosen to be h = 10−6 (Pe = 1), the finite element method approximates the convection-dominated problem (1.1.1) accurately again; see Figure 1.1(d). However, solving with mesh size of order O(ε) is not very feasible in practice, especially in higher dimensions. piecewise linear FEM, uniform mesh with h= 0.01  piecewise linear FEM, uniform mesh with h= 0.01  0.14  2 FEM solution  FEM solution 1.8  0.12 1.6 0.1  1.4 1.2 h  u  uh  0.08  0.06  1 0.8 0.6  0.04  0.4 0.02 0.2 0  0  0.1  0.2  0.3  0.4  0.5 x  0.6  0.7  0.8  0.9  0  1  (a) ε = 1, h = 10−2 , Pe = 10−2  0  0.1  0.2  0.3  0.4  0.5 x  0.6  0.7  0.8  0.9  1  (b) ε = 10−4 , h = 10−2 , Pe = 102 piecewise linear FEM, uniform mesh with h= 10−6  piecewise linear FEM, uniform mesh with h= 0.01 60  1 FEM solution  FEM solution 0.9  50 0.8 0.7  40  h  u  uh  0.6 30  0.5 0.4  20  0.3 0.2  10  0.1 0  0  0.1  0.2  0.3  0.4  0.5 x  0.6  0.7  0.8  0.9  (c) ε = 10−6 , h = 10−2 , Pe = 104  1  0  0  0.1  0.2  0.3  0.4  0.5 x  0.6  0.7  0.8  0.9  1  (d) ε = 10−6 , h = 10−6 , Pe = 1  Figure 1.1: The piecewise linear solutions for the convection-diffusion equation (1.1.1) on a uniform mesh. Various approaches to improve the performance of the standard finite element methods for convection-dominated problems have been proposed in the literature. Among these approaches are the streamline diffusion finite element methods (SDFEMs) [42, 43] and the related Streamline-Upwind 2  1.1. Motivation and objectives Petrov-Galerkin (SUPG) methods [28, 41], both introduced in the eighties. The main idea in the SDFEM is to introduce suitable stabilization terms that are obtained by using test functions that are upwinded in stream directions. However, these stabilization terms typically involve second-order derivatives of shape functions which are costly to evaluate, particularly for non-affinely mapped elements or for elements with curved boundaries. They also involve stabilization parameters whose proper choice is a delicate issue. It has been reported in many instances that these parameters can dramatically influence the accuracy of the discrete solutions. This sensitivity is particularly pronounced for the hp-version of the streamline diffusion finite element method [31]. The drawbacks of standard FEMs can also be overcome by discontinuous Galerkin (DG) methods. These methods were introduced in the seventies as non-standard discretization techniques for linear transport problems [49, 55]. They are based on discontinuous finite element spaces and make use of upwind techniques. As a result, DG methods are stable and robust for convection-dominated problems. This is illustrated in Figure 1.2(a) for the piecewise linear DG approximation of the one-dimensional convectiondiffusion problem (1.1.1) with ε = 10−4 and h = 0.01. Clearly, the DG approximation does not show spurious oscillations in contrast to Figure 1.1(b), except in the last two elements near x = 1. piecewise linear DGM, uniform mesh with h=0.01  piecewise linear DGM, uniform mesh with h=0.01  analytical solution DG solution  1  1.05  0.8  1  uh  uh  0.6  0.4  0.95  0.9  0.2  0  analytical solution DG solution  0.85  0  0.1  0.2  0.3  0.4  0.5 x  0.6  0.7  0.8  (a) DG approximation  0.9  1  0.8  0.85  0.9 x  0.95  1  (b) DG approximation near x = 1  Figure 1.2: The piecewise linear DG approximation for the convectiondiffusion equation (1.1.1) with ε = 10−4 , h = 0.01 and Pe = 100. A stable numerical method, however, is generally not enough to approximate convection-dominated problems effectively. This is because the solutions to such problems may have layers of small width where their gradients change rapidly. For example, in Figure 1.2(b), the DG discretization 3  1.1. Motivation and objectives is not accurate near x = 1 because the boundary layer is not sufficiently resolved with h = 0.01. A natural approach to overcome these difficulties is to use meshes that are locally refined in the vicinity of boundary layers. For example, this is achieved by the so-called Shishkin meshes [51, 52, 56, 68]. Roughly speaking, a Shishkin mesh is a piecewise structured mesh with anisotropic elements of high aspect ratio in the boundary layer region, with a judiciously chosen transition point. To use it effectively, an a-priori knowledge about the location and nature of the layers is necessary, which can be obtained in certain cases by employing techniques of asymptotic analysis. In this thesis, we are interested in using adaptive finite element methods to resolve layers and other solution singularities. These methods are designed to control and minimize the discretization errors by locally adapting the mesh sizes and polynomial degrees according to the features of the analytical solution. The main focus of our work is on hp-adaptive discontinuous Galerkin methods. In hp-adaptive algorithms, a combination of hand p-refinement is employed. The advantage of hp-methods over h-version methods (where a fixed polynomial degree is used) lies in the fact that exponential rates of convergence are obtained for problems with boundary layers and corner singularities; see the a-priori results in [9, 10, 33, 34, 62, 64, 63] and the references therein. At the heart of adaptive finite element methods are a-posteriori error estimators [3, 24, 25, 65]. They provide local estimates of the errors on each element, measured in a suitable norm, and indicate where elemental refinement/derefinement should be applied. Two standard properties are desirable for an a-posteriori error estimator, reliability and efficiency, guaranteeing that the error estimator tends to zero at the same asymptotic rate as the true error. For convection-diffusion problems, another desirable property is that of robustness. Robustness means that the upper and lower bounds are independent of the magnitudes of the diffusion and the convection (i.e., the Péclet number of the flow problem). Discontinuous Galerkin methods are naturally suited for use in adaptive finite element realizations. They can easily handle adaptive strategies since local refinement or derefinement of the grids and the local variation of polynomial degrees can be achieved immediately without taking into account continuity restrictions. For recent surveys on DG methods, we refer the reader to [18, 19, 20]. The main objectives of this thesis are the development, the numerical analysis and the practical implementation of h-adaptive and hp-adaptive discontinuous Galerkin methods for convection-diffusion problems, with a particular emphasis on the question of robustness.  4  1.2. Background  1.2  Background  We continue this introduction with an overview on recent developments concerning discontinuous Galerkin methods and a-posteriori error estimation for convection-diffusion problems. We do not strive to discuss all the existing literature in these fields, but only provide the theoretical background for this work.  1.2.1  Discontinuous Galerkin methods  In the seventies, discontinuous Galerkin methods were introduced in [49, 55] for the numerical discretization of the linear neutron transport equation and in [8, 11, 53] for elliptic problems where boundary conditions and inter elemental continuity constraints were enforced through the use of penalty terms. The interest in these methods was almost negligible until the introduction of the so-called Runge-Kutta discontinuous Galerkin (RKDG) methods for non-linear hyperbolic conservation laws in the late eighties and beginning of nineties; see the review article in [20] and the references therein. RKDG methods combine classical explicit Runge-Kutta time discretizations with spatial DG discretizations, which incorporate the ideas of numerical fluxes originally developed for high-resolution finite difference and finite volume methods. The resulting RKDG methods are non-linearly stable, high-order accurate and highly parallelizable schemes that can easily handle complicated geometries and boundary conditions. The success of RKDG methods for purely hyperbolic problems prompted several authors to extend them to convection-diffusion and other flow problems. In the end of the nineties, DG methods found their way into the main stream of computational mathematics and are now successfully applied to problems which they were not originally designed for (such as incompressible flow problems and elasticity problems); see the survey articles [17, 19, 23]. We also refer the reader to [18] for a historical review. The increasing popularity of the DG methods is due to several interesting properties. First of all, as shown in Section 1.1, discontinuous Galerkin methods are stable, accurate and robust for convection-dominated problems. Secondly, discontinuous Galerkin methods can easily handle irregularly refined meshes and variable approximation degrees, since refinement or derefinement of the grids can be achieved without taking into account continuity restrictions. DG methods are thus an natural choice for the realizations of hp-adaptive algorithms.  5  1.2. Background  1.2.2  A-posteriori error estimates for convection-diffusion problems  A-posteriori error control has always been a central issue in the development of numerical methods, and an enormous amount of literature can be found on this topic. In the context of finite element methods, we refer the reader to the monographs [3, 65] and the references therein. Here, we focus on explicit residual-based error estimators for convection-diffusion problems, and on the question of robustness. A robust estimator gives rise to upper and lower bounds of the error which is independent of the Péclet number. The first advance in this direction was made in [66] where an estimator for a conforming SUPG method was derived for which the ratio of the upper and lower bounds scales with the square root of the Péclet number. Other estimators that are almost robust can be found in [45] for conforming finite element methods, and in [4, 26] for a non-conforming finite element method. In 2005, a fully robust error estimator has been proposed in [67]. There, in addition to the energy norm, the error measure now also includes a dual norm of the convective derivative. Another approach to robust error estimation can be found in [57, 58], whereby the error in the convective term is evaluated in an interpolation norm of order 1/2. Most recently, the robustness of a-posteriori error estimates for discontinuous Galerkin finite element methods has been studied in [27] and our work [61] (see also Chapter 2). All the results above are concerned with h-version finite element methods. These methods are based on employing a fixed, usually low polynomial degree. As a consequence, adaptive h-version methods yield at most algebraic rates of convergence. This is in contrast to hp-version finite element methods, where the combination of h-refinement and p-refinement typically results in exponential rates of convergence, see, e.g., [62] and the references therein. In hp-adaptive algorithm, once an element has been flagged for refinement, a decision must be made whether the local mesh size or the local polynomial degree should be adjusted accordingly. The choice to perform either h-refinement or p-refinement is usually based on estimating the local smoothness of the analytical solution; see [37, 39]. A-posteriori error estimation is well developed for hp-adaptive DG methods for elliptic problems, see e.g. [24, 25, 35, 36, 40, 50], as well as for hyperbolic problems, see e.g. [38] and the references therein. Other estimators for hyperbolic problems that are based on superconvergence properties can be found in [1, 2]. A duality-based approach to hp-version error estimation on anisotropically refined meshes is developed in [29, 30].  6  1.3. Main results  1.3  Main results  In this section, we discuss our main results as presented in the thesis.  1.3.1  Model problem  Throughout the thesis, we consider the following convection-diffusion model problem: −ε∆u + a(x) · ∇u = f (x) in Ω, (1.3.1) with Ω being a domain in R2 or R3 . Here, the function a(x) is the prescribed velocity of a flow field and ε > 0 the diffusion coefficient. This problem is the higher-dimensional version of (1.1.1). The second-order term in this equation describes the diffusive processes while the first-order term describes transport phenomenon. The problem (1.3.1) is a central model problem for computational fluid dynamics. For example, the differential operator in (1.3.1) is an integral part of the linearized Navier-Stokes equations. Hence, any effective numerical method for these equations must rely on efficient solvers for (1.3.1). In our analysis, we assume the diffusion to be small, 0 < ε  1, while the convection term and the dimensions of the domain are of order one. Hence, the Péclet number of the problem is of the order 1/ε. In this regime and as illustrated in Section 1.1, boundary layers and internal layers may arise in the solution to (1.3.1); see [52, 56]. Additionally, corner singularities may also appear [62, 63]. Therefore, developing efficient finite element methods has been a longstanding challenge and is of considerable importance in computational mathematics.  1.3.2  Setting for error estimation  In this thesis, we develop and numerically test an approach to derive robust error estimators for adaptive DG methods for problem (1.3.1). These estimators give computable bounds η in terms of the data f , a, ε and the DG finite element approximation uh obtained on a mesh T on Ω. Thus, they provide (local) information on the errors between the solution u of (1.3.1) and its approximation uh . Throughout, the error is measured in terms of the following natural norm: ku − uh k = ku − uh kE + |u − uh |O .  (1.3.2)  7  1.3. Main results Here, k · kE is the energy norm associated with the diffusive term and | · |O a semi-norm associated with the convective term. This measure was first introduced in [67], and is also adopted in this thesis. Roughly speaking, we shall derive a-posteriori error estimators η with the following standard properties [3, 65]: X 2 Locality : η2 = ηK , (1.3.3) K  Reliability :  ku − uh k ≤ CR η,  (1.3.4)  Efficiency :  ku − uh k ≥ CE η.  (1.3.5)  Locality (1.3.3) ensures that η can be written as a sum of elemental estimators ηK . This requirement allows us to identify elements with large local errors. Reliability (1.3.4) means that the estimator always overestimates the numerical error in the given norm, up to the constant CR , and efficiency (1.3.5) that it does not overestimate the numerical error by too much. The constants CR and CE are ideally independent of the discretization parameters and close to one. However, for the residual-based approach pursued here, the constant CR is observed to be around 10 to 15 in some cases, thereby overestimating the true error by a significant factor. This is an inherent limitation of residual-based a-posteriori error estimation that can be overcame only by using more sophisticated approaches; see e.g., [27]. For convection-diffusion problems, an additional property is highly desirable and the main focus of this research, namely that of robustness: Robustness : CR and CE are independent of the Péclet number. (1.3.6) Robustness implies that the estimator η does not degenerate as ε tends to zero, and ensures that it can be used for nearly hyperbolic problems. All the estimators we derive in this thesis satisfy the four properties (1.3.3)-(1.3.6).  1.3.3  Implementation  We present several numerical examples to illustrate the practical performance of our estimators in adaptive refinement strategies. The implementations of the discontinuous Galerkin methods of Chapters 2, 3 and 5 are all based on the Deal.II finite element library [12, 13], while the computations in Chapter 4 are performed using the AptoFEM software package [32]. The non-symmetric sparse linear systems of equations arising from discretizations are solved by employing the UMFPACK package [21, 22], while the 8  1.3. Main results symmetric sparse linear systems are solved by exploiting the MUltifrontal Massively Parallel Solver (MUMPS); see [5, 6, 7]. For the hp-adaptive DG discretizations in Chapters 3, 4 and 5, once an element has been flagged for refinement, a decision must be made whether the local mesh size or the local polynomial degree should be adjusted accordingly. The choice to perform either h-refinement or p-refinement is based on estimating the local smoothness of the analytical solution. Here, we employ the hp-adaptive strategy developed in [39], where the local regularity of the analytical solution is estimated from truncated local Legendre expansions of the computed numerical solution; see also [37].  1.3.4  Robust a-posteriori error estimation  In Chapter 2, we outline a new approach to obtain robust a-posteriori error estimators in the setting of Section 1.3.2. We carry out the technical details for the h-version interior penalty DG method and show that the constants CR and CE are independent of the local mesh size and the diffusion parameter ε. This is the first fully robust a-posteriori error estimate for adaptive DG methods and was published in [61]. Let us show one of our numerical tests from Chapter 2 to illustrate the efficiency of our estimator in locating and resolving boundary layers. We consider the domain Ω = (0, 1)2 and select the right-hand side f and an appropriate boundary condition such that the solution of problem (1.3.1) has boundary layers of width O(ε) along the lines x = 1 and y = 1. Figure 1.3 depicts the adaptive meshes obtained by our method with a piecewise linear approximation order after 7 refinement steps for ε = 10−3 and ε = 2 · 10−4 , respectively. We observe strong mesh refinement near the boundary layers, as expected.  (a) ε = 10−3  (b) ε = 2 · 10−4  Figure 1.3: Adaptively generated h-version meshes after 7 refinement steps.  9  1.3. Main results In Chapter 3, we then extend our approach proposed in Chapter 2 to the context of the hp-version of the DG method and present the first fully robust hp-adaptive estimator for convection-diffusion problems, see also [70]. We make explicit the estimates (1.3.4) and (1.3.5) not only with respect to the elemental mesh sizes, but also with respect to the local approximation degrees. The reliability constant CR in (1.3.4) is shown to be independent of the diffusion parameter ε, the local mesh sizes and the polynomial degrees. The efficiency constant CE in (1.3.5) is proved to be independent of ε and the local mesh sizes, but depends weakly on the polynomial degrees. This suboptimality is a notorious difficulty for hp-methods and also appears in conforming methods; see [50]. (In [15], it has recently been shown that flux equilibrated error estimators are p-robust.) This loss is less dramatic for hp-version methods where exponential rates of convergence are achieved, which is the main interest of our work. We present a series of numerical examples where we use our error estimator as an error indicator in an hpadaptive refinement strategy. This refinement algorithm is based on local smoothness estimation of the analytic solution as developed in [39]. Once the local mesh size is of the same order as the width of the boundary or interior layers, both the energy error and the error indicator are observed to converge exponentially fast with respect to the number of degrees of freedom. For instance, here we compute the same example as before with hp-adaptive DG methods. Figure 1.4 shows the hp-adaptive meshes and polynomial degree distributions after 7 refinement steps for ε = 10−3 and ε = 2·10−4 . We observe that after initial strong h-refinement, p-refinement starts to dominate along the boundary layers. Away from the layers, the solution is almost linear and is approximated efficiently with low-order polynomials. In Figure 1.5, we compare the true energy error and the error estimate for h− and hp−adaptive methods. The h-version results are the same as those obtained in Figure 1.3. In the logarithmic scale in the plots, we evidently see that h-version methods give rise to algebraic rates of convergence (first order in this case), while hp-adaptive discretizations lead to exponential convergence in the number of degrees of freedom. Hence, comparing errors versus the number of degrees of freedom, the superiority of hp-adaptive methods over h-adaptive methods is clearly visible.  1.3.5  Averaging operator  The central ingredient to prove the reliability property (1.3.4) is the availability of an averaging operator. It allows us to split the error u − uh into two parts, a conforming part and a remainder. The conforming contribu10  1.3. Main results  (a) ε = 10−3  (b) ε = 2 · 10−4  Figure 1.4: Adaptively generated hp-version meshes after 7 refinement steps.  tion can then be dealt with using the technique of [67], while the remainder can be controlled using the stabilizing jump terms. This technique was first introduced in [44] and has been used in [16, 27, 36, 40, 61, 69, 70]. To apply this approach to hp-adaptive methods, we show new approximation bounds for the averaging operator on irregularly refined meshes and for varying polynomial degrees: if uh is a discontinuous finite element function, then there is an operator Ih mapping uh into a continuous finite element function such that X 2 (1.3.7) k∇h (uh − Ih uh )k2L2 (Ω) ≤ C1 p2E h−1 E k[[uh ]]kL2 (E) , E  kuh −  Ih uh k2L2 (Ω)  ≤ C2  X E  2 p−2 E hE k[[uh ]]kL2 (E) ,  (1.3.8)  with ∇h denoting the element-wise gradient operator. Here, E is an edge of the mesh T , pE the edge polynomial degree and hE the length of this edge. Finally, [[uh ]] denotes the jump of discontinuous functions over an edge. In [44], the H 1 -seminorm approximation result (1.3.7) was proved for an h-version averaging operator constructed on nonconforming meshes. In [36], the property (1.3.7) was established for an hp-version averaging operator on regular meshes. It was then extended in [40] to irregular meshes. In [16], the L2 -norm estimate (1.3.8) and the H 1 -seminorm estimate (1.3.7) were shown on regular meshes and for a fixed polynomial degree. The new contribution of our work is proving the estimates (1.3.7) and (1.3.8) in the case of irregular meshes consisting of parallelograms (hexahedron) and variable polynomial degrees, as is mandatory for hp-adaptivity. In our approach, the 11  1.3. Main results 1  10 0  10  0  10 −1  10  −1  10 −2  10  −2  10 −3  10  −3  10  −4  10  −6  10  −4  hp−Error Indicator hp−Energy Error h−Error Indicator h−Energy Error  −5  10  2  3  10  10  hp−Error Indicator hp−Energy Error h−Error Indicator h−Energy Error  10  −5  4  10  N  (a) ε = 10−3  10  4  10  5  10  6  10 N  7  10  8  10  (b) ε = 2 · 10−4  Figure 1.5: Comparison of h-adaptive and hp-adaptive DG methods.  continuous function Ih uh is piecewise polynomial on an auxiliary mesh Te , similarly to [40]. This auxiliary mesh is obtained from the original mesh T by eliminating its hanging nodes, as illustrated in Figure 1.6. The resulting mesh Te may still contain hanging nodes, but these are no longer essential for the construction of Ih .  =⇒  Figure 1.6: The construction of the auxiliary mesh Te from T . In Chapter 3 (Theorem 3.4.4), an hp-averaging operator satisfying (1.3.7) and (1.3.8) is explicitly constructed for two-dimensional 1-irregularly and isotropically refined meshes consisting of parallelograms. In Chapter 4 (Theorem 4.4.1), these results are then extended to 1-irregularly refined hexahedral meshes in three dimensions. The same construction can be used on anisotropic meshes. This is detailed in Chapter 5, where we analyze an hp-version averaging operator on two-dimensional anisotropically refined meshes with multiple hanging nodes on an elemental edge. In addition, we also allow for element-wise anisotropic polynomial degrees. 12  1.3. Main results  1.3.6  Diffusion problems in three dimensions  In Chapter 4, having the hp-version averaging operators available (cf. Theorem 4.4.1), we develop the energy norm a-posteriori error estimation for hp-version DG discretizations of three-dimensional elliptic boundary-value problems on 1-irregularly and isotropically refined hexahedral meshes. That is, we consider the model problem (1.3.1) with ε = 1 and a = 0. The error measure in (1.3.2) is now only the energy norm. As such, the problem is easier to analyze and we establish the upper and lower bounds as for the two-dimensional case considered in Chapter 3. We note that there is no fundamental obstacle to extend these results to the full convection-diffusion problem and a robust a-posteriori error estimator for three-dimensional convection-diffusion equations can be immediately obtained on isotropically refined meshes. The numerical tests presented in Chapter 4 demonstrate that applying our estimate as an error indicator in an hp-adaptive algorithm is efficient in capturing isotropic corner singularities at exponential convergence rates. To illustrate this, let Ω be the Fichera corner shown in Figure 1.7, and select the right-hand side f and an appropriate boundary condition such that the analytic solution has an isotropic corner singularity at the reentrant corner. In Figure 1.7, we show the mesh generated after 7 hp-adaptive refinement steps. We see that the mesh has been strongly and geometrically refined in the vicinity of the reentrant corner. Additionally, we see that the polynomial degrees have been increased away from the corner, as expected, since the underlying analytical solution is smooth in this region.  Figure 1.7: Adaptively generated hp-version mesh after 7 refinement steps.  13  1.3. Main results  1.3.7  Anisotropically refined meshes  The error estimates in Chapters 2–4 are all developed for isotropically refined meshes. However, it is well-known that boundary layers are most naturally resolved using anisotropic meshes [56]. In the context of the hp-version of the finite element method, they can in fact be numerically captured at exponential rates of convergence on boundary-fitted geometric meshes [62, 63, 64]. In addition, anisotropically and geometrically refined meshes are mandatory to achieve exponential rates of convergence for diffusion problems in generic polyhedral domains [59, 60]. This is due to the presence of edge singularities that have strong anisotropic features. Therefore, it is highly desirable to develop hp-adaptive DG methods on anisotropic elements. In Chapter 5, we extend our approach to a-posteriori error estimation for problem (1.3.1) on rectangular elements of arbitrarily high aspect ratios. We again set out to prove properties (1.3.3)–(1.3.5). However, the reliability constant CR in (1.3.4) now also depends on the introduction of the so-called alignment measure, which is a commonly used technique in anisotropic error estimation [46, 47]. In other words, the reliability bound (1.3.4) now is of the form ku − uh k ≤ CMT η, where C is a constant independent of the parameters of interest, η the estimator and MT the alignment measure. In order to achieve a reasonable resolution of a layer on anisotropically refined meshes, the aspect ratio of the elements depends on the width of the layer. Hence, since MT measures the anisotropy of the mesh, our estimator is not robust and property (1.3.6) cannot be proven. On the other hand, our numerical examples indicate that, as soon as a reasonable resolution of the layer is achieved, the alignment measure MT is of moderate size and the ratio of the error estimator and the true energy error is independent of the diffusion parameter ε and the mesh size. This means that, after a few refinement steps, our estimator is robust in practice. In Figure 1.8, we show an adaptively generated anisotropic mesh for the same problem as before. We see strong anisotropic refinement of the layers along x = 1 and y = 1. In Figure 1.9, we also demonstrate the superiority of using anisotropic refinement over isotropic refinement in terms of error versus numbers of degree of freedom N . An error of 10−4 is achieved with around N = 1752 = 30, 625 while the same resolution on isotropic meshes is obtained with N = 5002 = 250, 000.  14  1.4. Outline  Figure 1.8: Anisotropically generated hp-meshes after 7 refinement steps for ε = 2 · 10−4 . 1  10  Indicator (anisotropic) Error (anisotropic) Indicator (isotropic) Error (isotropic)  0  10  −1  10  −2  10  −3  10  −4  10  −5  10  100  150  200  250  300  350 N1/2  400  450  500  550  600  Figure 1.9: Convergence behavior for ε = 2 · 10−4 .  1.4  Outline  The outline of the thesis is as follows. In Chapter 2, we develop a new approach to obtain a robust a-posteriori error estimator for h-adaptive DG methods for two-dimensional convection-diffusion problems. In Chapter 3, we construct an hp-averaging operator on two-dimensional isotropic meshes and then employ it to derive a robust a-posteriori error estimator for hpversion DG methods for convection-diffusion problems. The construction of this averaging operator is extended to three dimensions in Chapter 4, and an energy norm error estimator for hp-DG methods for three-dimensional diffusion problems is presented. In Chapter 5, we extend our analysis in 15  1.4. Outline Chapter 3 to anisotropically refined meshes. An error estimator for hpDG methods for convection-diffusion problems is derived on anisotropically refined meshes. In this chapter, we also analyze an hp-version averaging operator on anisotropically refined meshes. Conclusions and an outline of future work are presented in Chapter 6.  16  1.5. Bibliography  1.5  Bibliography  [1] S. Adjerid and M. Baccouch. The discontinuous Galerkin method for two-dimensional hyperbolic problems Part II: A-posteriori error estimation. J. 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Anal., 2009. accepted for publication.  23  Chapter 2  An h-version a-posteriori error estimator 1 2.1  Introduction  One of the main difficulties in the finite element approximation of convection-diffusion equations is that solutions to these problems may have layers of small width where their gradients change extremely rapidly. Such layers appear as boundary layers near the outflow boundary of the domain, or as internal layers, caused by non-smooth data near the inflow boundary. The effective numerical resolution of these solution features requires adaptive finite element methods that are capable of locally refining the meshes in the vicinity of the layers and other singularities. At the heart of adaptive finite element methods are a-posteriori error estimators that provide information on the local error distribution. While there is a huge amount of literature available on error estimation for pure diffusion problems (here we only mention [1, 30] and the references therein), much fewer results can be found for convection-diffusion problems. Here, one is particularly interested in robust a-posteriori error estimators that yield upper and lower bounds for the error (measured in a suitable norm) that differ by a factor that is independent of the Péclet number of the convectiondiffusion problem at hand. Important advances in this direction were made in [31] where an estimator for a conforming SUPG method was derived for which the ratio of the upper and lower bounds scales with the square root of the Péclet number. Other estimators that are almost robust can be found in [23], still in a conforming setting, and in [2] for a non-conforming finite element method with face penalties. In the recent work [32], a fully robust error estimator has been proposed. There, in addition to the energy norm, the error measure now also includes a dual norm of the convective 1 A version of this chapter has been published. Schötzau, D. and Zhu, L. (2009) A robust a-posteriori error estimator for discontinuous Galerkin methods for convectiondiffusion equations. Applied Numerical Mathematics 59: 2236-2255.  24  2.1. Introduction derivative. Another approach to robust error estimation can be found in [28, 29], whereby the error in the convective term is evaluated in a interpolation norm of order 1/2. In this chapter, we propose and analyze a robust a-posteriori error estimator for discontinuous Galerkin discretizations of convection-diffusion problems. The estimator yields upper and lower bounds of the error measured in terms of the natural energy norm and a semi-norm associated with the convective terms. Our analysis is based on the approach in [19] where energy norm a-posteriori error estimates were developed for pure diffusion problems. In this approach, the error is decomposed into a conforming part and a remainder. The conforming contribution can then be dealt with using standard techniques, while the remainder can be controlled using the stabilizing jump terms. The same techniques were used in the related papers [16, 17, 18] on energy norm error estimation for discontinuous Galerkin discretizations of saddle point problems. The error measure used in our analysis includes a non-local norm similar to the one in [32]. Our numerical examples indicate that this error contribution is smaller than the energy error and of higher-order once the mesh is sufficiently refined. To the best of our knowledge, this is the first approach to robust error estimation for discontinuous Galerkin methods for convection-diffusion equations and reactiondiffusion equations. Other approaches to error estimation for discontinuous Galerkin applied to pure diffusion problems can be found in [7, 8, 9, 22] and the references therein. For L2 -norm and functional error estimation, we also mention [8, 15, 21, 27] and the references therein. The discontinuous Galerkin method proposed here is based on the upwind discretization of the transport terms, as originally introduced in [24, 26]. The diffusive terms are discretized using the classical interior penalty method; see [3, 25]. The resulting scheme is ideally suited for the numerical approximation of convection-diffusion equations. Indeed, it is known to be robust and stable in the hyperbolic limit, in contrast to standard Galerkin methods. For recent accounts on the state-of-the-art of discontinuous Galerkin methods we refer the reader to the articles in [4, 11, 10, 12] and the references therein. The outline of this chapter is as follows. In Section 2.2, we introduce the discontinuous Galerkin method for a convection-diffusion model problem. In Section 2.3, our a-posteriori error estimator is presented and discussed. We also consider the particular case of singularly perturbed reaction-diffusion equations. The proof of its reliability and efficiency is carried out in Section 2.4. In Section 2.5, we show a series of numerical tests. Finally, in Section 2.6 we present some concluding remarks. 25  2.2. Interior penalty discretization  2.2 2.2.1  Interior penalty discretization Model problem  We consider the convection-diffusion model problem: ( −ε∆u + a(x) · ∇u + b(x)u = f (x) u=0  in Ω, on Γ.  (2.2.1)  Here, Ω is a bounded Lipschitz polygon in R2 with boundary Γ = ∂Ω. The right-hand side f is a given function in L2 (Ω). We assume that the diffusion coefficient satisfies 0 < ε  1.  The coefficient functions a(x) and b(x) belong to W 1,∞ (Ω)2 and L∞ (Ω), respectively. Without loss of generality, we may assume that a and the size of the domain Ω are of order one so that ε−1 is the Péclet number of problem (2.2.1). We further assume that there is a constant β ≥ 0 such that 1 − ∇ · a(x) + b(x) ≥ β, 2  x ∈ Ω.  (2.2.2)  Finally, we suppose that there is a second constant c? ≥ 0 such that k − ∇ · a + bkL∞ (Ω) ≤ c? β.  (2.2.3)  The weak form of (2.2.1) is to find u ∈ H01 (Ω) such that Z Z  f v dx A(u, v) := ε∇u · ∇v + a · ∇uv + buv dx = Ω  Ω  for all v ∈ H01 (Ω). Under the above assumptions on the coefficients, this variational problem is uniquely solvable. Upon integration by parts of the convective term, we also have Z  A(u, v) = ε∇u · ∇v − au · ∇v + (b − ∇ · a)uv dx. Ω  Remark 2.2.1 If β = 0, we obtain from assumption (2.2.3) that b = ∇ · a. Hence, the convection-diffusion equation (2.2.1) can be written in the divergence form −ε∆u + ∇ · (au) = f. In this case, assumption (2.2.2) is satisfied provided that ∇ · a ≥ 0. 26  2.2. Interior penalty discretization  2.2.2  Discretization  To discretize (2.2.1), we consider regular and shape-regular meshes T = {K} that partition the computational domain Ω into open triangles and parallelograms. We define E(T ) to be the set of all edges of the mesh T . For the set of all interior edges we write EI (T ). The diameter of an element K and the length of an edge E are denoted by hK and hE , respectively. Furthermore, we write nK for the outward unit normal vector on the boundary ∂K of an element K. Throughout this thesis, the jumps and averages of piecewise smooth functions on two dimensional meshes are defined as follows. Let the edge E be shared by two neighboring elements K and K e . For a piecewise smooth function v, we denote by v|E its trace on E taken from inside K, and by v e |E the one taken inside K e . The average and jump of v across the edge E are then defined as 1 {{v}} = (v|E + v e |E ), 2  [[v]] = v|E nK + v e |E nK e .  Similarly, if q is piecewise smooth vector field, its average and (normal) jump across E are given by {{q}} =   1 q|E + q e |E , 2  [[q]] = q|E · nK + q e |E · nK e .  On a boundary edge E shared by Γ and K, we set accordingly {{q}} = q and [[v]] = vn, with n denoting the unit outward normal vector on Γ. We denote by Γin and Γout the inflow and outflow parts of Γ: Γin = {x ∈ Γ : a(x) · n(x) < 0},  Γout = {x ∈ Γ : a(x) · n(x) ≥ 0}.  Similarly, the inflow and outflow boundaries of an element K are defined by ∂Kin = {x ∈ ∂K : a(x)·nK (x) < 0},  ∂Kout = {x ∈ ∂K : a(x)·nK (x) ≥ 0}.  For an approximation order p ≥ 1, let now Vh be the finite element space Vh = { v ∈ L2 (Ω) : v|K ∈ Sp (K), K ∈ Th }, where Sp (K) is the space Pp (K) of polynomials of total degree ≤ p if K is a triangle, and the space Qp (K) of polynomials of degree ≤ p in each variable if K is a parallelogram. We consider the following discontinuous Galerkin method that is based on the original upwind discretization in [24, 26] for the convective term and 27  2.3. Robust a-posteriori error estimation on the classical interior penalty discretization in [3, 4, 25] for the Laplacian. It is given by: Find uh ∈ Vh such that Z f v dx (2.2.4) Ah (uh , v) = Ω  for all v ∈ Vh , with the bilinear Ah given by XZ (ε∇u · ∇v + a · ∇uv + buv) dx Ah (u, v) = K  K∈T  − +  X Z  E∈E(Th ) E  X  E∈E(T )  +  εγ hE  XZ  {{ε∇u}} · [[v]] ds − Z  E  ∂Kin \Γ  K∈T  X Z  E∈E(Th ) E  {{ε∇v}} · [[u]] ds  XZ  [[u]] · [[v]] ds −  ∂Kin ∩Γin  K∈T  a · nK uv ds  a · nK (ue − u)v ds.  Here, for a piecewise smooth function, the gradient operator ∇ is taken elementwise. The constant γ > 0 is the interior penalty parameter. To ensure the stability of the discontinuous Galerkin discretization, it has to be chosen sufficiently large, independently of the mesh size and the diffusion coefficient ε, see, e.g., [3, 4, 19]. Upon integration by parts of the convective term, we also have  XZ  Ah (u, v) = ε∇u · ∇v − au · ∇v + (b − ∇ · a)uv dx K  K∈T  − +  X Z  E∈E(Th ) E  X  E∈E(T )  +  XZ  K∈T  2.3  εγ hE  {{ε∇u}} · [[v]] ds − Z  E  ∂Kout \Γ  [[u]] · [[v]] ds +  X Z  E∈E(Th ) E  {{ε∇v}} · [[u]] ds  XZ  K∈T  ∂Kout ∩Γout  a · nK uv ds  a · nK u(v − v e ) ds.  Robust a-posteriori error estimation  In this section, we present and discuss our main results.  28  2.3. Robust a-posteriori error estimation  2.3.1  Norms  We begin by introducing the norm  X X k u k2E,T = εk∇uk2L2 (K) + βkuk2L2 (K) + K∈T  E∈E(T )  γε k[[u]]k2L2 (E) . (2.3.1) hE  It can be viewed as the energy norm associated with the discontinuous Galerkin discretization of the convection-diffusion problem (2.2.1). For q ∈ L2 (Ω)2 , we further define the semi-norm R Ω q · ∇v dx |q|? = sup . k v kE,T v∈H 1 (Ω)\{0} 0  Remark 2.3.1 The above semi-norm | · |? can be characterized by using a Helmholtz decomposition similar to the one in [14, Theorem 3.2]. We write q in the form q = ∇ϕ + q 0 , where ϕ ∈ H01 (Ω) solves Z Z ∇ϕ · ∇v dx = q · ∇v dx Ω  Ω  ∀ v ∈ H01 (Ω),  and q 0 = q − ∇ϕ is divergence-free in the sense that Z q 0 · ∇v dx = 0 ∀ v ∈ H01 (Ω). Ω  This decomposition is unique and orthogonal in L2 (Ω)2 . Thus, we observe that |q|? = 0 if and only if q = q 0 . Furthermore, if we introduce the norm R Ω ∇ϕ · ∇v dx kϕk? = sup , k v kE,T v∈H 1 (Ω)\{0} 0  we have that |q|? = kϕk? . We now define | u |2O,T  =  |au|2?  +  X   E∈E(T )  hE βhE + ε    k[[u]]k2L2 (E) .  (2.3.2)  The semi-norm |au|2? and the jump terms hE ε−1 k[[u]]k2L2 (E) will be used to bound the convective derivative, analogously to [32]. Here we note that hE ε−1 is the local mesh Péclet number, see also the discussion in Remark 2.3.5. Finally, the jump terms βhE k[[u]]|2L2 (E) are associated with the reaction term in the equation. 29  2.3. Robust a-posteriori error estimation  2.3.2  A robust a-posteriori error estimator  Next, we define our a-posteriori error estimator. To that end, we set 1  1  ρK = min{hK ε− 2 , β − 2 }, 1  1  1  ρE = min{hE ε− 2 , β − 2 }. 1  In the case β = 0, we set ρK = ε− 2 hK and ρE = ε− 2 hE . Let now uh be the discontinuous Galerkin approximation obtained by (2.2.4). Moreover, let fh , ah , and bh denote piecewise polynomial approximations in Vh to the right-hand side and the coefficient functions, respectively. For each element K ∈ T , we introduce a local error indicator ηK which is given by the sum of three terms 2 2 2 ηK = ηR + ηE + ηJ2K . K K  The first term ηRK is the interior residual defined by 2 ηR = ρ2K kfh + ε∆uh − ah · ∇uh − bh uh k2L2 (K) . K 2 The second term ηE is the edge residual given by K 1 X −1 2 ηE = ε 2 ρE k[[ε∇uh ]]k2L2 (E) . K 2 E∈∂K\Γ  The last term ηJK measures the jumps of the approximate solution uh and is defined by   1 X γε hE ηJ2K = + βhE + k[[uh ]]k2L2 (E) , 2 hE ε E∈∂K\Γ  X  γε hE + + βhE + k[[uh ]]k2L2 (E) . hE ε E∈∂K∩Γ  We also introduce a data approximation term by   Θ2K = ρ2K kf − fh k2L2 (K) + k(a − ah ) · ∇uh k2L2 (K) + k(b − bh )uh k2L2 (K) .  We then define the a-posteriori error estimator X 1 2 2 η= ηK .  (2.3.3)  K∈Th  The data approximation error is given by X 1 Θ= Θ2K 2 .  (2.3.4)  K∈T  30  2.3. Robust a-posteriori error estimation  2.3.3  Reliability and efficiency  In the following, we use the symbols . and & to denote bounds that are valid up to positive constants independent of the local mesh size and the diffusion coefficient ε. The constants will also be independent of γ, provided that γ ≥ 1. Our first main result states that, up to a constant and to the data approximation error, the estimator (2.3.3) gives rise to a reliable a-posteriori error bound. Theorem 2.3.2 Let u be the solution of (2.2.1) and uh ∈ Vh its DG approximation obtained by (2.2.4). Let the error estimator η be defined by (2.3.3), and the data approximation error Θ by (2.3.4). Then we have the a-posteriori error bound k u − uh kE,T + | u − uh |O,T . η + Θ. Our next theorem presents a lower bound for the error and shows the efficiency of the error estimator η. Theorem 2.3.3 Let u be the solution of (2.2.1) and uh ∈ Vh its DG approximation obtained by (2.2.4). Let the local error estimator η be defined by (2.3.3), and the data approximation error Θ by (2.3.4). Then we have the bound η . k u − uh kE,T + | u − uh |O,T + Θ. Remark 2.3.4 The reliability and efficiency constants in Theorem 2.3.2 and Theorem 2.3.3 are independent of the diffusion coefficient ε. Hence, the constants in the upper and lower bounds are independent of the Péclet number ε, up to data approximation errors. In this sense, the error estimator η in (2.3.3) is robust in the diffusion parameter ε. Remark 2.3.5 In our numerical tests in Section 2.5, we show that the non-standard error | u − uh |O,T is of at least the same order as the energy error k u − uh kE,T and even of higher-order, once the local mesh Péclet number is sufficiently small. Heuristically, this can be explained as follows. We p expect the error k u − uh kE,T to converge with the optimal order O(N − 2 ), where N is the number of degrees of freedom; cf. [4, 20]. We then have the bound 1 |a(u − uh )|? . √ ku − uh kL2 (Ω) . (2.3.5) ε 31  2.3. Robust a-posteriori error estimation If we now also assume that the L2 -error ku − uh kL2 (Ω) converges with the p  1  optimal rate O(N − 2 − 2 ), we obtain  1  N−2 √ −p |a(u − uh )|? . εN 2 . ε −1  The fraction N ε 2 is the local mesh Péclet number. Hence, |a(u − uh )|? is of at least the same order as the energy error, once the mesh Péclet number is sufficiently small. Similar arguments show that 1 X hE 1 N−2 √ −p 2 2 k[[u − uh ]]kL2 (E) . εN 2 , ε ε  E∈T  X  E∈T  βhE k[[u − uh ]]k2L2 (E)  1 2  .  p p 1 βN − 2 − 2 ,  where we have used that [[u]] = 0. Thus, the same conclusion as for |a(u − uh )|? can also be made for the error | u − uh |O,T .  2.3.4  A robust estimator for reaction-diffusion problems  Setting the convection coefficient a(x) to be zero in Theorems 2.3.2 and 2.3.3, we easily obtain a robust a-posteriori error estimator for singularly perturbed reaction-diffusion equations of the form ( −ε∆u + b(x) u = f in Ω, (2.3.6) u=0 on Γ, where 0 < ε  1. We assume that there are two constants β > 0 and c∗ ≥ 0, such that b(x) ≥ β for x ∈ Ω and kbkL∞ (Ω) ≤ c? β. For reaction-diffusion equations, the energy norm is now defined by  X X γε k u k2E,T = εk∇uk2L2 (K) + βkuk2L2 (K) + ( + βhE )k[[u]]k2L2 (E) . hE K∈T  E∈E(T )  The local error indicator ηK on every element K ∈ T becomes 2 2 2 = ηR + ηE + ηJ2K , ηK K K  32  2.3. Robust a-posteriori error estimation where 2 ηR = ρ2K kfh + ε∆uh − bh uh k2L2 (K) , K 1 X −1 2 ε 2 ρE k[[ε∇h uh ]]k2L2 (E) , ηE = K 2 E∈∂K\Γ  ηJ2K  1 = 2  X  E∈∂K\Γ  (  γε + βhE )k[[uh ]]k2L2 (E) + hE  X  (  E∈∂K∩Γ  γε + βhE )k[[uh ]]k2L2 (E) , hE  where ρK and ρE are defined by 1  ρK = hK ε− 2 ,  1  ρE = hE ε 2 .  Furthermore, since the convection term disappears in the data approximation term, we have,   Θ2K = ρ2K kf − fh k2L2 (K) + k(b − bh )uh k2L2 (K) ,  with uh being the discontinuous Galerkin approximation. We then employ the same notations as before and set X 1 2 2 , (2.3.7) η= ηK K∈Th  as well as Θ=  X  K∈T  Θ2K  1  2  .  (2.3.8)  As a corollary of Theorem 2.3.2 and Theorem 2.3.3, we have the following result for reaction-diffusion equations. Theorem 2.3.6 Let u be the solution of the reaction-diffusion equation (2.3.6) and uh ∈ Vh its DG approximation obtained by (2.2.4). Then we have the upper bound k u − uh kE,T . η + Θ, as well as the lower bound η . k u − uh kE,T + Θ, with η and Θ defined in (2.3.7) and (2.3.8), respectively.  33  2.4. Proofs  2.4  Proofs  In this section, we present the proofs of Theorem 2.3.2 and Theorem 2.3.3. We proceed in several steps.  2.4.1  Auxiliary forms and their properties  The discontinuous Galerkin form Ah (u, v) is not well-defined for functions u, v in H01 (Ω). In [19], this difficulty has been overcome by the use of a suitable lifting operator. Here, we present a different and new approach where we split the discontinuous Galerkin form into several parts. More precisely, we introduce the auxiliary forms  XZ  ε∇u · ∇v + (b − ∇ · a)uv dx, Dh (u, v) = (2.4.1) K  K∈T  Oh (u, v) = −  XZ  K∈T  +  K  XZ  ∂Kout \Γ  K∈T  Kh (u, v) = − Jh (u, v) =  au · ∇v dx +  X Z  E∈E(Th ) E  X  E∈E(T )  εγ hE  Z  X Z  K∈Th  a · nK uv ds  a · nK u(v − v e ) ds,  {{ε∇u}} · [[v]]ds −  E  ∂Kout ∩Γout  X Z  E∈E(Th ) E  (2.4.2) {{ε∇v}} · [[u]]ds, (2.4.3)  [[u]] · [[v]] ds.  (2.4.4)  Then, we set eh (u, v) = Dh (u, v) + Jh (u, v) + Oh (u, v). A  This form is well-defined for all u, v ∈ Vh + H01 (Ω). Obviously, we have eh (u, v) = A(u, v), A  (2.4.5)  eh (u, v) + Kh (u, v), Ah (u, v) = A  (2.4.6)  for all u, v ∈ H01 (Ω). Furthermore,  for all u, v ∈ Vh . As a consequence of (2.2.2) and (2.4.5), we have the following coercivity result. 34  2.4. Proofs Lemma 2.4.1 For any u ∈ H01 (Ω), we have eh (u, u) ≥ k u k2 . A E,T  Furthermore, the auxiliary forms are continuous. Lemma 2.4.2 There holds |Dh (u, v)| . k u kE,T k v kE,T ,  u, v ∈ Vh + H01 (Ω),  |Jh (u, v)| . k u kE,T k v kE,T ,  u, v ∈ Vh + H01 (Ω),  |Oh (u, v)| . |au|? k v kE,T ,  u ∈ Vh + H01 (Ω), v ∈ H01 (Ω).  Proof : The first claim follows from the Cauchy-Schwarz inequality and the bound in (2.2.3). The second is a straightforward consequence of the Cauchy-Schwarz inequality. The third one follows immediately from the definition of |au|? . 2 Lemma 2.4.3 For u ∈ Vh and v ∈ H01 (Ω) ∩ Vh we have X εγ 1 1 Kh (u, v) . γ − 2 k[[u]]k2L2 (E) 2 k v kE,T . hE E∈E(Th )  Proof : Since v ∈ H01 (Ω) ∩ Vh , we have X Z Kh (u, v) = − {{ε∇v}} · [[u]] ds. E∈E(Th ) E  Using the Cauchy-Schwarz inequality, the inverse estimate, −1  kvkL2 (∂K) . hK 2 kvkL2 (K) ,  v ∈ Sp (K),  and the shape-regularity of the mesh, we obtain X Z Kh (u, v) . |ε∇v||[[u]]| ds E∈E(T ) E 1  . γ− 2  X  E∈E(Th ) 1  . γ− 2  X  εhE  Z  E  |∇v|2 ds  εk∇vk2L2 (K)  K∈T  1 2  1 2  X  X  E∈E(T )  E∈E(Th )  εγ hE  Z  E  1 |[[u]]|2 ds 2  1 εγ k[[u]]k2L2 (E) 2 . hE  This yields the assertion.  2  Next, we show the following inf-sup condition. 35  2.4. Proofs Lemma 2.4.4 There is a constant C > 0 such that eh (u, v) A ≥ C > 0. u∈H01 (Ω)\{0} v∈H 1 (Ω)\{0} (k u kE,T + |au|? )k v kE,T inf  sup  0  Proof : Let u ∈ H01 (Ω) and θ ∈ (0, 1). Then there exists wθ ∈ H01 (Ω) such that Z au · ∇wθ dx ≥ θ|au|? . k wθ kE,T = 1, Oh (u, wθ ) = − Ω  From the continuity properties in Lemma 2.4.2, we obtain eh (u, wθ ) = Dh (u, wθ ) + Jh (u, wθ ) + Oh (u, wθ ) A ≥ θ|au|? − C1 k u kE,T k wθ kE,T  = θ|au|? − C1 k u kE,T , for a constant C1 > 0. Let us then define vθ = u +  k u kE,T wθ . 1 + C1  Obviously, k vθ kE,T ≤ (1 +  1 )k u kE,T . 1 + C1  By Lemma 2.4.1, A(u, u) ≥ k u k2E,T , so that eh (u, v) A v∈H 1 (Ω)\{0} k v kE,T sup  0  ≥ ≥ =  eh (u, vθ ) A k vθ kE,T  k u k2E,T + (1 + C1 )−1 k u kE,T (θ|au|? − C1 k u kE,T ) (1 +  1 1+C1 )k u kE,T  1 (k u kE,T + θ|au|? ). 2 + C1  Since θ ∈ (0, 1) and u ∈ H01 (Ω) are arbitrary, we obtain the inf-sup condition. 2  36  2.4. Proofs  2.4.2  Approximation operators  Let Vhc be the conforming subspace of Vh given by Vhc = Vh ∩ H01 (Ω). We denote by Ah : Vh → Vhc the approximation operator defined in [22, Theorem 2.2 and Theorem 2.3]; see also [19, Proposition 5.4] for an extension to the hp-version of the discontinuous Galerkin method. The following approximation result holds. Lemma 2.4.5 For any v ∈ Vh , we have X  K∈T  X  K∈T  kv − Ah vk2L2 (K) .  k∇(v − Ah v)k2L2 (K) .  X Z  E∈E(T ) E  X Z  E∈E(T ) E  hE |[[v]]|2 ds, 2 h−1 E |[[v]]| ds.  Moreover, we will make use of the Clément-type interpolant constructed in [32, Lemma 3.3] and the references therein. Lemma 2.4.6 There exists an interpolation operator Ih : H01 (Ω) → {ϕ ∈ C(Ω) : ϕ|K ∈ S1 (K), ∀ K ∈ T , ϕ = 0 on Γ}, that satisfies k Ih v kE,T . k v kE,T and X  K∈T  X  E∈E(T )  2 ρ−2 K kv − Ih vkL2 (K)  1  2 ε 2 ρ−1 E kv − Ih vkL2 (E)  1 2  1 2  . k v kE,T , . k v kE,T ,  for any v ∈ H01 (Ω).  2.4.3  Proof of Theorem 2.3.2  We are now ready to prove Theorem 2.3.2. Following [19], we decompose the discontinuous Galerkin solution into a conforming part and a remainder. That is, we write uh = uch + urh , 37  2.4. Proofs where uch = Ah uh ∈ Vhc , with Ah the approximation operator from Lemma 2.4.5. The remainder is then given by urh = uh − uch . By the triangle inequality we obtain k u − uh kE,T + | u − uh |O,T  ≤ k u − uch kE,T + | u − uch |O,T + k urh kE,T + | urh |O,T . (2.4.7)  Next, we prove that both the continuous error u − uch and the remainder urh can be bounded by the error estimator. We proceed in several steps. Lemma 2.4.7 There holds k urh kE,T + | urh |O,T . η. Proof : Since [[urh ]] = [[uh ]], we have  X k urh k2E,T + | urh |2O,T = εk∇urh k2L2 (K) + βkurh k2L2 (K) + |aurh |2? K∈T  +   X  γε hE + βhE + k[[uh ]]k2L2 (E) hE ε  E∈E (T )  .  X  K∈T  (εk∇urh k2L2 (K) + βkurh k2L2 (K) ) + |aurh |2? +  X  ηJ2K .  K∈T  Hence, only the volume terms and the expression involving the | · |? seminorm need to be bounded further. Lemma 2.4.5 yields X X εγ X ε k∇urh k2L2 (K) . γ −1 k[[uh ]]k2L2 (E) . γ −1 ηJ2K , hE K∈T  β  K∈T  E∈E(T )  X  K∈T  kurh k2L2 (K) .  X  E∈E(T )  βhE k[[uh ]]k2L2 (E) .  X  ηJ2K .  K∈T  To estimate |aurh |? , we apply Lemma 2.4.5 once more, and obtain X hE X 1 |aurh |2? . kurh k2L2 (Ω) . k[[uh ]]k2L2 (E) . ηJ2K . ε ε E∈E(T )  K∈T  This finishes the proof.  2  Lemma 2.4.8 For any v ∈ H01 (Ω), we have Z eh (uh , v − Ih v) . (η + Θ) k v kE,T . f (v − Ih v) dx − A Ω  Here, Ih is the interpolant introduced in Lemma 2.4.6.  38  2.4. Proofs Proof : Set T =  Z  Ω  eh (uh , v − Ih v). f (v − Ih v) dx − A  Integration by parts immediately yields XZ (f + ε∆uh − a · ∇uh − buh )(v − Ih v) dx T = K∈T  − +  K  XZ  ∂K  K∈T  ε∇uh · nK (v − Ih v) ds  XZ  a · nK (ueh − uh )(v − Ih v) ds  ∂Kin \Γ  K∈T  = T1 + T2 + T3 . In the term T1 , we first add and subtract the data approximations terms. This gives XZ T1 = (fh + ε∆uh − ah · ∇uh − bh uh )(v − Ih v) dx K∈T  +  K  XZ  K∈T  K   (f − fh ) − (a − ah ) · ∇uh − (b − bh )uh (v − Ih v) dx.  Using the Cauchy-Schwarz inequality and Lemma 2.4.6 yields T1 .  X  2 ηR K  +  Θ2K  K∈Th  .  X  2  K∈T  K∈Th  X  1 X  1 X 2  K∈T  2 (ηR + Θ2K ) K  K∈T  2 ρ−2 K kv − Ih vkL2 (K) 2 ρ−2 K kv − Ih vkL2 (K)  1  2  1 2  1 2  k v kE,T .  Next, if we rewrite the term T2 in terms of the jumps of ε∇uh , we obtain X Z T2 = − [[ε∇uh ]](v − Ih v) ds. E∈EI (T ) E  39  2.4. Proofs Hence, the Cauchy-Schwarz inequality and Lemma 2.4.6 yield X  1 X 1 −1 1 1 ε 2 ρE kv − Ih vk2L2 (E) 2 ε− 2 ρE k[[ε∇uh ]]k2L2 (E) 2 T2 . E∈E(T )  E∈EI (Th )  .  X  2 ηE K  K∈T  1 2  k v kE,T .  In order to bound T3 , we use the Cauchy-Schwarz inequality, Lemma 2.4.6, 1 and the fact that ρE ≤ hK ε− 2 : X X 1 1 1 1 2 2 T3 . ε− 2 ρE k[[uh ]]k2L2 (E) 2 ε 2 ρ−1 E kv − Ih vkL2 (E) E∈E(T )  .  X  E∈Eh (T )  ηJ2K  K∈T    1 2  k v kE,T . 2  This finishes the proof. Lemma 2.4.9 There holds: k u − uch kE,T + | u − uch |O,T . η + Θ.  Proof : Note that | u − uch |O,T = |a(u − uch )|? . Then the inf-sup condition yields: k u − uch kE,T + |a(u − uch )|? .  eh (u − uc , v) A h . k v kE,T v∈H 1 (Ω)\{0} sup  (2.4.8)  0  The properties (2.4.5) and (2.4.6) allow us to conclude that, for any v ∈ H01 (Ω), Z c e eh (uc , v) Ah (u − uh , v) = f v dx − A h Ω  =  Z  Ω  =  Z  Ω  f v dx − Dh (uch , v) − Jh (uch , v) − Oh (uch , v)  eh (uh , v) + Dh (ur , v) + Jh (ur , v) + Oh (ur , v). f v dx − A h h h  From the discontinuous Galerkin method in (2.2.4), we have Z eh (uh , Ih v) + Kh (uh , Ih v), f Ih v dx = Ah (uh , Ih v) = A Ω  40  2.4. Proofs where Ih is the operator introduced in Lemma 2.4.6. Therefore, A(u − uch , v) = T1 + T2 + T3 , where T1 =  Z  Ω  eh (uh , v − Ih v), f (v − Ih v) dx − A  T2 = Dh (urh , v) + Jh (urh , v) + Oh (urh , v), T3 = Kh (uh , Ih v). The estimate in Lemma 2.4.8 yields T1 . (η + Θ) k v kE,T .  Similarly, the continuity result in Lemma 2.4.2 and the approximation properties in Lemma 2.4.7 give T2 . (k urh kE,T + |aurh |? )k v kE,T ≤ ηk v kE,T . Finally, we use Lemma 2.4.3 to bound the term T3 . We obtain T3 . γ  − 12  X  K∈T  !1 2  ηJ2K  k Ih v kE,T . γ  − 12  X  !1 2  ηJ2K  K∈T  k v kE,T .  Here, we have also used the k · kE,T -stability of the operator Ih from Lemma 2.4.6. This completes the proof. 2 The proof of Theorem 2.3.2 now immediately follows from Lemma 2.4.7, Lemma 2.4.9 and (2.4.7).  2.4.4  Proof of Theorem 2.3.3  To prove Theorem 2.3.3, we use the same arguments as in [32]. To that end, for any interior edge E ∈ EI (T ), we denote by wE the union of the two elements that share it. Furthermore, we denote by ψK and ψE the bubble functions constructed and defined in [32, p. 1771]. The function ψK belongs to H01 (K), while ψE is in H01 (wE ). We have kψK kL∞ (K) = 1,  kψE kL∞ (E) = 1.  (2.4.9)  41  2.4. Proofs In the following, we denote by (·, ·)K and (·, ·)E the inner products in L2 (K) and L2 (E), respectively. Furthermore, for a set of elements D, we denote by k · kE,D the local energy norm  X k u k2E,D = εk∇uk2L2 (K) + βkuk2L2 (K) . K∈D  We also set kuk2L2 (D) = and define  Z  f (x) dx =  D  X  K∈D  kuk2L2 (K) ,  XZ  K∈D  f (x) dx.  K  The following result holds; cf. [32, Lemma 3.6]. Lemma 2.4.10 We have kvk2L2 (K) . (v, ψK v)K , k ψK v kE,K  . ρ−1 K kvkL2 (K) ,  kσk2L2 (E) . (σ, ψE σ)E , 1  1  1  −1  kψE σkL2 (wE ) . ε 4 ρE2 kσkL2 (E) , k ψE σ kE,wE  . ε 4 ρE 2 kσkL2 (E) ,  (2.4.10) (2.4.11) (2.4.12) (2.4.13) (2.4.14)  for any element K, edge E, and polynomials v and σ defined on elements and faces, respectively. In the last two inequalities, the polynomial σ defined on E is extended to R2 in a canonical fashion. To prove Theorem 2.3.3, we first note that, since [[u]] = 0, we have X  K∈T  ηJ2K  1 2  . k u − uh kE,T + | u − uh |O,T .  (2.4.15)  Hence, we only need to show the efficiency of the indicators ηRK and ηEK , respectively. This will be done in the next two lemmas. Lemma 2.4.11 There holds X 1 2 2 ηR . k u − uh kE,T + | u − uh |O,T + Θ. K K∈T  42  2.4. Proofs Proof : Let K be an element in T . We define R|K = (fh + ε∆uh − ah · ∇uh − bh uh )|K , and set W |K = ρ2K RψK . By inequality (2.4.10) in Lemma 2.4.10, X X X 2 2 2 ηR = ρ kRk . (R, ρ2K ψK R)K 2 K L (K) K K∈T  K∈T  =  X  K∈T  (R, W )K =  K∈T  X  K∈T  (fh + ε∆uh − ah · ∇uh − bh uh , W )K .  Since the exact solution satisfies (f + ε∆u − a · ∇u − bu)|K = 0, we obtain, by integration by parts and addition and subtraction of the exact data,  X X 2 ε(∇(u − u), ∇W ) − (a(u ηR . − u ), ∇W ) K K h h K K∈T  K∈T  +  X  K∈T  +  ((b − ∇ · a)(u − uh ), W )K  X   K∈Th  (fh − f ) + (a − ah ) · ∇uh + (b − bh )uh , W    K  .  Here, we have also used that W |∂K = 0. Then, by the Cauchy-Schwarz inequality, the bound in (2.2.3), and the definitions of | · |O,T and the data approximation error Θ, we obtain X  2 ηR . k u − uh kE,T + | u − uh |O,T + Θ K K∈T  ×  X  K∈Th  2 k W k2E,K + ρ−2 K kW kL2 (K)  1 2  .  By (2.4.11) and (2.4.9), we have the following estimates k W k2E,K . ρ2K kRk2L2 (K) ,  2 2 2 ρ−2 K kW kL2 (K) . ρK kRkL2 (K) .  This yields X  K∈T    X 1 2 2 2 ηR . k u − u k + | u − u | + Θ ηR , h E,T h O,T K K  which shows the assertion.  K∈T  2 43  2.4. Proofs Lemma 2.4.12 There holds X 1 2 2 . k u − uh kE,T + | u − uh |O,T + Θ. ηE K K∈T  Proof : We set τ=  X  1  ε− 2 ρE [[ε∇uh ]]ψE .  E∈EI (T )  By (2.4.12) in Lemma 2.4.10 and the fact that [[ε∇u]] = 0 on interior edges, we obtain X X X 2 ([[ε∇(uh − u)]], τ )E . ([[ε∇u ]], τ ) = ηE . E h K K∈T  E∈EI (T )  E∈EI (T )  After integration by parts over each of the two elements of wE , we have X X Z ([[ε∇(uh −u)]], τ )E = (ε(∆uh −∆u)τ +ε(∇uh −∇u)·∇τ )dx. E∈EI (T ) wE  E∈EI (T )  Using the differential equation and approximating the data, we obtain X X Z 2 ηEK . (fh + ε∆uh − ah · ∇uh − buh )τ dx E∈EI (T ) wE  K∈T  Z  X  +  E∈EI (T ) wE  +  Z  X    E∈EI (T ) wE   (a · ∇(uh − u) + b(uh − u))τ + ε(∇uh − ∇u) · ∇τ dx     (f − fh ) + (ah − a) · ∇uh + (bh − b)uh τ dx  Integration by parts over wE of the convection term a · ∇(uh − u) yields, X 2 ηE . T1 + T2 + T3 + T4 + T5 , K K∈T  where X  T1 =  Z  E∈EI (T ) wE  T2 =  X  Z  E∈EI (T ) wE  T3 = −  X  Z  (fh + ε∆uh − ah · ∇uh − buh )τ dx,  (−∇ · a + b)(uh − u)τ + ε(∇uh − ∇u) · ∇τ dx,  E∈EI (T ) wE  a(uh − u) · ∇τ dx, 44  2.4. Proofs X  T4 =  Z  E∈EI (T ) E  X  T5 =  a · [[uh ]]τ ds,  Z  E∈EI (T ) wE     (f − fh ) + (ah − a) · ∇uh + (bh − b)uh τ dx.  The Cauchy-Schwarz inequality, the shape-regularity of the mesh and Lemma 2.4.11 yield  X 1 2 2 ρ−2 T1 . k u − uh kE,T + | u − uh |O,T + Θ . E kτ kL2 (wE ) E∈EI (T )  By inequality (2.4.13) in Lemma 2.4.10 we obtain, X X 1 1 2 2 2 2 ρ−2 . ηE , E kτ kL2 (wE ) K K∈T  E∈EI (T )  so that T1 . k u − uh kE,T + | u − uh |O,T + Θ   X  2 ηE K  K∈T  1 2  .  Using the shape regularity of the mesh and inequality (2.4.14), the term T2 can be bounded by X X 1 1 2 2 . T2 . k u − uh kE,T ηE k τ k2E,wE 2 . k u − uh kE,T K K∈T  E∈EI (T )  For the term T3 , we use the previous estimate and obtain X X 1 1 2 2 T3 . | u − uh |O,T ηE . k τ k2E,wE 2 . | u − uh |O,T K K∈Th  E∈EI (T )  Since the support of ψE intersects the support of at most two other ψE ’s and since kψE kL∞ (E) = 1, we have T4 .  X  E∈EI (T )  1  ε− 2 ρE k[[uh ]]k2L2 (E)  1 2  X  E∈EI (T )  1  2 ε 2 ρ−1 E kτ kL2 (E)  1 2  .  1  Then, due to ρE ≤ hE ε− 2 , we obtain T4 .  X  E∈EI (T )  1 hE k[[uh ]]k2L2 (E) 2 ε  X  K∈Th  2 ηE K  1  2  . | u − uh |O,T  X  K∈T  2 ηE K  1 2  .  45  2.5. Numerical experiments Finally, the data error term T5 can be bounded by X  T5 . Θ  K∈Th  2 ρ−2 K kτ kL2 (K)  1 2  .Θ  X  2 ηE K  K∈Th  1 2  . 2  This finishes the proof.  The proof of Theorem 2.3.3 now follows from (2.4.15), Lemma 2.4.11, and Lemma 2.4.12, respectively.  2.5  Numerical experiments  In this section, we present a set of numerical examples where we use η in (2.3.3) as an error indicator in an adaptive refinement strategy. Our implementation of the discontinuous Galerkin method (2.2.4) is based on the Deal.II finite element library [5, 6]. In all the examples presented below, we construct adaptively refined mesh sequences by marking elements for refinement or derefinement according to the size of the local indicators ηK , with refinement and derefinement fractions set to 25% and 10%, respectively. We begin all the tests with a uniform square mesh of 16×16 elements. Refinement and derefinement are done so that the resulting meshes are at least 1-irregular. We set the stabilization parameter to γ = 10p2 , where p is the polynomial degree. This is the standard choice in hp-version discontinuous Galerkin methods, see, e.g., [19]. In all the examples, the data approximation error Θ is of higher order and is neglected.  2.5.1  Example 1  In this example, we take Ω = (0, 1)2 in R2 , and choose a = (1, 1)> and b ≡ 0. This implies that β = 0. Further, we set u = 0 on Γ = ∂Ω, and select the right-hand side f so that the analytical solution to (2.2.1) is given by u(x, y) =   e x−1 ε −1 1  e− ε − 1  +x−1   e y−1 ε −1 1  e− ε − 1   +y−1 .  The solution is smooth, but has boundary layers at x = 1 and y = 1; their widths are both of order O(ε). This problem is well-suited to test whether the estimator η is able to pick up the steep gradients near these boundaries. In Figure 2.1, we show the performance of our estimator η for piecewise linear elements (p = 1) and for ε = 1, ε = 10−2 , and ε = 10−4 . In the upper row of subfigures, we plot the ”true” energy error k u − uh kE,T and 46  2.5. Numerical experiments 1  the value of the estimator η against N − 2 , with N denoting the number of degrees of freedom in each refinement step. These curves are labeled ”ERR” and ”EST”, respectively. The estimator always overestimates the true energy error, in agreement with Theorem 2.3.2. Asymptotically, the curves 1 are straight lines, indicating convergence of the optimal order O(N − 2 ). The asymptotic regime is achieved immediately for the diffusion-dominated problem where ε = 1. For the smaller values of ε, it is achieved once the boundary layers are sufficiently resolved. Motivated by Remark 2.3.5, in the 1 curve ”DERR”, we also calculate the error ε− 2 ku − uh kL2 (Ω) , which is an upper bound for |a(u − uh )|? , cf. equation (2.3.5). We can see that this error is of higher order for ε = 1. For the smaller values of ε, it is initially of the same order as the energy error, but then decreases faster once the boundary layers arePapproximated well enough. The same behavior is ob served for the error E∈E(T ) βhE + hE ε−1 k[[uh ]]k2L2 (E) , which is computed in the curve ”JERR”; see also Remark 2.3.5. In the second row of subfigures in Figure 2.1 we show the ratio of the estimator and the true energy error. It stays bounded between 6 and 7, uniformly in ε, as predicted by Theorem 2.3.2 and Theorem 2.3.3. Based on the above observations, here we have neglected the error component | u − uh |O,T . Finally, we note that, for ε small, the effectivity ratio is substantially smaller in the pre-asymptotic regime. In Figure 2.2, we show the same plots for piecewise quadratic elements (p = 2). Qualitatively, we observe the same behavior as before, but obtain convergence of the order O(N −1 ) in the asymptotic regime. The ratio of the estimator and the true energy error is independent of ε, and in the range of 11. As is common in residual-based error estimation, the true error is now overestimated by a significant factor. This limitation of residual-based error estimation can only be overcame by using more sophisticated approaches such as the guaranteed estimators in [13]. Finally, in Figure 2.3, we show the adaptive meshes after 7 refinement steps, both for p = 1 and p = 2 and the same values of ε. Obviously, we observe strong mesh refinement near the lines y = 1 and x = 1, indicating that the estimator correctly recognizes boundary layers and is able to resolve them in convection-dominated regimes. Recall that, in the case where ε = 1, the problem is diffusion-dominated and no boundary layers are present.  2.5.2  Example 2  Let us next consider an example with an internal layer and with variable coefficients. In the domain Ω = (−1, 1)2 , we set a(x, y) = (−x, y)> and 47  2.5. Numerical experiments 1  −1  1  10  10  10  −2  10  0  10  0  10 −3  10  −1  10  −1  10  −4  10  −2  10 −5  10  −2  10 −3  10 −6  10  −3  10  −4  EST ERR DERR JERR  −7  10  EST ERR DERR JERR  10  −8  EST ERR DERR JERR  −5  10  −4  10 1  10  2  3  10  10  10 1  10  N1/2  2  10  3  10  2  10  N1/2  (b) ε = 10−2  (a) ε = 1 8  8  7  7  7  6  6  6  5  5  5  4  4  4  3  3  3  ratio 2  10  N1/2  (d) ε = 1  4  10  (c) ε = 10−4  8  2 1 10  3  10  N1/2  ratio 3  10  2 1 10  2  10  ratio 3  10  2 2 10  N1/2  (e) ε = 10−2  3  10  4  10  N1/2  (f) ε = 10−4  Figure 2.1: Example 1: Convergence behavior for ε = 1, 10−2 , 10−4 and p = 1.  b ≡ 0. We choose f and the inhomogeneous Dirichlet boundary conditions such that the solution to (2.2.1) is given by x u(x, y) = erf( √ )(1 − y 2 ). 2ε Rx 2 Here, erf(x) is the error function defined by erf(x) = √2π 0 e−t dt. For small values of ε, the solution u has an internal layer around x = 0, whose √ width is of order O( ε). We use standard DG terms to incorporate the inhomogeneous boundary conditions, and modify the error indicator η correspondingly. For details we refer the reader to [19]. In Figure 2.4, the numerical results for this example are depicted for the values ε = 10−2 and ε = 10−3 , and for both linear and quadratic approximations. In the asymptotic regime, we observe linear and quadratic p convergence rates of the optimal order O(N − 2 ) for both the energy error and the estimator. The additional curves ”DERR” and ”JERR” show the same quantities as in Example 1 associated with the error | u − uh |O,T . They are not dominant and clearly smaller than the energy error. In Figure 2.4 48  2.5. Numerical experiments 1  −3  10  10  1  10 −4  10  0  10  0  10 −5  10  −1  10  −1  10  −6  10  −2  10 −7  10  −2  10 −3  10 −8  10  −3  10  −4  10  −9  10  EST ERR DERR JERR  −10  10  −4  EST ERR DERR JERR  −5  10  −11  −5  10 3  10  4  5  10  6  10  10  10 3  10  4  5  10  N  6  10  10  4  10  (b) ε = 10−2 12  11  11  11  10  10  10  9  9  9  8  8  8  7  7  5  10  (d) ε = 1  ratio 6  10  8  10  7  ratio  N  7  10  (c) ε = 10−4  12  4  6  10  N  12  10  5  10  N  (a) ε = 1  6 3 10  EST ERR DERR JERR  10  −6  10  6 3 10  4  5  10  ratio 6  10  10  N  6 4 10  5  10  6  10  7  10  8  10  N  (e) ε = 10−2  (f) ε = 10−4  Figure 2.2: Example 1: Convergence behavior for ε = 1, 10−2 , 10−4 and p = 2.  we further show the ratio of the estimator and the energy error, resulting in values that are bounded independently of ε. Figure 2.5 shows the adaptive meshes after 7 refinement steps, both for p = 1 and p = 2. We clearly see strong mesh refinement along x = 0, indicating that the estimator η is effective in locating the internal layer there.  2.5.3  Example 3  Next, let us consider an example with a boundary layer with width of order O(ε). In the domain Ω = (−1, 1)2 , we take a = (0, 1)T , b ≡ 0. Then we select f ≡ 0 and Dirichlet boundary conditions such that the analytical solution to (2.2.1) is given by u(x, y) = x  1−e  y−1 ε 2  1 − e− ε  .  As ε goes to zero, there exists a boundary layer near y = 1, with width of order O(ε). 49  2.5. Numerical experiments  (a) ε = 1, p = 1  (b) ε = 10−2 , p = 1  (c) ε = 10−4 , p = 1  (d) ε = 1, p = 2  (e) ε = 10−2 , p = 2  (f) ε = 10−4 , p = 2  Figure 2.3: Example 1: Adaptively generated meshes after 7 refinement steps. In Figure 2.6, we show the convergence behavior for ε = 10−2 , 10−4 and p = 1, 2. We observe, as in previous two examples, linear and quadratic p convergence rates of the optimal order O(N − 2 ) for both the energy error and the estimator. The additional curves “DERR” and “JERR” associated with the error | u − uh |O,T go to zero faster than the energy error. We also observe that the ratio of the energy error and the estimator is bounded independently of ε. In Figure 2.7, we show the adaptive meshes after 7 refinement steps, both for p = 1 and p = 2. We obtain the strong mesh refinement near y = 1 as expected.  2.5.4  Example 4  Next, we test the estimator for a problem with convection that is not aligned with the mesh. We take Ω = (−1, 1)2 , a = (− sin π6 , cos π6 )> and b = f = 0,  50  2.5. Numerical experiments 0  0  10  10  8 ε=10−2 ε=10−3 7  −1  −1  10  10  6 −2  −2  10  10  5  −3  4  −3  10  10  3 −4  −4  10  10 EST ERR DERR JERR  EST ERR DERR JERR  −5  2  −5  10  10 2  3  10  1 2  10  3  10  2  10  N1/2  3  10  10  N1/2  (a) ε = 10−2 , p = 1  N1/2  (b) ε = 10−3 , p = 1  −1  (c) p = 1  −1  10  10  12 ε=10−2 ε=10−3  11.5 −2  10  −2  10  11 −3  10  10.5  −3  10  10  −4  10 −4  10  9.5 −5  10  9 −5  10  8.5  −6  10  8  −6  EST ERR DERR JERR  10  EST ERR DERR JERR  −7  10  −7  7.5  −8  10  10 3  10  4  10  5  10  6  10  7  10  4  10  5  (d) ε = 10−2 , p = 2  6  10  N  10  7  10  7 3 10  4  5  10  10  N  6  10  7  10  N  (e) ε = 10−3 , p = 2  (f) p = 2  Figure 2.4: Example 2: Convergence behavior for ε = 10−2 , 10−3 and p = 1, 2. and consider the boundary conditions u=0 1−y u = tanh( ) ε  1 x u= tanh( ) + 1 2 ε  on x = −1 and y = 1, on x = 1, on y = −1.  The boundary condition is almost discontinuous near the point √ (0, −1) and √ causes u to have an internal layer of width O( ε) along y + 3x = −1, with values u = 0 to the left and u = 1 to the right, as well as a boundary layer along the outflow boundary. There is no exact solution available to this problem. Again, we incorporate the inhomogeneous boundary conditions as described in [19]. In Figure 2.8, we plot the values of η for ε = 10−2 , 10−4 and p = 1 p and p = 2 against N − 2 , respectively. We also indicate the minimum mesh size achieved through the adaptive refinement. We observe that, when the minimum mesh size is of order O(ε), i.e., the local mesh Péclet number is 51  2.5. Numerical experiments  (a) ε = 10−2 , p = 1  (b) ε = 10−3 , p = 1  (c) ε = 10−2 , p = 2  (d) ε = 10−3 , p = 2  Figure 2.5: Example 2: Adaptively generated meshes after 7 refinement steps.  p  of order one, the error estimator converges with the optimal rate O(N − 2 ). Figure 2.9 depicts the adaptive meshes after 7 refinement steps. The layers are resolved, with mesh refinement being more pronounced for ε = 10−4 .  2.5.5  Example 5  This example is known as the double-glazing problem. We take Ω = (−1, 1)2 , a = (2y(1 − x2 ), −2x(1 − y 2 ))> and b = f = 0. The boundary conditions are: u = tanh( u=0  1−y ) ε  on x = 1, on x = −1, y = ±1.  Note that ∇ · a = 0, so that the conditions (2.2.2) and (2.2.3) are satisfied. Again, the almost discontinuous boundary conditions lead to boundary layers near the corners.  52  2.5. Numerical experiments 1  1  10  10  8  7  0  10  0  10  6 −1  10  −1  10  5 −2  10  4  −2  10 −3  10  3 −3  10  −4  EST ERR DERR JERR  10  EST ERR DERR JERR  −5  2  10 1  2  10  3  10  2  10  ε=10−2 ε=10−4  −4  10  3  10  N1/2  1 1 10  4  10  10  2  N1/2  (a) ε = 10−2 , p = 1  3  10  4  10  10  N1/2  (b) ε = 10−4 , p = 1  (c) p = 1  1  10  11 1  10 0  10  10 0  10 −1  10  9 −1  −2  10  10  8  −3  10  7  −2  10  −4  10  6 −3  10 −5  10  5 −4  EST ERR DERR JERR  −6  10  EST ERR DERR JERR  10  −7  4  10 3  4  10  10  5  10  6  10  N  (d) ε = 10−2 , p = 2  7  10  4  10  ε=10−2 ε=10−4  −5  10  5  10  6  10  7  10  N  (e) ε = 10−4 , p = 2  8  10  3 2 10  3  10  4  10  5  10  6  10  7  10  8  10  N  (f) p = 2  Figure 2.6: Example 3: Convergence behavior for ε = 10−2 , 10−4 and p = 1, 2. Figure 2.10 shows the numerical results for this example. Asymptotically, the optimal convergence orders are attained for the estimator, once the smallest mesh size is of order O(ε) and the boundary layers are resolved. The adaptive meshes are shown in Figure 2.11. Strong mesh refinement near the boundary is observed.  2.5.6  Example 6  In the last example, we verify Theorem 2.3.6 numerically. We consider the reaction-diffusion equation ( (2.3.6) in Ω = (0, 1)2 with b = 2, and select f such that the analytical solution to ( 2.3.6) is given by √ √ √ ! √ ! ex/ ε + e(1−x)/ ε ey/ ε + e(1−y)/ ε √ √ u(x, y) = 1 − 1− . 1 + e1/ ε 1 + e1/ ε As ε tends to zero, there are boundary layers near Γ with width of order √ O( ε). Figure 2.12 shows the numerical results for this example. We get a convergence behavior for the energy error that is similar to the convectiondiffusion examples, that is, the error converges with the optimal order 53  2.5. Numerical experiments  (a) ε = 10−2 , p = 1  (b) ε = 10−4 , p = 1  (c) ε = 10−2 , p = 2  (d) ε = 10−4 , p = 2  Figure 2.7: Example 3: Adaptively generated meshes after 7 refinement steps.  2  2  10  10 ε=10−2  ε=10−2  ε=10−4  ε=10−4 1  10  hmin=3.45e−04 1  hmin=3.45e−04  10  hmin=1.73e−04 0  hmin=1.73e−04  10  h =2.21e−02  min  10  =8.63e−05  min  hmin=8.63e−05 h 0  h  −1  10  hmin=2.21e−02  =1.10e−02  min  hmin=1.10e−2  −2  10  hmin= 5.52e−03 −1  h  10  =5.52e−3  min  −3  10  −2  −4  10  10 1  10  2  3  10  10  N1/2  4  10  3  10  4  10  5  6  10  10  7  10  8  10  N  Figure 2.8: Example 4: Convergence behavior with ε = 10−2 , 10−4 and p = 1 (left), 2 (right).  54  2.5. Numerical experiments  (a) ε = 10−2 , p = 1  (b) ε = 10−4 , p = 1  (c) ε = 10−2 , p = 2  (d) ε = 10−4 , p = 2  Figure 2.9: Example 4: The adaptively generated meshes after 7 refinement steps.  1  10 ε=10−2  1  10  ε=10−2  −3  −3  ε=10  ε=10 0  10  0  10  hmin=2.76e−3  −1  10 hmin=2.76e−3  hmin=1.38e−3  hmin=2.21e−2  hmin=1.38e−3 h  −1  10  h  =2.21e−2  hmin=6.91e−4  −2  10  =6.91e−4  min  hmin=1.10e−2  min  hmin=1.10e−2 hmin=5.52e−3  hmin=5.52e−3  −3  10  −2  10  −4  10 1  10  2  3  10  10  N1/2  4  10  3  10  4  10  5  10  6  10  7  10  N  Figure 2.10: Example 5: Convergence behavior for ε = 10−2 , 10−3 and p = 1 (left), 2 (right).  55  2.6. Conclusions  (a) ε = 10−2 , p = 1  (b) ε = 10−3 , p = 1  (c) ε = 10−2 , p = 2  (d) ε = 10−3 , p = 2  Figure 2.11: Example 5: Adaptively generated meshes after 7 refinement steps. p  O(N − 2 ). The ratio of the energy error and the estimator is bounded by 7 as p = 1, and by 11 as p = 2. Figure 2.13 is the adaptive mesh after 7 refinement steps. The mesh is refined strongly near the boundary of Ω, as expected.  2.6  Conclusions  In this chapter, we have derived a robust a-posteriori error estimator for a convection-diffusion equation. The estimator yields upper and lower bounds for the error measured in terms of the energy norm and a semi-norm associated with the convective term of the equation. The constants in these bounds are independent of the Péclet number of the problem; in this sense the estimator is robust in convection-dominated regimes. Our numerical results indicate that the estimator is effective in locating and resolving interior and boundary layers. Once the local mesh Péclet number is of order one, the energy error converges with optimal order, and is dominating the error | u − uh |O,T related to convection. In fact, we observe numerically that the 56  2.6. Conclusions error indicator is robust, reliable and efficient for estimating the error in the energy norm. In all our examples the effectivity ratio of the estimator and the energy error is around 6 for p = 1, and increases to around 11 for p = 2. 0  10  8 EST ERR  EST ERR −1  10  7  6 −1  10  5  −2  10  4  −2  10  3  2  ε=10−2 ε=10−4  −3  −3  10  10 1  2  10  3  10  1  10  2  10  3  10  N1/2  1 1 10  4  10  10  2  N1/2  (a) ε = 10−2 , p = 1 −1  4  10  10  N1/2  (b) ε = 10−4 , p = 1  (c) p = 1  −1  10  3  10  10  11  EST ERR  EST ERR 10 −2  10 −2  10  9 −3  10  8 −3  10  7  −4  10  6  −4  10  −5  10  5  ε=10−2 ε=10−4  −5  −6  10  10 3  10  4  10  5  10  6  10  N  (d) ε = 10−2 , p = 2  7  10  4  10  5  6  10  10  N  (e) ε = 10−4 , p = 2  7  10  4 3 10  4  10  5  10  6  10  7  10  N  (f) p = 2  Figure 2.12: Example 6: Convergence behavior for ε = 10−2 , 10−4 and p = 1, 2.  57  2.6. Conclusions  (a) ε = 10−2 , p = 1  (b) ε = 10−4 , p = 1  (c) ε = 10−2 , p = 2  (d) ε = 10−4 , p = 2  Figure 2.13: Example 6: Adaptively generated meshes after 7 refinement steps.  58  2.7. Bibliography  2.7  Bibliography  [1] M. Ainsworth and J.T. Oden. A-posteriori Error Estimation in Finite Element Analysis. Wiley-Interscience Series in Pure and Applied Mathematics. Wiley, New York, 2000. [2] L. El Alaoui, A. Ern, and E. Burman. A-priori and a-posteriori analysis of non-conforming finite elements with face penalty for advectiondiffusion equations. IMA J. Numer. Anal., 27:151–171, 2007. [3] D.N. Arnold. An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal., 19:742–760, 1982. [4] D.N. Arnold, F. Brezzi, B. Cockburn, and L.D. Marini. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal., 39:1749–1779, 2002. [5] W. Bangerth, R. Hartmann, and G. Kanschat. Differential Equations Analysis Library, Technical http://www.dealii.org.  deal.II Reference.  [6] W. Bangerth, R. Hartmann, and G. Kanschat. deal.II — a general purpose object oriented finite element library. ACM Trans. Math. Software, 33:24:1–24:27, 2007. [7] R. Becker, P. Hansbo, and M.G. Larson. Energy norm a-posteriori error estimation for discontinuous Galerkin methods. Comput. Methods Appl. Mech. Engrg., 192:723–733, 2003. [8] R. Becker, P. Hansbo, and R. Stenberg. A finite element method for domain decomposition with non-matching grids. Modél. Math. Anal. Numér., 37:209–225, 2003. [9] R. Bustinza, G. Gatica, and B. Cockburn. An a-posteriori error estimate for the local discontinuous Galerkin method applied to linear and nonlinear diffusion problems. J. Sci. Comp., 22:147–185, 2005. [10] B. Cockburn. Discontinuous Galerkin methods for convectiondominated problems. In T. Barth and H. Deconink, editors, High-Order Methods for Computational Physics, volume 9 of Lect. Notes Comput. Sci. Engrg., pages 69–224. Springer–Verlag, 1999.  59  BIBLIOGRAPHY [11] B. Cockburn, G.E. Karniadakis, and C.-W. Shu, editors. Discontinuous Galerkin Methods. Theory, Computation and Applications, volume 11 of Lect. Notes Comput. Sci. Engrg. Springer–Verlag, Heidelberg, 2000. [12] B. Cockburn and C.-W. Shu. Runge–Kutta discontinuous Galerkin methods for convection–dominated problems. J. Sci. Comp., 16:173– 261, 2001. [13] A. Ern, A.F. Stephansen, and M. Vohralı́k. Guaranteed and robust discontinuous Galerkin a-posteriori error estimates for convectiondiffusion-reaction problems. J. Comput. Appl. Math., 234:114–130, 2010. [14] V. Girault and P.-A. Raviart. Finite Element Methods for NavierStokes Equations: Theory and Algorithms, volume 5 of Springer Series in Computational Mathematics. Springer–Verlag, 1986. [15] K. Harriman, P. Houston, B. Senior, and E. Süli. hp–Version discontinuous Galerkin methods with interior penalty for partial differential equations with nonnegative characteristic form. In C.-W. Shu, T. Tang, and S.-Y. Cheng, editors, Recent Advances in Scientific Computing and Partial Differential Equations, volume 330 of Contemporary Mathematics, pages 89–119. AMS, 2003. [16] P. Houston, D. Schötzau, and T. Wihler. hp-Adaptive discontinuous Galerkin finite element methods for the Stokes problem. In P. Neittaanmäki, T. Rossi, S. Korotov, E. Oñate, J. Périaux, and D. Knörzer, editors, Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering, Volume II, 2004. [17] P. Houston, D. Schötzau, and T. Wihler. Energy norm a-posteriori error estimation for mixed discontinuous Galerkin approximations of the Stokes problem. J. Sci. Comp., 22:357–380, 2005. [18] P. Houston, D. Schötzau, and T. Wihler. An hp-adaptive discontinuous Galerkin FEM for nearly incompressible linear elasticity. Comput. Methods Appl. Mech. Engrg., 195:3224–3246, 2006. [19] P. Houston, D. Schötzau, and T. P. Wihler. Energy norm a-posteriori error estimation of hp-adaptive discontinuous Galerkin methods for elliptic problems. Math. Models Methods Appl. Sci., 17:33–62, 2007. [20] C. Johnson and J. Pitkäranta. 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Über ein Variationsprinzip zur Lösung von Dirichlet Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Math. Abh. Sem. Univ. Hamburg, 36:9–15, 1971. [26] W.H. Reed and T.R. Hill. Triangular mesh methods for the neutron transport equation. Technical Report Tech. Report LA-UR-73-479, Los Alamos Scientific Laboratory, Los Alamos, New Mexico, 1973. [27] B. Rivière and M.F. Wheeler. A-posteriori error estimates for a discontinuous Galerkin method applied to elliptic problems. Comput. Math. Appl., 46:141–163, 2003. [28] G. Sangalli. A uniform analysis of nonsymmetric and coercive linear operators. SIAM J. Math. Anal., 36:2033–2048, 2005. [29] G. Sangalli. Robust a-posteriori estimator for advection-diffusionreaction problems. Math. Comp., 77:41–70, 2008. [30] R. Verfürth. A Review of A-Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, Stuttgart, 1996. [31] R. Verfürth. A-posteriori error estimators for convection-diffusion equations. Numer. Math., 80:641–663, 1998. [32] R. Verfürth. Robust a-posteriori error estimates for stationary convection-diffusion equations. SIAM J. Numer. Anal., 43:1766–1782, 2005. 61  Chapter 3  An hp-version a-posteriori error estimator 2 3.1  Introduction  It is well-known that solutions to convection-diffusion equations may have boundary or internal layers of small width where their gradients change extremely rapidly. One way to efficiently approximate convection-diffusion problems is to use adaptive finite element methods that are capable of locally refining the meshes in the vicinity of these layers. The decision when to refine an element is usually based on a-posteriori estimates of the errors (or functionals thereof). For excellent surveys on adaptive finite elements and a-posteriori error estimations, we refer to [1, 34]. The design of robust a-posteriori error estimates has attracted a lot of attention recently. Here, by robustness we mean that the estimates yield upper and lower bounds for the errors measured in suitable norms that differ by a factor that is independent of the Péclet number of the problem. Several robust and semi-robust a-posteriori estimates can now be found in the literature. For conforming and mixed finite element methods, we refer to the recent papers [30, 31, 35, 36, 37] and the references therein. The robustness of a-posteriori error estimates for non-conforming and discontinuous Galerkin finite element methods has been studied in [2, 15, 16, 32] and the references therein. However, all the papers above are concerned with h-version finite element methods. These methods are based on employing a fixed, usually low polynomial degree. As a consequence, adaptive h-version methods yield at most algebraic rates of convergence. This is in contrast to hp-version finite element methods, where the combination of h-refinement and p-refinement typically results in exponential rates of convergence, see, e.g., [33] and the references therein. 2  A version of this chapter has been accepted for publication. Zhu, L. and Schötzau, D. (2009) A robust a-posteriori error estimate for hp-adaptive DG methods for convectiondiffusion equations. IMA Journal of Numerical Analysis.  62  3.1. Introduction Discontinuous Galerkin methods are naturally suited for realizing hpadaptivity. Indeed, being based on discontinuous finite element spaces, these methods can easily deal with irregularly refined meshes and locally varying polynomial degrees. For recent accounts on the state-of-the-art of DG methods we refer to reader to [4, 11, 12, 20] and the references therein. Several approaches to energy norm error estimation for DG methods applied to elliptic problems can be found in the literature; see [7, 8, 24]. Extensions to hp-version DG methods have been successfully developed in the recent papers [18, 19, 23]. In this chapter, we extend the h-version technique proposed in Chapter 2 to the hp-version of the DG method and derive a robust a-posteriori error estimate for convection-diffusion equations. Similarly to [36], we introduce as an error measure the natural energy norm and a dual norm associated with the convection. In this measure, we derive upper and lower bounds of the errors, which are explicit in the local mesh sizes and polynomial degrees. The constants in the upper and lower bounds are independent of the local mesh size, although the one in the lower bound weakly depends on the polynomial degrees. More importantly, they are independent of the Péclet number of the problem; hence, our estimate is robust. In our analysis, the error is decomposed into a conforming part and a remainder using an averaging operator as in the approaches of [19, 24]. The conforming contribution of the error can be dealt with using standard techniques, while the remainder is shown to be controlled by the jump. A major ingredient of our analysis is a new L2 -norm approximation property for the hp-version averaging operator. In [19], an optimal H 1 -seminorm approximation property was established on regular meshes without hanging nodes. It was then extended in [23] to irregular meshes. In [10], optimal L2 -norm (and H 1 -seminorm) estimates were proven on regular meshes and for fixed polynomial degrees. We extend the L2 -norm estimate to the case of 1-irregular meshes consisting of parallelograms and variable polynomial degrees. Similarly to [23], we also use an auxiliary mesh underlying the possibly irregular computational mesh. However, to obtain the L2 -norm estimate, we also allow the auxiliary mesh to be 1-irregular. We present a series of numerical tests where we use our a-posteriori error estimator as an error indicator in an hp-adaptive algorithm. To decide whether to apply h- or p-refinement of marked elements, we employ the smoothness estimation strategy developed in [21, 22]. Our numerical examples indicate that our algorithm is effective in locating and resolving boundary layers. Moreover, we observe that both the energy error and the error indicator converge exponentially once the local mesh size is sufficiently 63  3.2. Interior penalty discretization small. The outline of the rest of this chapter is as follows. In Section 3.2, we introduce hp-adaptive discontinuous Galerkin methods for a convectiondiffusion model problem. In Section 3.3, we state and discuss our robust a-posteriori error estimate. The proof of this estimate is carried out in Sections 3.4 and 3.5. In Section 3.6, we present a series of numerical tests that illustrate the theoretical results. Finally, in Section 3.7, we end with concluding remarks.  3.2  Interior penalty discretization  In this section, we introduce an hp-adaptive interior penalty discontinuous Galerkin finite element method for the discretization of convection-diffusion equations.  3.2.1  Model problem  We consider the convection-diffusion model problem: −ε∆u + a(x) · ∇u = f (x) u=0  in Ω, on Γ.  (3.2.1)  Here, Ω is a bounded Lipschitz polygon in R2 with boundary Γ = ∂Ω. The parameter ε > 0 is the (constant) diffusion coefficient, the vector-valued function a(x) a given flow field, and the function f (x) a generic right-hand side in L2 (Ω). The coefficient a(x) is assumed to belong to W 1,∞ (Ω)2 and to satisfy ∇·a=0 in Ω. (3.2.2) Without loss of generality, we shall assume that kakL∞ (Ω) and the length scale of Ω are one so that ε−1 is the Péclet number of the problem. The standard weak form of the convection-diffusion equation in (3.2.1) is to find u ∈ H01 (Ω) such that Z Z  A(u, v) = f v dx ∀ v ∈ H01 (Ω). (3.2.3) ε∇u · ∇v + a · ∇uv dx = Ω  Ω  Under assumption (3.2.2), the variational problem (3.2.3) is uniquely solvable.  64  3.2. Interior penalty discretization  3.2.2  Discretization  Throughout, we assume that the computational domain Ω can be partitioned into shape-regular (sequences of) meshes T = {K} of parallelograms K. b = (−1, 1)2 Each element K ∈ T is the image of the reference square K b → K. As usual, we denote under an affine elemental mapping FK : K by hK the diameter of K. We store the elemental diameters in the mesh size vector h = {hK : K ∈ T }. We will make use of the following sets of vertices and edges. For an element K ∈ T , we denote by N (K) the set of its four vertices. A node ν of a finite element mesh T is the vertex of at least one element K ∈ T . The node ν is called an interior node if ν ∈ / Γ; similarly, it is a boundary node if ν ∈ Γ. We denote by NI (T ), NB (T ) the sets of interior and boundary nodes, respectively, and set N (T ) = NI (T ) ∪ NB (T ). Further, we denote by E(K) the set of its four elemental edges. If the intersection E = ∂K ∩ ∂K 0 of two elements K, K 0 ∈ T is a proper line segment (and not a single point), we call E an interior edge of T . The set of all interior edges is denoted by EI (T ). Analogously, if the intersection E = ∂K ∩ Γ of an element K ∈ T and Γ is a proper line segment, we call E a boundary edge of T . The set of all boundary edges of T is denoted by EB (T ). Moreover, we set E(T ) = EI (T ) ∪ EB (T ). We denote by hE the length of an edge E ∈ E(K) or E ∈ E(T ). In our analysis, we allow for 1-irregularly refined meshes T where each elemental edge E ∈ E(K) may contain at most one hanging node located in the middle of it. That is, we either have E ∈ E(T ) or E can be written as E = E1 ∪ E2 , for two edges E1 and E2 in E(T ) that satisfy hE1 = hE2 = hE /2. With each element K of a mesh T , we associate a polynomial degree pK ≥ 1, introduce the degree vector p = { pK : K ∈ T }, and set |p| = maxK∈T pK . We assume that p is of bounded local variation. That is, there is a constant % ≥ 1 independent of the particular mesh in a sequence of meshes such that, for any pair of neighboring elements K, K 0 ∈ T , we have %−1 ≤ pK /pK 0 ≤ %. For E ∈ E(T ), we introduce the edge polynomial degree pE by ( max{pK , pK 0 }, E = ∂K ∩ ∂K 0 ∈ EI (T ), pE = (3.2.4) pK , E = ∂K ∩ Γ ∈ EB (T ). For a partition T of Ω and a degree vector p on T , we then define the hp-version discontinuous Galerkin finite element space by b K ∈ T }, Sp (T ) = { v ∈ L2 (Ω) : v|K ◦ FK ∈ QpK (K),  (3.2.5) 65  3.3. Robust a-posteriori error estimates b denoting the set of all polynomials on the reference square K b with Qp (K) of degree less or equal than p in each variable. We now consider the following discontinuous Galerkin method for the approximation of the convection-diffusion problem (3.2.1): Find uhp ∈ Sp (T ) such that Z f v dx (3.2.6) Ahp (uhp , v) = Ω  for all v ∈ Sp (T ), with the bilinear form Ahp given by XZ (ε∇u · ∇v + a · ∇uv) dx Ahp (u, v) = K∈T  −  K  X Z  E∈E(T ) E  {{ε∇u}} · [[v]] ds −  X Z  E∈E(T ) E  {{ε∇v}} · [[u]] ds  XZ X εγp2 Z E a · nK uv ds [[u]] · [[v]] ds − + hE E K∈T ∂Kin ∩Γin E∈E(T ) XZ + a · nK (ue − u)v ds. K∈T  ∂Kin \Γ  Here, for a piecewise smooth function, the gradient operator ∇ is taken element by element. The constant γ > 0 is the interior penalty parameter. To ensure the stability and well-posedness of the discontinuous Galerkin discretization, it is well-known that it has to be chosen sufficiently large, independently of h, p and ε. Finally, we denote by Γin and ∂Kin the inflow parts of Γ and K ∈ T , respectively: Γin = { x ∈ Γ : a(x) · n(x) < 0 },  ∂Kin = { x ∈ ∂K : a(x) · nK (x) < 0 }.  Remark 3.2.1 The discretization in (3.2.6) is based on the original upwind discretization in [25, 29] for the convective term and the classical symmetric interior penalty discretization in [3, 4, 28] for the diffusion term.  3.3  Robust a-posteriori error estimates  In this section, our main results are presented and discussed.  3.3.1  Norms  We begin by introducing the norms in which the errors are measured. First, we introduce the following energy norm associated with the discontinuous 66  3.3. Robust a-posteriori error estimates Galerkin discretization of the diffusive term: X k v k2E,T = εk∇vk2L2 (K) + ejumpp,T (v)2 , K∈T  ejumpp,T (v)2 =  X γεp2 E k[[v]]k2L2 (E) . hE  (3.3.1)  E∈E(T )  Next, we define |q|? =  R  q · ∇v dx k v kE,T  Ω  sup v∈H01 (Ω)\{0}  ∀ q ∈ L2 (Ω)2 .  Analogously to [32, 36], we introduce the following semi-norm associated with the discretization of the convection term: | v |O,T = |av|2? + ojumpp,T (v)2 , X hE ojumpp,T (v)2 = k[[v]]k2L2 (E) . εpE  (3.3.2)  E∈E(T )  Notice that hE ε−1 is the local mesh Péclet number.  3.3.2  A robust a-posteriori error estimate  Let now uhp ∈ Sp (T ) be the discontinuous Galerkin approximation obtained by (3.2.6). Moreover, let fhp and ahp denote piecewise polynomial approximations in Sp (T ) to the right-hand side f and the flow field a, respectively. For each element K ∈ T , we introduce the following local error indicator ηK which is given by the sum of the three terms 2 2 2 ηK = ηR + ηE + ηJ2K . K K  (3.3.3)  The first term ηRK is the interior residual defined by 2 2 2 ηR = ε−1 p−2 K hK kfhp + ε∆uhp − ahp · ∇uhp kL2 (K) . K  The second term ηEK is the edge residual given by 2 ηE = K  1 2  X  E∈∂K\Γ  2 ε−1 p−1 E hE k[[ε∇uhp ]]kL2 (E) .  67  3.3. Robust a-posteriori error estimates The last residual ηJK measures the jumps of the approximate solution uhp :  2 3  γ εpE hE 1 X + k[[uhp ]]k2L2 (E) ηJ2K = 2 hE εpE E∈∂K\Γ  X  γ 2 εp3 hE E + + k[[uhp ]]k2L2 (E) . hE εpE E∈∂K∩Γ  We also introduce the local data approximation term   2 2 2 Θ2K = ε−1 p−2 h kf − f k + k(a − a ) · ∇u k 2 2 hp hp hp K L (K) L (K) . K  We then introduce the global error estimator and data approximation error X X 2 η2 = ηK , Θ2 = Θ2K . (3.3.4) K∈T  K∈T  In Chapters 3, 4 and 5, we use the symbols . and & to denote bounds that are valid up to positive constants independently of the mesh sizes, the polynomial degree distributions and ε. Theorem 3.3.1 (Reliability) Let u be the solution of (3.2.1) and uhp ∈ Sp (T ) its DG approximation obtained by (3.2.6). Let the error estimator η and the data approximation error Θ be defined by (3.3.4). Then we have the a-posteriori error bound k u − uhp kE,T + | u − uhp |O,T . η + Θ. Remark 3.3.2 The power γ 2 p3E in ηJK is slightly suboptimal with respect to the one used in the jump terms of the energy norm (3.3.1). This suboptimality is due to the possible presence of hanging nodes in T . Indeed, for conforming meshes, the conforming hp-version Clément interpolant constructed in [26] can be employed in our proof; see also [19]. As a consequence, Theorem 3.3.1 holds true with the following version of ηJK :   γεp2E 1 X hE 2 η JK = + k[[uhp ]]k2L2 (E) 2 hE εpE E∈∂K\Γ  X  γεp2 hE E + + k[[uhp ]]k2L2 (E) . hE εpE E∈∂K∩Γ  On the other hand, the numerical results in Section 3.6 indicate that the two versions of ηJK yield practically identical results on 1-irregularly refined square meshes. 68  3.4. Proofs Our next theorem derives a lower bound for the error measured in terms of the energy norm and the semi-norm | · |O,T . For p-independence in both the upper and lower bounds, special weighting techniques seem to be necessary which we do not pursue in this chapter; see [9]. Here, we only present a weakly p-dependent lower bound for the a-posteriori error estimator ηK defined above. Theorem 3.3.3 (Efficiency) Let u be the solution of (3.2.1) and uhp ∈ Sp (T ) its DG approximation obtained by (3.2.6). Let the error estimator η and the data approximation error Θ be defined by (3.3.4). Then for any δ ∈ (0, 12 ) we have the bound 1  η . |p|δ+1 k u − uhp kE,T + |p|2δ+1 | u − uhp |O,T + |p|2δ+ 2 Θ. As the constants in the upper and lower bounds in Theorem 3.3.1 and Theorem 3.3.3 are independent of the Péclet number of (3.2.1), the estimator η is robust.  3.4  Proofs  In this section, we present the proofs of Theorems 3.3.1 and 3.3.3.  3.4.1  Stability and auxiliary forms  The following inf-sup condition for the continuous form A is the crucial stability result in our analysis. It holds with an absolute constant, which can be immediately inferred from the proof of Lemma 2.4.4. Lemma 3.4.1 Assume (3.2.2). Then we have inf  u∈H01 (Ω)\{0}  sup v∈H01 (Ω)\{0}  (k u kE,T  A(u, v) 1 ≥ . + |au|? ) k v kE,T 3  Next, we split the discontinuous Galerkin form Ahp into two parts, see Chapter 2, and define XZ X εγp2 Z E e Ahp (u, v) = (ε∇u · ∇v + a · ∇uv) dx + [[u]] · [[v]] ds h E E K∈T K E∈E(T ) XZ XZ − a · nK uv ds + a · nK (ue − u)v ds, K∈T  Khp (u, v) = −  ∂Kin ∩Γin  X Z  E∈E(T ) E  K∈T  {{ε∇u}} · [[v]] ds −  ∂Kin \Γ  X Z  E∈E(T ) E  {{ε∇v}} · [[u]] ds. 69  3.4. Proofs We shall use the above auxiliary forms to express both the continuous form A in (3.2.3) and the discontinuous Galerkin form Ahp in (3.2.6). Indeed, we have  3.4.2  ehp (u, v), A(u, v) = A ehp (u, v) + Khp (u, v), Ahp (u, v) = A  u, v ∈ H01 (Ω),  u, v ∈ Sp (T ).  (3.4.1) (3.4.2)  Auxiliary meshes  We shall make use of an auxiliary 1-irregular mesh Te of parallelograms, similarly to the approach in [23], which is obtained from T as follows. Let K ∈ T . If all four elemental edges are edges of the mesh T , that is, if E(K) ⊆ E(T ), we leave K untouched. Otherwise, at least one of the elemental edges of K contains a hanging node. In this case, we replace K by the four parallelograms obtained from bisecting the elemental edges of K. This construction is illustrated in Figure 3.1. Clearly, the mesh Te is a refinement of T ; it is also shape-regular and 1-irregular. We denote by ER (T ) the set of edges in E(T ) that have been refined in the above process. We denote by NA (Te ) the set of vertices in N (Te ) and EA (Te ) the set of edges in E(Te ) which are inside an element K of T , respectively. Moreover, we write R(K) for the elements in Te that are inside K. If K is unrefined, R(K) = {K}. Otherwise, the set R(K) consists of four newly created elements.  =⇒  Figure 3.1: The construction of the auxiliary mesh Te from T .  Next, we introduce the following auxiliary discontinuous Galerkin finite element space on the mesh Te : b K e ∈ Te }, Spe (Te ) = { v ∈ L2 (Ω) : v|Ke ◦ FKe ∈ QpKe (K),  e is defined by pKe = pK , where the auxiliary polynomial degree vector p e for K ∈ R(K). Thus, we clearly have the inclusion Sp (T ) ⊆ Spe (Te ). In complete analogy to (3.3.1) and (3.3.2), the energy and convective norms 70  3.4. Proofs associated with the auxiliary mesh Te are given by X 2 εk∇vk2L2 (K) k v k2E,Te = e + ejumpp e ,Te (v) , | v |2O,Te  =  e Te K∈ |av|2?  (3.4.3)  2  + ojumppe ,Te (v) ,  where the auxiliary edge polynomial degrees pEe for the jump terms over Te are defined as in (3.2.4), using the auxiliary degrees pKe . Obviously, we have k v kE,T = k v kE,Te ,  | v |O,T = | v |O,Te ,  for all v ∈ H01 (Ω). Furthermore, the following result holds. Lemma 3.4.2 Let v ∈ Spe (Te ) + H01 (Ω) be such that [[v]]|E = [[w]]|E for all E ∈ E(Te ), for a function w ∈ Sp (T ) + H01 (Ω). Then we have ejumpp,T (w) . ejumppe ,Te (v) . ejumpp,T (w),  ojumpp,T (w) . ojumppe ,Te (v) . ojumpp,T (w). Proof : Since w ∈ Sp (T ) + H01 (Ω), we have that [[v]]|E = [[w]]|E = 0 over newly created edges in EA (Te ). To look at the jumps over refined edges, let E ∈ ER (T ). We have E = E1 ∪ E2 with E1 and E2 in E(Te ) and hE1 = hE2 = hE /2. Thus hE2 1 hE hE1 k[[v]]k2L2 (E1 ) + k[[v]]k2L2 (E2 ) = k[[w]]k2L2 (E) . εpE1 εpE2 2 εpE  We conclude that ojumppe ,Te (v)2 =  X  E∈E(T )∩E(Te )  hE k[[w]]k2L2 (E) + εpE  X  E∈ER (T )  1 hE k[[w]]k2L2 (E) . 2 εpE  This readily implies the desired conclusion for the convective jumps. The equivalence for the diffusive jumps follows completely analogously. 2 As a consequence of this result, we also have the following estimate. Lemma 3.4.3 For v ∈ Sp (T ) + H01 (Ω), we have the bounds k v kE,T . k v kE,Te ,  | v |O,T . | v |O,Te . 71  3.4. Proofs Proof : Clearly, we have X  εk∇vk2L2 (K) =  K∈T  X  εk∇vk2L2 (K) e  e Te K∈  for v ∈ Sp (T ) + H01 (Ω). Applying Lemma 3.4.2 with w = v yields ejumpp,T (v) . ejumppe ,Te (v),  ojumpp,T (v) . ojumppe ,Te (v). 2  The assertion now follows readily.  3.4.3  Averaging operator  Our analysis is based on an hp-version averaging operator that allows us to approximate discontinuous functions by continuous ones, analogously to the one used in [10, 19, 23]. For the h-version of the finite element method, we also refer to [15, 24]. To define this operator, we let Spec (Te ) be the conforming subspace of Spe (Te ) given by S c (Te ) = Spe (Te ) ∩ H 1 (Ω). 0  e p  Theorem 3.4.4 (Averaging operator) There is operator Ihp : Sp (T ) → Spec (Te ) that satisfies X  e Te K∈  k∇(v − Ihp v)k2L2 (K) e .  X  e Te K∈  kv −  Ihp vk2L2 (K) e  X  E∈E(T )  .  X  E∈E(T )  2 p2E h−1 E k[[v]]kL2 (E) ,  (3.4.4)  2 p−2 E hE k[[v]]kL2 (E) .  (3.4.5)  The detailed proof of Theorem 3.4.4 will be presented in Section 3.5. Remark 3.4.5 In [19], an hp-version averaging operator has been constructed that satisfies (3.4.4) on regular meshes without hanging nodes. This was then extended in [23] to irregular meshes. In [10, Lemma 3.2], the estimates in (3.4.4) and (3.4.5) have been proven on regular meshes and for fixed polynomial degrees. Theorem 3.4.4 extends (3.4.5) to the case of 1-irregular meshes and variable polynomial degrees.  3.4.4  Proof of Theorem 3.3.1  Following [19, 32], we decompose the discontinuous Galerkin solution into a conforming part and a remainder: uhp = uchp + urhp ,  (3.4.6) 72  3.4. Proofs where uchp = Ihp uhp ∈ Spec (Te ) ⊂ H01 (Ω), with Ihp the approximation operator from Theorem 3.4.4. The remainder is then given by urhp = uhp − uchp = uhp − Ihp uhp ∈ Spe (Te ). By Lemma 3.4.3 and the triangle inequality, we obtain k u − uhp kE,T + | u − uhp |O,T  . k u − uhp kE,Te + | u − uhp |O,Te  . k u − uchp kE,Te + | u − uchp |O,Te + k urhp kE,Te + | urhp |O,Te  (3.4.7)  = k u − uchp kE,T + | u − uchp |O,T + k urhp kE,Te + | urhp |O,Te . In a series of lemmas, we now prove that both the continuous error u − uchp and the remainder urhp can be bounded by the estimator η and the data approximation term Θ. Lemma 3.4.6 There holds k urhp kE,Te + | urhp |O,Te . η. Proof : Since [[urhp ]]|E = [[uhp ]]|E for all E ∈ E(Te ) and uhp ∈ Sp (T ), the definition of the jump residual ηJK and Lemma 3.4.2 yield k urhp k2E,Te + | urhp |2O,Te X r 2 r 2 r 2 = εk∇urhp k2L2 (K) e + |auhp |? + ejumpp e ,Te (uhp ) + ojumpp e ,Te (uhp ) e Te K∈  .  X  e Te K∈  r 2 εk∇urhp k2L2 (K) e + |auhp |? +  X  ηJ2K .  K∈T  Hence, only the volume terms and |aurhp |? need to be bounded further. Since pE ≥ 1, Theorem 3.4.4 yields ε  X  e Te K∈  −1 k∇urhp k2L2 (K) e .γ  X εγp2 X E k[[uhp ]]k2L2 (E) . γ −1 ηJ2K . hE K∈T  E∈E(T )  To estimate |aurhp |? , we again use Theorem 3.4.4 and the fact that pE ≥ 1, |aurhp |2? . .   1 X r 2 kak2L∞ (K) ku k hp L2 (K) e e ε e Te K∈  X  E∈E(T )  X hE k[[uhp ]]k2L2 (E) . ηJ2K . 2 εpE K∈T  73  3.4. Proofs 2  This finishes the proof.  Next, we recall the following standard hp-version approximation result from [23, Lemma 3.7]: For any v ∈ H01 (Ω), there exists a function vhp ∈ Sp (T ) such that p2K pK kv − vhp k2L2 (K) + k∇(v − vhp )k2L2 (K) + kv − vhp k2L2 (∂K) . k∇vk2L2 (K) , 2 hK hK (3.4.8) for any K ∈ T . Lemma 3.4.7 For any v ∈ H01 (Ω), we have Z ehp (uhp , v − vhp ) + Khp (uhp , vhp ) . (η + Θ) k v kE,T . f (v − vhp ) dx − A Ω  Here, vhp ∈ Sp (T ) is the hp-interpolant of v in (3.4.8).  Proof : Integration by parts of the diffusive volume terms readily yields Z ehp (uhp , v − vhp ) + Khp (uhp , vhp ) = T1 + T2 + T3 + T4 + T5 , f (v − vhp ) dx − A Ω  where  T1 =  XZ  K  K∈T  T2 = − T3 = − T4 =  (f + ε∆uhp − a · ∇uhp )(v − vhp ) dx,  X  Z  E∈EI (T ) E  X Z  E∈E(T ) E  X  Z  K∈T ∂K \Γ in  +  T5 = −  X  [[ε∇uhp ]]{{v − vhp }} ds,  {{ε∇vhp }} · [[uhp ]] ds,  a · nK (uhp − uehp )(v − vhp ) ds  Z  K∈T ∂K ∩Γ in in X εγp2 E E∈E(T )  hE  a · nK uhp (v − vhp ) ds, Z  E  [[uhp ]] · [[v − vhp ]] ds.  74  3.4. Proofs To bound T1 , we first add and subtract the data approximations. From the weighted Cauchy-Schwarz inequality and the approximation properties in (3.4.8), we then readily obtain T1 .  X  2 (ηR + Θ2K ) K  K∈T  1 2  k v kE,T .  Similarly, by the Cauchy-Schwarz inequality and (3.4.8), we have X X 1 1 2 2 2 2 εpE h−1 ε−1 p−1 T2 . E kv − vhp kL2 (E) E hE k[[ε∇uhp ]]kL2 (E) E∈EI (T )  E∈EI (T )  X  .  2 ηE K  K∈T  1 2  k v kE,T .  To estimate T3 , we employ the Cauchy-Schwarz inequality, the hp-version trace inequality and the H 1 -stability of vhp from (3.4.8). This results in X  T3 .  E∈E(T )  .  X  2 εp2E h−1 E k[[uhp ]]kL2 (E)  ηJ2K  K∈T  1 X 2  K∈T  1 X 2  K∈T  εk∇vhp k2L2 (K)  1 2  2 εp−2 K hK k∇vhp kL2 (∂K)  .  X  K∈T  ηJ2K  1  2  1 2  k v kE,T .  For T4 , we apply again the Cauchy-Schwarz inequality and (3.4.8) to get X 1 1 X 2 2 2 2 εpE h−1 T4 . ε−1 p−1 E kv − vhp kL2 (E) E hE k[[uhp ]]kL2 (E) E∈E(T )  E∈E(T )  .  X  ηJ2K  K∈T  1  2  k v kE,T .  Finally, we have X 1 2 2 T5 . εγ 2 p3E h−1 k[[u ]]k 2 hp L (E) E E∈E(T )  .  X  K∈T  ηJ2K  1 2  X  E∈E(T )  2 εpE h−1 E kv − vhp kL2 (E)  1 2  k v kE,T .  The above estimates for the terms T1 through T5 imply the assertion.  2  Lemma 3.4.8 There holds: k u − uchp kE,T + | u − uchp |O,T . η + Θ. 75  3.4. Proofs Proof : Since u − uchp ∈ H01 (Ω), we have | u − uchp |O,T = |a(u − uchp )|? . Then the inf-sup condition in Lemma 3.4.1 yields: k u − uchp kE,T + | u − uchp |O,T .  A(u − uchp , v)  sup v∈H01 (Ω)\{0}  k v kE,T  .  (3.4.9)  To bound (3.4.9), let v ∈ H01 (Ω). Then, property (3.4.1) shows that Z Z ehp (uc , v). f v dx − A f v dx − Ahp (uchp , v) = A(u − uchp , v) = hp Ω  Ω  By employing the fact that v ∈ H01 (Ω) and integrating by parts the convection term, one can readily see that ehp (uc , v) = A ehp (uhp , v) + R, A hp  with R=  XZ  e Te K∈  e K   −ε∇urhp + aurhp · ∇v dx.  Furthermore, from the DG method in (3.2.6) and property (3.4.2), we have Z ehp (uhp , vhp ) + Khp (uhp , vhp ), f vhp dx = A Ω  where vhp ∈ Sp (T ) is the hp-version interpolant of v introduced in (3.4.8). Combining the above results yields Z c ehp (uhp , v − vhp ) + Khp (uhp , vhp ) − R. A(u − uhp , v) = f (v − vhp ) dx − A Ω  The estimate in Lemma 3.4.7 now shows that  |A(u − uchp , v)| . (η + Θ) k v kE,T + |R|.  (3.4.10)  It remains to bound |R|. From the Cauchy-Schwarz inequality, the definition of the norm | · |? , the conformity of v and Lemma 3.4.6, one readily obtains   |R| . k urhp kE,Te + | urhp |O,Te k v kE,T . ηk v kE,T . (3.4.11) Equations (3.4.9)–(3.4.11) imply the desired result.  2  The proof of Theorem 3.3.1 now immediately follows from the inequality (3.4.7), Lemma 3.4.6 and Lemma 3.4.8. 76  3.4. Proofs  3.4.5  Proof of Theorem 3.3.3  We first introduce the following bubble functions. On the reference element b = (−1, 1)2 , we define the weight function Ψ b (b b K x) = dist(b x, ∂ K). For K −1 an arbitrary element K ∈ T , we Rset ΨK = cK ΨKb ◦ FK , where cK is a scaling factor chosen such that K (ΨK − 1) dx = 0. Similarly, on the reference interval Ib = (−1, 1), we define the weight function ΨIb(b x) = 1 − x b2 . −1 For an interior edge E, we let ΨE = cE ΨIb ◦ FE , where FE is the affine R transformation that maps Ib onto E and cE is chosen such that E (ΨE − 1) ds = 0. Next, we show the efficiency of ηRK , ηEK and ηJK , respectively. Lemma 3.4.9 Under the assumptions of Theorem 3.3.3, there holds X 1 1 2 ( ηR ) 2 . |p| k u − uhp kE,T + |p| |a(u − uhp )|? + |p|δ+ 2 Θ. K K∈T  Proof : For any element K ∈ T , we set vK = ε−1 (fhp + ε∆uhp − ahp · ∇uhp )|K ΨαK , where α ∈ ( 12 , 1]. Applying the inverse inequality from [27, Theorem 2.5], we obtain kfhp + ε∆uhp − ahp · ∇uhp kL2 (K)  −α/2  α/2  . pαK k(fhp + ε∆uhp − ahp · ∇uhp )ΨK kL2 (K) = εpαK kvK ΨK  This leads to X 2 ηR . S2 K  with  K∈T  S2 =  X  kL2 (K) .  −α/2 2 kL2 (K) .  h2K εkvK ΨK p2α−2 K  K∈T  Since the exact solution satisfies (f + ε∆u − a · ∇u)|K = 0, we obtain, by integration by parts and insertion of the data a and f , Z X 2α−2 2 2 S = pK hK (fhp + ε∆uhp − ahp · ∇uhp )vK dx K∈T  =  X  p2α−2 h2K K  K∈T  +  X  K∈T  p2α−2 h2K K  Z  Z  K  K  K  (ε∇(u − uhp ) − a(u − uhp )) · ∇vK dx 1  α  1  −α  (((fhp − f ) + (a − ahp ) · ∇uhp )ε− 2 ΨK2 )(ε 2 vK ΨK 2 )dx. 77  3.4. Proofs Here, we have also used that vK |∂K = 0. From the proof of [27, Lemma 3.4], we have −α/2 2−α k∇vK kL2 (K) . h−1 kL2 (K) . K pK kvK ΨK By the Cauchy-Schwarz inequality, the definition of the dual norm and the data approximation error Θ, we obtain S 2 . S (|p|k u − uhp kE,T + |p||a(u − uhp )|? + |p|α Θ) . Therefore, X 2 ( ηR )1/2 . |p| k u − uhp kE,T + |p| |a(u − uhp )|? + |p|α Θ. K K∈T  2  Choosing δ = α − 1/2 finishes the proof. For any edge E ∈ E(T ), we define the sets wE = { K1 , K2 ∈ T : E = ∂K1 ∩∂K2 },  e ∈ T ∪ Te : E ∈ E(K) e }. w eE = { K  For simplicity, we also use the notation wE and w eE to denote the domain formed by the elements in wE and in w eE , respectively.  Lemma 3.4.10 Under the assumptions of Theorem 3.3.3, there holds X 1 1 2 ( ηE ) 2 . |p|δ+1 k u − uhp kE,T + |p|2δ+1 | u − uhp |O,T + |p|2δ+ 2 Θ. K K∈T  Proof : Let E = ∂K1 ∩ ∂K2 be an interior edge shared by two elements K1 , K2 ∈ T . For α ∈ (1/2, 1], set τE = [[∇uhp ]]ΨαE . We construct a bubble function ψE over wE . Case 1: Suppose that none of the end points of E is a hanging node. That is, we have E ∈ E(K1 ) ∩ E(K2 ). Lemma 2.6 of [27] then ensures the existence of a function ψE ∈ H01 (wE ) with ψE |E = τE , ψE |∂wE = 0 and 1/2  −α/2  kψE kL2 (wE ) . hE p−1 E kτE ΨE k∇ψE kL2 (wE ) .  kL2 (E) , −1/2 −α/2 hE pE kτE ΨE kL2 (E) .  (3.4.12) (3.4.13)  Case 2: Suppose that one of the end points of E is a hanging node of T ; without loss of generality, we may assume it is a hanging node of K1 . e 1 ∈ Te , such that In this case, wE is concave, and there exists an element K 78  3.4. Proofs e 1 ( K1 and K e 1 ∩ K2 = E. Thus w e 1 ∪ K2 ( wE . By Lemma 2.6 of K eE = K [27] we can find a function ψeE ∈ H01 (w eE ) with ψeE |E = τE , ψeE |∂ weE = 0 and −α/2  kψeE kL2 (weE ) . hE p−1 E kτE ΨE 1/2  −1/2  k∇ψeE kL2 (weE ) . hE  kL2 (E) ,  −α/2  pE kτE ΨE  kL2 (E) .  Now define the function ψE on wE by ψE = ψeE on w eE , and by zero oth1 erwise. Thus, we have ψE ∈ H0 (wE ) with ψE |E = τE , ψE |∂wE = 0, and (3.4.12)–(3.4.13) also hold. In both cases above, we now proceed as follows. Applying again the inverse inequality from [27, Lemma 2.4], we get α  −α  k[[∇uhp ]]kL2 (E) . pαE k[[∇uhp ]]ΨE2 kL2 (E) = pαE kτE ΨE 2 kL2 (E) . Therefore, X 2 ηE . S2 K  with  K∈T  S2 =  X  E∈EI (T )  (3.4.14)  −α  hE εkτE ΨE 2 k2L2 (E) . p2α−1 E  Since [[ε∇u]] = 0 on interior edges, integration by parts over wE yields Z Z [[ε∇(uhp − u)]]τE ds = ε(∆uhp − ∆u)ψE + ε(∇uhp − ∇u) · ∇ψE dx, E  wE  where ∆uhp and ∇uhp are understood piecewise. Using the differential equation, approximating the data and integrating by parts the convective term, we readily obtain S 2 = T1 + T2 + T3 + T4 + T5 , with T1 =  X  Z  E∈EI (T ) wE  T2 =  X  Z  E∈EI (T ) wE  T3 =  X  Z  E∈EI (T ) wE  T4 =  X  Z  E∈EI (T ) E  T5 =  X  Z  p2α−1 hE (fhp + ε∆uhp − ahp · ∇uhp )ψE dx, E p2α−1 hE (ε∇uhp − ε∇u) · ∇ψE dx, E 2α−1 pE hE a(u − uhp ) · ∇ψE dx,  p2α−1 hE a · [[uhp ]]τE ds, E  E∈EI (T ) wE    2α−1 pE hE (f − fhp ) + (ahp − a) · ∇uhp ψE dx. 79  3.4. Proofs The Cauchy-Schwarz inequality, Lemma 3.4.9 and inequality (3.4.12) yield  1 T1 . S |p|α− 2 |p| k u − uhp kE,T + |p| |a(u − uhp )|? + |p|α Θ . Similarly, we obtain  1  T2 . S |p| 2 +α k u − uhp kE,T , as well as  1  T3 . S |p| 2 +α |a(u − uhp )|? .  To bound T4 , we first notice that kΨE kL∞ (E) = cE ≡ 32 . By (3.4.14) and the definition of semi-norm | · |O,T , we conclude that X 1 1 1 1 −1 3α− 1 −α T4 . ε− 2 hE2 pE 2 k[[uhp ]]kL2 (E) ε 2 hE2 pE 2 kτE ΨE 2 kL2 (E) E∈EI (T )  . S |p|2α | u − uhp |O,T .  Finally, the data error term T5 can be bounded by X 1 T5 . ε− 2 p−1 E hE k(f − fhp ) + (ahp − a) · ∇uhp kL2 (wE ) E∈EI (T )  1  × ε 2 p2α E kψE kL2 (wE )  1  . S |p|α− 2 Θ.  Combining the above bounds for T1 through T5 , we obtain   1 1 S 2 . S |p|α+ 2 k u − uhp kE,T + |p|2α | u − uhp |O,T + |p|2α− 2 Θ . Thus,  X  K∈T  !1 2  2 ηE K  1  1  . |p|α+ 2 k u − uhp kE,T + |p|2α | u − uhp |O,T + |p|2α− 2 Θ.  Choosing δ = α − 1/2 implies the assertion.  2  Since the jumps of u vanish over the edges, we also have the following result. Lemma 3.4.11 Under the assumptions of Theorem 3.3.3, there holds X 1 ( ηJ2K )1/2 . |p| 2 k u − uhp kE,T + | u − uhp |O,T . K∈T  The proof of Theorem 3.3.3 now follows from Lemmas 3.4.9, 3.4.10 and 3.4.11. 80  3.5. Proof of Theorem 3.4.4  3.5  Proof of Theorem 3.4.4  In this section, we prove the result of Theorem 3.4.4.  3.5.1  Polynomial basis functions  We begin by introducing hp-version basis functions. To that end, let Ib = (−1, 1) be the reference interval. We denote by Zbp = { zb0p , · · · , zbpp } the b Recall that zbp = −1 and zbpp = Gauss-Lobatto nodes of order p ≥ 1 on I. 0 p p 1. We denote by Zbint = { zb1p , · · · , zbp−1 } the interior Gauss-Lobatto nodes b Let now E ∈ E(K) be an edge of an element K. The of order p on I. p b nodes in Z can be affinely mapped onto E and we denote by Z p (E) = { z0E,p , · · · , zpE,p } the Gauss-Lobatto nodes of order p on E. The points p (E) = z0E,p and zpE,p coincide with the two end points of E. The set Zint E,p E,p { z1 , · · · , zp−1 } denotes the interior Gauss-Lobatto points of order p. We write Pp (E) for the space of all polynomials of degree less or equal than p on E and define Ppint (E) = { q ∈ Pp (E) : q(z0E,p ) = q(zpE,p ) = 0 },  p (E) }. Ppnod (E) = { q ∈ Pp (E) : q(z) = 0, z ∈ Zint  By construction, we have Pp (E) = Ppint (E) ⊕ Ppnod (E). For an element K and p ≥ 1, we now define basis functions for polynomials of the form v ∈ Qp (K),  v|E ∈ PpE (E),  E ∈ E(K),  (3.5.1)  where 1 ≤ pE ≤ p is the edge polynomial degree associated with E ∈ E(K). As usual, we shall divide the basis functions into interior, edge and vertex basis functions. b = (−1, 1)2 . We denote its We first consider the reference element K b1 , . . . , E b4 and its four vertices by νb1 , . . . , νb4 , numbered as in four edges by E p Figure 3.2. Let {ϕ bi }0≤i≤p be the Lagrange basis functions associated with p the nodes Zbp . We denote by {b zi,j = (b zip , zbjp )}1≤i,j≤p the interior tensorb The interior basis functions are then product Gauss-Lobatto nodes on K. given by b int,p (b Φ x1 , x b2 ) = ϕ bpi (b x1 ) ϕ bpj (b x2 ), i,j  1 ≤ i, j ≤ p − 1.  81  3.5. Proof of Theorem 3.4.4 b3 E  νb4  1  0.8  νb3  0.6  0.4  0.2  b4 E  0  b2 E  −0.2  −0.4  −0.6  −0.8  −1  νb1−1  −0.8  −0.6  −0.4  −0.2  0  0.2  0.4  0.6  0.8  1  b1 E  νb2  Figure 3.2: Reference element with variable edge polynomial degrees: p = 5, pEb1 = 2, pEb2 = 3, pEb3 = 4, pEb4 = 1.  b1 in Figure 3.2 with edge deNext, we consider as an example the edge E b1 are gree pEb1 . The edge basis functions for E b1 ,p b E E  b Φ i  1  b ,p E  pEb  (b x1 , x b2 ) = ϕ bi  1  x2 ), (b x1 ) ϕ bp0 (b  i = 1, · · · , pEb1 − 1.  b 1 Eb1 vanishes on E b2 , E b3 and E b4 . The other edge basis functions Note that Φ i are defined analogously. Finally, we consider the vertex νb1 , which is shared b1 and E b4 ; see Figure 3.2. We then introduce the associated vertex basis by E function pEb pEb b νb1 (b Φ b2 ) = ϕ b0 1 (b x1 ) ϕ b0 4 (b x2 ). b x1 , x K  b2 and E b3 . The vertex basis functions associated with the It vanishes on E b are defined analogously. This completes the definition other vertices of K b of the shape functions on the reference element K. For an arbitrary parallelogram K, shape functions Φ on K can be defined b by using the pull-back map Φ = Φ b ◦ F −1 , from the analogous ones on K K E giving rise to shape functions ΦνK , ΦE,p and Φint,p on K. Therefore, a i i,j polynomial v of the form (3.5.1) can be expanded into v(x) =  X  ν∈N (K)  +  v(ν) ΦνK (x) X  +  X  pX E −1  E v(ziE,pE ) ΦE,p (x) i  E∈E(K) i=1  cij Φint,p i,j (x),  1≤i, j≤p−1  with expansion coefficients cij . 82  3.5. Proof of Theorem 3.4.4 We will make use of the following estimates, see Lemma 3.1 of [10]. Lemma 3.5.1 There holds: (i) For a function vb ∈ Qp (Ibd ), d = 1, 2, that vanishes at the interior Gauss-Lobatto nodes on Ibd , there holds kb v k2 2 bd . p−2 kb v k2 2 bd . L (I )  L (∂ I )  b is shared by two edges E bn (ii) If the vertex νb of the reference element K ν b bm , the associated vertex basis function Φ b can be bounded by and E b K  b νb k 2 b . p−1 p−1 . kΦ b L (K) b b K En Em  3.5.2  Extension operators  b ∈ E(K) b be an eleNext, we define extension operators over edges. Let E b E b We define L b p by mental edge of the reference element K. b Eb : P int (E) b −→ Qp (K), b L p p  qb(x) 7−→  p−1 X i=1  b b b E,p qb(b ziE,p )Φ i (x).  (3.5.2)  b Eb (b b and LEb (b By construction, L b on E, p q) = q p q ) vanishes in all the interior p b and on the other tensor-product Gauss-Lobatto nodes {b zi,j }1≤i,j≤p−1 of K b From [10, Lemma 3.1], we have the following inequality. three edges of K. b Eb introduced in (3.5.2) satLemma 3.5.2 The linear extension operator L p isfies −1 b Eb (b kL q kL2 (E) b . p kb b . p q )kL2 (K) Now consider an arbitrary element K ∈ T and fix an edge E ∈ E(K). If E contains no hanging node in T (i.e., E ∈ E(T )), we define the extension int operator LE p,K (q) : Pp (E) → Qp (K) by −1 E LE p,K (q) = [Lp (q ◦ FK )] ◦ FK , b  q ∈ Ppint (E).  (3.5.3)  If E contains a hanging node located in the middle of it, E can be written as E = E1 ∪ E2 for two edges E1 and E2 in E(T ). We then partition K into two parallelograms, K = K1 ∪ K2 , by connecting the hanging node on E with the midpoint of the edge opposite to E, as illustrated in Figure 3.3 (left). For any q1 ∈ Ppint (E1 ) and q2 ∈ Ppint (E2 ), we define the extension operator LE p,K (q1 , q2 ) by E1 E2 LE p,K (q1 , q2 ) = Lp,K1 (q1 ) + Lp,K2 (q2 ),  (3.5.4) 83  3.5. Proof of Theorem 3.4.4 E2 1 with LE p,K1 and Lp,K2 given in (3.5.3). E By definition, the extensions LE p,K (q) and Lp,K (q1 , q2 ) are continuous in E E K and satisfy LE p,K (q)|E = q, Lp,K (q1 , q2 )|E1 = q1 and Lp,K (q1 , q2 )|E2 = q2 . E Moreover, LE p,K (q) and Lp,K (q1 , q2 ) both vanish on the other edges of E(K).  E3 ν4 K1  ν3  E1 E4  E K2  E2 ν1  e4 E E1  νe4  νe1  e3 E  e K  e1 E  νe3  E2 e2 E  νe2 ν 2  Figure 3.3: Left: Partition of K into K1 and K2 . Right: Element K and e ∈ R(K). K  3.5.3  Decomposition of functions in Sp (T )  We shall now decompose functions in Sp (T ), similarly to [19, Proposition 5.4]. For any edge E ∈ E(T ) ∪ E(Te ), we set e ∈w pE = min{ pKe : K eE }.  (3.5.5)  e K e ∈ Te , belongs to E(T ) ∪ E(Te ). Notice that an elemental edge E in E(K), e ∈ Te , equation (3.5.5) defines the elemental edge polynoHence, for any K mial degrees as used in (3.5.1). Furthermore, we denote by vK the restriction of a piecewise smooth function v to an element K ∈ T ∪ Te . Let v ∈ Sp (T ). Firstly, we introduce a (nodal) interpolant v nod ∈ Spe (Te ). e ∈ R(K), we will construct the restriction For each element K ∈ T and K nod nod e v e of v to K such that K  nod e vK e ∈ QpK (K),  nod vK e |E ∈ PpE (E),  e E ∈ E(K),  (3.5.6)  with pE given in (3.5.5). To define v nod e , we distinguish two cases. K Case 1: If R(K) = {K} (i.e., if K is unrefined), the interpolant v nod e = K nod vK is simply defined by X nod vK (x) = vK (ν) ΦνK (x). (3.5.7) ν∈N (K)  84  3.5. Proof of Theorem 3.4.4 Case 2: If R(K) consists of four newly created elements, we define v nod e K e ∈ R(K) separately. To do so, fix K e ∈ R(K). Without on each element K loss of generality, we may consider the situation shown in Figure 3.3 (right), ei }4 and where {Ei }4i=1 and {νi }4i=1 denote the edges and vertices of K, {E i=1 e Notice that here we have νe2 = ν2 and νe4 ∈ NA (Te ). {e νi }4i=1 the ones of K. e3 and E e4 are in EA (Te ) and p e = p e = p e = pK . Let us Furthermore, E E3 E4 K e At the now define the value of v nod at the edge and vertex nodes of K. e K  e3 and E e4 , we set interior nodes of E nod vK e (z) = vK (z),  p  p  e E e3 ) ∪ Z Ee4 (E e4 ). z ∈ Zint 3 (E int  (3.5.8)  Similarly, we set v nod e2 and ν = νe4 . e (ν) = vK (ν) for the vertices ν = ν K  e It remains to define the values of v nod e on the nodes of the edges E1 and K e2 , as well as on νe1 and νe3 . We only consider νe1 and E e1 (the construction E e for νe3 and E2 is completely analogous). If νe1 ∈ N (T ) (i.e., νe1 is a hanging node in T ), then we define nod vK e (z) = 0,  p  e E e1 ), z ∈ Zint 1 (E  nod ν1 ) = vK (e ν1 ). vK e (e  (3.5.9)  On the other hand, if νe1 ∈ / N (T ), then E1 ∈ E(T ) and ν1 ∈ N (T ). In this case, we interpolate the values of the nodal interpolant over the long e1 . That is, we set edge E1 at the Gauss-Lobatto nodes on E ν1 ν2 nod vK e (z) = vK (ν1 ) ΦK (z)+vK (ν2 ) ΦK (z),  p  e E e1 )∪{e z ∈ Zint 1 (E ν1 }. (3.5.10)  With the nodal values of v nod e constructed (3.5.8)–(3.5.10), we have K  nod vK e (x) =  X  nod ν vK e (ν) ΦK e (x) +  e ν∈N (K)  X  pE −1   X  e i=1 E∈E(K)  E,pE  nod vK e (zi  E,pE  )Φi   (x) .  This finishes the construction of the interpolant of v nod . Notice that v nod ∈ Spe (Te ) is continuous over edges E ∈ EA (Te ) and satisfies nod vK (ν) − vK (ν) = 0,  ν ∈ N (T ) located on ∂K,  (3.5.11)  e ∈w E ∈ E(T ), K eE .  (3.5.12)  as well as nod nod vK e |E ∈ PpE (E),  85  3.5. Proof of Theorem 3.4.4 Secondly, we construct a function v edge ∈ Spe (Te ) related to the edge degrees of freedom. To do so, fix an element K ∈ T . For any edge E ∈ E(K), E by we define vK ( nod LE E ∈ E(T ), pK ,K ((vK − vK )|E ), E vK = E nod nod LpK ,K ((vK − vK )|E1 , (vK − vK )|E2 ), E = E1 ∪ E2 , E1,2 ∈ E(T ), E with LE pK ,K (·) in (3.5.3) and LpK ,K (·, ·) in (3.5.4), respectively. We then edge define v elementwise as X edge E vK (x). vK (x) = E∈E(K)  Thirdly, we construct a function v int ∈ Spe (Te ) simply by setting elementwise edge int nod , K ∈T. vK = vK − vK − vK 1 int Notice that vK belongs to H0 (K). Hence, we have v int ∈ Spec (Te ). In conclusion, any function v ∈ Sp (T ) can be decomposed into three parts: v = v nod + v edge + v int , (3.5.13) nod edge int e with v , v and v in Spe (T ) as defined above.  3.5.4  Proof of Theorem 3.4.4  In this section, we outline the proof of Theorem 3.4.4. Some of the auxiliary results are postponed to Sections 3.5.5 and 3.5.6. Let v ∈ Sp (T ), we write v = v nod + v edge + v int , according to (3.5.13). We will define the averaging operator Ihp v in three parts: Ihp v = ϑnod + ϑedge + ϑint ,  (3.5.14)  with ϑnod , ϑedge , ϑint ∈ Spec (Te ). Since v int ∈ Spec (Te ), we simply take ϑint = v int . In Sections 3.5.5 and 3.5.6, we will further construct ϑnod and ϑedge such that the following two results hold true. Lemma 3.5.3 There is a conforming approximation ϑnod ∈ Spec (Te ) that satisfies X X Z nod nod 2 nod 2 kv − ϑ kL2 (K) p−2 ]] ds, e . E hE [[v E∈E(T ) E  e Te K∈  X  e Te K∈  k∇(v  nod  −ϑ  nod  )k2L2 (K) e  .  X Z  E∈E(T ) E  nod 2 p2E h−1 ]] ds. E [[v  86  3.5. Proof of Theorem 3.4.4 Lemma 3.5.4 There is a conforming approximation ϑedge ∈ Spec (Te ) that satisfies X Z X edge edge 2 2 nod 2 p−2 kv −ϑ kL2 (K) ]] )ds, e . E hE ([[v]] + [[v E∈E(T ) E  e Te K∈  X  e Te K∈  k∇(v  edge  −ϑ  edge  )k2L2 (K) e  .  X Z  E∈E(T ) E  2 nod 2 p2E h−1 ]] )ds. E ([[v]] + [[v  By the triangle inequality and Lemmas 3.5.3 and 3.5.4, we then obtain   X X Z −2 2 nod 2 h kv − Ihp vk2L2 (K) p [[v]] ds, . + [[v ]] e E E E∈E(T ) E  e Te K∈  X  e Te K∈  k∇(v − Ihp v)k2L2 (K) e .  X Z  E∈E(T ) E    2 nod 2 ds. [[v]] + [[v ]] p2E h−1 E  Theorem 3.4.4 now follows if we show that k[[v nod ]]k2L2 (E) . k[[v]]k2L2 (E) ,  E ∈ E(T ).  (3.5.15)  To prove (3.5.15), we denote by ν1 and ν2 the two end points of E ∈ E(T ). By the construction of v nod , the jump over E satisfies [[v nod ]](νi ) = [[v]](νi ),  i = 1, 2.  Since [[v nod ]] vanishes on all the interior Gauss-Lobatto nodes on E, item (i) in Lemma 3.5.1 and a scaling argument yield nod k[[v nod ]]k2L2 (E) . p−2 ]](ν1 )2 + [[v nod ]](ν2 )2 ) E hE ([[v  −2 2 2 2 = p−2 E hE ([[v]](ν1 ) + [[v]](ν2 ) ) . pE hE k[[v]]kL∞ (E) .  From [33, Theorem 3.92], we further have the inverse estimate 2 k[[v]]k2L∞ (E) . p2E h−1 E k[[v]]kL2 (E) .  This shows (3.5.15) and finishes the proof of Theorem 3.4.4, up to the proofs of Lemmas 3.5.3 and 3.5.4 which we present next.  87  3.5. Proof of Theorem 3.4.4  3.5.5  Proof of Lemma 3.5.3  Let v nod ∈ Spe (Te ) be the nodal interpolant in the decomposition (3.5.13). We now shall construct the conforming approximation ϑnod in Spec (Te ). For simplicity, we shall omit the superscript ”nod” and, in the sequel, write v for v nod and ϑ for ϑnod . For a node ν, we introduce the sets: e ∈ Te : ν ∈ N (K) e }, w(ν) e = {K  wE (ν) = { E ∈ E(T ) : ν ∈ E }.  e ∈ R(K). We proceed by distinguishing the same two Fix K ∈ T and K cases as in Subsection 3.5.3. e Then any elemental edge Case 1: If R(K) = {K}, we have K = K. e ∈ E(K) e belongs to E(T ) and we have v e | e ∈ P nod (E). e For any GaussE pEe K E e we define the value of ϑ(ν) by Lobatto node ν located on ∂ K, ϑ(ν) =   X −1  e vKe (ν), |w(ν)|  0,  ν ∈ NI (T ),  e w(ν) K∈ e  (3.5.16)  otherwise.  Here, |w(ν)| e denotes the cardinality of the set w(ν). e In the case considered, e by we have |w(ν)| e = 4 for ν ∈ NI (T ). Then we define ϑ on K X ϑ(x) = ϑ(ν) ΦνKe (x). (3.5.17) e ν∈N (K)  From (3.5.7) and (3.5.17), we have X kvKe − ϑkL2 (K) |vKe (ν) − ϑ(ν)| kΦνKe kL2 (K) e . e .  (3.5.18)  Analogously to [10, Page 1125], we have X −1/2 |vKe (ν) − ϑ(ν)| . pE hE k[[v]]kL2 (E) .  (3.5.19)  e ν∈N (K)  E∈wE (ν)  Hence, by combining (3.5.18), (3.5.19) and item (ii) in Lemma 3.5.1 (with scaling), we obtain Z X 2 2 kvKe − ϑkL2 (K) p−2 (3.5.20) e . E hE [[v]] ds. E∈{wE (ν)}ν∈N (K) e  E  88  3.5. Proof of Theorem 3.4.4 Case 2: If R(K) consists of four elements, we define ϑ on each elee ∈ R(K) separately, analogously to the construction of the nodal ment K interpolant in Subsection 3.5.3. Without loss of generality, we may again e3 , E e4 ∈ EA (Te ), the consider the case illustrated in Figure 3.3 (right). Since E e e function v is continuous over E3 and E4 . The values of ϑ on the edge nodes p e3 ) ∪ Z pKe (E e4 ), and the vertex νe4 are defined by z ∈ ZintKe (E int p e3 ) ∪ Z pKe (E e4 ), z ∈ ZintKe (E int  ϑ(z) = vKe (z),  ϑ(e ν4 ) = vKe (e ν4 ).  (3.5.21)  We further define the value of ϑ on the vertex νe2 by (3.5.16). e1 and E e2 , It remains to define the values of ϑ for the nodes on the edges E e1 (the construction for νe3 as well as for νe1 and νe3 . We only consider νe1 and E e and E2 is completely analogous). If νe1 ∈ N (T ), then νe1 is a hanging node of p e1 ∈ E(T ). Thus, v e | e ∈ P nod (E e1 ) ∪ {e e1 ). For any z ∈ Z Ee1 (E T and E ν1 }, pEe  K E1  int  1  the value of ϑ(z) is taken as in (3.5.16). On the other hand, if νe1 ∈ / N (T ), then E1 ∈ E(T ) and vK |E1 ∈ Ppnod (E ). We define ϑ(ν ) again by (3.5.16). 1 1 E1 Recall that ϑ(ν2 ) = ϑ(e ν2 ) has already been defined. Then we set ϑ(z) = ϑ(ν1 )ΦνK1 (z) + ϑ(ν2 )ΦνK2 (z),  e by setting Now we construct ϑ on K ϑ(x) =  X  ϑ(ν)ΦνKe (x)  e ν∈N (K)  +  X  p  e E e1 ) ∪ {e z ∈ Zint 1 (E ν1 }.  pEe −1   X  e e E,p E  ϑ(zi  e e E,p E  )Φi  e e i=1 E∈E( K)  (3.5.22)   (x) .  This completes the construction of ϑ. Clearly, ϑ ∈ Spec (Te ). We shall now derive an estimate analogous to (3.5.20). To do so, we e as follows: bound the difference between vKe and ϑ on K kvKe − ϑkL2 (K) e . .  X  e ν∈N (K)  X  e ν∈N (K)  X  kςEe kL2 (K) k vKe (ν) − ϑ(ν) ΦνKe kL2 (K) e + e e e E∈E( K)  X  −1 1/2 k vKe (ν) − ϑ(ν) ΦνKe kL2 (K) kςEe kL2 (E) e e +pe he  = T1 + T2 ,  K  K  e e E∈E( K)  (3.5.23)   e e e e  E,p e e PpEe −1  E,p E,p E E E with ςEe (x) = (z v ) − ϑ(z ) Φ (x) . For the second e i=1 i i K i inequality in (3.5.23), we have used estimate (i) in Lemma 3.5.1 and a scaling 89  3.5. Proof of Theorem 3.4.4 argument noticing that the function ςEe (x) vanishes at all the interior tensore and on the edges of K e that are different product Gauss-Lobatto nodes in K e from E. Let us first bound the term T1 in (3.5.23). If the node ν ∈ NA (Te ), then, by (3.5.21),  e vKe (ν) − ϑ(ν) ΦνKe (x) = 0, x ∈ K. (3.5.24) Furthermore, if the node ν belongs to N (T ), we apply estimate (ii) in Lemma 3.5.1 (with scaling) and an argument as in (3.5.19). We obtain  k vKe (ν) − ϑ(ν) ΦνKe kL2 (K) e .  X  E∈wE (ν)  1  2 p−1 E hE k[[v]]kL2 (E) .  (3.5.25)  Finally, if ν ∈ / N (T ) ∪ NA (Te ), then ν is the midpoint of an elemental edge, E ∈ E(K) ∩ E(T ). Denote the two end points of this edge E by ν1 and ν2 . In view of (3.5.10) and (3.5.22), we have |vKe (ν) − ϑ(ν)| ≤ |(vK (ν1 ) − ϑ(ν1 ))| + |(vK (ν2 ) − ϑ(ν2 ))|. Thus, as before, we obtain  k vKe (ν) − ϑ(ν) ΦνKe kL2 (K) e .  X  E∈wE (ν  E 1 )∪w (ν2 )  1  2 p−1 E hE k[[v]]kL2 (E) .  (3.5.26)  e as To combine the results in (3.5.24)-(3.5.26), we define the set N ? (K) e and first remove all the vertices belonging to follows. We start from N (K) e e with νe ∈ NA (T ). Then, any vertex νe ∈ N (K) / N (T ) ∪ NA (Te ) is replaced by the vertex ν ∈ N (K) which is on the same elemental edge of K as νe. e = For example, in the case shown in Figure 3.3 (right), we have N ? (K) { ν1 , νe2 , νe3 } if νe1 ∈ / N (T ) and νe3 ∈ N (T ). We also set e = { E ∈ wE (ν) : ν ∈ N ? (K) e }. E ? (K)  In conclusion, the term T1 is bounded by T1 .  X  e E∈E ? (K)  1  2 p−1 E hE k[[v]]kL2 (E) .  (3.5.27)  e ∈ E(T ) or E e ∈ EA (Te ), Next, let us estimate the term T2 in (3.5.23). If E by the constructions of v and ϑ, we clearly have kςEe kL2 (E) e = 0. Otherwise, 90  3.5. Proof of Theorem 3.4.4 e say νe1 , is a newly created node in Te and the one of the two end points of E, other one, νe2 , belongs to N (T ). Thus, we have 2 X  1  1  −1 2 2 p−1 e kL2 (E) e ≤ pe he e h e kςE K  K  K  K  i=1  1   k vKe (e νi ) − ϑ(e νi ) Φνeei kL2 (E) e K  2 + p−1 e − ϑkL2 (E) e e h e kvK  K  K  = T21 + T22 . Then there exists an elemental edge E ∈ E(K) such that νe1 is the midpoint of E. Denote the end points of E by ν1 and ν2 . Similarly to (3.5.25) and (3.5.26), we have X  T21 .  1  E∈wE (ν1 )∪wE (ν2 )  2 p−1 E hE k[[v]]kL2 (E) .  In view of (3.5.22), we proceed as in (3.5.19) and obtain 1  T22 . .  2 p−1 e hK ek K  2  X i=1  X    vK (νi ) − ϑ(νi ) ΦνKi kL2 (E) 1  E∈wE (ν1 )∪wE (ν2 )  2 p−1 E hE k[[v]]kL2 (E) .  Combining the above results shows that T2 .  X  1  e E∈E ? (K)  2 p−1 E hE k[[v]]kL2 (E) .  The bounds for T1 and T2 in (3.5.27) and (3.5.28) yield X Z 2 2 p−2 kvKe − ϑkL2 (K) e . E hE [[v]] ds. e E∈E ? (K)  (3.5.29)  E  This finishes the discussion of Case 2. Thus, by the key estimates in (3.5.20) and (3.5.29), we have X Z 2 e ∈ Te . kvKe − ϑk2L2 (K) . p−2 K e E hE [[v]] ds, e E∈E ? (K)  (3.5.28)  (3.5.30)  E  91  3.5. Proof of Theorem 3.4.4 This proves the first inequality in Lemma 3.5.3. Moreover, by the inverse inequality 2 −1 k∇vkL2 (K) e . pK e , e h e kvkL2 (K) K  see [33], we obtain from (3.5.30) k∇(vKe −  ϑ)k2L2 (K) e  .  X  e E∈E ? (K)  Z  E  e ∈ Te , v ∈ Spe (Te ), K 2 p2E h−1 E [[v]] ds,  e ∈ Te , K  (3.5.31)  (3.5.32)  which shows the second assertion in Lemma 3.5.3.  3.5.6  Proof of Lemma 3.5.4  Fix an element K ∈ T and let E be an elemental edge in E(K). We define E as follows: If E ∈ E (T ), we set the function WK B  E nod WK = LE pK ,K (vK − vK )|E ,  0 e with the extension operator LE pK ,K (·) in (3.5.3). If E ∈ EI (T ), let K in T be 0 the element such that E is also an elemental edge of K , that is, E ∈ E(K 0 ). Denote by K 0 the element which has the lower polynomial degree of the elements K and K 0 , i.e., K 0 = K if pK ≤ pK 0 and K = K 0 otherwise. We E by define WK  E nod WK = LE pK ,K (vK 0 − vK 0 )|E .  with LE pK ,K (·) in (3.5.3). In the case where E contains a hanging node, E is partitioned into E = E1 ∪ E2 with E1 , E2 ∈ EI (T ). There exist two elements K 0 , K 00 ∈ T such that E1 ∈ E(K 0 ) and E2 ∈ E(K 00 ). Denote by K 0 the element that has the lower polynomial degree of K and K 0 , and by K 00 E by the element that has the lower degree of K and K 00 . We now define WK  E nod nod WK = LE pK ,K (vK 0 − vK 0 )|E1 , (vK 00 − vK 00 )|E2 ,  with LE pK ,K (·, ·) in (3.5.4). P E, Then we define ϑedge elementwise by setting ϑedge |K = E∈E(K) WK E defined above. Clearly, the function ϑedge belongs to S c (T e ). Next, with WK e p we prove the approximation properties of Lemma 3.5.4. By Lemma 3.5.2  92  3.6. Numerical experiments (with a scaling argument), we have X X X kv edge − ϑedge k2L2 (K) e =  K∈T K∈R(K) e  e Te K∈  .  X  X  K∈T E∈E(K)  .  X  X  K∈T E∈E(K)  .  X  kv edge − ϑedge k2L2 (K) e   nod E 2 kLE pK ,K (vK − vK )|E − WK kL2 (K)  nod E 2 p−2 K hK k(vK − vK )|E − WK |E kL2 (E)  X Z  K∈T E∈E(K) E  2 nod 2 p−2 ]] ) ds. E hE ([[v]] + [[v  This completes the proof of the first assertion of Lemma 3.5.4; the second one follows again from the first one by using the inverse inequality in (3.5.31).  3.6  Numerical experiments  In this section, we present a series of numerical examples where we use η in (3.3.4) as an error indicator in an hp-adaptive isotropic refinement strategy. Our implementation of the DG method (3.2.6) is based on the Deal.II finite element library [5, 6]. The non-symmetric sparse linear systems of equations are solved by using the UMFPACK package [13, 14]. In all the examples, the hp-adaptive meshes are constructed by first marking the elements for refinement and derefinement according to the size of the local error indicator ηK in (3.3.3), with refinement and derefinement fractions set to 25% and 10%, respectively. Once an element has been flagged for refinement, a decision must be made whether the local mesh size or the local polynomial degree should be adjusted accordingly. The choice to perform either h-refinement or p-refinement is based on estimating the local smoothness of the analytical solution. Here, we employ the hp-adaptive strategy developed in [22], where the local regularity of the analytical solution is estimated from truncated local Legendre expansions of the computed numerical solution; see also [21]. For an element K flagged for refinement, the smoothness estimation algorithm consists of the following three steps (here, we assume K is a square for simplifying the notations): STEP 1: We denote by x? and y? the midpoint of K in x- and y-direction respectively. Expand uh (x, y? )|K and uh (x? , y)|K into Legendre series in each direction with coefficients ai and bi , 0 ≤ i ≤ pK , separately. 93  3.6. Numerical experiments STEP 2: Set mi = log(|ai |) and ni = log(|bi |), 0 ≤ i ≤ pK . Then evaluate the quantities PK PK 2 pi=0 imi − pK pi=0 mi lx = 6 , 2 (pK + 1)((pK + 1) − 1) PK PK 2 pi=0 ini − pK pi=0 ni ly = 6 . 2 (pK + 1)((pK + 1) − 1) STEP 3: If exp(−lx ) and exp(−ly ) are both smaller than a given threshold value θ, 0 ≤ θ ≤ 1, (meaning that u(x, y) is relatively smooth on K), we increase the polynomial degree on K by one; otherwise, we refine the element K isotropically into four elements by bisecting the elemental edges of K. In this algorithm, we set θ = 0.7 and additional refinement might be performed to ensure that the meshes are 1-irregular. In all our tests, we set the stabilization parameter to γ = 10. The approximate right-hand side fhp is taken as the L2 -projection of f onto Sp (T ). Moreover, since the flow field a is either constant or linear, we simply choose ahp = a in ηRK . We numerically reproduce solutions that are analytic over the computational domain, although they have steep gradients along boundary and internal layers. In all our examples, we observe prefinement to be dominating once the local mesh size is sufficiently resolved. Based on the a-priori error analysis for p-version methods in [33], we thus 1 plot all computed quantities against N 2 in a logarithmic scale, with N = dim(Sp (T )).  3.6.1  Example 1  In this example, we take Ω = (0, 1)2 , choose a = (1, 1) and select the right-hand side f so that the analytical solution to the convection-diffusion problem (3.2.1) is given by u(x1 , x2 ) =   e x1ε−1 − 1 1  e− ε   e x2ε−1 − 1  + x1 − 1 + x − 1 . 2 1 e− ε − 1 −1  The solution is smooth, but has boundary layers at x1 = 1 and x2 = 1; their widths are both of order O(ε). This problem is well-suited to test whether the indicator η is able to pick up the steep gradients near these boundaries. We begin this test with a uniform mesh of 16 × 16 elements and the uniform polynomial degree pK = 1. In Figure 3.4(a), we show the performance 94  3.6. Numerical experiments of our hp-adaptive algorithm for ε = 10−3 . In the curves labeled “Error Indicator” and “Energy Error”, we see that the indicator η always overestimates the true energy error k u − uhp kE,T , in agreement with Theorem 3.3.1. Additionally, the convergence lines using hp-refinement are (roughly) straight on a linear-log scale, which indicates that exponential convergence is attained for this problem. In the curve “L2 Error”, we calculate the error 1 ε− 2 ku − uh kL2 (Ω) , which is an upper bound for |a(u − uh )|? . We see that this error is at least of the same order as the energy error. The same behavP 1/2 −1 p−1 h k[[u ]]k2 ior is observed for the error (ε shown ) h L2 (E) E∈E(T ) E E in ”Jump Error”. Finally, in the curve labeled ”Theta”, we calculate an approximation to the data error Θ in (3.3.4). This is done by using a Gauss-Legendre quadrature rule of order pK + 3 to approximate Θ2K = 2 2 ε−1 p−2 K hK kf − fhp kL2 (K) on each element K. The data approximation error Θ is of almost three orders of magnitude smaller than η. In Figure 3.4(b), we compare the true energy error and the error estimate for h− and hp−adaptive methods. Here, the hp-refinement result is identical as shown in Figure 3.4(a), while the h-refinement employs the error estimate stated in Chapter 2 for piecewise linear finite elements, with the fixed fraction strategy with refinement and derefinement set to 25% and 10%, respectively. The superiority of hp-adaptive methods can be clearly observed in the comparison. In Figure 3.4(c), we compare the true energy error and the error indicator generated by our hp-adaptive algorithm using the indicator η in (3.3.4) (denoted by p3 in the figure) and the corresponding one outlined in Remark 3.3.2 (denoted by p2 in the figure). As in [23], we observe that the two error indicators give rise to almost identical results. In Figure 3.4(d), we plot the ratios of the indicator and the true energy error. It stays around 8, 1 uniformly in N 2 . In Figure 3.5, we show the same plots for ε = 2 · 10−4 . Qualitatively, we observe the same behavior as before. Together with Figure 3.4(c), we see that the ratio of the indicator and the true energy error oscillates around 8 for both ε = 10−3 and ε = 2 · 10−4 , in agreement with Theorems 3.3.1 and 3.3.3. In Figure 3.6, we show the meshes and polynomial degree distribution after 7 hp-adaptive refinement steps. We observe that the p-refinement is dominating once the local mesh size is of order O(ε), the order of the width of the boundary layer. The p-refinement is concentrated around the boundary layers. Away from the layers, the solution is almost linear and can be approximated efficiently with low-order polynomials.  95  3.6. Numerical experiments (a)  (b)  0  0  10  10  −1  10  −1  10  −2  10  −2  10 −3  10  −3  10 −4  10  −4  10  −5  10  Error Indicator Energy Error L2 Error Jump Error Theta  −6  10  −7  10  100  150  hp−Error Indicator hp−Energy Error h−Error Indicator h−Energy Error  −5  10  −6  200 N1/2  250  300  10  2  3  10  4  10  10  N  (c)  (d) 10  0  10  9 8  −1  10  7 −2  10  6 −3  10  5 4  −4  10  Error Indicator (p3)  3  Energy Error (p3)  −5  10  2  Error Indicator (p )  ratio (p3)  2  Energy Error (p2)  ratio(p2)  −6  10  100  150  200 N1/2  250  300  1  100  150  200 N1/2  250  300  Figure 3.4: Example 1: Convergence behavior for ε = 10−3 .  3.6.2  Example 2  Next, we consider an example with an internal layer and with variable coefficients. In the domain Ω = (−1, 1)2 , we take a(x1 , x2 ) = (−x1 , x2 ). We choose f and the inhomogeneous Dirichlet boundary conditions such that the solution to (3.2.1) is given by Z x 2 x1 2 2 with erf(x) = √ e−t dt. u(x1 , x2 ) = erf( √ )(1 − x2 ), π 0 2ε For small values of ε, the solution u has an internal layer around x1 = 0, √ whose width is of order O( ε). We use standard DG terms to incorporate the inhomogeneous boundary conditions, and modify the error indicator η correspondingly. For details we refer to [19]. We begin this test with a 96  3.6. Numerical experiments (a)  (b)  1  10 0  10  0  10 −1  10  −1  10  −2  10  −3  10  −2  10  −4  10  −3  10 −5  10  −6  10  −7  10  200  Error Indicator Energy Error L2 Error Jump Error Theta  hp−Error Indicator hp−Energy Error h−Error Indicator h−Energy Error  −4  10  −5  300  400 N1/2  500  600  10  4  10  5  10  (c)  1  6  10 N  7  8  10  10  (d)  10  11 10  0  10  9 8  −1  10  7 −2  10  6 5  −3  10  4 Error Indicator (p3)  −4  10  3  3  Energy Error (p ) Error Indicator (p2)  −5  10  200  300  400 N1/2  ratio (p3)  2  Energy Error (p2) 500  600  1 200  ratio(p2) 300  400 N1/2  500  600  Figure 3.5: Example 1: Convergence behavior for ε = 2 · 10−4 . uniform mesh of 8 × 8 elements and the uniform polynomial degree pK = 2. In Figure 3.7 and Figure 3.8, the numerical results for this example are shown for the values ε = 10−3 and ε = 5 · 10−6 , respectively. We plot the same quantities as in Example 1. For ε = 10−3 , we observe exponential convergence rates for both the energy error and the indicator. The curves associated with the convection and approximation errors are of the same order as the energy error. In particular, the jump error related to convection in the curve ”Jump Error” and Θ in ”Theta” are clearly below the energy error. If we now decrease the value of ε to ε = 5 · 10−6 , the jump related to the convection term depicted in ”Jump Error” is dominating the estimator η. (Recall that the error plotted in the curve ”L2 Error” is only an upper bound for the error |a(u − uh )|? and can overestimate η.) Nevertheless, exponential 97  3.6. Numerical experiments  (a) ε = 10−3 , hp-refinement  (b) ε = 2 · 10−4 , hp-refinement  Figure 3.6: Example 1: Adaptively generated meshes after 7 refinement steps. (a)  −1  (b)  10  12 Error Indicator Energy Error L2 Error Jump Error Theta  −2  10  ratio 11 10 9  −3  10  8 7  −4  10  6 5  −5  10  4 3  −6  10  2 −7  10  50  60  70  80  90 N1/2  100  110  120  1 50  60  70  80  90  100  110  120  N1/2  Figure 3.7: Example 2: Convergence behavior for ε = 10−3 .  convergence rates are observed for all quantities. This illustrates the fact that the estimator η is not robust in estimating the energy error alone; the inclusion of the dual norm in the error measure is essential. This is further reflected in the plots at the right-hand sides of Figures 3.7 and 3.8 where we show the ratio of the indicator and the energy error. While for ε = 10−3 the values are between 8 and 12, they clearly increase for ε = 5 · 10−5 . Initially, they also strongly oscillate. Again, this is due to the fact that we do not include the dual norm in the error measure. Figure 3.9 shows the hp-adaptive meshes and polynomial degree distributions after 9 refinement and 15 refinement steps, both for ε = 10−3 and ε = 5 · 10−6 . We observe that the mesh refinement stops once the local mesh √ size is of order O( ε) and p-refinement starts to take over in the vicinity of 98  3.6. Numerical experiments (a)  0  (b)  10  40 Error Indicator Energy Error L2 Error Jump Error Theta  −1  10  ratio 35  30  25  −2  10  20 −3  10  15  10 −4  10  5 −5  10  210  220  230  240  250  260  270  280  290  0 200  300  N1/2  210  220  230  240  250 N1/2  260  270  280  290  300  Figure 3.8: Example 2: Convergence behavior for ε = 5 · 10−6 . the layer, which is much more pronounced for ε = 5 · 10−6 .  3.6.3  Example 3  Finally, we consider a problem with convection not aligned with the mesh. We take Ω = (−1, 1)2 , a = (− sin π6 , cos π6 ), f = 0 and consider the boundary conditions u = 0 on x1 = −1 and x2 = 1, as well as u = tanh(  1 − x2 ) ε  on x1 = 1,  u=   x1 1 tanh( ) + 1 2 ε  on x2 = −1.  The boundary data is almost discontinuous near the point (0, −1)√and causes √ u to have an internal layer of width O( ε) along the line x2 + 3x1 = −1, with values u = 0 to the left and u = 1 to the right, as well as a boundary layer along the outflow boundary. There is no exact solution available to this problem. We test this problem with ε = 10−3 and start the algorithm for pK = 2 on a uniform mesh of 16 × 16 elements. In Figures 3.10, we plot the values of the indicator η for ε = 10−3 . We also present the comparison between the estimate η for the hp- and hadaptive methods, where the h-adaptive algorithm is still based on employing the error estimate in Chapter 3 for piecewise quadratic finite elements and the fixed refinement and derefinement fraction setting to 25% and 10% respectively. We observe almost exponential convergence for η. Figure 3.11 depicts the adaptive meshes after 7 refinement steps. Since the solution is almost constant away from the layers, p-refinement is again concentrated along the layers.  99  3.7. Conclusions  (a) ε = 10−3 , 9 hp-adaptive refinements  (b) ε = 10−3 , 15 hp-adaptive refinements  (c) ε = 5 · 10−6 , 9 hp-adaptive refinements (d) ε = 5 · 10−6 , 15 hp-adaptive refinements  Figure 3.9: Example 2: Adaptively generated meshes after 9 and 15 refinement steps.  3.7  Conclusions  We have derived a robust a-posteriori error estimate for hp-adaptive discontinuous Galerkin methods for convection-diffusion equations on 1-irregular parallelogram meshes. The constants in the reliability and efficiency bounds are independent of the Péclet number ε of the equation, and hence the estimate is fully robust. We have applied our estimate as an error indicator in an hp-adaptive refinement algorithm. Our numerical results indicate that the indicator is effective in locating and resolving interior and boundary layers. Once the local mesh size is of the same order as the width of the boundary or interior layer, both the energy error and the error indicator are observed to converge exponentially. The results of this chapter can be extended to problems with zero-order 100  3.7. Conclusions (a)  1  (b)  1  10  10  hp−Error Indicator h−Error Indicator  hp−Error Indicator  0  0  10  10  −1  −1  10  10  −2  −2  10  10  −3  −3  10  10  −4  10  100  −4  150  200  250 N1/2  300  350  400  10  2  10  3  10 N1/2  4  10  Figure 3.10: Example 3: Convergence behavior for ε = 10−3 .  Figure 3.11: Example 3: Adaptively generated meshes after 7 refinement steps. term; see [32] or Chapter 2 for the h-version of the DG method. In this case, the flow field a does not necessarily need to be divergence-free. Instead, an assumption on the coefficient functions as in [32, 36] is sufficient. The a-posteriori analysis presented in this chapter is based on the availability of an averaging operator as in Theorem 3.4.4. 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A-posteriori error estimates for lowest-order mixed finite element discretizations of convection-diffusion-reaction equations. SIAM J. Numer. Anal., 45:1570–1599, 2007.  105  Chapter 4  Diffusion problems in three dimensions 3 4.1  Introduction  In this chapter we develop the energy norm a-posteriori error estimation for hp-adaptive discontinuous Galerkin (DG) discretizations of the following model diffusion equation in three dimensions: −∆u = f (x) u=0  in Ω ⊂ R3 , on Γ.  (4.1.1)  Here, Ω is a bounded Lipschitz polyhedron in R3 with boundary Γ = ∂Ω. The right-hand side f (x) is a given function in L2 (Ω). The standard weak formulation of (4.1.1) is to find u ∈ H01 (Ω) such that Z Z A(u, v) ≡ ∇u · ∇v dx = f v dx ∀ v ∈ H01 (Ω). (4.1.2) Ω  Ω  DG methods are ideally suited for realizing hp-adaptivity for secondorder boundary-value problems, an advantage that has been noted early on in the recent development of these methods; see, for example, [6, 13, 19, 27, 28, 34] and the references therein. Indeed, working with discontinuous finite element spaces easily facilitates the use of variable polynomial degrees and local mesh refinement techniques on possibly irregularly refined meshes – the two key ingredients for hp-adaptive algorithms. The development of energy-norm error estimation for hp-adaptive DG methods for elliptic boundary-value problems was initiated in [18] where a residual-based hp-version error estimator was derived for regular meshes of triangular and quadrilateral elements on two-dimensional domains. It 3  A version of this chapter has been accepted for publication. Zhu, L., Giani S., Houston P. and Schötzau, D. (2010) Energy norm a-posteriori error estimation for hp-adaptive discontinuous Galerkin methods for elliptic problems in three dimensions. Mathematical Models and Methods in Applied Sciences.  106  4.1. Introduction was verified numerically that the resulting hp-adaptive algorithm achieves exponential rates of convergence for problems with piecewise smooth data. In [23], a similar approach was presented for quasi-linear second-order problems in two dimensions. By using an underlying auxiliary mesh, it was possible to also analyze the case of irregular meshes. Another technique to deal with irregular meshes was proposed in [35] where hp-version a-posteriori error estimates for two-dimensional convection-diffusion equations were derived that are robust in the Péclet number of the problem. In this chapter, we extend the two-dimensional analysis presented in [18] to 1-irregularly, isotropically refined affine hexahedral meshes in three space dimensions. We propose an energy norm error estimator which gives rise to global upper and local lower bounds of the error measured in the natural DG energy norm. As in [18], the ratio of these error bounds is independent of the local mesh sizes and weakly depends on the local polynomial degrees. Crucial in our analysis is the use of an averaging operator which allows us to approximate a discontinuous finite element function by a continuous one. Operators of this type were originally introduced in [24] for the energy norm a-posteriori error analysis of DG methods for elliptic problems. The same operators have been employed in the papers [11, 17, 18, 23, 31, 35], both for h- and hp-version DG methods. Here, we follow the approach of [35] and extend the analysis there to three space dimensions. By doing so, we also obtain an optimal L2 -norm error bound for the averaging operator on irregular meshes which is of interest on its own. We use our estimators as error indicators in hp-adaptive computations and present a set of numerical experiments. We first test the resulting algorithms for problems with smooth solutions. Then we also show the performance of our method for a problem in the classical Fichera polyhedron, with a solution that has an isotropic singularity at the reentrant corner. In both cases, our numerical results indicate that exponential rates of convergence are achieved with respect to the number of degrees of freedom. We emphasize that our analysis and techniques of proof are valid only for isotropically refined elements. In light of the hp-version a-priori error analysis for elliptic boundary-value problems presented in [29, 30], anisotropic refinement is essential for resolving edge and edge-corner singularities with exponential rates of convergence. The extension of our results to anisotropic elements (and anisotropic polynomial spaces) is studied in the following chapter. The outline of the rest of this chapter is as follows. In Section 4.2, we introduce the hp-adaptive DG discretization of the model problem stated 107  4.2. Discontinuous Galerkin discretization of a diffusion problem in (4.1.1). In Section 4.3, we present our energy norm a-posteriori error estimate and discuss its reliability and local efficiency. The reliability proof shall be presented in Section 4.4. As an analysis tool, we use a new hpversion averaging operator that is analyzed in Section 4.6. In Section 4.7, we present a series of numerical tests that verify the theoretical results. Finally, in Section 4.8, we end with some concluding remarks.  4.2  Discontinuous Galerkin discretization of a diffusion problem  In this section, we introduce an hp-version interior penalty DG finite element method for the discretization of (4.1.1).  4.2.1  Meshes and traces  Throughout, we assume that the computational domain Ω can be partitioned into shape-regular and affine sequences of meshes T = {K} of hexahedra K. b = (−1, 1)3 under an Each element K ∈ T is the image of the cube K b → K. As usual, we denote by hK the affine elemental mapping TK : K diameter of K. We store the elemental diameters in the mesh size vector h = { hK : K ∈ T }. For an element K ∈ T , we make use of the following sets of elemental faces: the set F(K) consists of the six elemental faces of K. We further denote by FB (K) the elemental faces of K that lie on Γ, and by FI (K) the set of interior faces; thereby, we have that F(K) = FB (K) ∪ FI (K). In order to be able to deal with irregular meshes, we also need to define the faces of a mesh T . We refer to F as an interior mesh face of T if F = ∂K ∩ ∂K 0 for two neighboring elements K, K 0 ∈ T whose intersection has a positive surface measure. The set of all interior mesh faces is denoted by FI (T ). Analogously, if the intersection F = ∂K ∩ Γ of the boundary of an element K ∈ T and Γ is of positive surface measure, we refer to F as a boundary mesh face of T . The set of all boundary mesh faces of T is denoted by FB (T ) and we set F(T ) = FI (T ) ∪ FB (T ). The diameter of a face F is denoted by hF . We allow for 1-irregularly refined meshes T defined as follows. Let K be an element of T and F an elemental face in F(K). Then F may contain at most one hanging node located in the center of F and at most one hanging node in the middle of each elemental edge of F . That is, we have that F is either a mesh face belonging to F(T ) or F can be written as F = ∪4i=1 Fi , 108  4.2. Discontinuous Galerkin discretization of a diffusion problem with four mesh faces Fi ∈ F(T ), i = 1, . . . , 4, of diameter hFi = hF /2, respectively. Next, let us define the jumps and averages of piecewise smooth functions across faces of the mesh T . To that end, let the interior face F ∈ FI (T ) be shared by two neighboring elements K and K e where the superscript e stands for ”exterior”. For a piecewise smooth function v, we denote by v|F the trace on F taken from inside K, and by v e |F the one taken from inside K e . The average and jump of v across the face F are then defined as 1 {{v}} = (v|F + v e |F ), 2  [[v]] = v|F nK + v e |F nK e .  Here, nK and nK e denote the unit outward normal vectors on the boundary of elements K and K e , respectively. Similarly, if q is piecewise smooth vector field, its average and (normal) jump across F are given by {{q}} =   1 q|F + q e |F , 2  [[q]] = q|F · nK + q e |F · nK e .  On a boundary face F ∈ FB (T ), we accordingly set {{q}} = q and [[v]] = vn, with n denoting the unit outward normal vector on Γ. The other trace operators will not be used on boundary faces and are thereby left undefined.  4.2.2  Finite element spaces  We begin by introducing polynomial spaces on elements and faces. To that end, let K ∈ T be an element. We set b }, Qp (K) = { v : K → R : v ◦ TK ∈ Qp (K)  (4.2.1)  Qp (F ) = { v : F → R : v ◦ TK |F ∈ Qp (Fb) },  (4.2.2)  b denoting the set of tensor product polynomials on the reference with Qp (K) b of degree less than or equal to p in each coordinate direction on element K b K. In addition, if F ∈ F(K) is a face of K and Fb the corresponding one on b we define the reference element K, where Qp (Fb) denotes the set of tensor product polynomials on Fb of degree less than or equal to p in each coordinate direction on Fb. To define hp-version finite element spaces, we assign a polynomial degree pK ≥ 1 with each element K of the mesh T . We then introduce the degree vector p = { pK : K ∈ T }. We assume that p is of bounded local variation, 109  4.2. Discontinuous Galerkin discretization of a diffusion problem that is, there is a constant % ≥ 1, independent of the mesh T sequence under consideration, such that %−1 ≤ pK /pK 0 ≤ %  (4.2.3)  for any pair of neighboring elements K, K 0 ∈ T . For a mesh face F ∈ F(T ), we introduce the face polynomial degree pF by ( max{pK , pK 0 }, if F = ∂K ∩ ∂K 0 ∈ FI (T ), pF = (4.2.4) pK , if F = ∂K ∩ Γ ∈ FB (T ). For a partition T of Ω and a polynomial degree vector p on T , we define the hp-version DG finite element space by Sp (T ) = { v ∈ L2 (Ω) : v|K ∈ QpK (K), K ∈ T }.  4.2.3  (4.2.5)  Interior penalty discretization  We now consider the following interior penalty DG discretization for the numerical approximation of the diffusion problem (4.1.1): find uhp ∈ Sp (T ) such that Z (4.2.6) Ahp (uhp , v) = f v dx ∀ v ∈ Sp (T ). Ω  The bilinear form Ahp (u, v) is given by  XZ X Z  Ahp (u, v) = ∇u · ∇v dx − {{∇u}} · [[v]] + {{∇v}} · [[u]] ds K∈T  +  X  K  F ∈F (T )  γp2F hF  F ∈F (T ) F  Z  F  [[u]] · [[v]] ds,  where the gradient operator ∇ is defined elementwise. The parameter γ > 0 is the interior penalty parameter. The method in (4.2.6) is a straightforward extension of the classical (symmetric) interior penalty method introduced in [4, 26] to the context of the hp-version finite element method; see also [5, 19, 34] and the references therein. Remark 4.2.1 The stability and well-posedness of the DG method (4.2.6) follow from the same arguments as those employed in [34, Proposition 3.8] used to analyze the scheme in two-dimensions: there is a threshold parameter γ0 > 0, independent of h and p, such that for γ ≥ γ0 the formulation (4.2.6) possesses a unique solution uhp ∈ Sp (T ). 110  4.3. Energy norm a-posteriori error estimates  4.3  Energy norm a-posteriori error estimates  In this section, we present and discuss our main results.  4.3.1  Energy norm and residuals  We measure the error in the following energy norm associated with the DG formulation (4.2.6): k u k2E,T =  X  K∈T  k∇uk2L2 (K) +  X  F ∈F (T )  γp2F k[[u]]k2L2 (F ) . hF  (4.3.1)  To introduce our energy norm indicator, let uhp ∈ Sp (T ) be the DG approximation obtained by (4.2.6). Moreover, we denote by fhp a piecewise polynomial approximation in Sp (T ) of the right-hand side f . For each element K ∈ T , we introduce the following local error indicator ηK which is given by the sum of the three terms 2 2 ηK = ηR + ηF2 K + ηJ2K . K  (4.3.2)  The first term ηRK is the interior residual defined by 2 2 2 ηR = p−2 K hK kfhp + ∆uhp kL2 (K) . K  The second term ηFK is the face residual given by ηF2 K =  1 2  X  F ∈FI (K)  2 p−1 F hF k[[∇uhp ]]kL2 (F ) .  The last residual ηJK measures the jumps of the approximate solution uhp and is defined as ηJ2K =  1 2  X  F ∈FI (K)  γ 2 p3F k[[uhp ]]k2L2 (F ) + hF  X  F ∈FB (K)  γ 2 p3F k[[uhp ]]k2L2 (F ) . hF  We also introduce the local data approximation term 2 2 Θ2K = p−2 K hK kf − fhp kL2 (K) .  (4.3.3)  Summing up the local error indicators, we introduce the global a-posteriori error estimator !1 2 X 2 η= ηK . (4.3.4) K∈T  111  4.3. Energy norm a-posteriori error estimates Similarly, we define the global data approximation term !1 2 X 2 Θ= . ΘK  (4.3.5)  K∈T  4.3.2  Reliability  Our first theorem states that, up to a constant and to data approximation, the estimator η in (4.3.4) gives rise to a reliable a-posteriori error bound. In this result and in the sequel, we shall use the symbols . and & to denote bounds that are valid up to positive constants independent of h and p. Theorem 4.3.1 Let u be the solution of (4.1.1) and uhp ∈ Sp (T ) its DG approximation obtained by (4.2.6) with γ ≥ γ0 . Let the error estimator η be defined by (4.3.4) and the data approximation error Θ by (4.3.5). Then we have the a-posteriori error bound k u − uhp kE,T . η + Θ. The detailed proof of Theorem 4.3.1 will be given in Section 4.4. It is similar to the one given in Chapter 3 for two-dimensional convectiondiffusion equations. Crucial in our proof, however, is the use of a threedimensional averaging operator, whose hp-version approximation properties will be introduced in Theorem 4.4.1 and proven in Section 4.6. Remark 4.3.2 As for the two-dimensional cases analyzed in [23, 35], the penalization of the jump terms in the interior penalty form Ahp (u, v) is of the order p2F h−1 F on each face, while the corresponding weight in the jump residual ηJK is of the different order p3F h−1 F . This suboptimality with respect to the powers of pF is due to the possible presence of hanging nodes in the underlying mesh T . Indeed, on meshes without irregular nodes, Theorem 4.3.1 holds true with the following (optimal) jump residual: ηbJ2K =  1 2  X  F ∈FI (K)  γ 2 p2F k[[uhp ]]k2L2 (F ) + hF  X  F ∈FB (K)  γ 2 p2F k[[uhp ]]k2L2 (F ) ; hF  see also Remark 4.4.3 below. The associated estimator ηb is then given by X 2 2 2 ηb2 = ηbK with ηbK = ηR + ηF2 K + ηbJ2K . (4.3.6) K K∈T  Our numerical experiments in Section 4.7 indicate that the indicators η and ηb yield almost identical results on 1-irregular meshes.  112  4.3. Energy norm a-posteriori error estimates  4.3.3  Efficiency  In our next result, we present a local lower bound for the error measured in the energy norm. As for many residual-based hp-version a-posteriori error estimates, reliability and efficiency bounds, which are uniform in p, are not readily available; cf. [18, 25] and the references therein. We thus restrict ourselves to stating a weakly p-dependent local lower bound for ηK defined in (4.3.2). We note that our numerical results indicate that exponential rates of convergence are obtained for both smooth and non-smooth solutions; in this context, the p-suboptimality is less relevant. For an element K ∈ T , we introduce the patch of neighboring elements as wK = {K 0 ∈ T : ∂K 0 ∩ ∂K ∈ F(T )}. (4.3.7) The local energy norm over wK is defined by k u k2E,wK = Similarly, we set  X  K 0 ∈wK  k∇uk2L2 (K 0 ) +   ΘwK =   X  X  F ∈F (K)  K 0 ∈wK  γp2F k[[u]]k2L2 (F ) . hF  1/2  Θ2K 0   .  (4.3.8)  (4.3.9)  With this notation the following result holds. Theorem 4.3.3 Let u be the solution of (4.1.1) and uhp ∈ Sp (T ) its DG approximation obtained by (4.2.6) with γ ≥ γ0 . Let the local error estimators ηK be defined by (4.3.2) and the local data approximation error ΘK by (4.3.3). Then, for any δ ∈ (0, 21 ), we have the local upper bound 2δ+ 12  ηK . pδ+1 K k u − uhp kE,wK + pK  ΘwK .  The proof of Theorem 4.3.3 will be presented in Section 4.5. It is similar to the corresponding results in two dimensions in [18, 23, 35]. However, while the proofs there are based on the bubble function technique introduced in [25], here we employ a simple tensor-product argument based on a result for squares in [9]. Remark 4.3.4 As in the two-dimensional case considered in [18], our error estimator can be extended to the Poisson problem with the inhomogeneous  113  4.4. Proof of Theorem 4.3.1 boundary condition u = g on Γ for g ∈ H 1/2 (Γ). In this case, the local error indicators ηK have to be modified by redefining the jump estimators ηJK as ηJ2K =  1 2  X  F ∈FI (K)  γ 2 p3F k[[uhp ]]k2L2 (F ) + hF  X  F ∈FB (K)  γ 2 p3F kuhp − ghp k2L2 (F ) , hF  where ghp is a polynomial approximation of the boundary datum g. In this setting, Theorem 4.3.1 and Theorem 4.3.3 still hold with the inclusion of an additional data-oscillation term on the boundary; see [18] for details.  4.4  Proof of Theorem 4.3.1  In this section, we present the proof of Theorem 4.3.1. To this end, we proceed in the following steps.  4.4.1  Edges and nodes  We begin by introducing the following sets associated with nodes. We denote by N (K) the set of eight vertices of an element K ∈ T , and by N (F ) the set of the four vertices of a face F in F(T ). We then introduce the set of all mesh nodes by [ N (T ) = N (K). K∈T  We write N (T ) = NI (T ) ∪ NB (T ), where NI (T ) and NB (T ) are the sets of interior and boundary mesh nodes, respectively. Next, we introduce the following sets of edges. We denote E(K) the set of the twelve elemental edges of an element K ∈ T , and by E(F ) the set of the four edges of a mesh face F ∈ F(T ). We call E an edge of the mesh T if E = ∂F ∩ ∂F 0 is a line segment given by the intersection of two faces F, F 0 in F(T ) in such a way that its midpoint is not a mesh node of N (T ). We denote by E(T ) the set of all mesh edges of T . The length of an edge E is denoted by hE .  4.4.2  Auxiliary meshes  As in Chapter 3, we shall make use of an auxiliary 1-irregular mesh Te of affine hexahedra. We construct the auxiliary mesh Te from the mesh T as follows. Let K ∈ T . If all twelve elemental edges are edges of the mesh T , that is, if E(K) ⊆ E(T ) (in this case, we have also F(K) ⊆ F(T )), we leave K untouched. Otherwise, at least one of the elemental edges of K 114  4.4. Proof of Theorem 4.3.1 contains a hanging node. In this case, we replace K by the eight hexahedral elements obtained from bisecting the elemental edges of K; see Chapter 3 for an illustration of the analogous construction in two dimensions. Clearly, the mesh Te is a refinement of T ; it is also shape-regular and 1-irregular. More importantly, the hanging nodes of T are not hanging nodes of Te anymore. In the following, we shall write R(K) for the elements in Te that are inside K. That is, if K is unrefined, we have R(K) = {K}. Otherwise R(K) consists of eight newly created elements. We denote by FR (T ) the set of mesh faces in F(T ) that have been refined in the construction of Te . Furthermore, we denote by FH (T ) the set of faces in FR (T ) that have at least one hanging node of T on their edges, and by FN (T ) the ones that have no hanging nodes of T on their edges. The sets of nodes, edges and faces of the auxiliary mesh Te are denoted by N (Te ), E(Te ) and F(Te ), respectively; these sets are defined in an analogous manner to the corresponding sets introduced for the mesh T . We then define the following subsets of N (Te ), E(Te ) and F(Te ): NA (Te ) = { νe ∈ N (Te ) : ∃K ∈ T such that νe is inside K }, e ∈ E(Te ) : ∃K ∈ T such that E e is inside K }, EA (Te ) = { E  FA (Te ) = { Fe ∈ F(Te ) : ∃K ∈ T such that Fe is inside K }.  We then introduce the following auxiliary DG finite element space on the mesh Te : b K e ∈ Te }, Sep (Te ) = { v ∈ L2 (Ω) : v|Ke ◦ TKe ∈ QpKe (K),  where the auxiliary polynomial degree vector e p is defined by pKe = pK for e b onto K. e We clearly have K ∈ R(K) and TKe is the affine mapping from K the following inclusion: Sp (T ) ⊆ Sep (Te ). (4.4.1) In analogy with (4.3.1), the energy norm associated with Te is defined by k u k2E,Te =  X  e Te K∈  k∇uk2L2 (K) e +  X  Fe∈F (Te )  γp2e F  hFe  k[[u]]k2L2 (Fe) ,  (4.4.2)  where the auxiliary face polynomial degrees pFe for the jump terms over Te are defined as in (4.2.4), but using the auxiliary degrees pKe . 115  4.4. Proof of Theorem 4.3.1  4.4.3  Averaging operator  Our analysis is based on an hp-version averaging operator that allows us to approximate discontinuous functions by continuous ones. Analogous operators are used in the hp-version approaches presented in [11, 18, 23, 35]. For the h-version of the DG method, we also refer the reader to [15, 24] and the references therein. To state our result, let Sepc (Te ) be the conforming subspace of Sep (Te ) given by Sepc (Te ) = Sep (Te ) ∩ H01 (Ω).  (4.4.3)  Theorem 4.4.1 There exists an averaging operator Ihp : Sp (T ) → Sepc (Te ) that satisfies X X 2 k∇(v − Ihp v)k2L2 (K) . (4.4.4) p2F h−1 e F k[[v]]kL2 (F ) , e Te K∈  F ∈F (T )  X  e Te K∈  kv − Ihp vk2L2 (K) e .  X  F ∈F (T )  2 p−2 F hF k[[v]]kL2 (F ) .  (4.4.5)  The explicit construction of Ihp and the detailed proof of properties (4.4.4)– (4.4.5) are presented in Section 4.6. Remark 4.4.2 The result in Theorem 4.4.1 generalizes several hp-version approximation results of the same type to three dimensions. The analyses in [18, 23] showed the H 1 -norm estimate (4.4.4) on two-dimensional regular and irregular meshes, respectively. In [11, Lemma 3.2], both estimates in (4.4.4) and (4.4.5) were proven for regular two-dimensional meshes and a fixed polynomial degree. In [35], these results have been extended to twodimensional 1-irregular meshes and variable polynomial degrees; see also Chapter 3. Remark 4.4.3 We emphasize that for partitions with no irregular nodes, the auxiliary mesh Te coincides with T . In this case, Theorem 4.4.1 holds true directly on the original mesh T .  4.4.4  Proof of Theorem 4.3.1  To prove Theorem 4.3.1, we follow [18, 31] and decompose the DG solution uhp into a conforming part and a remainder: uhp = uchp + urhp , 116  4.4. Proof of Theorem 4.3.1 where uchp = Ihp uhp ∈ Sepc (Te ) ⊂ H01 (Ω) is defined using the averaging operator Ihp in Theorem 4.4.1. The remainder urhp is given by urhp = uhp − uchp ∈ Sep (Te ). Analogously to Lemma 3.4.3, one can show that k u − uhp kE,T . k u − uhp kE,Te .  Therefore, by the triangle inequality, k u − uhp kE,T . k u − uchp kE,Te + k urhp kE,Te . Finally, since u − uchp ∈ H01 (Ω), we have k u − uchp kE,T = k u − uchp kE,Te . As the starting point of our proof, we thus obtain the following inequality: k u − uhp kE,T . k u − uchp kE,T + k urhp kE,Te .  (4.4.6)  We first show that k urhp kE,Te in (4.4.6) can be bounded by the error estimator η. Lemma 4.4.4 Under the foregoing assumptions, the following upper bound holds k urhp kE,Te . η. Proof : Recall from (4.4.2) that k urhp k2E,Te  =  X  e Te K∈  k∇urhp k2L2 (K) e  +  X  Fe∈F (Te )  γp2e F  hFe  k[[urhp ]]k2L2 (Fe) .  Since uhp ∈ Sp (T ) and [[urhp ]]|F = [[uhp ]]|F for all F ∈ F(Te ), an argument similar to 3.4.2 allows us to bound the jump terms by X  Fe∈F (Te )  γp2e F  hFe  k[[urhp ]]k2L2 (Fe) . γ −1  X  F ∈F (T )  X γ 2 p2F k[[uhp ]]k2L2 (F ) . γ −1 ηJ2K , hF K∈T  where we have also used the fact that pF ≥ 1. To bound the volume terms, we apply Theorem 4.4.1 and the last bound in the previous argument. This results in the estimate X  e Te K∈  −2 k∇urhp k2L2 (K) e .γ  This completes the proof.  X  F ∈F (T )  X γ 2 p2F k[[uhp ]]k2L2 (F ) . γ −2 ηJ2K . hF K∈T  2 117  4.4. Proof of Theorem 4.3.1 To bound k u − uchp kE,T in (4.4.6), we shall make use of the following two auxiliary forms: X γp2 Z XZ F ∇u · ∇v dx + [[u]] · [[v]] ds, Dhp (u, v) = h F K F K∈T F ∈F (T ) X Z X Z {{∇v}} · [[u]] ds. {{∇u}} · [[v]] ds − Khp (u, v) = − F ∈F (T ) F  F ∈F (T ) F  The form Dhp (u, v) is well-defined for u, v ∈ Sp (T ) + H01 (Ω), whereas the term Khp (u, v) is only well-defined for discrete functions u, v ∈ Sp (T ). Furthermore, we have A(u, v) = Dhp (u, v)  ∀ u, v ∈ H01 (Ω),  (4.4.7)  as well as Ahp (u, v) = Dhp (u, v) + Khp (u, v)  ∀ u, v ∈ Sp (T ).  (4.4.8)  We also recall the standard hp-version approximation result from [23, Lemma 3.7]: For any v ∈ H01 (Ω), there exists a function vhp ∈ Sp (T ) such that 2 2 p2K h−2 K kv − vhp kL2 (K) . k∇vkL2 (K) ,  k∇(v − vhp )k2L2 (K) . k∇vk2L2 (K) ,  (4.4.9)  2 2 pK h−1 K kv − vhp kL2 (∂K) . k∇vkL2 (K) ,  for any element K ∈ T . Next, we prove the following auxiliary estimate. Lemma 4.4.5 For any v ∈ H01 (Ω), we have Z f (v − vhp ) dx − Dhp (uhp , v − vhp ) + Khp (uhp , vhp ) . (η + Θ) k v kE,T . Ω  Here, vhp ∈ Sp (T ) is the hp-version approximation of v defined in (4.4.9). Proof : For notational convenience, let us set Z f (v − vhp ) dx − Dhp (uhp , v − vhp ) + Khp (uhp , vhp ). T = Ω  118  4.4. Proof of Theorem 4.3.1 By writing out the forms Dhp and Khp , integrating by parts the volume terms and manipulating the resulting expressions, we readily obtain X γp2 Z XZ F (f + ∆uhp )(v − vhp ) dx − [[uhp ]] · [[v − vhp ]] ds T = h F F K∈T K F ∈F (T ) X Z X Z {{∇vhp }} · [[uhp ]] ds [[∇uhp ]]{{v − vhp }} ds − − F ∈F (T ) F  F ∈FI (T ) F  ≡ T1 + T2 + T3 + T4 . To bound term T1 , we first add and subtract the approximation fhp to f : XZ XZ T1 = (fhp + ∆uhp )(v − vhp ) dx + (f − fhp )(v − vhp ) dx. K  K∈T  K∈T  K  Using the approximation properties (4.4.9) and the Cauchy-Schwarz inequality shows that X 1   21  X 2 −2 2 2 2 2 T1 . ηR + Θ kv − v k p h 2 hp K K K L (K) K K∈T  .  X  K∈T  2 ηR + Θ2K K  K∈T   1  2  k v kE,T .  For term T2 , we again exploit the Cauchy-Schwarz inequality to conclude that 1  X 1  X 2 2 −1 2 2 T2 ≤ k[[u ]]k k[[v − v ]]k . γ 2 p3F h−1 p h F F hp L2 (F ) hp L2 (F ) F F ∈F (T )  F ∈F (T )  Thus, by the shape-regularity of the meshes, the bounded variation the approximation properties (4.4.9) and property (4.2.3) of the polynomial degrees, we get the bound X 1 2 2 ηJK k v kE,T . T2 . K∈T  Similarly, term T3 can be bounded by  X 1  2 2 T3 ≤ p−1 h k[[∇u ]]k F hp L2 (F ) F F ∈FI (T )  .  X  K∈T  ηF2 K    1 2  X  F ∈FI (T )  2 pF h−1 F k{{v − vhp }}kL2 (F )  1 2  k v kE,T . 119  4.4. Proof of Theorem 4.3.1 Finally, for term T4 , we use the Cauchy-Schwarz inequality, the shaperegularity of the meshes, and the bounded variation property (4.2.3) of the polynomial degrees, to obtain T4 . γ  −1   X  F ∈F (T )  2 γ 2 p2F h−1 F k[[uhp ]]kL2 (F )  1  X 2  K∈T  2 p−2 K hK k∇vhp kL2 (∂K)  1  2  .  From the standard hp-version inverse trace inequality, see [32], we conclude that X 1  X 1 2 2 −1 2 2 T4 . γ η JK k∇vhp kL2 (K) . K∈T  K∈T  From the approximation properties in (4.4.9) it follows that X X X k∇vhp k2L2 (K) . k∇(v − vhp )k2L2 (K) + k∇vk2L2 (K) . k v k2E,T . K∈T  K∈T  K∈T  Hence, T4 . γ −1  X  ηJ2K  K∈T  1 2  k v kE,T .  The above bounds for terms T1 , T2 , T3 , and T4 now imply the assertion. 2 We are now ready to bound k u − uchp kE,T in (4.4.6). Lemma 4.4.6 Under the foregoing assumptions, the following upper bound holds k u − uchp kE,T . η + Θ. Proof : Since u − uchp ∈ H01 (Ω), we have that ku −  uchp kE,T  =  A(u − uchp , v) k v kE,T  ,  (4.4.10)  where v = u − uhp . To bound the right-hand side of (4.4.10), we note that, by (4.1.2) and property (4.4.7), Z Z c c A(u − uhp , v) = f v dx − A(uhp , v) = f v dx − Dhp (uchp , v). Ω  Ω  One can now readily see that Dhp (uchp , v) = Dhp (uhp , v) + R, 120  4.5. Proof of Theorem 4.3.3 with R=−  XZ  e Te K∈  e K  ∇urhp · ∇v dx.  Here, we have also used that the jumps of v vanish. Furthermore, from the DG method in (4.2.6) and property (4.4.8), we have Z f vhp dx = Dhp (uhp , vhp ) + Khp (uhp , vhp ), Ω  where vhp ∈ Sp (T ) is the hp-version approximation of v in (4.4.9). Combining these results, we thus arrive at Z c f (v − vhp ) dx − Dhp (uhp , v − vhp ) + Khp (uhp , vhp ) − R, A(u − uhp , v) = Ω  The estimate in Lemma 4.4.5 now yields |A(u − uchp , v)| . (η + Θ) k v kE,T + |R|.  (4.4.11)  It remains to bound |R|; from Lemma 4.4.4 and the Cauchy-Schwarz inequality, we readily obtain |R| . k urhp kE,Te k v kE,T . ηk v kE,T . The desired result now follows from (4.4.10), (4.4.11) and (4.4.12).  (4.4.12) 2  The proof of Theorem 4.3.1 readily follows from (4.4.6), Lemma 4.4.4 and Lemma 4.4.6.  4.5  Proof of Theorem 4.3.3  In this section, we prove Theorem 4.3.3.  4.5.1  Inverse estimates  We first establish the following p-version inverse estimates. Let Ibd = (−1, 1)d 1 ≤ d ≤ 3, be the unit hypercube in d dimensions. Analogously to (4.2.1)– (4.2.2), we write Qp (Ibd ) for the space of the polynomials on Ibd of degree less or equal than p in each variable.  b d (x) be the bubble function on Ibd given Lemma 4.5.1 For 1 ≤ d ≤ 3, let Ψ by b d (x) = Πd (1 − x2 ). Ψ (4.5.1) i=1 i 121  4.5. Proof of Theorem 4.3.3 Let α, β ∈ R satisfy −1 < α < β and δ ∈ [0, 1]. Then there exist positive constants C1 , C2 and C3 such that for all polynomials φ ∈ Qp (Ibd ), p ∈ N, we have the inverse estimates Z Z 2 2 b |φ|2 dx, (4.5.2) Ψd |∇φ| dx ≤ C1 p IbZd Ibd Z b α |φ|2 dx ≤ C2 p2(β−α) b β |φ|2 dx, Ψ Ψ (4.5.3) d d bd bd Z I ZI b 2δ |∇φ|2 dx ≤ C3 p2(2−δ) b δ |φ|2 dx. Ψ Ψ (4.5.4) d d Ibd  Ibd  In addition, if φ = 0 on ∂ Ibd , then Z Z |∇φ|2 dx ≤ C1 p2 Ibd  Ibd  b d )−1 |φ|2 dx. (Ψ  (4.5.5)  Proof : For d = 1, the results in (4.5.2)–(4.5.5) can be found in [10, 9]. Using a tensor-product argument, they can be readily extended to d = 2 and d = 3. 2  4.5.2  Polynomial extension over faces  Next, we establish the following polynomial extension result, cf. [25, Lemma 2.6] for two-dimensional elements (including triangles). To that end, let b be an elemental face of the reference element K. b Starting from Fb ∈ F(K) b 2 in (4.5.1) and using an affine transformation from Ib2 onto Fb, we can Ψ b b on Fb. readily define a face bubble function Ψ F  Lemma 4.5.2 For α ∈ (1/2, 1], there exists Cα > 0 such that for any polynomial φ ∈ Qp (Fb) of degree p ≥ 1 and any δ ∈ (0, 1] there is an extension b satisfying vb b = φ · Ψ b α on Fb and vb b = 0 on ∂ K b \ Fb, as well as vbFb ∈ H 1 (K) b F F F  b α/2 2 kb vFb k2L2 (K) b ≤ Cα δkφΨ b kL2 (Fb) ,  (4.5.6)  F  2(2−α) b α/2 k2 k∇vFb k2L2 (K) + δ −1 )kφΨ . b ≤ Cα (δp b L2 (Fb) F  (4.5.7)  Proof : Without loss of generality, we may assume that the face Fb is given by Fb = (−1, 1) × (−1, 1) × {−1}. 122  4.5. Proof of Theorem 4.3.3 b b (x1 , x2 , −1) = Ψ b 2 (x1 , x2 ). We now define the exIn this case, we have Ψ F tension vFb by vFb (x1 , x2 , x3 ) = 2−2α φ(x1 , x2 )ΨFb (x1 , x2 )α (1 − x3 )α e−  1+x3 2δ  .  Ψαb F  Obviously, we have vbFb |Fb = φ · and vFb |∂ K\ b Fb = 0. To prove (4.5.6), we note that Ψ2α ≤ Ψαb on Fb. Therefore, a direct computation shows that Fb F Z α 2 −2α α 2 kvFb kL2 (K) kφΨFb kL2 (Fb) (1−x3 )2α e−(1+x3 )/δ dx3 ≤ Cα δkφΨ b2 k2L2 (Fb) . b =4 F  Ib  To show (4.5.7), we proceed as follows:   α−1 2 α 2 k∂x1 vFb k2L2 (K) b ≤ Cα k∂x1 φΨFb kL2 (Fb) + kφΨ b kL2 (Fb) δ F  α/2 F  ≤ Cα δp2(2−α) kφΨ b k2L2 (Fb) ,  where we used Lemma 4.5.1 twice (for d = 2). The estimate for ∂x2 vFb can be derived analogously. For ∂x3 vFb , we have Z 1 1 − x3 2(α−1) − x3 +1 2 α 2 ) e δ dx3 k∂x3 vFb kL2 (K) ( b ≤ Cα kφΨFb kL2 (Fb) 2 −1  Z 1 1 − x3 2α − x3 +1 1 ( + 2 ) e δ dx3 4δ −1 2 α/2 F  ≤ Cα p−2α kφΨ b k2L2 (Fb) [δ −1+2α + δ −1 ],  where we estimated the integral Z  1  (  −1  Z  1 − x3 2(α−1) − x3 +1 ) e δ dx3 2 1−2δ  R1  1−x3 2(α−1) − e −1 ( 2 )  1 − x3 2(α−1) −(x3 +1)/δ = ( ) e dx3 + 2 −1 Z 1−2δ 2(α−1) ≤δ e−(x3 +1)/δ dx3  Z  1  1−2δ  (  2(x3 +1) δ  dx3 as follows:  1 − x3 2(α−1) −(x3 +1)/δ ) e dx3 2  −1  Z  2  2δ−2 2α−1 δ 2α−1 1 e δ δ + e−(x3 +1)/δ dx3 + 2α − 1 δ 1−2δ 2 2 ≤ δ 2(α−1) δ + δ 2α−1 + (e2 − 1)δ 2α−1 2α − 1 2α − 1 ≤ Cα δ −1+2α .  This implies (4.5.7).  2 123  4.5. Proof of Theorem 4.3.3  4.5.3  Proof of Theorem 4.3.3  We are now ready to prove Theorem 4.3.3. b 3 ◦ T −1 , where Ψ b 3 is the reference For an element K ∈ T , we set ΨK = Ψ K bubble defined in (4.5.1). For an elemental face F ∈ F(K) corresponding to b we similarly define ΨF = Ψ b b ◦ (TK |F )−1 , where Ψ b b is the face Fb ∈ F(K), F F bubble function defined before Lemma 4.5.1. In the next three lemmas, we show the efficiency of the error indicators ηRK , ηFK and ηJK , respectively. Lemma 4.5.3 Under the assumptions of Theorem 4.3.3, there holds δ+ 1  ηRK . pK k∇(u − uhp )kL2 (K) + pK 2 ΘK . Proof : For any element K ∈ T , we set vK = (fhp + ∆uhp )|K ΨαK , where α ∈ (1/2, 1]. Applying the inverse inequality in (4.5.3) with a simple scaling argument, we obtain α/2  kfhp + ∆uhp kL2 (K) . pαK k(fhp + ∆uhp )ΨK kL2 (K) −α/2  = pαK kvK ΨK  kL2 (K) .  This leads to 2 2 ηR . SK K  with  −α/2 2 kL2 (K) .  2 SK = p2α−2 h2K kvK ΨK K  (4.5.8)  Since the exact solution satisfies (f + ∆u)|K = 0, we obtain Z 2 2 SK = p2α−2 h (fhp + ∆uhp )vK dx K K K Z 2 = p2α−2 h (∆(uhp − u) + (fhp − f )) vK dx K K K Z Z α/2 −α/2 2α−2 2 = pK hK ( ∇(u − uhp ) · ∇vK dx + (fhp − f )ΨK (vK ΨK ) dx). K  K  Here, we have also used integration by parts and the fact that vK |∂K = 0. Following the proof of [25, Lemma 3.4], we have −α/2  2−α k∇vK kL2 (K) . h−1 K pK kvK ΨK  kL2 (K) . 124  4.5. Proof of Theorem 4.3.3 By the Cauchy-Schwarz inequality and the definition of the data approximation term ΘK , we conclude that  1 −α/2 2 SK . pK k∇(u − uhp )kL2 (K) + pαK ΘK p2α−2 h2K kvK ΨK k2L2 (K) 2 K  . pK k∇(u − uhp )kL2 (K) + pαK ΘK SK .  Therefore, by this inequality and (4.5.8), we have  ηRK . pK k∇(u − uhp )kL2 (K) + pαK ΘK .  2  Choosing δ = α − 1/2 finishes the proof.  For a mesh face F ∈ F(T ), we define wF = { K1 , K2 ∈ T : F = ∂K1 ∩ ∂K2 }, e ∈ T ∪ Te : F ∈ F(K) e }. w eF = { K  (4.5.9)  For simplicity, we also use the notation wF and w eF to denote the domain formed by the elements in wF and in w eF , respectively. Lemma 4.5.4 Under the assumptions of Theorem 4.3.3, there holds 2δ+ 12  ηFK . pδ+1 K k u − uhp kE,wK + pK  ΘwK .  Proof : Let F = ∂K1 ∩ ∂K2 be an interior face shared by two elements K1 , K2 ∈ T . For any α ∈ (1/2, 1], set τF = [[∇uhp ]]ΨαF . Next, we construct a bubble function ψF on wF . We distinguish the following two cases. Case 1: Suppose that F ∈ F(K1 ) ∩ F(K2 ). Lemma 4.5.2 with δ = 1/p2F and a scaling argument then ensure the existence of vF ∈ H01 (wF ) with vF |F = τF , vF |∂wF = 0 such that 1/2  −α/2  kvF kL2 (wF ) . hF p−1 F kτF ΨF −1/2  k∇vF kL2 (wF ) . hF  kL2 (F ) ,  −α/2  pF kτF ΨF  kL2 (F ) .  (4.5.10) (4.5.11)  Case 2: Otherwise, without loss of generality, we may assume F ∈ F(K2 ), but F ∈ / F(K1 ). In this case, wF is concave, and there exists e 1 ∈ Te , such that K e 1 ( K1 and F ∈ F(K e 1 ) ∩ F(K2 ). Thus, an element K e w eF = K1 ∪ K2 ( wF . As before, by Lemma 4.5.2 (with δ = 1/p2F ) and a scaling argument, we can then find a function veF ∈ H01 (w eF ) with veF |F = τF , veF |∂ weF = 0 and 1/2  −α/2  ke vF kL2 (weF ) . hF p−1 F kτF ΨF −1/2  k∇e vF kL2 (weF ) . hF  kL2 (F ) ,  −α/2  pF kτF ΨF  kL2 (F ) . 125  4.5. Proof of Theorem 4.3.3 We then define the function vF on wF by ( veF on w eF , vF = 0 otherwise.  Thus, vF ∈ H01 (wF ) with vF |F = τF , vF |∂wF = 0 and the crucial inequalities (4.5.10) and (4.5.11) also hold in this case. In both cases above, we now proceed as follows. Applying again the inverse inequality (4.5.3) and scaling, we get −α  α  k[[∇uhp ]]kL2 (F ) . pαF k[[∇uhp ]]ΨF2 kL2 (F ) = pαF kτF ΨF 2 kL2 (F ) . Therefore, 2 ηF2 K . SK  X  2 SK =  with  −α  F ∈FI (K)  hF kτF ΨF 2 k2L2 (F ) . p2α−1 F  (4.5.12)  Since [[∇u]] = 0 on interior edges, integration by parts over wF yields Z −α/2 2 kτF ΨF kL2 (F ) = [[∇(uhp − u)]]τF ds F  =  X Z  K∈wF  K  (∆uhp − ∆u)vF + (∇uhp − ∇u) · ∇vF dx.  Using the differential equation and approximating the data, we obtain X X Z 2 SK = h (fhp + ∆uhp )ψF dx p2α−1 F F K∈wF  F ∈FI (K)  +  X  hF p2α−1 F  F ∈FI (K)  +  X  p2α−1 hF F  F ∈FI (K)  K  X Z  K∈wF  K  K∈wF  K  X Z  (∇uhp − ∇u) · ∇ψF dx (f − fhp )ψF dx  = T1 + T2 + T3 . To bound T1 , we use the Cauchy-Schwarz inequality, inequality (4.5.10), the results of Lemma 4.5.3 and the bounded variation property of p in (4.2.3). This readily yields X 1 α− 1 −α T1 . pK 2 (pK k u − uhp kE,wK + pαK ΘwK )( p2α−1 hF kτF ΨF 2 kL2 (F ) ) 2 F F ∈FI (K)  α− 12  . pK  (pK k u − uhp kE,wK + pαK ΘwK ) SK . 126  4.6. Proof of Theorem 4.4.1 Similarly, the term T2 can be bounded by   X T2 . k u − uhp kE,wK  hF p2α−1 k∇ψF kL2 (wF )  F F ∈FI (K)    . k u − uhp kE,wK   X  1 +α 2  pF  F ∈FI (K)  1 +α 2  . pK  k u − uhp kE,wK SK .    α− 1 −α hF pF 2 kτF ΨF 2 kL2 (F )  1 2  Finally, we estimate the data error term T3 as follows: X 2α T3 . p−1 F hF kf − fhp kL2 (wF ) pF kψF kL2 (wF ) F ∈FI (K)    . ΘwK  α− 12  . pK  X  F ∈FI (K)  1 2  −α 2  kτF ΨF hF p2α−1 p2α−1 F F  ΘwK SK .  k2L2 (F )   Combining the above bounds for T1 through T3 , we obtain α+ 21  SK . pK  2α− 12  k u − uhp kE,wK + pK  ΘwK .  By (4.5.12), we conclude that α+ 21  ηFK . pK  2α− 12  k u − uhp kE,wK + pK  ΘwK .  Choosing δ = α − 1/2 leads to the assertion.  2  Lemma 4.5.5 Under the assumptions of Theorem 4.3.3, there holds 1 2 η JK . p K k u − uhp kE,wK .  Proof : This follows from the fact that the jump of u vanishes over all faces.2 The proof of Theorem 4.3.3 now immediately follows from Lemma 4.5.3, Lemma 4.5.4 and Lemma 4.5.5.  4.6  Proof of Theorem 4.4.1  In this section, we prove the result of Theorem 4.4.1. 127  4.6. Proof of Theorem 4.4.1  4.6.1  Polynomial basis functions  As in the proof of Theorem 3.4.4, we begin by introducing polynomial basis functions. To that end, let Ib = (−1, 1) be the reference interval. We denote b = { zbp , · · · , zbpp } the Gauss-Lobatto nodes of order p ≥ 1 on I. b by Zbp (I) 0 p p p p p b = { zb , · · · , zb } Recall that zb0 = −1 and zbp = 1. We denote by Zbint (I) 1 p−1 b the interior Gauss-Lobatto nodes of order p on I. Now let E ∈ E(K) be an elemental edge of K ∈ T . The nodes in Zbp can be affinely mapped onto E and we denote by Z p (E) = { z0E,p , · · · , zpE,p } the Gauss-Lobatto nodes of order p on E. The points z0E,p and zpE,p coincide p E,p with the two end points of E. The set Zint (E) = { z1E,p , · · · , zp−1 } denotes the interior Gauss-Lobatto points of order p. We write Pp (E) for the space of all polynomials of degree less than or equal to p on E and define Ppint (E) = { q ∈ Pp (E) : q(z0E,p ) = q(zpE,p ) = 0 },  p (E) }. Ppnod (E) = { q ∈ Pp (E) : q(z) = 0, z ∈ Zint  By construction, we have Pp (E) = Ppint (E) ⊕ Ppnod (E). On the reference square Ib2 = (−1, 1)2 , we define the tensor-product p Gauss-Lobatto nodes of order p by Zbp (Ib2 ) = { zbi,j = (b zip , zbjp ) }0≤i,j≤p . These nodes can be affinely mapped onto an elemental face F ∈ F(K) of K ∈ T F,p and we define Z p (F ) = { zi,j }0≤i,j≤p to be the Gauss-Lobatto nodes of p F,p order p on F . Furthermore, we write Zint (F ) = { zi,j }1≤i,j≤p−1 for the interior Gauss-Lobatto points on F . We also define  Qint pF (F ) = { q ∈ QpF (F ) : q = 0 on ∂F }. Similarly, we define the interior Gauss-Lobatto nodes of order p on the p b by Zbp (K) b = {b = (b zip , zbjp , zbkp )}1≤i,j,k≤p−1 . For an reference element K zi,j,k int element K ∈ T and a polynomial degree p ≥ 1, we denote its interior Gaussp K,p K,p Lobatto points by Zint (K) = { zi,j,k }1≤i,j,k≤p−1 . Here, the points zi,j,k are p the affine mappings of zbi,j,k onto the element K. Suppose now that we are given edge and face polynomial degrees 1 ≤ pE ≤ p and 1 ≤ pF ≤ p, associated with the elemental edges E ∈ E(K) and elemental faces F ∈ F(K). We assume that pE ≤ pF for E ∈ E(F ). We shall define basis functions for polynomials v ∈ Qp (K) with the restriction that v|E ∈ PpE (E),  E ∈ E(K),  v|F ∈ QpF (F ),  F ∈ F(K).  (4.6.1)  128  4.6. Proof of Theorem 4.4.1  νb8  νb7  νb5  b12 E  νb6  νb4  νb3  νb1  b5 E  Fb6  b11 E  b8 E b4 E  b10 E  b9 E b3 E  b6 E  b7 E  b2 E  b1 E  νb2  (a) Numbering of nodes  Fb4  (b) Numbering of edges  Fb3  Fb2  Fb1  b x3 b x2  Fb5  b x1  (c) Numbering of faces  b with the numbering of faces, edges and Figure 4.1: Reference element K vertices. As usual, we shall divide the basis functions into interior, face, edge and b = vertex basis functions. We first consider the reference element K = K 3 b b b b (−1, 1) . We denote its faces by F1 , . . . F6 , its edges by E1 , . . . , E12 and its vertices by νb1 , . . . , νb8 , numbered as in Figure 4.1. Let {ϕ bpi }0≤i≤p be the b Lagrange basis functions associated with the Gauss-Lobatto nodes Zbp (I) b on I. The interior basis functions are then b int,p (b x3 ), b2 , x b3 ) = ϕ bpi (b x1 ) ϕ bpj (b x2 ) ϕ bpk (b Φ i,j,k x1 , x  1 ≤ i, j, k ≤ p − 1.  Next, we define the face basis functions exemplary for the face Fb1 in Figure 4.1 with face polynomial degree pFb1 . They are given by Fb ,pFb  b 1 Φ i,j  1  pFb  (b x1 , x b2 , x b3 ) = ϕ bi  1  Fb1 ,pFb  pFb  x2 ) ϕ bj 1 (b x3 ), (b x1 ) ϕ bp0 (b  1 ≤ i, j ≤ pFb1 − 1.  1 b Note that Φ vanishes on Fb2 through Fb6 . The other face basis functions i,j are then defined analogously. To define the edge basis functions, we consider b1 in Figure 4.1 with edge degree p b . The edge basis exemplary the edge E E1 b1 are functions for E  b1 ,p b E E  b Φ i  1  pEb  (b x1 , x b2 , x b3 ) = ϕ bi b1 ,p b E E  b Note that Φ i  1  1  pFb  pFb  (b x1 ) ϕ b0 5 (b x2 ) ϕ b0 1 (b x3 ),  i = 1, . . . , pEb1 − 1.  vanishes on all the other edges and on the faces Fb2 , Fb3 , Fb ,pFb  1 Fb4 and Fb6 . Moreover, it vanishes on the interior nodes {b zi,j  1  pFb −1  1 }i,j=1 and  129  4.6. Proof of Theorem 4.4.1 Fb ,p  p  −1  5 F b b F 5 {b zi,j 5 }i,j=1 of the faces Fb1 and Fb5 , respectively. The other edge basis functions are then defined analogously. Finally, we consider the vertex νb1 , b1 , E b4 and E b5 ; see Figure 4.1. The associated which is shared by the edges E vertex basis function is then defined by  p  p  p  b b b E E E b νb1 (b x3 ). x2 ) ϕ b0 5 (b x1 ) ϕ b0 4 (b b2 , x b3 ) = ϕ b0 1 (b Φ b x1 , x  K  b can be The vertex basis functions associated with the other vertices of K defined analogously. This completes the definition of the shape functions on b the reference element K. For an arbitrary element K, the basis functions Φ on K can be defined b by the pull-back map TK : Φ(x1 , x2 , x3 ) = from the analogous ones on K −1 E F b , ΦF,p and Φ ◦ TK (x1 , x2 , x3 ), giving rise to shape functions ΦνK , ΦE,p i i,j Φint,p i,j,k on K. Therefore, a polynomial v ∈ Qp (K) satisfying (4.6.1) can be expanded in the following form: v(x) =  X  v(ν) ΦνK (x)  X  pX E −1  E v(ziE,pE ) ΦE,p (x) i  E∈E(K) i=1  ν∈N (K)  +  +  X  pX F −1  F ∈F (K) i,j=1  F cFi,j ΦF,p i,j (x) +  X  ci,j,k Φint,p i,j,k (x),  1≤i,j,k≤p−1  with coefficients cFi,j and ci,j,k . In the sequel, we will make use of the following two estimates for polynomials, which are proven in Lemma 3.1 of [11]; see also [35]. Lemma 4.6.1 For an element K, we have the following estimates: (i) If v ∈ QpK (K) vanishes at the interior tensor-product Gauss-Lobatto nodes of K, then there holds 2 kvk2L2 (K) . hK p−2 K kvkL2 (∂K) .  (ii) If the vertex ν of K is shared by the elemental edges Ei , Ej and Ek , then the vertex basis function ΦνK can be bounded by 3/2  −1 −1 kΦνK kL2 (K) . hK p−1 Ei pEj pEk .  (iii) Let the elemental face F be spanned by the two elemental edges Ei and Ej . Suppose that the vertex ν is given by the intersection of Ei and Ej . Then the vertex basis ΦνK can be bounded by −1 kΦνK kL2 (F ) . hK p−1 Ei pEj .  130  4.6. Proof of Theorem 4.4.1  4.6.2  Edge extension operators  In this section, we define extension operators over an edge E. To that end, fix an element K ∈ T . We discuss three cases where we shall employ edge extensions. First, if E ∈ E(K) is an elemental edge of K without a hanging node, we define the edge extension operator LE p by LE p,K  :  Ppint (E)  −→ Qp (K),  q(x) 7−→  p−1 X  q(ziE,p )ΦE,p i (x).  (4.6.2)  i=1  Second, if the edge E ∈ E(K) contains a hanging node located in the middle of E, then E = E1 ∪ E2 for two mesh edges E1 and E2 in E(T ). In this case, we partition K into two parallelepipeds, K = K1 ∪ K2 , by connecting the hanging node on E with the midpoint of the edge parallel to E, as illustrated in Figure 4.2. For q1 ∈ Ppint (E1 ) and q2 ∈ Ppint (E2 ), we then define the extension operator LE p,K (q1 , q2 ) by E1 E2 LE p,K (q1 , q2 ) = Lp,K1 (q1 ) + Lp,K2 (q2 ),  (4.6.3)  E2 1 with LE p,K1 (·) and Lp,K2 (·) given in (4.6.2). The third case arises if the edge E belongs to the space  EF (K) = { E ∈ E(T ) : E is inside F }  (4.6.4)  for an elemental face F ∈ F(K). That is, E ∈ EF (K) is one of the four mesh edges whose intersection is a hanging node located in the middle of F . This situation is depicted in Figure 4.3. In this case, we partition K = ∪4i=1 Ki into four elements, as illustrated in Figure 4.3. If E is shared by K1 and K2 and if q ∈ Ppint (E), the extension LE p,K (q) is then defined by E E LE p,K (q) = Lp,K1 (q) + Lp,K2 (q),  (4.6.5)  E with LE p,K1 and Lp,K2 given in (4.6.2) and extended by zero to the other two elements. By construction, the extension operators LE p,K (q) in (4.6.2), (4.6.5) and E Lp,K (q1 , q2 ) in (4.6.3) are continuous on K and satisfy  LE p,K (q)|E = q,  LE p,K (q1 , q2 )|E1 = q1 ,  LE p,K (q1 , q2 )|E2 = q2 .  E Moreover, LE p,K (q) and Lp,K (q1 , q2 ) vanish at the interior Gauss-Lobatto p nodes in Zint (K), on the other edges of E(K) and the elemental faces in F(K) not containing E. From [11, Lemma 3.1], we have the following inequalities.  131  4.6. Proof of Theorem 4.4.1  K1  K2  E1 K1 E1  E4 uE  u E2  E3  K3  K2 E2  K4  Figure 4.3: Case 3: The mesh edges Ei belong to EF (K) for the elemental face F . The element K is then divided into four elements.  Figure 4.2: Case 2: The elemental edge E ∈ E(K) has a hanging node located in its midpoint.  Lemma 4.6.2 The linear edge extension operators LE p introduced above satisfy −2 kLE p,K (q)kL2 (K) . p hK kqkL2 (E) ,  E ∈ E(K),  −2 kLE p,K (q)kL2 (K) . p hK kqkL2 (E) ,  E ∈ EF (K), F ∈ F(K),  −2 kLE p,K (q1 , q2 )kL2 (K) . p hK  4.6.3  2 X i=1  kqi kL2 (Ei ) ,  E ∈ E1 ∪ E2 , E1 , E2 ∈ E(T ).  Face extension operators  Next, we define extension operators over faces. To that end, fix an element K ∈ T and let F ∈ F(K) be an elemental face of K. Again, we shall discuss three cases of face extensions. First, if there is no hanging node of T located on F (i.e., F ∈ F(T ) ∩ F(Te ) or F ∈ FN (T )), we define LFp,K by LFp,K : Qint p (F ) −→ Qp (K),  q(x) 7−→  p−1 X  F,p q(zi,j )ΦF,p i,j (x).  (4.6.6)  i,j=1  Second, if F has a hanging node in its midpoint (i.e., F ∈ / F(T )), we 4 write F as F = ∪i=1 Fi , for four faces Fi ∈ F(T ). We then partition K into four parallelepipeds, K = ∪4i=1 Ki , as illustrated in Figure 4.4. For polynoF mials qi ∈ Qint p (Fi ), i = 1, . . . , 4, we define the operator Lp,K (q1 , q2 , q3 , q4 ) 132  4.6. Proof of Theorem 4.4.1 by LFp,K (q1 , q2 , q3 , q4 )  =  4 X  i LFp,K (qi ), i  (4.6.7)  i=1  i with LE p,Ki , i = 1, . . . , 4, given in (4.6.6). Third, if F contains a hanging node located on one of its elemental edges (i.e., F ∈ FH (T )), we divide F into four faces F1 , . . . , F4 ∈ F(Te ) and again partition K into four parallelepipeds, K = ∪4i=1 Ki , as shown in Figure 4.5. We denote by νc the center of F . If q ∈ Qp (F ) with q = 0 on ∂F , we define the extension operator LFp,K (q) by  LFp,K (q) =  4 X  i LFp,K (q|Fi ), i  (4.6.8)  i=1  where, for 1 ≤ i ≤ 4, i LFp,K (q|Fi ) i  =  p−1 X  Fi ,p )ΦFk,li ,p q(zk,l  F4 K1 F1  νc q(zkE,p )ΦE,p k + q(νc )ΦKi .  E∈E(Fi ) k=1  k,l=1  K4  +  p−1 X X  K3  K4  F3 u  F4 K2  K1 F1  F2  Figure 4.4: Case 2: Partition of K associated with the partition of face F .  K3 F3 u K2  F2  Figure 4.5: Case 3: Partition of K associated with the partition of face F .  By definition, the face extensions LFp,K (q) defined in (4.6.7), (4.6.8) and LFp,K (q1 , q2 , q3 , q4 ) in (4.6.7) are continuous on K and satisfy LFp,K (q)|F = q,  LFp,K (q1 , q2 , q3 , q4 )|Ei = qi ,  i = 1, . . . , 4.  Moreover, LFp,K (q) and LFp,K (q1 , q2 , q3 , q4 ) both vanish in the interior Gaussp Lobatto nodes in Zint (K) and on the elemental faces of K not equal to F . From [11, Lemma 3.1], we have the following inequalities. 133  4.6. Proof of Theorem 4.4.1 Lemma 4.6.3 The linear face extension operators LFp,K introduced above satisfy 1/2  kLFp,K (q)kL2 (K) . p−1 hK kqkL2 (F ) , 1/2  kLFp,K (q)kL2 (K) . p−1 hK kqkL2 (F ) , 1/2  kLFp,K (q1 , . . . , q4 )kL2 (K) . p−1 hK  4.6.4  F ∈ F(T ) ∩ F(Te ) or F ∈ FN (T ), F ∈ FH (T ),  4 X i=1  kqi kL2 (Fi ) ,  F = F1 ∪ F2 ∪ F3 ∪ F4 , and F1 , . . . , F4 ∈ F(T ).  Decomposition of functions in Sp (T )  We shall now decompose functions in Sp (T ), in a similar manner to the construction in Section 3.5.3. To this end, we first define the minimal edge and face degrees. For an edge E ∈ E(T )∪E(Te ) and a face F ∈ F(T )∪F(Te ), we set e ∈ T ∪ Te , E ∈ E(K) e }, pE = min{ pKe : K (4.6.9) e ∈ T ∪ Te , F ∈ F(K) e }. pF = min{ pKe : K  Let v ∈ Sp (T ). We denote by vK the restriction of v to an element K ∈ T ∪ Te . We decompose v into a nodal, edge, face and interior part, respectively: v = v nod + v edge + v face + v int ,  (4.6.10)  with v nod , v edge , v face and v int in Sep (Te ) introduced below. Nodal part  First, we construct the nodal part v nod ∈ Sep (Te ) in (4.6.10). For each element e e ∈ R(K), we will construct the restriction v nod of v nod to K K ∈ T and K  e (note that pK = p e ) and such that v nod (K) e e ∈ QpK K  e K  K  nod vK e |E  ∈ PpE (E),  e E ∈ E(K),  nod vK e |F ∈ PpF (F ),  e F ∈ F(K),  with pE and pF given in (4.6.9). To define v nod e , we distinguish the following K two cases. Case 1: If R(K) = {K} (i.e., if K is unrefined), the interpolant v nod e = K nod vK is simply defined by X nod vK (x) = vK (ν) ΦνK (x). (4.6.11) ν∈N (K)  134  4.6. Proof of Theorem 4.4.1 Case 2: If R(K) consists of eight newly created elements, we define v nod e K e ∈ R(K) separately. To do so, fix K e ∈ R(K). Without on each element K loss of generality, we may consider the situation shown in Figure 4.6, where ej and Fek the vertices, edges and faces of K, e respectively, we denote by νei , E numbered as in Figure 4.1. Similarly, we denote by νi , Ej and Fk the vertices, edges and faces of K, respectively. In this configuration, notice that we have νe8 ∈ NA (Te ), Fe3 , Fe4 , Fe6 ∈ e8 , E e11 , E e12 ∈ EA (Te ). Hence, the polynomial degrees are FA (Te ), as well as E given by i ∈ {3, 4, 6}, j ∈ {8, 11, 12}.  pFei = pEej = pKe = pK ,  e At the Let us now define the value of v nod at the nodes located on ∂ K. e K ej for i ∈ {3, 4, 6} and j ∈ {8, 11, 12}, we interior nodes shared by Fei and E set pEe  pFe  nod vK e (z) = vK (z),  ej )}j∈{8,11,12} . (4.6.12) z ∈ {Zinti (Fei )}i∈{3,4,6} ∪ {Zint j (E  Similarly, we set v nod e2 and ν = νe8 . e (ν) = vK (ν) for the vertices ν = ν K  E9  ν5  E5 e5 E  νe5  ν6 e9 E  νe6  E6  e Fe1 E6  νe1 E e1 νe2 = ν2 E1 e ∈ R(K). Figure 4.6: The element K is refined into 8 elements K ν1  e e e It remains to define v nod e on the nodes located on the faces F1 , F2 and F5 K (excluding the vertex νe2 ). We only consider Fe1 (the construction for Fe2 and Fe5 is completely analogous); see Figure 4.6. If Fe1 ∈ F(T ), then we have ei ∈ E(Fe1 ) for i ∈ {1, 5, 6, 9} belong to νe1 , νe5 , νe6 ∈ N (T ). The four edges E 135  4.6. Proof of Theorem 4.4.1 E(T ). For i ∈ {1, 5, 6, 9} and j ∈ {1, 5, 6}, we define pFe  nod vK e (z) = 0, nod vK νj ) = vK (e νj ). e (e  pEe  ei )} e z ∈ Zint1 (Fe1 ) ∪ {Zint i (E , Ei ∈E(Fe1 )  (4.6.13) (4.6.14)  Otherwise, if Fe1 ∈ / F(T ), then the large elemental face F1 belongs to FR (T ). Moreover, we have that either F1 ∈ FN (T ) or F1 ∈ FH (T ). We distinguish these two subcases. First, if F1 ∈ FN (T ), then there is no hanging node of T located on F1 or any edge of F1 , and we have pFe1 = pF1 . In this case, we interpolate the values of the nodal interpolant over the face F1 at the Gauss-Lobatto nodes on Fe1 . That is, we define X nod vK vK (ν) ΦνK (z), (4.6.15) e (z) = ν∈N (F1 )  p  p e e e ∪ Z Fe1 (Fe1 ) ∪ {e νi }i∈{1,5,6} . for all z ∈ {ZintEe (E)} int E∈E(F1 ) Second, if F1 ∈ FH (T ), then νe5 ∈ / N (T ), but νe1 and νe6 may or may not belong to N (T ). We define the value of v nod at the nodes located on e K Fe1 for this case as follows. First, noticing that pEe5 = pEe9 = pFe1 = pF1 and νe2 ∈ N (T ), we set nod vK e (z) = 0,  pF pF e5 ) ∪ Z pF1 (E e9 ) ∪ {e z ∈ Zint1 (Fe1 ) ∪ Zint1 (E ν5 }, int  (4.6.16)  e e Next, we define the values of v nod e on the nodes of the edges E1 and E6 , as K e1 (the construction well as on the nodes νe1 and νe6 . We only consider νe1 and E e6 is completely analogous). If νe1 ∈ N (T ) (i.e., νe1 is a hanging for νe6 and E node in T ), then we define nod vK e (z) = 0,  p  e E e1 ), z ∈ Zint 1 (E  nod vK ν1 ) = vK (e ν1 ). e (e  (4.6.17)  If νe1 ∈ / N (T ), then we have E1 ∈ E(T ) and ν1 ∈ N (T ). In this case, pEe1 = pE1 , and we interpolate the values of the nodal interpolant over the e1 . That is, we set long edge E1 at the Gauss-Lobatto nodes on E ν1 ν2 nod vK e (z) = vK (ν1 ) ΦK (z)+vK (ν2 ) ΦK (z),  p  e E e1 )∪{e z ∈ Zint 1 (E ν1 }. (4.6.18)  136  4.6. Proof of Theorem 4.4.1 With the nodal values of v nod e constructed in (4.6.12)-(4.6.18), we have K  nod vK e (x)  =  X  nod vK e (ν)  ΦνKe (x)  e ν∈N (K)  +  X  pF −1   X  e i,j=1 F ∈F (K)  +  X  pE −1   X  e i=1 E∈E(K)  E,pE  nod vK e (zi  E,pE  )Φi  (x)     F,pF nod F,pF vK e (zi,j )Φi,j (x) .  This finishes the construction of the interpolant v nod . Notice that v nod ∈ Sep (Te ); it is continuous over faces F ∈ FA (Te ) and over edges inside faces F ∈ F(T ). Moreover, it satisfies nod vK (ν) − vK (ν) = 0,  ν ∈ N (T ) located on ∂K,  and nod nod vK e |E ∈ PpE (E),  with w eE defined by Edge part  e ∈w E ∈ E(T ), K eE ,  e ∈ T ∪ Te : E ∈ E(K) e }, w eE = { K  ∀E ∈ E(T ).  Second, we construct the edge function v edge ∈ Sep (Te ) in the decomposition (4.6.10). To do so, fix an element K ∈ T . For an edge E on ∂K, we E by define vK  E nod E ∈ E(K) ∩ E(T ),   LpK ,K ((vK − vK )|E ),  E nod E E ∈ EF (K), F ∈ F(K), LpK ,K ((vK − vK )|E ), vK =    LE ((v − v nod )| , (v − v nod )| ), E = E ∪ E , E ∈ E(T ), pK ,K  K  K  E1  K  K  E2  1  2  1,2  with LE pK ,K (·) defined for Case 1 in (4.6.2) or for Case 3 in (4.6.5), and E LpK ,K (·, ·) for Case 2 in (4.6.3), respectively. We then define v edge on each element as: X X X edge E E vK (x) = vK (x) + vK (x). E∈E(K)  F ∈F (K) E∈EF (K)  137  4.6. Proof of Theorem 4.4.1 Face part Third, we construct the face function v face ∈ Sep (Te ) in (4.6.10). Fix an element K ∈ T and let F be an elemental face in F(K). If F ∈ F(T ), we F by define vK F vK  =  (  nod − v edge )| ), LFpK ,K ((vK − vK F K  F ∈ / FH (T ),  nod − LFpK ,K ((vK − vK  F ∈ FH (T ),  edge vK )|F ),  with LFpK ,K (·) defined for Case 1 in (4.6.6) and for Case 3 in (4.6.8). Otherwise, there exists four faces Fi ∈ F(T ), i = 1, . . . , 4, such that F = ∪4i=1 {Fi }. F by We define vK edge edge F nod nod )|F1 , . . . , (vK − vK )|F4 ), vK = LFpK ,K ((vK − vK − vK − vK  with LFpK ,K (·, ·, ·, ·) defined for Case 2 in (4.6.7). We then define v face elementwise as X face F vK (x) = vK (x). F ∈F (K)  4.6.5  Interior part  Finally, the interior function v int ∈ Sep (Te ) in (4.6.10) is simply obtained by setting on each element edge face int nod − vK , vK = vK − vK − vK  K ∈T.  int belongs to H 1 (K). Hence, we have v int ∈ S c (T e ). Notice that vK 0 e p  4.6.6  Proof of Theorem 4.4.1  In this section, we outline the proof of Theorem 4.4.1. Some of the auxiliary results are postponed to Sections 4.6.7, 4.6.7 and 4.6.7. For v ∈ Sp (T ), we write v = v nod + v edge + v face + v int , according to (4.6.10). We shall define the averaging operator Ihp v in four parts: Ihp v = ϑnod + ϑedge + ϑface + ϑint , with ϑnod , ϑedge , ϑface , ϑint ∈ Sepc (Te ).  (4.6.19)  Since v int ∈ Sepc (Te ), we simply  take ϑint = v int . Below we further construct ϑnod , ϑedge , and ϑface such that the following three approximation results hold.  138  4.6. Proof of Theorem 4.4.1 Proposition 4.6.4 (i) Nodal approximation: There is a conforming approximation ϑnod ∈ Sepc (Te ) that satisfies: X  e Te K∈  X  e Te K∈  kv  k∇(v  nod  nod  −  ϑnod k2L2 (K) e  −ϑ  nod  )k2L2 (K) e  X  .  F ∈F (T )  X  .  F ∈F (T )  hF p2F  p2F hF  Z  [[v nod ]]2 ds,  F  Z  (4.6.20) [[v  nod 2  ]] ds.  F  (ii) Edge approximation: There is a conforming approximation ϑedge ∈ that satisfies:  Sepc (Te )  X  e Te K∈  X  e Te K∈  kv edge − ϑedge k2L2 (K) e .  k∇(v  edge  −ϑ  edge  )k2L2 (K) e  .  X  F ∈F (T )  X  F ∈F (T )  hF p2F p2F hF  Z  ([[v]]2 + [[v nod ]]2 ) ds,  F  Z  (4.6.21) 2  ([[v]] + [[v  nod 2  ]] ) ds.  F  (iii) Face approximation: There is a conforming approximation ϑface ∈ that satisfies:  Sepc (Te )  X  e Te K∈  X  kv  k∇(v  e Te K∈  face  face  −  ϑface k2L2 (K) e  −ϑ  face  )k2L2 (K) e  .  X  F ∈F (T )  .  X  F ∈F (T )  hF p2F p2F hF  Z  ([[v]]2 + [[v nod ]]2 ) ds,  F  Z  (4.6.22) 2  ([[v]] + [[v  nod 2  ]] ) ds.  F  By the triangle inequality and Proposition 4.6.4, we then obtain X X 2 nod 2 kv − Ihp vk2L2 (K) p−2 ]]kL2 (F ) ), e . F hF (k[[v]]kL2 (F ) + k[[v F ∈F (T )  e Te K∈  X  e Te K∈  k∇(v − Ihp v)k2L2 (K) e .  X  F ∈F (T )  2 nod 2 p2F h−1 ]]kL2 (F ) ). F (k[[v]]kL2 (F ) + k[[v  Hence, Theorem 4.4.1 follows if we show that k[[v nod ]]k2L2 (F ) . k[[v]]k2L2 (F ) ,  F ∈ F(T ).  (4.6.23)  To prove (4.6.23), we define the set NT (F ) = { ν ∈ N (T ) : ν is located on ∂F },  F ∈ F(T ). 139  4.6. Proof of Theorem 4.4.1 By the construction of v nod , the jump over F satisfies [[v nod ]](ν) = [[v]](ν),  ν ∈ NT (F ).  If F ∈ F(T ) ∩ F(Te ) or F ∈ FN (T ), then we have N (F ) = NT (F ). Lemma 4.6.1(iii) and the bounded local variation of p in (4.2.3) yield k[[v nod ]]kL2 (F ) .  X  ν∈N (F )  |[[v nod (ν)]]|kΦνK kL2 (F ) . p−2 F hF  max |[[v nod ]](ν)|,  ν∈NT (F )  with K one of the elements of which F is an elemental face. Otherwise, we have F ∈ FH (T ). In this case, F is divided into four faces e Fi ∈ F(Te ), i = 1, . . . , 4, and the middle points of the elemental edges of F may or may not belong to N (T ). This situation is the same as the one discussed for the two-dimensional case in Section 3.5.5 (Case 2). Thus, proceeding as in the corresponding proof of Lemma 3.5.3, we obtain from (4.2.3) and the construction of v nod that k[[v nod ]]kL2 (F ) =  4 X i=1  k[[v nod ]]kL2 (Fei ) . p−2 F hF  max |[[v nod ]](ν)|.  ν∈NT (F )  Thus, for any face F ∈ F(T ), we have k[[v nod ]]kL2 (F ) . p−2 F hF  max |[[v nod ]](ν)| = p−2 F hF  ν∈NT (F )  max |[[v]](ν)|.  ν∈NT (F )  Without loss of generality, we suppose that |[[v nod ]](ν)| reaches its maximum at the vertex ν1 , an end point of an edge E ∈ E(T ) which lies on ∂F . From [32, Theorem 3.92], [11, Lemma 3.1] and (4.2.3), we further have the inverse estimate −1/2  max k[[v]](ν)k = k[[v]](ν1 )k . pE hE  ν∈NT (F )  k[[v]]kL2 (E) . p2F h−1 F k[[v]]kL2 (F ) .  This, together with the bounded local variation of p in (4.2.3), implies (4.6.23). To complete the proof of Theorem 4.4.1, it remains now to prove Proposition 4.6.4, which will be undertaken in the next section.  4.6.7  Proof of Proposition 4.6.4  In this section, we present the proofs of the three approximation results in Proposition 4.6.4.  140  4.6. Proof of Theorem 4.4.1 Nodal approximation Let v nod ∈ Sep (Te ) be the nodal part of v ∈ Sp (T ) in the decomposition (4.6.10). We shall now construct the conforming approximation ϑnod in Sepc (Te ). For simplicity, we shall omit the superscript “nod” and, in the  sequel, write v for v nod and ϑ for ϑnod . We introduce the sets: e ∈ Te : ν ∈ N (K) e }, w(ν) e = {K  wF (ν) = { F ∈ F(T ) : ν ∈ F }.  e ∈ R(K). We proceed by distinguishing the same two Fix K ∈ T and K cases as in Subsection 4.6.4. e Then, any elemental face Case 1: If R(K) = {K}, we have K = K. e e F ∈ F(K) belongs to F(T ) and we have vKe |Fe ∈ QpFe (Fe). Moreover, any e ∈ E(K) e belongs to E(T ) and v e | e ∈ P nod (E). e For any elemental edge E pEe K E e we define the value of ϑ(ν) by Gauss-Lobatto node ν located on ∂ K, ϑ(ν) =   X −1  e vKe (ν), |w(ν)|  0,  ν ∈ NI (T ),  e w(ν) K∈ e  (4.6.24)  otherwise.  Here, |w(ν)| e denotes the cardinality of the set w(ν). e Note that we have |w(ν)| e e by: = 8 for ν ∈ NI (T ). Then we define ϑ on K X (4.6.25) ϑ(x) = ϑ(ν) ΦνKe (x). e ν∈N (K)  From (4.6.11) and (4.6.25), we have X kvKe − ϑkL2 (K) |vKe (ν) − ϑ(ν)| kΦνKe kL2 (K) e . e .  (4.6.26)  Analogously to [11, Pages 1125-1126], we conclude that X |vKe (ν) − ϑ(ν)| . p2F h−1 F k[[v]]kL2 (F ) .  (4.6.27)  e ν∈N (K)  F ∈wF (ν)  Hence, by combining (4.6.26), (4.6.27), Lemma 4.6.1(ii) and the bounded variation property of p in (4.2.3), we obtain kvKe − ϑkL2 (K) e .  X  F ∈{wF (ν)}ν∈N (K) e  1/2  p−1 F hF k[[v]]kL2 (F ) .  (4.6.28)  141  4.6. Proof of Theorem 4.4.1 Case 2: If R(K) consists of eight elements, we define ϑ on each elee ∈ R(K) separately, analogously to the construction of the nodal ment K interpolant in Subsection 4.6.4. Without loss of generality, we may again consider the case illustrated in Figure 4.6. Since the faces Fe3 , Fe4 , Fe6 belong to FA (Te ), the function v is continuous over them. The values of ϑ on the p p ej )}j∈{8,11,12} and the vertex νe8 face nodes z ∈ {ZintKe (Fei )}i∈{3,4,6} ∪ {ZintKe (E are defined by ϑ(e ν8 ) = vKe (e ν8 ) and ϑ(z) = vKe (z),  p p ej )}j∈{8,11,12} . (4.6.29) z ∈ {ZintKe (Fei )}i∈{3,4,6} ∪ {ZintKe (E  We further define the value of ϑ on the vertex νe2 by (4.6.24). It remains to define the values of vKe on the nodes located on the faces Fe1 , Fe2 and Fe5 , excluding the vertex νe2 . We only consider Fe1 (the construction for Fe2 and Fe5 is completely analogous); see Figure 4.6. If Fe1 ∈ F(T ), then pFe p for any Gauss-Lobatto node on Fe1 , z ∈ Z 1 (Fe1 ) ∪ {Z E (E)} e ∪ {ν1 } ∪ int  int  E∈E(F1 )  {ν5 } ∪ {ν6 }, the value of ϑ(z) is taken as in (4.6.24). Otherwise, if Fe1 ∈ / F(T ), then F1 ∈ FR (T ) and F1 belongs to FN (T ) or FH (T ). We distinguish these two subcases. First, if F1 ∈ FN (T ), we define ϑ(ν), ν ∈ N (F1 ), by (4.6.24). Then we interpolate the values of the nodal interpolant over the face F1 at the Gauss-Lobatto nodes on Fe1 . That is, we set X ϑ(z) = ϑ(ν) ΦνK (z), (4.6.30) ν∈N (F1 )  pFe  p  e e e ∪ Z 1 (Fe1 ) ∪ {e for z ∈ {ZintEe (E)} νi }i∈{1,5,6} . int E∈E(F1 ) Second, if F1 ∈ FH (T ), then νe5 ∈ / N (T ), but νe1 and νe6 may or may not belong to N (T ). We first define ϑ(z) = 0,  pF  pF  pF  e5 ) ∪ Z 1 (E e9 ) ∪ {e z ∈ Zint1 (Fe1 ) ∪ Zint1 (E ν5 }, int  (4.6.31)  e1 and E e6 , as Next, we define the values of ϑ on the nodes of the edges E e1 (the definition for νe6 and well as on νe1 and νe6 . We only consider νe1 and E e E6 is completely analogous). If νe1 ∈ N (T ) (i.e., νe1 is a hanging node of pEe e1 ) ∪ {e T ), then we define ϑ(z) for z ∈ Zint 1 (E ν1 } by (4.6.24). If νe1 ∈ / N (T ), then E1 ∈ E(T ) and ν1 ∈ N (T ). We define ϑ(ν1 ) again by (4.6.24). Recall e1 , we that ϑ(ν2 ) = ϑ(e ν2 ) has already been defined. Then, for the nodes on E set ϑ(z) = ϑ(ν1 )ΦνK1 (z) + ϑ(ν2 )ΦνK2 (z),  p  e E e1 ) ∪ {e ν1 }. z ∈ Zint 1 (E  (4.6.32) 142  4.6. Proof of Theorem 4.4.1 e by setting Now we construct ϑ on K ϑ(x) =  X  ϑ(ν) ΦνKe (x) +  e ν∈N (K)  +  X  pF −1   X  e i,j=1 F ∈F (K)  pE −1   X  X  E,pE  ϑ(zi  E,pE  )Φi  (x)  e i=1 E∈E(K)     F,p F,p ϑ(zi,j F )Φi,j F (x) .  (4.6.33)  This completes the construction of ϑ. It can be readily seen that ϑ ∈ Sepc (Te ). We shall now derive an estimate analogous to (4.6.28) for Case 2. To do e as follows: so, we estimate the difference between vKe and ϑ on K X X X kvKe − ϑkL2 (K) kςνekL2 (K) kςEe kL2 (K) kςFe kL2 (K) e . e + e + e , e νe∈N (K)  e Fe∈F (K)  e e E∈E( K)  (4.6.34)  with  e ςνe(x) = vKe (e ν ) − ϑ(e ν ) ΦνK e (x),  ςEe (x) = ςFe (x) =  pEe −1   X  i=1 pFe −1   X  ) − ϑ(zi  Fe,pFe  ) − ϑ(zi,j  vKe (zi,j  i,j=1  e e E,p E  e e E,p E  vKe (zi  Fe,pFe   e  E,p ) Φi Ee (x) ,    Fe,p ) Φi,j Fe (x) .  Proceeding as in the two-dimensional proof in Lemma 3.5.3, we obtain the following estimates. First, we have that kςνekL2 (K) e ∈ NA (Te ) and e = 0 for ν kςνekL2 (K) e .  X  F ∈wF (e ν)  1/2  p−1 F hF k[[v]]kL2 (F ) ,  νe ∈ N (T ).  Second, for νe ∈ / N (T ), we have  X 1 −1 2  p h ∃ E ∈ E(K), νe is inside E,  F F k[[v]]kL2 (F ) ,   F ∈{wF (ν)}ν∈∂E kςνekL2 (K) X 1 e .   p−1 hF2 k[[v]]kL2 (F ) , ∃ F ? ∈ F(K), νe inside F ? .  F  F ∈{wF (ν)}ν∈N (F ? )  e ∈ EA (Te ) or if E e ∈ Similarly, for ςEe in (4.6.34), we have that ςEe = 0 if E ? e ∈ EF ? (K) for a face EF ? (K) for a face F ∈ FH (T ) ∩ F(K). Moreover, if E  143  4.6. Proof of Theorem 4.4.1 F ? ∈ FN (T ) ∩ F(K), we have kςEe kL2 (K) e .  X  1/2  F ∈{wF (ν)}ν∈N (F ? )  p−1 F hF k[[v]]kL2 (F ) .  e ⊆ E, we For the situation when there exists an edge E ∈ E(T ) such that E have X 1/2 kςEe kL2 (K) p−1 e . F hF k[[v]]kL2 (F ) . F ∈{wF (ν)}ν∈∂E  e e Now we only need to bound kςFe kL2 (K) e in (4.6.34) for any face F ∈ F(K). If Fe ∈ F(T ) or Fe ∈ FA (Te ), by the construction of v and ϑ, we have kςFe kL2 (K) e = 0. Otherwise, there exist a face F ∈ F(K) such that F ∈ FR (T ) and Fe is obtained by refining F . Without loss of generality, we may again consider the case illustrated in Figure 4.6, with the faces F and Fe discussed being F1 and Fe1 , respectively. If F1 ∈ FH (T ), then kςFe1 kL2 (K) e = 0. Otherwise, F1 ∈ FN (T ). Since ςFe1 vanishes at all the interior tensor-product e and on the faces of Z pKe (K) e that are different Gauss-Lobatto nodes in K int from Fe1 , we obtain from Lemma 4.6.1(i) and the construction of v and ϑ that 1/2 K  −1 kςFe1 kL2 (K) e . p e h e kςFe1 kL2 (Fe1 ) K  1/2 K  . p−1 e he K  kvKe − ϑkL2 (Fe1 ) +  X  ei ∈E(Fe1 ) E  1 2  1 2  −1 . p−1 e − ϑkL2 (F1 ) + pK hK ( K hK kvK  kςEei kL2 (Fe1 ) +  X  X  νej ∈N (Fe1 )  kςEei kL2 (Fe1 ) +  ei ∈E(Fe1 ) E  kςνej kL2 (Fe1 )  X    kςνej kL2 (Fe1 ) )  νej ∈N (Fe1 )  ≡ T1 + T2 . Using (4.6.27), Lemma 4.6.1(iii) and (4.2.3), we get X 1/2 T1 . p−1 h kςν kL2 (F1 ) K K ν∈N (F1 )  1/2  . p−1 K hK  X  ν∈N (F1 )  .  X  (|vK (ν) − ϑ(ν)| kΦνK kL2 (F1 ) )  F ∈{wF (ν)}ν∈N (F1 )  1/2  p−1 F hF k[[v]]kL2 (F ) .  144  4.6. Proof of Theorem 4.4.1 In an analogous manner to the two-dimensional proof in Lemma 3.5.3, term T2 is bounded by X 1/2 T2 . p−1 F hF k[[v]]kL2 (F ) . F ∈{wF (ν)}ν∈N (F1 )  Hence, ςFe in (4.6.34) can be bounded by kςFe kL2 (K) e .  X  1/2  F ∈{wF (ν)}ν∈N (F1 )  p−1 F hF k[[v]]kL2 (F ) .  e as To combine the bounds for ςνe, ςEe and ςFe , we define the set N ? (K) e and first remove all the vertices belonging to follows. We start from N (K) e e with νe ∈ NA (T ). Then, any vertex νe ∈ N (K) / N (T ) ∪ NA (Te ) is replaced by the vertex ν ∈ N (K) which lies on the same elemental edge of K as νe; see Section 3.5.5. We also set Thus, we have  e = { F ∈ wF (ν) : ν ∈ N ? (K) e }. F ? (K)  kvKe − ϑkL2 (K) e .  X  e F ∈F ? (K)  1/2  p−1 F hF k[[v]]kL2 (F ) .  (4.6.35)  This completes the discussion of Case 2. By the key estimates in (4.6.28) and (4.6.35), we have in both cases above X 1/2 e ∈ Te . K (4.6.36) kvKe − ϑkL2 (K) p−1 e . F hF k[[v]]kL2 (F ) , e F ∈F ? (K)  This proves the first inequality in (4.6.20). Moreover, by the inverse inequality, 2 −1 k∇vkL2 (K) e . pK e , e h e kvkL2 (K) K  e ∈ Te , v ∈ Sep (Te ), K  see [12], we obtain from (4.6.36) and (4.2.3) X −1/2 pF hF k[[v]]kL2 (F ) , k∇(vKe − ϑ)kL2 (K) e . e F ∈F ? (K)  e ∈ Te , K  (4.6.37)  (4.6.38)  which shows the second assertion in the nodal approximation result (4.6.20). 145  4.6. Proof of Theorem 4.4.1 Edge approximation For any edge E ∈ E(T ), we define the set wE = { K ∈ T : E ⊂ ∂K }.  (4.6.39)  Fix an element K ∈ T . First, we consider an elemental edge E ∈ E(K) and E as follows: if E ∈ E (T ), we set define the function WK B  E nod WK = LE pK ,K (vK − vK )|E ,  with the extension operator LE pK ,K (·) defined in (4.6.2). 0 If E ∈ EI (T ), let K ∈ wE be the element which has the lowest polynomial degree in the set wE defined in (4.6.39); see Section 3.5.6. We define E by WK  E nod WK = LE pK ,K (vK 0 − vK 0 )|E ,  with LE pK ,K (·) defined in (4.6.2). In the case where E contains a hanging node, E is partitioned into E = E1 ∪ E2 with E1 , E2 ∈ EI (T ), cf. Figure 4.2. Denote by K 0 ∈ wE1 and K 00 ∈ wE2 the elements in T which have the lowest polynomial degree in the E by set wE1 and wE2 , respectively; see Section 3.5.6. We now define WK  E nod nod WK = LE pK ,K (vK 0 − vK 0 )|E1 , (vK 00 − vK 00 )|E2 ,  with LE pK ,K (·, ·) in (4.6.3). E is given Next, for an edge E ∈ EF (K), F ∈ F(K), the function WK analogously. Let K 0 ∈ wE be the element which has the lowest polynomial E by degree in the set wE . We define WK  E nod WK = LE pK ,K (vK 0 − vK 0 )|E ,  with LE pK ,K (·) given in (4.6.5). Then we define ϑedge elementwise by setting X X X E ϑedge |K = WK + E∈E(K)  E WK ,  F ∈F (K) E∈EF (K)  E defined above. Clearly, the function ϑedge belongs to S c (T e ). By with WK e p employing Lemma 4.6.2 and proceeding as in Section 3.5.6, the approximation property (4.6.21) can be readily derived.  146  4.6. Proof of Theorem 4.4.1 Face approximation Fix an element K ∈ T and let F be an element face in F(K). We define F as follows: if F ∈ F (T ), we set the function WK B  edge F nod WK = LFpK ,K (vK − vK − vK )|F ,  with LFpK ,K (·) defined in (4.6.6). If F ∈ FI (T ), let K 0 in T be the neighboring element such that F ∈ F(K) ∩ F(K 0 ). Denote by K 0 the element which F has the lower polynomial degree of the elements K and K 0 . We define WK by  edge nod F WK = LFpK ,K (vK 0 − vK 0 − v 0 )|F , K  with LFpK ,K (·) defined in (4.6.8) if F ∈ FH (T ) (see Figure 4.5) and in (4.6.6) otherwise. If F contains a hanging node in the center, F is partitioned into F = ∪4i=1 Fi with Fi ∈ F(T ), i = 1, . . . , 4, cf. Figure 4.4. There exist four elements Ki ∈ T such that Fi ∈ F(Ki ). Denote by K i the element that has F the lower polynomial degree of K and Ki , i = 1, . . . , 4. We now define WK by  edge edge F nod nod )|F1 , . . . , (vK 4 − vK )|F4 , WK = LFpK ,K (vK 1 − vK − vK − vK 1 4 1  4  with LFpK ,K (·, ·, ·, ·) defined in (4.6.7). Next, we prove the face approximation property (4.6.22). By the local bounded variation of p (4.2.3), Lemma 4.6.3 and the polynomial trace inequality (see [32]), we have X X X kv face − ϑface k2L2 (K) = kv face − ϑface k2L2 (K) e e e Te K∈  .  X  X  K∈T F ∈F (K)  .  X  X  K∈T F ∈F (K)  .  X  X  K∈T F ∈F (K)  .  X  X  K∈T F ∈F (K)  K∈T K∈R(K) e   edge nod F 2 )|F − WK kLFpK ,K (vK − vK − vK kL2 (K)  edge nod F 2 p−2 K hK k(vK − vK − vK )|F − WK |F kL2 (F ) 2 nod 2 p−2 ]]kL2 (F ) + k[[v edge ]]k2L2 (F ) ) F hF (k[[v]]kL2 (F ) + k[[v 2 nod 2 p−2 ]]kL2 (F ) ) + F hF (k[[v]]kL2 (F ) + k[[v  X  e Te K∈  k[[v edge ]]k2L2 (K) e .  This, together with the edge approximation result (4.6.21) completes the proof of the first assertion of (4.6.22); the second one follows again from the first one by using the inverse inequality in (4.6.37). 147  4.7. Numerical experiments  4.7  Numerical experiments  In this section, we present a series of numerical examples to demonstrate the practical performance of the proposed a-posteriori error estimator η derived in Theorem 4.3.1 within an automatic hp-adaptive refinement procedure which is based on 1-irregular hexahedral elements. In each of the examples shown in this section the DG solution uhp defined by (4.2.6) is computed with the interior penalty parameter γ equal to 10. All computations have been performed using the AptoFEM software package (see [16], for details). Additionally, the resulting system of linear equations is solved by exploiting the MUltifrontal Massively Parallel Solver (MUMPS), see [1, 2, 3], for example. The hp-adaptive meshes are constructed by first marking the elements for refinement according to the size of the local error indicators ηK ; this is achieved by employing the fixed fraction strategy, see [21], with refinement fraction 25%. Note that in the present chapter, we do not employ any derefinement of the underlying hp-meshes. Once an element K ∈ T has been flagged for refinement, a decision must be made whether the local mesh size hK or the local degree pK of the approximating polynomial should be adjusted accordingly. The choice to perform either h- or p-refinement is based on estimating the local smoothness of the (unknown) analytical solution. To this end, we employ the hp-adaptive strategy developed in [22], where the local regularity of the analytical solution is estimated from truncated local Legendre expansions of the computed numerical solution; see, also, [14, 20]. Here, the emphasis will be on investigating the asymptotic sharpness of the proposed a-posteriori error bound on a sequence of nonuniform hpadaptively refined 1-irregular meshes. To this end, we shall compare the estimator η derived in Theorem 4.3.1, which is slightly suboptimal (by a 1/2 factor of pF ) in the face polynomial order pF , with the indicator ηb discussed in Remark 4.3.2; we note that the derivation of the latter precludes the use of hanging nodes, at least theoretically. Indeed, here we shall show that despite the loss of optimality in the polynomial degree, both indicators perform extremely well on hp-refined meshes, in the sense that the effectivity index, which is defined as the ratio of the a-posteriori error bound and the energy norm of the actual error, is roughly constant on all of the meshes employed. Moreover, our numerical experiments indicate that both a-posteriori error indicators give rise to very similar quantitative results. For simplicity, as in [7], we set the constant C arising in Theorem 4.3.1 equal to one; in general, to ensure the reliability of the error estimator, this constant must be determined numerically for the underlying problem at hand. In all of 148  4.7. Numerical experiments 0  −1  10  10  10  h−Refinement hp−Refinement  9  −1  10  −2  10  8 −2  10  −4  10  −5  10  −6  10  −7  10  15  20  25  5  30  35  40  45  50  −5  10  2  0 0  −4  10  4  1  −3  10  6  3  Error Estimator (p2) True Error (p2) Error Estimator (p3) True Error (p3)  10  ku − uhpkE,T  Effectivity  7  −3  10  −6  10  p2 p3  −7  2  4  6  8  10  10  20  40  Mesh Number  (DOF)1/3  (a)  (b)  60  80  100  120  140  160  (DOF)1/3  (c)  Figure 4.7: Example 1. (a) Comparison of the actual and estimated energy norm of the error with respect to the (third root of the) number of degrees of freedom with hp-adaptive mesh refinement; (b) Effectivity indices; (c) Comparison of the actual error with h- and hp-adaptive mesh refinement. our experiments, the data-approximation terms in the a-posteriori bound stated in Theorem 4.3.1 will be neglected. For both the error estimators η and ηb, inhomogeneous boundary conditions are incorporated as discussed in Remark 4.3.4.  4.7.1  Example 1  In this example, we let Ω be the unit cube (0, 1)3 in R3 ; further, we select f and an appropriate inhomogeneous boundary condition, so that the analytical solution to (4.1.1) is given by u(x1 , x2 , x3 ) = sin(πx1 ) cos(πx2 ) cos(πx3 ). In Figure 4.7(a) we present a comparison of the actual and estimated energy norm of the error versus the third root of the number of degrees of freedom in the finite element space Sp (T ) on a linear-log scale, for the sequence of meshes generated by our hp-adaptive algorithm using the indicator η stated in Theorem 4.3.1 (denoted by p3 in the figure) and ηb outlined in Remark 4.3.2 (denoted by p2 ). Here, we observe that the two error indicators perform in a very similar manner: in each case the error bound over-estimates the true error by a (reasonably) consistent factor. From Figure 4.7(b), we see that the computed effectivity indices lie in the range 5–9; in particular, we note that although there is some initial growth in the effectivity indices as the hp-mesh is refined, these numbers seem to settle at approximately 8 as the adaptive refinement strategy proceeds. Additionally, from Figure 4.7(a) we observe that after an initial transient, the convergence 149  4.7. Numerical experiments  (a)  (b)  Figure 4.8: Example 1. Finite element mesh after 8 adaptive refinements, with 440 elements and 100578 degrees of freedom: (a) hp-mesh; (b) Threeslice of the hp-mesh. lines using hp-refinement are (roughly) straight on a linear-log scale, which indicates that exponential convergence is attained for this smooth problem, as we would expect. In Figure 4.7(c), we present a comparison between the actual energy norm of the error employing both h- and hp-mesh refinement; here, the hp-refinement is based on employing the error indicator stated in Theorem 4.3.1. In the former case, the DG solution uhp is computed using triquadratic elements, i.e., pK = 2; here, the adaptive algorithm is again based on employing the fixed fraction strategy, with the refinement fraction set to 25%, without any derefinement. From Figure 4.7(c), we clearly observe the superiority of employing a grid adaptation strategy based on exploiting hp-adaptive refinement: on the final mesh, the energy norm of the error using hp-refinement is around four orders of magnitude smaller than the corresponding quantity computed when h-refinement is employed alone. In Figure 4.8 we show the mesh generated using the proposed hp-version a-posteriori error indicator stated in Theorem 4.3.1 after 8 hp-adaptive refinement steps. For clarity, we also show the three-slice of the hp-mesh centered at the centroid of the computational domain Ω. Here, we observe that some h-refinement of the mesh has been performed in the vicinity of steep gradients present in the analytical solution situated in the interior of Ω. Within this region, the polynomial degree is between 4–5. Away from this region, the hp-adaptive algorithm increases the degree of the approximating polynomial where the analytical solution is extremely smooth. 150  4.7. Numerical experiments 1  25  10  h−Refinement hp−Refinement 20  Effectivity  ku − uhpkE,T  0  10  15  10  −1  10  −2  10  Error Estimator (p2) True Error (p2) Error Estimator (p3) True Error (p3)  10  12  14  5 p2 p3 16  18  20  22  0 0  1  −1  2  3  4  5  6  7  8  9  10  10  Mesh Number  1/4  (DOF)  (a)  (b)  15  20  25  30  (DOF)1/4  (c)  Figure 4.9: Example 2. (a) Comparison of the actual and estimated energy norm of the error with respect to the (fourth root of the) number of degrees of freedom with hp-adaptive mesh refinement; (b) Effectivity indices; (c) Comparison of the actual error with h- and hp-adaptive mesh refinement.  4.7.2  Example 2  In this section, we let Ω be the Fichera corner (−1, 1)3 \ [0, 1)3 , and select f and an appropriate inhomogeneous boundary condition for u so that u(x1 , x2 , x3 ) = (x21 + x22 + x23 )q/2 , where q is a real number. We note that for q > −1/2, the analytical solution u to (4.1.1) satisfies u ∈ H 1 (Ω); cf. [8], for example. In this section we set q = −1/4; in this case u possesses typical (isotropic) singular behavior that solutions of elliptic boundary-value problems exhibit in the vicinity of reentrant corners in the computational domain. The most general type of singularity involving anisotropic edge singularities will be treated elsewhere. Figure 4.9(a) shows the history of the actual and estimated energy norm of the error on each of the meshes generated by our hp-adaptive algorithm using both the indicator η in Theorem 4.3.1 (denoted by p3 in the figure) and ηb in Remark 4.3.2 (denoted by p2 ). Here, we have plotted the errors versus the fourth root of the number of degrees of freedom in the finite element space Sp (T ) on a linear-log scale; the fourth root of the number of degrees of freedom is chosen empirically based on the fact that the singularity is isotropic; we also refer to the two–dimensional hp-version a-priori error analysis performed in [34]. We point out that for general (anisotropic) edge singularities in 3D, the fifth root of the degrees of freedom should be considered; cf. [33]. As in the previous example, we observe that the two error indicators perform in a very similar manner, though for this non-smooth example the loss 151  4.8. Conclusions in optimality in the jump indicator in the estimator stated in Theorem 4.3.1 does lead to a slight increase in the effectivity indices in comparison with indicator ηb in (4.3.6). Indeed, from Figure 4.9(b) we observe that the effectivity indices for both a-posteriori bounds do slowly grow as the hp-mesh is refined. Additionally, from Figure 4.9(a) we observe exponential convergence of the energy norm of the error using both estimators with hp-refinement; indeed, on a linear-log scale, the convergence lines are, on average, straight. Figure 4.9(c) highlights the superiority of employing hp-adaptive refinement in comparison with h-refinement. Indeed, although on the final mesh, the energy norm of the error using the hp-refinement indicator stated in Theorem 4.3.1 is only around a factor 2 smaller than the corresponding quantity when h-refinement is employed alone, based on using triquadratic elements, we can clearly see that an excessively large number of degrees of freedom will be required to simply ensure that ku − uhp kE,T is less than 10−1 when using the fixed-order h-refinement strategy. In Figure 4.10 we show the mesh generated using the local error indicators ηK stated in Theorem 4.3.1 after 7 hp-adaptive refinement steps. Here, we see that the h-mesh has been refined in the vicinity of the re-entrant corner located at the origin. Additionally, we see that the polynomial degrees have been increased away from the re-entrant corner located at the origin, since the underlying analytical solution is smooth in this region.  4.8  Conclusions  In this chapter, we derived an a-posteriori error estimator for hp-adaptive discontinuous Galerkin methods for elliptic problems on 1-irregularly, isotropically refined meshes in three dimensions. The estimator yields upper and lower bounds for the error measured in terms of the natural energy norm. We applied our estimate as an error indicator for energy norm error estimation in an hp-adaptive refinement algorithm. Our numerical results show that the indicator is efficient in locating and resolving isotropic corner singularities at exponential convergence rates. In our analysis, we employed the approximation properties of the threedimensional hp-version averaging operator in Theorem 4.4.1. This theorem allows us to also extend the analysis in Chapter 3 to three dimensions. Hence, a robust a-posteriori error estimator for hp-adaptive DG discretizations of three-dimensional stationary convection-diffusion equations can be immediately obtained on isotropically refined meshes. However, due to the presence of edge singularities in the exact solution 152  4.8. Conclusions  (a)  (b) Figure 4.10: Example 2. Finite element mesh after 7 adaptive refinements, with 686 elements and 197670 degrees of freedom: (a) hp-mesh; (b) Threeslice of the hp-mesh. of the three dimensional equations, anisotropic geometric mesh refinement is necessary to achieve exponential convergence rates for hp-adaptive DG algorithm. Therefore, it is desirable to develop hp-adaptive DG methods on anisotropically refined meshes. This is detailed in Chapter 5.  153  4.9. Bibliography  4.9  Bibliography  [1] P. R. Amestoy, I. S. Duff, J. Koster, and J.-Y. L’Excellent. A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM Journal on Matrix Analysis and Applications, 23:15–41, 2001. [2] P. R. Amestoy, I. S. Duff, and J.-Y. L’Excellent. Multifrontal parallel distributed symmetric and unsymmetric solvers. Comput. Methods Appl. Mech. Eng., 184:501–520, 2000. [3] P. R. Amestoy, A. Guermouche, J.-Y. L’Excellent, and S. Pralet. Hybrid scheduling for the parallel solution of linear systems. Parallel Computing, 32:136–156, 2006. [4] D.N. Arnold. An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal., 19:742–760, 1982. [5] D.N. Arnold, F. Brezzi, B. Cockburn, and L.D. Marini. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal., 39:1749–1779, 2002. [6] C.E. Baumann and J.T. Oden. A discontinuous hp-finite element method for convection-diffusion problems. Comput. Methods Appl. Mech. Engrg., 175:311–341, 1999. [7] R. Becker, P. Hansbo, and M.G. Larson. Energy norm a-posteriori error estimation for discontinuous Galerkin methods. 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Karniadakis, and C.-W. Shu, editors, Discontinuous Galerkin Methods: Theory, Computation and Applications, volume 11 of Lect. Notes Comput. Sci. Engrg., pages 3–50. Springer–Verlag, 2000. [14] T. Eibner and J. M. Melenk. An adaptive strategy for hp-FEM based on testing for analyticity. Comp. Mech., 39:575–595, 2007. [15] A. Ern and A.F. Stephansen. A-posteriori energy-norm error estimates for advection-diffusion equations approximated by weighted interior penalty methods. J. Comp. Math., 26:488–510, 2008. [16] S. Giani, E.J.C. Hall, and P. Houston. AptoFEM: Users manual. Technical report, University of Nottingham. In preparation. [17] P. Houston, D. Schötzau, and T. Wihler. An hp-adaptive discontinuous Galerkin FEM for nearly incompressible linear elasticity. Comput. Methods Appl. Mech. Engrg., 195:3224–3246, 2006. [18] P. Houston, D. Schötzau, and T. P. Wihler. Energy norm a-posteriori error estimation of hp-adaptive discontinuous Galerkin methods for elliptic problems. Math. 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Houston, E. Süli, and T. Wihler. A-posteriori error analysis of hpversion discontinous Galerkin finite-element methods for second-order quasi-linear elliptic PDEs. IMA J. Numer. Anal., 28:245–273, 2008. [24] O. A. Karakashian and F. Pascal. A-posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems. SIAM J. Numer. Anal., 41:2374–2399, 2003. [25] J.M. Melenk and B.I. Wohlmuth. On residual-based a-posteriori error estimation in hp-FEM. Adv. Comp. Math., 15:311–331, 2001. [26] J. Nitsche. Über ein Variationsprinzip zur Lösung von Dirichlet Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Math. Abh. Sem. Univ. Hamburg, 36:9–15, 1971. [27] I. Perugia and D. Schötzau. An hp-analysis of the local discontinuous Galerkin method for diffusion problems. J. Sci. Comp., 17:561–571, 2002. [28] B. Rivière, M.F. Wheeler, and V. Girault. Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems, Part I. Computational Geosciences, 3:337–360, 1999. [29] D. Schötzau, C. Schwab, and T. Wihler. hp-DGFEM for second-order elliptic problems in polyhedra I: Stability and quasioptimality on geometric meshes. submitted, 2010. [30] D. Schötzau, C. Schwab, and T. Wihler. hp-DGFEM for second-order elliptic problems in polyhedra II: Exponential convergence. submitted, 2010. [31] D. Schötzau and L. Zhu. A robust a-posteriori error estimator for discontinuous Galerkin methods for convection-diffusion equations. Appl. Numer. Math., 59:2236–2255, 2009. [32] C. Schwab. p- and hp-Finite Element Methods: Theory and Application to Solid and Fluid Mechanics. Oxford University Press, Oxford, 1998.  156  BIBLIOGRAPHY [33] C. Schwab. hp-FEM for fluid flow simulation. In T. Barth and H. Deconink, editors, High-Order Methods for Computational Physics, volume 9 of Lect. Notes Comput. Sci. Engrg., pages 325–438. Springer– Verlag, 1999. [34] T. Wihler, P. Frauenfelder, and C. Schwab. Exponential convergence of the hp-DGFEM for diffusion problems. Comput. Math. Appl., 46:183– 205, 2003. [35] L. Zhu and D. Schötzau. A robust a-posteriori error estimate for hpadaptive DG methods for convection-diffusion equations. IMA J. Numer. Anal., 2009. accepted for publication.  157  Chapter 5  Anisotropic meshes 4 5.1  Introduction  It is well-known that solutions to convection-diffusion problems develop boundary or internal layers. These layers appear as exponential layers near outflow boundaries or as parabolic layers near discontinuous inflow conditions. The finite element resolution of such layers is achieved most naturally using anisotropically refined meshes, where the aspect ratio of the elements (i.e., the ratio of the radii of the circumscribed and inscribed circles) is unbounded [1, 22]. There is a huge amount of literature on finite element methods using anisotropic elements. A natural approach is based on the so-called Shishkin meshes [20, 22], which are piecewise structured meshes with anisotropic elements in the boundary layers with a properly chosen transition point. Today, many theoretical aspects of anisotropic finite element methods are well-understood [1, 2, 3] and much effort has been undertaken to incorporate anisotropy into h-version adaptive techniques; see e.g., [10, 17, 18, 21] and the references therein. An h-version approach to error estimation of discontinuous Galerkin (DG) methods has been recently proposed in [7] for the Stokes problem. We are mainly interested in hp-version finite element methods. They have the advantage that boundary layers can be numerically resolved at exponential rates of convergence on boundary-fitted geometric meshes [26, 27, 28]. In addition, anisotropic elements can be used to capture edge singularities of diffusion problems in generic polyhedra at exponential rates of convergence [23, 24]. In [11, 12], a duality-based a-posteriori error estimator was recently proposed for hp-adaptive DG methods for convection-diffusion problems on anisotropically refined meshes with anisotropically enriched elemental polynomial degrees. 4  A version of this chapter will be submitted for publication. Zhu, L. and Schötzau, D. An a-posteriori error estimate for hp-adaptive DG methods for convection-diffusion equations on anisotropically refined meshes.  158  5.2. Interior penalty discretization In this chapter, we extend our approach to residual-based error estimation for hp-adaptive DG methods for convection-diffusion equations on anisotropically refined rectangular meshes with anisotropic polynomial degree orders. We present an estimator η which yields upper and lower bounds of the error measured in terms of the energy norm and a dual norm associated with the convective term. The constant in the lower bound is independent of the diffusion and the mesh size, but weakly depending on the polynomial degrees, as the error estimators derived in previous chapters. In the upper bound, we use an alignment measure as in the approach of [7, 17]. Since this measure depends in principle on the anisotropy of the meshes, our estimator is not robust in contrast to the ones proposed in Chapters 3 and 4. On the other hand, our numerical examples indicate that, as soon as a reasonable resolution of the layer is achieved, the alignment measure is of moderate size and the ratio of the error estimator and the true energy error is independent of the diffusion parameter ε and the mesh size. This means that, after a few refinement steps, our estimator is robust in practice. Our analysis is again based on the use of an averaging operator, as in [13, 16] and our work [29, 30]. Here, we extend our results in Chapters 3 and 4 to anisotropically refined meshes with anisotropic polynomial degrees. The outline of the rest of this chapter is as follows. In Section 5.2, we introduce hp-adaptive discontinuous Galerkin methods for a convectiondiffusion model problem using anisotropically refined rectangular elements. In Section 5.3, we state and discuss our a-posteriori error estimate. The proof of this estimate is carried out in Sections 5.4 and 5.5. In Section 5.6, we present a series of numerical tests that illustrate the theoretical results. Finally, in Section 5.7, we end with concluding remarks.  5.2  Interior penalty discretization  In this section, we introduce an hp-adaptive interior penalty discontinuous Galerkin finite element method for the discretization of convection-diffusion equations on anisotropically refined meshes. To avoid excessive use of constants, the abbreviation x ∼ y stands for c1 x ≤ y ≤ c2 x, respectively, with positive constants c1 and c2 independent of any parameters.  159  5.2. Interior penalty discretization  5.2.1  Model problem  We consider the convection-diffusion model problem: −ε∆u + a(x) · ∇u = f (x)  in Ω,  u=0  (5.2.1)  on Γ.  Here, Ω is a bounded Lipschitz polyhedral domain in R2 , with boundary Γ = ∂Ω. The parameter ε > 0 is the (constant) diffusion coefficient, the vectorvalued function a(x) a given flow field, and the function f (x) a generic righthand side in L2 (Ω). The coefficient a(x) is assumed to belong to W 1,∞ (Ω)2 and to satisfy ∇·a=0 in Ω. (5.2.2) Without loss of generality, we shall assume that kakL∞ (Ω) and the length scale of Ω are one so that ε−1 is the Péclet number of the problem. The standard weak form of the convection-diffusion equation in (5.2.1) is to find u ∈ H01 (Ω) such that Z Z  A(u, v) = ε∇u · ∇v + a · ∇uv dx = f v dx ∀ v ∈ H01 (Ω). (5.2.3) Ω  Ω  Under assumption (5.2.2), the variational problem (5.2.3) is uniquely solvable.  5.2.2  Discretization  Let T be a subdivision of the polygonal domain Ω ⊂ R2 into disjoint elements, which for simplicity we assume to be rectangles. Each element K ∈ T is the image of the reference square under an affine elemental mapb → K; see Figure 5.1. ping FK : K x b2  6  (−1, −1)  b K  (1, 1)  x2  FK  R -x b1  6  A   A A v2K  K *v 1    K K A 1 h2KA  A hK - x1  Figure 5.1: Mapping of the element K. For an element K ∈ T , we denote by N (K) the set of its four vertices. A node ν of a finite element mesh T is the vertex of at least one element 160  5.2. Interior penalty discretization K ∈ T . The node ν is called an interior node if ν ∈ / Γ; similarly, it is a boundary node if ν ∈ Γ. We denote by NI (T ), NB (T ) the sets of interior and boundary nodes, respectively, and set N (T ) = NI (T ) ∪ NB (T ). We denote by E(K) the set of the four elemental edges of an element K. If the intersection E = ∂K ∩ ∂K 0 of two elements K, K 0 ∈ T is a proper line segment (and not a single point), we call E an interior edge of T . The set of all interior edges is denoted by EI (T ). Analogously, if the intersection E = ∂K ∩ Γ of an element K ∈ T and Γ is a proper line segment, we call E a boundary edge of T . The set of all boundary edges of T is denoted by EB (T ). Moreover, we set E(T ) = EI (T ) ∪ EB (T ).  5.2.3  Mesh sizes  For each element K ∈ T , we define the two anisotropic vectors v1K and v2K , as shown in Figure 5.1. These vectors reflect the two anisotropic directions of element K. The lengths of these two vectors are denoted by h1K and h2K , respectively: h1K = length(v1K ), h2K = length(v2K ). Further, we set hmin,K = min{h1K , h2K },  hmax,K = max{h1K , h2K }.  We then define the matrix MK = [v1K , v2K ].  (5.2.4)  Note that MK is orthogonal and M> K MK =   1 2 (hK )  (h2K )2    .  Given an edge E ∈ E(T ), for any element K ∈ T , if E ∈ E(K) or E is a part of an elemental edge of K, we define a local function of the edge E: 3−i h⊥ E,K = hK ,  if E is parallel to viK , i = 1, 2.  Moreover, for any E ∈ EI (T ), we assume that ⊥ h⊥ E,K ∼ hE,K 0 ,  E = K ∩ K 0 , K, K 0 ∈ T .  (5.2.5)  Note that this assumption does not bound the aspect ratios of elements. For any edges E, E 0 ∈ E(K) and E ∩ E 0 6= ∅, hE /hE 0 could be significantly large. 161  5.2. Interior penalty discretization If E ∈ E(K) and E is parallel to viK , i = 1, 2, we define hE = hiK ,  i = 1, 2.  For any edge E ∈ E(T ), we further set ( ⊥ min{h⊥ E ∈ EI (T ), E = ∂K ∩ ∂K 0 , ⊥ E,K , hE,K 0 }, hE = h⊥ E ∈ EB (T ), E = ∂K ∩ Γ. E,K , In our analysis, we allow for irregularly refined meshes T , where each elemental edge E ∈ E(K) may contain at most M hanging nodes on it. We assume that the M hanging nodes on any elemental edge E are fixed as follows. Let Ib = (−1, 1) be the reference interval. Then we take M points, b = { yb1 , · · · , ybM }. Let now −1 < yb1 < yb2 < · · · < ybM < 1, on Ib and set H b can be affinely E ∈ E(K) be an edge of an element K. The nodes of H E E mapped onto E and we denote by H(E) = { y1 , · · · , yM } the possible locations of the hanging nodes on E. In other words, any hanging node on E belongs to H(E). Setting yb0 = −1 and ybM +1 = 1, we further assume there exists a positive constant κ ≤ 1 independent of the mesh T , such that |b yi+1 − ybi | ≥ 2κ for i = 0, 1, · · · , M . Under this assumption, if the edge E 0 ∈ E(T ) is also a part of the elemental edge E (i.e., E 0 ⊂ E), we have κ ≤ hE 0 /hE ≤ 1.  Next, we define hmin,E by setting ( min{hmin,K , hmin,K 0 }, hmin,E = hmin,K ,  (5.2.6)  E ∈ EI (T ), E = ∂K ∩ ∂K 0 , E ∈ EB (T ), E = ∂K ∩ Γ,  Note that our assumptions (5.2.5) and (5.2.6) imply that for any edge E ∈ E(T ) and any element K ∈ T , if E ∈ E(K) or E is a part of one element edge of K, we have ⊥ h⊥ E ∼ hE,K ,  5.2.4  hmin,E ∼ hmin,K .  (5.2.7)  Polynomial degrees  In our analysis, we allow for anisotropic elemental polynomial degrees such that the polynomial degrees in each space direction are different from each other. With each element K of a mesh T , we associate a set of two polynomial degrees pK = ( p1K , p2K ) with p1K , p2K ≥ 1. We set pmin,K = min{p1K , p2K },  pmax,K = max{p1K , p2K }. 162  5.2. Interior penalty discretization We then define two local polynomial degrees associated with an elemental edge E ∈ E(K), pE,K = piK ,  if E is parallel to viK , i = 1, 2,  3−i p⊥ E,K = pK ,  if E is parallel to viK , i = 1, 2.  We define a polynomial degree vector p = { pK : K ∈ T } and set |p| = max pmax,K . K∈T  Moreover, we assume that p is of bounded local variation. That is, for any pair of neighboring elements K, K 0 ∈ T sharing an edge E ∈ T , we have ⊥ p⊥ E,K ∼ pE,K 0 ,  pE,K ∼ pE,K 0 .  (5.2.8)  For an edge E ∈ E(T ), we introduce two sets of edge polynomial degrees pE and p⊥ E by: ( max{pE,K , pE,K 0 }, E ∈ EI (T ), E = ∂K ∩ ∂K 0 , pE = pE,K , E ∈ EB (T ), E = ∂K ∩ Γ, ( (5.2.9) ⊥ E ∈ EI (T ), E = ∂K ∩ ∂K 0 , max{p⊥ ⊥ E,K , pE,K 0 }, pE = E ∈ EB (T ), E = ∂K ∩ Γ. p⊥ E,K ,  5.2.5  Bilinear form  For a partition T of Ω and a degree vector p on T , we then define the hp-version discontinuous Galerkin finite element space by b K ∈ T }, Sp (T ) = { v ∈ L2 (Ω) : v|K ◦ FK ∈ QpK (K),  b denoting the set of all polynomials on the reference square K b of with QpK (K) 1 2 degree less or equal than pK and pK in the two space directions, respectively. We now consider the following discontinuous Galerkin method for the approximation of the convection-diffusion problem (5.2.1): Find uhp ∈ Sp (T ) such that Z Ahp (uhp , v) = f v dx (5.2.10) Ω  163  5.3. A-posteriori error estimates for all v ∈ Sp (T ), with the bilinear form Ahp given by XZ (ε∇u · ∇v + a · ∇uv) dx Ahp (u, v) = K  K∈T  −  X Z  E∈E(T ) E  X Z  +  E∈E(T ) E  +  XZ  K∈T  {{ε∇u}} · [[v]] ds −  X Z  E∈E(T ) E  {{ε∇v}} · [[u]] ds  2 XZ εγ(p⊥ E) [[u]] · [[v]] ds − a · nK uv ds h⊥ ∂K ∩Γ in in E K∈T  ∂Kin \Γ  a · nK (ue − u)v ds.  Here, for a piecewise smooth function, the gradient operator ∇ is taken element by element. As in Chapter 3, we denote by Γin and ∂Kin the inflow parts of Γ and K ∈ T , respectively: Γin = { x ∈ Γ : a(x) · n(x) < 0 },  ∂Kin = { x ∈ ∂K : a(x) · nK (x) < 0 }.  The constant γ > 0 is the interior penalty parameter. It is shown that the above anisotropic DG discretization (5.2.10) is stable provided the constant γ is chosen sufficiently large; see, e.g., [11, 23] and the references therein.  5.3  A-posteriori error estimates  In this section, our main results are presented and discussed.  5.3.1  Norms  We begin by introducing the norms in which the errors are measured. First, we introduce the following energy norm associated with the discontinuous Galerkin discretization of the diffusive term: X k v k2E,T = εk∇vk2L2 (K) + ejumpp,T (v)2 , K∈T  ejumpp,T (v)2 =  X εγ(p⊥ )2 E k[[v]]k2L2 (E) . h⊥ E  (5.3.1)  E∈E(T )  Next, we define |q|? =  sup v∈H01 (Ω)\{0}  R  q · ∇v dx k v kE,T  Ω  ∀ q ∈ L2 (Ω)2 . 164  5.3. A-posteriori error estimates Analogously to Chapter 3, the semi-norm associated with the discretization of the convection term is now given by: | v |O,T = |av|2? + ojumpp,T (v)2 , 2 ⊥ X X εh⊥ h⊥ (5.3.2) E,K pE,K E,K pE,K  ojumpp,T (v)2 = + k[[v]]k2L2 (E) . 2 2 hmin,K εpmin,K K∈T E∈E(K)  Notice that the definition of ojumpp,T (v) is different from the one in Chapter 3.  5.3.2  An a-posteriori error estimate  Let now uhp ∈ Sp (T ) be the discontinuous Galerkin approximation obtained by (5.2.10). Moreover, let fhp and ahp denote piecewise polynomial approximations in Sp (T ) to the right-hand side f and the flow field a, respectively. For each element K ∈ T , we introduce the following local error indicator ηK which is given by the sum of the three terms 2 2 2 ηK = ηR + ηE + ηJ2K . K K  (5.3.3)  The first term ηRK is the interior residual defined by 2 2 2 ηR = ε−1 p−2 min,K hmin,K kfhp + ε∆uhp − ahp · ∇uhp kL2 (K) . K  The second term ηEK is the edge residual given by 2 ηE K  1 = 2  X  E∈∂K\Γ  h2min,K p⊥ E,K εp2min,K h⊥ E,K  k[[ε∇uhp ]]k2 .  The last residual ηJK measures the jumps of the approximate solution uhp : ! Z ⊥ 5 2 εγ 2 (p⊥ εh⊥ h⊥ 1 X E,K pE,K E,K ) E,K pE,K 2 η JK = + [[uhp ]]2 ds + 2 2 ⊥ p2 2 h εp h min,K min,K E,K min,K E∈∂K\Γ E ! 5 2 ⊥ X Z εγ 2 (p⊥ εh⊥ h⊥ E,K ) E,K pE,K E,K pE,K + + + [[uhp ]]2 ds. 2 2 ⊥ p2 h εp h E min,K min,K E,K min,K E∈∂K∩Γ  Note that, for anisotropic meshes, the jump residual term is different from the one defined in Chapter 3, while it coincides with the result in  165  5.3. A-posteriori error estimates Chapter 3 for isotropically refined meshes. We also introduce the local data approximation term   2 2 2 Θ2K = ε−1 p−2 + k(a h kf − f k − a ) · ∇u k 2 2 hp hp hp min,K L (K) L (K) . min,K We then introduce the global error estimator and data approximation error X X 2 η2 = ηK , Θ2 = Θ2K . (5.3.4) K∈T  K∈T  In the following, we use the symbols . and & to denote bounds that are valid up to positive constants independently of the mesh sizes, the polynomial degree distributions and ε. To show the reliability of the error estimator in (5.3.4), we first define a so-called alignment measure M(v, T ), which was originally introduced in [17]; see also [7, 18]. Definition 5.3.1 (Alignment measure) Let v ∈ H 1 (Ω) be an arbitrary non-constant function and T be a family of triangulations of Ω. The alignment measure M(v, T ) is then defined by P 2 1/2 ( K∈T h−2 min,K kMK ∇vkL2 (K) ) M(v, T ) = . k∇vkL2 (Ω) We now state the following result. Theorem 5.3.2 (Reliability) Let u be the solution of (5.2.1) and uhp ∈ Sp (T ) its DG approximation obtained by (5.2.10). Let the error estimator η and the data approximation error Θ be defined by (5.3.4). Then we have the a-posteriori error bound k u − uhp kE,T + | u − uhp |O,T . M(v, T )(η + Θ). Here, v ∈ H01 (Ω) is the test function such that the inf-sup condition (5.4.10) holds true. Remark 5.3.3 The alignment measure M(v, T ) arises from the anisotropic interpolation estimates; see [17]. From [17], we know that 1 ≤ M(v, T ) . max K∈T  hmax,K . hmin,K  For isotropic meshes one obtains that the alignment measure is of order 1. The same is achieved for anisotropic meshes aligned with the function v. 166  5.4. Proofs Therefore, the alignment measure is not too much of an obstacle for reliable error estimation if the anisotropic mesh refinement is aligned with boundary layers or internal layers. This is numerically confirmed in Section 5.6. In all our numerical tests, we observe that the alignment measure is of moderate size, as soon as a reasonable resolution of the layer is achieved, and the ratio of the error estimate and the true energy error is independent of the mesh size. We refer the reader to [17, 18] for a more in-depth discussion on the alignment measure. Our next theorem derives a lower bound for the error measured in terms of the energy norm and the semi-norm | · |O,T . Theorem 5.3.4 (Efficiency) Let u be the solution of (5.2.1) and uhp ∈ Sp (T ) its DG approximation obtained by (5.2.10). Let the error estimator η and the data approximation error Θ be defined by (5.3.4). Then for any δ ∈ (0, 21 ) we have the bound 3  3  η . |p|δ+ 2 k u − uhp kE,T + |p|2δ+ 2 | u − uhp |O,T + |p|2δ+1 Θ. As in Chapters 3 and 4, the efficiency bound is suboptimal with respect to the polynomial degree due to the use of inverse estimates which are suboptimal in the spectral order; see [19]. Remark 5.3.5 All the results presented here can be naturally generalized to the three dimensional convection-diffusion problems by exploiting analogous arguments to those presented in the sequel; see Chapters 3 and 4 for the detail of the extension strategy.  5.4  Proofs  In this section, we present the proofs of Theorems 5.3.2 and 5.3.4.  5.4.1  Stability and auxiliary forms  The following inf-sup condition for the continuous form A is the crucial stability result in our analysis. It holds with an absolute constant, which can be immediately inferred from the proof of Lemma 2.4.4. Lemma 5.4.1 Assume (5.2.2). Then we have inf  u∈H01 (Ω)\{0}  sup v∈H01 (Ω)\{0}  (k u kE,T  A(u, v) 1 ≥ . + |au|? ) k v kE,T 3 167  5.4. Proofs Next, we split the discontinuous Galerkin form Ahp into two parts, see Chapter 2, and define XZ X Z εγ(p⊥ )2 E ehp (u, v) = A [[u]] · [[v]]ds (ε∇u · ∇v + a · ∇uv) dx + ⊥ h K E E K∈T E∈E(T ) XZ XZ − a · nK uv ds + a · nK (ue − u)v ds, K∈T  Khp (u, v) = −  ∂Kin ∩Γin  X Z  E∈E(T ) E  K∈T  {{ε∇u}} · [[v]] ds −  ∂Kin \Γ  X Z  E∈E(T ) E  {{ε∇v}} · [[u]] ds.  We shall use the above auxiliary forms to express both the continuous form A in (5.2.3) and the discontinuous Galerkin form Ahp in (5.2.10). Indeed, we have  5.4.2  ehp (u, v), A(u, v) = A ehp (u, v) + Khp (u, v), Ahp (u, v) = A  u, v ∈ H01 (Ω),  u, v ∈ Sp (T ).  (5.4.1) (5.4.2)  Auxiliary meshes  We shall make use of an auxiliary irregular mesh Te of rectangles, similarly to the approach in Chapter 4, which is obtained from T as follows. Let K ∈ T . If all four elemental edges are edges of the mesh T , that is, if E(K) ⊆ E(T ), we leave K untouched. Otherwise, at least one of the elemental edges of K, say E, contains hanging nodes. Suppose the hanging nodes on the edge E are yiE , i ∈ {1, · · · , M }. In this case, we refine K by connecting yiE 0 −1 E −1 E 0 and yiE along the curve FK (FK (yi ), FK (yi )) with E 0 being the opposite elemental edge to the edge E. Here, ν1 , ν2 denotes by the line connecting the nodes ν1 and ν2 . This construction is illustrated in Figure 5.2. Clearly, the mesh Te is a refinement of T ; it is also irregular. We denote by ER (T ) the set of edges in E(T ) that have been refined in the above process. We denote by NA (Te ) the set of vertices in N (Te ) and EA (Te ) the set of edges in E(Te ) which are inside an element K of T , respectively. Moreover, we write R(K) for the elements in Te that are inside K. If K is unrefined, R(K) = {K}. Otherwise, the set R(K) consists of at most (M + 1)2 newly created elements. Next, we introduce the following auxiliary discontinuous Galerkin finite element space on the mesh Te : b K e ∈ Te }, Spe (Te ) = { v ∈ L2 (Ω) : v|Ke ◦ FKe ∈ QpKe (K),  168  5.4. Proofs s s s  s s s  s s s s s  s s  s s  s s s s s  s s s s s  =⇒  s  s s s c c c s s s  s s s s s  s s s s s  s s s s s  s c c s  s s  c c  c s  Figure 5.2: The construction of the auxiliary mesh Te from T .  e is defined by pKe = pK , where the auxiliary polynomial degree vector p e for K ∈ R(K). Thus, we clearly have the inclusion Sp (T ) ⊆ Spe (Te ). In complete analogy to (5.3.1) and (5.3.2), the energy and convective norms associated with the auxiliary mesh Te are given by X 2 k v k2E,Te = εk∇vk2L2 (K) e + ejumpp e ,Te (v) , e Te K∈ (5.4.3) 2 2 2 | v |O,Te = |av|? + ojumppe ,Te (v) , where the auxiliary edge polynomial degrees for the jump terms over Te are defined as in (5.2.9), using the auxiliary degrees pKe . Obviously, we have k v kE,T = k v kE,Te ,  | v |O,T = | v |O,Te ,  for all v ∈ H01 (Ω). Furthermore, analogously to Lemmas 3.4.2 and 3.4.3, one can show the following results. Lemma 5.4.2 Let v ∈ Spe (Te ) + H01 (Ω) be such that [[v]]|E = [[w]]|E for all E ∈ E(Te ), for a function w ∈ Sp (T ) + H01 (Ω). Then we have ejumpp,T (w) . ejumppe ,Te (v) . ejumpp,T (w),  ojumpp,T (w) . ojumppe ,Te (v) . ojumpp,T (w). Lemma 5.4.3 For v ∈ Sp (T ) + H01 (Ω), we have the bounds k v kE,T . k v kE,Te ,  | v |O,T . | v |O,Te .  169  5.4. Proofs  5.4.3  Averaging operator  Our analysis is based on an hp-version averaging operator on anisotropic meshes T that allows us to approximate discontinuous functions by continuous ones, analogously to the one used in Chapters 3 and 4. To define this operator, we let Spec (Te ) be the conforming subspace of Spe (Te ) given by Spec (Te ) = Spe (Te ) ∩ H01 (Ω). We then have the following approximation result. Theorem 5.4.4 (Averaging operator) There is operator Ihp : Sp (T ) → Spec (Te ) that satisfies X  e Te K∈  X  e Te K∈  kv − Ihp vk2L2 (K) e .  k∇(v − Ihp v)k2L2 (K) e .  X Z  −2 ⊥ 2 (p⊥ E ) hE [[v]] ds,  (5.4.4)  −2 2 p2E h⊥ E hmin,E [[v]] ds.  (5.4.5)  E∈E(T ) E  X Z  E∈E(T ) E  The detailed proof of Theorem 5.4.4 will be presented in Section 5.5. Remark 5.4.5 Fixing the polynomial degrees in the inequality (5.4.5), we obtain immediately the H 1 -seminorm estimation for the h-version averaging operators in [7, Theorem 4.4].  5.4.4  Proof of Theorem 5.3.2  Following [13, 25], we decompose the discontinuous Galerkin solution into a conforming part and a remainder: uhp = uchp + urhp ,  (5.4.6)  where uchp = Ihp uhp ∈ Spec (Te ) ⊂ H01 (Ω), with Ihp the approximation operator from Theorem 5.4.4. The remainder is then given by urhp = uhp − uchp = uhp − Ihp uhp ∈ Spe (Te ). By Lemma 5.4.3 and the triangle inequality, we obtain k u − uhp kE,T + | u − uhp |O,T  . k u − uhp kE,Te + | u − uhp |O,Te  . k u − uchp kE,Te + | u − uchp |O,Te + k urhp kE,Te + | urhp |O,Te  (5.4.7)  = k u − uchp kE,T + | u − uchp |O,T + k urhp kE,Te + | urhp |O,Te . 170  5.4. Proofs In a series of lemmas, we now prove that both the continuous error u − uchp and the remainder urhp can be bounded by the estimator η and the data approximation term Θ. Lemma 5.4.6 There holds k urhp kE,Te + | urhp |O,Te . η. Proof : Since [[urhp ]]|E = [[uhp ]]|E for all E ∈ E(Te ) and uhp ∈ Sp (T ), the definition of the jump residual ηJK and Lemma 5.4.2 yield k urhp k2E,Te + | urhp |2O,Te X r 2 r 2 r 2 = εk∇urhp k2L2 (K) e + |auhp |? + ejumpp e ,Te (uhp ) + ojumpp e ,Te (uhp ) e Te K∈  .  X  e Te K∈  r 2 εk∇urhp k2L2 (K) e + |auhp |? +  X  ηJ2K .  K∈T  Hence, only the volume terms and |aurhp |? need to be bounded further. Theorem 5.4.4, the assumptions (5.2.7) and (5.2.8) yield X X Z X 2 2 ⊥ −2 r 2 [[u ]] ds . ηJ2K . . εp h h ε k∇uhp kL2 (K) E E min,E hp e E∈E(T ) E  e Te K∈  K∈T  To estimate |aurhp |? , we again use Theorem 5.4.4, the fact that p⊥ E ≥ 1, the assumptions (5.2.7) and (5.2.8),  1 X r 2 |aurhp |2? . kak2L∞ (K) e e kuhp kL2 (K) ε e Te K∈  .  X  E∈E(T )  X h⊥ 2 E k[[u ]]k ηJ2K . 2 (E) . hp L ⊥ ε(pE )2 K∈T  2  This finishes the proof.  To prove Theorem 5.3.2, we also need the following interpolation error estimations. Lemma 5.4.7 (Local interpolation error bounds) For any function v ∈ H01 (Ω), there exists a function vhp ∈ Sp (T ) such that p2min,K kv − vhp k2L2 (K) . kMK ∇vk2L2 (K) ,  X  E∈E(K)  kMK ∇(v − vhp )k2L2 (K) . kMK ∇vk2L2 (K) ,  2 h⊥ E,K pmin,K  p⊥ E,K  kv −  vhp k2L2 (E)  .  (5.4.8)  kMK ∇vk2L2 (K) , 171  5.4. Proofs for any K ∈ T . Proof : The first and the second inequalities can be easily derived following the approach in [14, Theorem 3.3] and the scaling argument; see also the proof of [26, Lemma 4.67, Corollary 4.68, Corollary 4.69]. Now consider one of the elemental edge E. By the following trace inequality, kv − vhp k2L2 (E) .  1 kMK ∇(v − vhp )kL2 (K) kv − vhp kL2 (K) h⊥ E,K 1 + ⊥ kv − vhp k2L2 (K) , hE,K  the Cauchy-Schwarz inequality, the fact that p⊥ E,K ≥ 1 and the results in the first and the second inequalities, we obtain kv −  vhp k2L2 (E)  p⊥ 1 E,K 2 . ⊥ kMK ∇(v − vhp )kL2 (K) + ⊥ kv − vhp k2L2 (K) ⊥ hE,K pE,K hE,K p⊥ 1 E,K 2 . ⊥ kMK ∇vkL2 (K) + ⊥ 2 kMK ∇vk2L2 (K) ⊥ hE,K pE,K hE,K pmin,K .  p⊥ E,K 2 h⊥ E,K pmin,K  kMK ∇vk2L2 (K) , 2  which shows the third inequality.  The global interpolation error estimation follows from Lemma 5.4.7 and the definition of the alignment measure. Lemma 5.4.8 (Global interpolation error bounds) For any function v ∈ H01 (Ω), we have X p2min,K  h2min,K  kv − vhp k2L2 (K) . M(v, T )2 k∇vk2L2 (Ω) ,  K∈T 2 X X h⊥ E,K pmin,K kv ⊥ h2 p E,K min,K K∈T E∈∂K  (5.4.9) −  vhp k2L2 (E)  . M(v, T )  2  k∇vk2L2 (Ω) .  We then have the following result. Lemma 5.4.9 For any v ∈ H01 (Ω), we have Z ehp (uhp , v−vhp )+Khp (uhp , vhp ) . M(v, T ) (η + Θ) k v kE,T . f (v−vhp )dx−A Ω  Here, vhp ∈ Sp (T ) is the hp-interpolant of v in Lemma 5.4.7.  172  5.4. Proofs Proof : Integration by parts of the diffusive volume terms readily yields Z ehp (uhp , v − vhp ) + Khp (uhp , vhp ) = T1 + T2 + T3 + T4 + T5 , f (v − vhp ) dx − A Ω  where  T1 =  XZ  K∈T  T2 = − T3 = − T4 =  (f + ε∆uhp − a · ∇uhp )(v − vhp ) dx,  K  Z  X  E∈EI (T ) E  X Z  E∈E(T ) E  Z  X  K∈T ∂K \Γ in  +  {{ε∇vhp }} · [[uhp ]] ds,  a · nK (uhp − uehp )(v − vhp ) ds  Z  X  [[ε∇uhp ]]{{v − vhp }} ds,  K∈T ∂K ∩Γ in in  T5 = −  X Z  E∈E(T ) E  a · nK uhp (v − vhp ) ds,  2 εγ(p⊥ E) [[uhp ]] · [[v − vhp ]] ds. h⊥ E  To bound T1 , we first add and subtract the data approximations. From the weighted Cauchy-Schwarz inequality and the approximation properties in (5.4.9), we then readily obtain X 1 2 2 2 T1 . M(v, T ) (ηR + Θ ) k v kE,T . K K K∈T  Similarly, by the Cauchy-Schwarz inequality and (5.4.9), we have T2 .  X  X  h2min,K p⊥ E,K  εp2min,K h⊥ E,K K∈T E∈∂K\Γ ×  . M(v, T )  X  X  k[[ε∇uhp ]]k2L2 (E)  εp2min,K h⊥ E,K  h2 p⊥ K∈T E∈∂K\Γ min,K E,K  X  K∈T  2 ηE K  1 2  1 2  kv − vhp k2L2 (E)  1  2  k v kE,T .  173  5.4. Proofs To estimate T3 , we employ the Cauchy-Schwarz inequality, the trace inequality in [6, Lemma 3.1] for anisotropic polynomial degrees. This results in Z X Z ε(p⊥ )2 1 X X 1 εh⊥ 2 2 E E 2 2 T3 . [[u ]] ds |∇v | ds hp hp ⊥ ⊥ 2 hE (p ) E E E K∈T E∈E(K) E∈E(T ) X X X 1 1  1 εk∇vhp k2L2 (K) 2 . . ηJ2K 2 ηJ2K 2 k v kE,T . K∈T  K∈T  K∈T  For T4 , we apply again the Cauchy-Schwarz inequality and (5.4.9) to get T4 .  X  X  h2min,K p⊥ E,K  εp2min,K h⊥ E,K K∈T E∈E(K) ×  . M(v, T )  X  X  ηJ2K  K∈T  T5 .  X  . M(v, T )  1  2  3 p2min,K (h⊥ E,K )  X  ηJ2K  X  K∈T  1  2  2  kv − vhp k2L2 (E)  k[[uhp ]]k2L2 (E)  εp2min,K h⊥ E,K  h2min,K p⊥ E,K K∈T E∈E(K)  X  1  1 2  k v kE,T .  5 εγ 2 h2min,K (p⊥ E,K )  K∈T E∈E(K)  ×  εp2min,K h⊥ E,K  h2min,K p⊥ E,K K∈T E∈E(K)  Finally, we have X  X  k[[uhp ]]k2L2 (E)  1  2  kv − vhp k2L2 (E)  1 2  k v kE,T .  The above estimates for the terms T1 through T5 imply the assertion.  2  Lemma 5.4.10 There holds: k u − uchp kE,T + | u − uchp |O,T . M(v, T )(η + Θ). Proof : The proof is the same as the one for Lemma 3.4.8; we write the detail here just for reader’s convenience. Since u − uchp ∈ H01 (Ω), we have | u − uchp |O,T = |a(u − uchp )|? . Then the inf-sup condition in Lemma 5.4.1 yields: k u − uchp kE,T + | u − uchp |O,T .  sup v∈H01 (Ω)\{0}  A(u − uchp , v) k v kE,T  .  (5.4.10) 174  5.4. Proofs To bound (5.4.10), let v ∈ H01 (Ω). Then, property (5.4.1) shows that Z Z c c ehp (uc , v). f v dx − A f v dx − Ahp (uhp , v) = A(u − uhp , v) = hp Ω  Ω  By employing the fact that v ∈ H01 (Ω) and integrating by parts the convection term, one can readily see that  with R=  ehp (uc , v) = A ehp (uhp , v) + R, A hp XZ  e Te K∈  e K   −ε∇urhp + aurhp · ∇v dx.  Furthermore, from the DG method in (5.2.10) and property (5.4.2), we have Z ehp (uhp , vhp ) + Khp (uhp , vhp ), f vhp dx = A Ω  where vhp ∈ Sp (T ) is the hp-version interpolant of v in Lemma 5.4.7. Combining the above results yields Z ehp (uhp , v − vhp ) + Khp (uhp , vhp ) − R. f (v − vhp ) dx − A A(u − uchp , v) = Ω  The estimate in Lemma 5.4.9 now shows that  |A(u − uchp , v)| . M(v, T ) (η + Θ) k v kE,T + |R|.  (5.4.11)  It remains to bound |R|. From the Cauchy-Schwarz inequality, the definition of the norm | · |? , the conformity of v and Lemma 5.4.6, one readily obtains   (5.4.12) |R| . k urhp kE,Te + | urhp |O,Te k v kE,T . ηk v kE,T . Equations (5.4.10)–(5.4.12) imply the desired result.  2  The proof of Theorem 5.3.2 now immediately follows from the inequality (5.4.7), Lemma 5.4.6 and Lemma 5.4.10.  5.4.5  Proof of Theorem 5.3.4  b1 In the proof of Theorem 5.3.4, we shall make use of the bubble functions, Ψ b and Ψ2 , constructed in Lemma 4.5.1. For an arbitrary element K ∈ T , we −1 set ΨK = Ψ2 ◦ FK ; for an interior edge E, we let ΨE = Ψ1 ◦ FE−1 , where FE is the affine transformation that maps the interval [−1, 1] onto E. Now we are ready to show the efficiency of ηRK , ηEK and ηJK , respectively. 175  5.4. Proofs Lemma 5.4.11 Under the assumptions of Theorem 5.3.4, there holds X 1 1 2 ( ηR ) 2 . |p| k u − uhp kE,T + |p| |a(u − uhp )|? + |p|δ+ 2 Θ. K K∈T  Proof : For any element K ∈ T , we set vK = ε−1 (fhp + ε∆uhp − ahp · ∇uhp )|K ΨαK , where α ∈ ( 12 , 1]. Applying the inverse inequality (4.5.3), we obtain kfhp + ε∆uhp − ahp · ∇uhp kL2 (K) α  α/2  . (p1K p2K ) 2 k(fhp + ε∆uhp − ahp · ∇uhp )ΨK kL2 (K) α  −α/2  = ε(p1K p2K ) 2 kvK ΨK  This leads to X 2 ηR . S2 K K∈T  with S 2 =  X  K∈T  kL2 (K) . −α/2 2 kL2 (K) .  εh2min,K (p1K p2K )α /p2min,K kvK ΨK  Since the exact solution satisfies (f + ε∆u − a · ∇u)|K = 0, we obtain, by integration by parts and insertion of the data a and f , Z X 2 2 1 2 α 2 S = hmin,K (pK pK ) /pmin,K (fhp + ε∆uhp − ahp · ∇uhp )vK dx K∈T  =  X  K∈T  +  h2min,K (p1K p2K )α /p2min,K  Z  K  K  Z  K  (ε∇(u − uhp ) − a(u − uhp )) · ∇vK dx   α 1 1 −α (((fhp − f ) + (a − ahp ) · ∇uhp )ε− 2 ΨK2 )(ε 2 vK ΨK 2 )dx .  Here, we have also used that vK |∂K = 0. From the proof of [19, Lemma 3.4], we have (p1 p2 )2−α −α/2 k∇vK k2L2 (K) . K2 K kvK ΨK k2L2 (K) . hmin,K By the Cauchy-Schwarz inequality, the definition of the dual norm and the data approximation error Θ, we obtain S 2 . S (|p|k u − uhp kE,T + |p||a(u − uhp )|? + |p|α Θ) . Therefore, X 2 ( ηR )1/2 . |p| k u − uhp kE,T + |p| |a(u − uhp )|? + |p|α Θ. K K∈T  176  5.4. Proofs Choosing δ = α − 1/2 finishes the proof.  2  For any edge E ∈ E(T ), we define the sets wE = { K1 , K2 ∈ T : E = ∂K1 ∩ ∂K2 }, e ∈ T ∪ Te : E ∈ E(K) e }. w eE = { K  (5.4.13)  For simplicity, we also use the notation wE and w eE to denote the domain formed by the elements in wE and in w eE , respectively.  Lemma 5.4.12 Under the assumptions of Theorem 5.3.4, there holds X p  1 1 2 ( ηE ) 2 . DT |p|δ+1 k u − uhp kE,T +|p|2δ+1 | u − uhp |O,T +|p|2δ+ 2 Θ , K K∈T  with DT given by  DT := max K∈T  pmax,K . pmin,K  (5.4.14)  Proof : Let E = ∂K1 ∩ ∂K2 be an interior edge shared by two elements K1 , K2 ∈ T . For α ∈ (1/2, 1], set τE = [[∇uhp ]]ΨαE . We construct a bubble function ψE over wE . Case 1: Suppose that none of the end points of E is a hanging node. That is, we have E ∈ E(K1 ) ∩ E(K2 ). Lemma 2.6 of [19] (see also Lemma 4.5.2 for the case in three dimensions) then ensures the existence of a function ψE ∈ H01 (wE ) with ψE |E = τE , ψE |∂wE = 0 and −α/2 2 kL2 (E) , −α/2 2 −2 2 kL2 (E) . h⊥ E hmin,E pE kτE ΨE  −2 kψE k2L2 (wE ) . h⊥ E pE kτE ΨE  k∇ψE k2L2 (wE ) .  (5.4.15) (5.4.16)  Case 2: Suppose that one of the end points of E is a hanging node of T ; without loss of generality, we may assume it is a hanging node of K1 . e 1 ∈ Te , such that In this case, wE is concave, and there exists an element K e 1 ( K1 and K e 1 ∩ K2 = E. Thus w e 1 ∪ K2 ( wE . By Lemma 2.6 K eE = K 1 e of [19] we can find a function ψE ∈ H0 (w eE ) with ψeE |E = τE , ψeE |∂ weE = 0 and −α/2 2 kL2 (E) , −α/2 2 −2 2 h⊥ kL2 (E) . E hmin,E pE kτE ΨE  −2 kψE k2L2 (wE ) . h⊥ E pE kτE ΨE  k∇ψE k2L2 (wE ) .  Now define the function ψE on wE by ψE = ψeE on w eE , and by zero oth1 erwise. Thus, we have ψE ∈ H0 (wE ) with ψE |E = τE , ψE |∂wE = 0, and (5.4.15)–(5.4.16) also hold. 177  5.4. Proofs In both cases above, we now proceed as follows. Applying again the inverse inequality from Lemma 4.5.1 with d = 1, we get α  −α  k[[∇uhp ]]kL2 (E) . kpαE [[∇uhp ]]ΨE2 kL2 (E) = kpαE τE ΨE 2 kL2 (E) .  (5.4.17)  Therefore, X  2 ηE K  .S  2  2  with S =  X  X  Z  K∈T E∈∂K\Γ E  K∈T  ε  h2min,K p⊥ E,K p2min,K h⊥ E,K  2 −α p2α E τE ΨE ds.  Since [[ε∇u]] = 0 on interior edges, integration by parts over wE yields Z Z ε(∆uhp − ∆u)ψE + ε(∇uhp − ∇u) · ∇ψE dx, [[ε∇(uhp − u)]]τE ds = E  wE  where ∆uhp and ∇uhp are understood piecewise. Using the differential equation, approximating the data and integrating by parts the convective term, we readily obtain S 2 = T1 + T2 + T3 + T4 + T5 , with T1 =  X  X  Z  h2min,K p⊥ E,K  K∈T E∈∂K\Γ wE  T2 =  X  X  Z  h2min,K p⊥ E,K  K∈T E∈∂K\Γ wE  T3 =  X  X  Z  T4 =  T5 =  X  Z  p2min,K h⊥ E,K h2min,K p⊥ E,K  p2 h⊥ K∈T E∈∂K\Γ E min,K E,K X  X  Z  p2α E (ε∇uhp − ε∇u) · ∇ψE dx,  p2min,K h⊥ E,K h2min,K p⊥ E,K  K∈T E∈∂K\Γ wE  X  p2α E (fhp + ε∆uhp − ahp · ∇uhp )ψE dx,  p2min,K h⊥ E,K  h2min,K p⊥ E,K  K∈T E∈∂K\Γ wE  p2min,K h⊥ E,K  p2α E a(u − uhp ) · ∇ψE dx,  p2α E a · [[uhp ]]τE ds,   p2α (f − f ) + (a − a) · ∇u hp hp ψE dx. hp E  The Cauchy-Schwarz inequality, Lemma 5.4.11 and inequality (5.4.15) yield p  1 T1 . S DT |p|α− 2 |p| k u − uhp kE,T + |p| |a(u − uhp )|? + |p|α Θ .  Similarly, we obtain  T2 . S  p 1 DT |p|α+ 2 k u − uhp kE,T ,  178  5.4. Proofs as well as T3 . S  p 1 DT |p|α+ 2 |a(u − uhp )|? .  By (5.4.17) and the definition of semi-norm | · |O,T , we conclude that X X Z h2 p⊥ 1 2 4α min,K E,K 2 [[u ]] ds pE 2 T4 . hp ⊥ εp h E min,K E,K K∈T E∈∂K\Γ  ×  X  X  Z  K∈T E∈∂K\Γ E  ε  h2min,K p⊥ E,K p2min,K h⊥ E,K  . S |p|2α | u − uhp |O,T .  2 −α p2α E τE ΨE ds  1 2  Finally, the data error term T5 can be bounded by X X Z 1 h2min,K p⊥ 2 E,K 2α 2 −α T5 . ε 2 pE τE ΨE dx ⊥ wE pmin,K hE,K K∈T E∈∂K\Γ  ×  X  .S  X  Z  K∈T E∈∂K\Γ wE  2α−1 2 p⊥ hmin,K E,K pE  pE  εp2min,K  |(f − fhp ) + (ahp − a) · ∇uhp |2 dx  p 1 DT |p|α− 2 Θ.  1 2  Combining the above bounds for T1 through T5 , we obtain   p 1 1 S 2 . S DT |p|α+ 2 k u − uhp kE,T + |p|2α | u − uhp |O,T + |p|2α− 2 Θ . Thus, X  K∈T  !1 2  2 ηE K  1  1 1 . DT2 |p|α+ 2 k u − uhp kE,T + |p|2α | u − uhp |O,T + |p|2α− 2 Θ .  Choosing δ = α − 1/2 implies the assertion.  2  Since the jumps of u vanish over the edges, we also have the following result. Lemma 5.4.13 Under the assumptions of Theorem 5.3.4, there holds X 1 ( ηJ2K )1/2 . DT |p| 2 k u − uhp kE,T + | u − uhp |O,T . K∈T  with DT given in (5.4.14). The proof of Theorem 5.3.4 now follows from Lemmas 5.4.11, 5.4.12, 5.4.13 and the fact that 1 ≤ DT ≤ |p|. 179  5.5. Proof of Theorem 5.4.4  5.5  Proof of Theorem 5.4.4  In this section, we prove the result of Theorem 5.4.4.  5.5.1  Polynomial basis functions  We begin by introducing hp-version basis functions. To that end, let Ib = (−1, 1) be the reference interval. We denote by Zbp = { zb0p , · · · , zbpp } the b Recall that zbp = −1 and zbpp = Gauss-Lobatto nodes of order p ≥ 1 on I. 0 p p 1. We denote by Zbint = { zb1p , · · · , zbp−1 } the interior Gauss-Lobatto nodes b Let now E ∈ E(K) be an edge of an element K. The of order p on I. p b nodes in Z can be affinely mapped onto E and we denote by Z p (E) = { z0E,p , · · · , zpE,p } the Gauss-Lobatto nodes of order p on E. The points p (E) = z0E,p and zpE,p coincide with the two end points of E. The set Zint E,p E,p { z1 , · · · , zp−1 } denotes the interior Gauss-Lobatto points of order p. We write Pp (E) for the space of all polynomials of degree less or equal than p on E and define Ppint (E) = { q ∈ Pp (E) : q(z0E,p ) = q(zpE,p ) = 0 },  p (E) }. Ppnod (E) = { q ∈ Pp (E) : q(z) = 0, z ∈ Zint  By construction, we have Pp (E) = Ppint (E) ⊕ Ppnod (E). For an element K, we now define basis functions for polynomials of the form v ∈ QpK (K), v|E ∈ PpE (E), E ∈ E(K), (5.5.1) where 1 ≤ pE ≤ pE,K is the edge polynomial degree associated with E ∈ E(K). As usual, we shall divide the basis functions into interior, edge and vertex basis functions. b = (−1, 1)2 . We denote its We first consider the reference element K b1 , . . . , E b4 and its four vertices by νb1 , . . . , νb4 , numbered as four edges by E p in Figure 5.3. Let {ϕ bi }0≤i≤p be the Lagrange basis functions associated 1  2  p p p with the nodes Zbp . We denote by {b zi,jKb = (b zi Kb , zbj Kb )}, 1 ≤ i ≤ p1b − 1, K b 1 ≤ j ≤ p2 − 1, the interior tensor-product Gauss-Lobatto nodes on K. b K  Note that pEb1 ,Kb = pEb3 ,Kb = p1b and pEb2 ,Kb = pEb4 ,Kb = p2b . The interior basis K K functions are then given by 1  2  p p b int,pKb (b Φ x1 , x b2 ) = ϕ bi Kb (b x1 ) ϕ bj Kb (b x2 ), i,j  1 ≤ i ≤ p1Kb − 1, 1 ≤ j ≤ p2Kb − 1.  180  5.5. Proof of Theorem 5.4.4 νb4  b3 E  1  νb3  0.8 0.6 0.4 0.2  b2 E  b4 0 E  −0.2 −0.4 −0.6 −0.8 −1 −1  νb1  −0.8  −0.6  −0.4  −0.2  0  0.2  0.4  0.6  0.8  b1 E  1  νb2  Figure 5.3: Reference element with variable edge polynomial degrees: pKb = (5, 4), pEb1 = 2, pEb2 = 3, pEb3 = 4, pEb4 = 1.  b1 in Figure 5.3 with edge degree Next, we consider exemplarily the edge E b1 are pEb1 . The edge basis functions for E b1 ,p b E E  b Φ i  1  b ,p E  pEb  (b x1 , x b2 ) = ϕ bi  1  p2  i = 1, · · · , pEb1 − 1.  (b x1 ) ϕ b0K (b x2 ),  b 1 Eb1 vanishes on E b2 , E b3 and E b4 . The other edge basis functions Note that Φ i are defined analogously. Finally, we consider the vertex νb1 , which is shared b1 and E b4 ; see Figure 5.3. We then introduce the associated vertex basis by E function pEb pEb b νb1 (b Φ b2 ) = ϕ b0 1 (b x1 ) ϕ b0 4 (b x2 ). b x1 , x K  b2 and E b3 . The vertex basis functions associated with the It vanishes on E b other vertices of K are defined analogously. This completes the definition b of the shape functions on the reference element K. For an arbitrary parallelogram K, shape functions Φ on K can be defined b by using the pull-back map Φ = Φ b ◦ F −1 , from the analogous ones on K K E,p int,p giving rise to shape functions ΦνK , Φi E and Φi,j K on K. Therefore, a polynomial v of the form (5.5.1) can be expanded into v(x) =  X  ν∈N (K)  +  v(ν) ΦνK (x) + X  X  pX E −1  E v(ziE,pE ) ΦE,p (x) i  E∈E(K) i=1 K cij Φint,p (x), i,j  1≤i≤p1K −1 1≤j≤p2K −1  181  5.5. Proof of Theorem 5.4.4 with expansion coefficients cij . Following the proof of Lemma 3.1 of [6], we have the following estimates: Lemma 5.5.1 There holds: b that vanishes at the interior Gauss-Lobatto (i) For a function vb ∈ Qp (I) b nodes on I, there holds −2 kb v k2L2 (I) v (−1)2 + vb(1)2 ). b . p (b  b that vanishes at the interior Gauss(ii) For a function vb ∈ QpKb (K) b there holds Lobatto nodes on K, kb v k2L2 (K) b .  4 X i=1  −2 (p⊥ v k2L2 (Eb ) . b ,K b ) kb E i  i  b is shared by two edges E bn (iii) If the vertex νb of the reference element K ν b b b and Em , the associated vertex basis function Φ b can be bounded by K  b νb k 2 b . p−1 p−1 . kΦ b L (K) b b K En Em  5.5.2  Extension operators  b ∈ E(K) b be an eleNext, we define extension operators over edges. Let E b b We define L b E by mental edge of the reference element K. pb K  b int bE b b L (K), p b : Pp b b (E) −→ QpK b K  E,K  pE, b K b −1  qb(x) 7−→  X i=1  b b b b b b E,p K b E,pE, qb(b zi E,K )Φ (x). i  (5.5.2)  b b bE b and LE By construction, L (b q ) = qb on E, (b q ) vanishes in all the interior pK pK b b pK b b and on the other three tensor-product Gauss-Lobatto nodes {b zi,j } of K b From [6, Lemma 3.1], we have the following inequality. edges of K.  b bE Lemma 5.5.2 The linear extension operator L introduced in (5.5.2) satpK b isfies ⊥ −1 b Eb (b q kL2 (E) kL b . (pE, b . p b q )kL2 (K) bK b ) kb K  182  5.5. Proof of Theorem 5.4.4 Now consider an arbitrary element K ∈ T and fix an edge E ∈ E(K). If E contains no hanging node in T (i.e., E ∈ E(T )), we define the extension int operator LE pK ,K (q) : PpE,K (E) → QpK (K) by −1 E LE pK ,K (q) = [LpK (q ◦ FK )] ◦ FK , b  q ∈ Ppint (E). E,K  (5.5.3)  If E contains N hanging nodes on it, N ≤ M , E can be written as E = E0 ∪ · · · ∪ EN for N + 1 edges E0 , · · · , EN in E(T ). We then partition K into N + 1 parallelograms, K = K0 ∪ · · · ∪ KN , by connecting the hanging nodes on E with the corresponding points of the edge opposite to E, as illustrated in Figure 5.4. For any qi ∈ Ppint (Ei ), i = 0, · · · , N , we define the extension E,K E operator LpK ,K (q0 , · · · , qN ) by LE pK ,K (q0 , · · · , qN ) =  N X  i LE pK ,Ki (qi ),  (5.5.4)  i=0  i with LE pK ,Ki , i = 0, · · · , N , given in (5.5.3). By definitions, the extensions E LE pK ,K (q) and LpK ,K (q0 , · · · , qN ) are continuous in K and satisfy  LE pK ,K (q)|E = q,  and LE pK ,K (q0 , · · · , qN )|Ei = qi ,  i = 0, · · · , N.  E Moreover, LE pK ,K (q) and LpK ,K (q0 , · · · , qN ) both vanish on the other edges of E(K).  s c  c s  s  K0 p p p p p  KN  E0  s  s  p p p E p p  EN  s  Figure 5.4: Partition of K into K1 , · · · , KN .  183  5.5. Proof of Theorem 5.4.4  5.5.3  Decomposition of functions in Sp (T )  We shall now decompose functions in Sp (T ), similarly to [30]. For any edge E ∈ E(T ) ∪ E(Te ), we set e ∈w pE = min{ pE,Ke : K eE },  (5.5.5)  e K e ∈ with w eE defined in (5.4.13). Notice that an elemental edge E in E(K), e ∈ Te , equation (5.5.5) defines Te , belongs to E(T ) ∪ E(Te ). Hence, for any K the elemental edge polynomial degrees as used in (5.5.1). Furthermore, we denote by vK the restriction of a piecewise smooth function v to an element K ∈ T ∪ Te . Let v ∈ Sp (T ). Firstly, we introduce a (nodal) interpolant v nod ∈ Spe (Te ). e ∈ R(K), we will construct the restriction For each element K ∈ T and K nod nod e such that v e of v to K K  nod e vK e ∈ QpK (K),  nod vK e |E ∈ PpE (E),  e E ∈ E(K),  (5.5.6)  with pE given in (5.5.5). To define v nod e , we distinguish two cases. K Case 1: If R(K) = {K} (i.e., if K is unrefined), the interpolant v nod e = K nod vK is simply defined by X nod vK (x) = vK (ν) ΦνK (x). (5.5.7) ν∈N (K)  Case 2: If R(K) consists of multiple newly created elements, we define e ∈ R(K) separately. To do so, fix K e ∈ R(K). on each element K v nod e K e located in Without loss of generality, we may consider the situation that K the corner of K; see Figure 5.5 for illustration. The constructions of v nod e for K e locates inside K (see Figure 5.6) or shares an edge of K the cases when K (see Figure 5.7) are analogous. Here {Ei }4i=1 and {νi }4i=1 denote the edges e Notice that here we ei }4 and {e and vertices of K, {E νi }4i=1 the ones of K. i=1 e e e4 are in EA (Te ) and have νe2 = ν2 and νe4 ∈ NA (T ). Furthermore, E3 and E pEe3 = pE1 ,K = pE3 ,K , pEe4 = pE2 ,K = pE4 ,K . Let us now define the value of e e e v nod e at the edge and vertex nodes of K. At the interior nodes of E3 and E4 , K we set pEe p nod e3 ) ∪ Z Ee4 (E e4 ). vK z ∈ Zint 3 (E (5.5.8) e (z) = vK (z), int Similarly, we set v nod e2 and ν = νe4 . e (ν) = vK (ν) for the vertices ν = ν K  184  5.5. Proof of Theorem 5.4.4 ν4 s c  E4 ν1 s  E3  c c  e4 E  νe4  e1 cν  e3 E e K  sν3 ν4 s  c  c  c  c  νe3 E E e4 2 4 E e2 E c  e1 νe2 sν ν s E 1 2  E1  e ∈ R(K) Figure 5.5: K at the corner of K.  E3 c  νe4  e1 cν c  e3 E e K  e1 E  νe3  c  e2 E  νe2 c c  sν3 ν4 s  c  E c 3  c  c  c  c  E2 E4 c  c  sν2 ν1 s  E1  e ∈ R(K) Figure 5.6: K locates inside K.  νe4  sν3  e3 E  c  νe3 e4 e2 E2 e E E K e1 νe2 c c cν e1 E c c sν2 E1  e ∈ R(K) Figure 5.7: K shares an edge of K.  e1 It remains to define the values of v nod on the nodes of the edges E e K e2 , as well as on νe1 and νe3 . We only consider νe1 and E e1 (the construction and E e for νe3 and E2 is completely analogous). If νe1 ∈ N (T ) (i.e., νe1 is a hanging node in T ), then we define nod vK e (z) = 0,  p  e E e1 ), z ∈ Zint 1 (E  nod vK ν1 ) = vK (e ν1 ). e (e  (5.5.9)  On the other hand, if νe1 ∈ / N (T ), then E1 ∈ E(T ) and ν1 ∈ N (T ). In this case, we interpolate the values of the nodal interpolant over the long e1 . That is, we set edge E1 at the Gauss-Lobatto nodes on E ν1 ν2 nod vK e (z) = vK (ν1 ) ΦK (z)+vK (ν2 ) ΦK (z),  p  e E e1 )∪{e z ∈ Zint 1 (E ν1 }. (5.5.10)  With the nodal values of v nod e constructed (5.5.8)–(5.5.10), we have K  nod vK e (x)  =  X  nod vK e (ν)  ΦνKe (x)  e ν∈N (K)  +  X  pE −1   X  e i=1 E∈E(K)  E,pE  nod vK e (zi  E,pE  )Φi   (x) .  This finishes the construction of the interpolant of v nod . Notice that v nod ∈ Spe (Te ) is continuous over edges E ∈ EA (Te ) and satisfies nod vK (ν) − vK (ν) = 0,  ν ∈ N (T ) located on ∂K,  (5.5.11)  e ∈w E ∈ E(T ), K eE .  (5.5.12)  as well as nod nod vK e |E ∈ PpE (E),  185  5.5. Proof of Theorem 5.4.4 Secondly, we construct a function v edge ∈ Spe (Te ) related to the edge degrees of freedom. To do so, fix an element K ∈ T . For any edge E ∈ E(K), E by we define vK E vK  =   E nod  LpK ,K ((vK − vK )|E ),  E ∈ E(T ),  nod nod  LE pK ,K ((vK − vK )|E0 , · · · , (vK − vK )|EN ),  E = E0 ∪ · · · ∪ EN , E0 , · · · , EN ∈ E(T ),  E with LE pK ,K (·) in (5.5.3) and LpK ,K (·, · · · , ·) in (5.5.4), respectively. We then define v edge elementwise as X edge E vK (x) = vK (x). E∈E(K)  Thirdly, we construct a function v int ∈ Spe (Te ) simply by setting elementwise edge int nod , K ∈T. vK = vK − vK − vK  int belongs to H 1 (K). Hence, we have v int ∈ S c (T e ). Notice that vK 0 e p In conclusion, any function v ∈ Sp (T ) can be decomposed into three parts: v = v nod + v edge + v int , (5.5.13)  with v nod , v edge and v int in Spe (Te ) as defined above.  5.5.4  Proof of Theorem 5.4.4  In this section, we outline the proof of Theorem 5.4.4. Some of the auxiliary results are postponed to Sections 5.5.5 and 5.5.6. Let v ∈ Sp (T ), we write v = v nod + v edge + v int , according to (5.5.13). We will define the averaging operator Ihp v in three parts: Ihp v = ϑnod + ϑedge + ϑint ,  (5.5.14)  with ϑnod , ϑedge , ϑint ∈ Spec (Te ). Since v int ∈ Spec (Te ), we simply take ϑint = v int . In Sections 5.5.5 and 5.5.6, we will further construct ϑnod and ϑedge such that the following two results hold true.  186  5.5. Proof of Theorem 5.4.4 Lemma 5.5.3 There is a conforming approximation ϑnod ∈ Spec (Te ) that satisfies X Z X −2 ⊥ nod 2 (p⊥ ]] ds, kv nod − ϑnod k2L2 (K) . E ) hE [[v e E∈E(T ) E  e Te K∈  X  e Te K∈  k∇(v nod − ϑnod )k2L2 (K) e .  X Z  E∈E(T ) E  −2 nod 2 p2E h⊥ ]] ds. E hmin,E [[v  Lemma 5.5.4 There is a conforming approximation ϑedge ∈ Spec (Te ) that satisfies X Z X −2 ⊥ 2 nod 2 ]] ) ds, (p⊥ . kv edge − ϑedge k2L2 (K) E ) hE ([[v]] + [[v e E∈E(T ) E  e Te K∈  X  e Te K∈  k∇(v edge − ϑedge )k2L2 (K) e .  X Z  E∈E(T ) E  −2 2 nod 2 ]] ) ds. p2E h⊥ E hmin,E ([[v]] + [[v  By the triangle inequality and Lemmas 5.5.3 and 5.5.4, we then obtain   X X Z ⊥ −2 ⊥ 2 nod 2 kv − Ihp vk2L2 (K) . ds, (p ) h [[v]] + [[v ]] E E e E∈E(T ) E  e Te K∈  X  e Te K∈  k∇(v − Ihp v)k2L2 (K) e .  X Z  E∈E(T ) E    −2 2 nod 2 ds. [[v]] + [[v ]] p2E h⊥ h E min,E  Theorem 5.4.4 now follows if we show that k[[v nod ]]k2L2 (E) . k[[v]]k2L2 (E) ,  E ∈ E(T ).  (5.5.15)  To prove (5.5.15), we denote by ν1 and ν2 the two end points of E ∈ E(T ). By the construction of v nod , the jump over E satisfies [[v nod ]](νi ) = [[v]](νi ),  i = 1, 2.  [[v nod ]]  Since vanishes on all the interior Gauss-Lobatto nodes on E, item (i) in Lemma 5.5.1 and a scaling argument yield nod k[[v nod ]]k2L2 (E) . p−2 ]](ν1 )2 + [[v nod ]](ν2 )2 ) E hE ([[v  −2 2 2 2 = p−2 E hE ([[v]](ν1 ) + [[v]](ν2 ) ) . pE hE k[[v]]kL∞ (E) .  From [26, Theorem 3.92], we further have the inverse estimate 2 k[[v]]k2L∞ (E) . p2E h−1 E k[[v]]kL2 (E) .  This, together with the local bounded variation (5.2.8), show (5.5.15) and finish the proof of Theorem 5.4.4, up to the proofs of Lemmas 5.5.3 and 5.5.4 which we present next. 187  5.5. Proof of Theorem 5.4.4  5.5.5  Proof of Lemma 5.5.3  Let v nod ∈ Spe (Te ) be the nodal interpolant in the decomposition (5.5.13). We now shall construct the conforming approximation ϑnod in Spec (Te ). For simplicity, we shall omit the superscript ”nod” and, in the sequel, write v for v nod and ϑ for ϑnod . For a node ν, we introduce the sets: e ∈ Te : ν ∈ N (K) e }, w(ν) e = {K  wE (ν) = { E ∈ E(T ) : ν ∈ E }.  e ∈ R(K). We proceed by distinguishing the same two Fix K ∈ T and K cases as in Section 5.5.3. e Then any elemental edge Case 1: If R(K) = {K}, we have K = K. e ∈ E(K) e belongs to E(T ) and we have v e | e ∈ P nod (E). e For any GaussE pEe K E e we define the value of ϑ(ν) by Lobatto node ν located on ∂ K, ϑ(ν) =   X −1  e vKe (ν), |w(ν)|  0,  ν ∈ NI (T ),  e w(ν) K∈ e  (5.5.16)  otherwise.  Here, |w(ν)| e denotes the cardinality of the set w(ν). e In the case considered, e by we have |w(ν)| e = 4 for ν ∈ NI (T ). Then we define ϑ on K X ϑ(x) = ϑ(ν) ΦνKe (x). (5.5.17) e ν∈N (K)  From (5.5.7) and (5.5.17), we have X kvKe − ϑkL2 (K) |vKe (ν) − ϑ(ν)| kΦνKe kL2 (K) e . e .  (5.5.18)  Analogously to [6, Page 1125], we have X −1/2 |vKe (ν) − ϑ(ν)| . pE hE k[[v]]kL2 (E) .  (5.5.19)  e ν∈N (K)  E∈wE (ν)  Hence, by combining (5.5.18), (5.5.19), item (iii) in Lemma 5.5.1 (with scaling), the local bounded variations (5.2.5) and (5.2.8), we obtain Z X 2 −2 ⊥ 2 kvKe − ϑkL2 (K) (p⊥ (5.5.20) E ) hE [[v]] ds. e . E∈{wE (ν)}ν∈N (K) e  E  188  5.5. Proof of Theorem 5.4.4 Case 2: If R(K) consists of multiple elements, we define ϑ on each elee ∈ R(K) separately, analogously to the construction of the nodal inment K terpolant in Section 5.5.3. Without loss of generality, we may again consider the case illustrated in Figure 5.5. The analysis for the situations illustrated e3 , E e4 ∈ EA (Te ), in Figures 5.6 and 5.7 is analogous and omitted here. Since E e e the function v is continuous over E3 and E4 . The values of ϑ on the edge pEe ,K p e e3 ) ∪ Z Ee4 ,Ke (E e4 ), and the vertex νe4 are defined by nodes z ∈ Zint 3 (E int ϑ(z) = vKe (z),  pEe  pEe  e3 ) ∪ Z 4 (E e4 ), z ∈ Zint 3 (E int e ,K  e ,K  ϑ(e ν4 ) = vKe (e ν4 ). (5.5.21)  We further define the value of ϑ on the vertex νe2 by (5.5.16). e1 and E e2 , It remains to define the values of ϑ for the nodes on the edges E e as well as for νe1 and νe3 . We only consider νe1 and E1 (the construction for νe3 e2 is completely analogous). If νe1 ∈ N (T ), then νe1 is a hanging node and E p e1 ∈ E(T ). Thus, v e | e ∈ P nod (E e1 ). For any z ∈ Z Ee1 (E e1 )∪{e of T and E ν1 }, pEe  K E1  int  1  the value of ϑ(z) is taken as in (5.5.16). On the other hand, if νe1 ∈ / N (T ), then E1 ∈ E(T ) and vK |E1 ∈ Ppnod (E ). We define ϑ(ν ) again by (5.5.16). 1 1 E1 Recall that ϑ(ν2 ) = ϑ(e ν2 ) has already been defined. Then we set ϑ(z) = ϑ(ν1 )ΦνK1 (z) + ϑ(ν2 )ΦνK2 (z),  e by setting Now we construct ϑ on K ϑ(x) =  X  ϑ(ν)ΦνKe (x)  e ν∈N (K)  +  X  p  e E e1 ) ∪ {e z ∈ Zint 1 (E ν1 }.  pEe −1   X  e e E,p E  ϑ(zi  e e E,p E  )Φi  e e i=1 E∈E( K)  (5.5.22)   (x) .  This completes the construction of ϑ. Clearly, ϑ ∈ Spec (Te ). We shall now derive an estimate analogous to (5.5.20). To do so, we e as follows: bound the difference between vKe and ϑ on K X X  kvKe − ϑk2L2 (K) k vKe (ν) − ϑ(ν) ΦνKe k2L2 (K) kςEe k2L2 (K) e e . e + .  X  e ν∈N (K)  e ν∈N (K)  e e E∈E( K)  X  −2 ⊥ 2 k vKe (ν) − ϑ(ν) ΦνKe k2L2 (K) + (p⊥ e kL2 (E) e eK e ) hE, eK e kςE e E,  = T1 + T2 ,  e e E∈E( K)  (5.5.23)   e e e e  E,p e e PpEe −1  E,p E,p E E E with ςEe (x) = (z v ) − ϑ(z ) Φ (x) . For the second e i i=1 i i K inequality in (5.5.23), we have used estimate (ii) in Lemma 5.5.1 and a 189  5.5. Proof of Theorem 5.4.4 scaling argument noticing that the function ςEe (x) vanishes at all the interior e and on the elemental edges of K e tensor-product Gauss-Lobatto nodes in K e that are different from E. Let us first bound the term T1 in (5.5.23). If the node ν ∈ NA (Te ), then, by (5.5.21),  e vKe (ν) − ϑ(ν) ΦνKe (x) = 0, x ∈ K. (5.5.24) Furthermore, if the node ν belongs to N (T ), we apply estimate (iii) in Lemma 5.5.1 (with scaling) and an argument as in (5.5.19). We obtain X Z  −2 ⊥ 2 (p⊥ (5.5.25) k vKe (ν) − ϑ(ν) ΦνKe k2L2 (K) . E ) hE [[v]] ds. e E∈wE (ν)  E  Finally, if ν ∈ / N (T ) ∪ NA (Te ), then ν is a hanging node of an elemental edge, E ∈ E(K) ∩ E(T ). Denote the two end points of this edge E by ν1 and ν2 . In view of (5.5.10) and (5.5.22), we have |vKe (ν) − ϑ(ν)| ≤ |(vK (ν1 ) − ϑ(ν1 ))| + |(vK (ν2 ) − ϑ(ν2 ))|. Thus, as before, we obtain  k vKe (ν) − ϑ(ν) ΦνKe k2L2 (K) e .  X  E∈wE (ν1 )∪wE (ν2 )  Z  −2 ⊥ 2 (p⊥ E ) hE [[v]] ds. (5.5.26)  E  e as To combine the results in (5.5.24)-(5.5.26), we define the set N ? (K) e follows. We start from N (K) and first remove all the vertices belonging to e with νe ∈ NA (Te ). Then, any vertex νe ∈ N (K) / N (T ) ∪ NA (Te ) is replaced by the vertex ν ∈ N (K) which is on the same elemental edge of K as νe. For e = { ν1 , νe2 , νe3 } if example, in the case shown in Figure 5.5, we have N ? (K) νe1 ∈ / N (T ) and νe3 ∈ N (T ). We also set e = { E ∈ wE (ν) : ν ∈ N ? (K) e }. E ? (K)  In conclusion, the term T1 is bounded by X Z −2 ⊥ 2 T1 . (p⊥ E ) hE [[v]] ds. e E∈E ? (K)  (5.5.27)  E  e ∈ E(T ) or E e ∈ EA (Te ), Next, let us estimate the term T2 in (5.5.23). If E by the constructions of v and ϑ, we clearly have kςEe kL2 (E) e = 0. Otherwise, 190  5.5. Proof of Theorem 5.4.4 e say νe1 , is a newly created node in Te and the one of the two end points of E, other one, νe2 , belongs to N (T ). Thus, we have −2 ⊥ 2 ⊥ −2 ⊥ (p⊥ e kL2 (E) eK e ) hE, eK e kςE e ≤ (pE, eK e ) hE, eK e E,  2 X i=1   k vKe (e νi ) − ϑ(e νi ) Φνeei k2L2 (E) e K  −2 ⊥ 2 + (p⊥ e − ϑkL2 (E) eK e ) hE, eK e kvK e E,  := T21 + T22 . Then there exists an elemental edge E ∈ E(K) such that νe1 is on E. Denote the end points of E by ν1 and ν2 . Similarly to (5.5.25) and (5.5.26), we have Z X −2 ⊥ 2 (p⊥ T21 . E ) hE [[v]] ds. E∈wE (ν1 )∪wE (ν2 )  E  In view of (5.5.22), we proceed as in (5.5.19) and obtain −2 ⊥ T22 . (p⊥ eK ek eK e ) hE, E,  .  X  E∈wE (ν1 )∪wE (ν2 )  2  X    vK (νi ) − ϑ(νi ) ΦνKi k2L2 (E)  i=1 Z −2 ⊥ 2 (p⊥ E ) hE [[v]] ds. E  Note that, in the above estimates for T21 and T22 , we have also used the local bounded variations (5.2.5) and (5.2.8). Combining the above results shows that X Z −2 ⊥ 2 T2 . (p⊥ (5.5.28) E ) hE [[v]] ds. e E∈E ? (K)  E  The bounds for T1 and T2 in (5.5.27) and (5.5.28) yield X Z 2 −2 ⊥ 2 (p⊥ kvKe − ϑkL2 (K) E ) hE [[v]] ds. e . e E∈E ? (K)  (5.5.29)  E  Note that the inequality (5.5.29) holds true for the situations illustrated in Figure 5.6 and 5.7 following the above approach. This finishes the discussion of Case 2. Thus, by the key estimates in (5.5.20) and (5.5.29), we have X Z 2 −2 ⊥ 2 e ∈ Te . (p⊥ K (5.5.30) kvKe − ϑkL2 (K) E ) hE [[v]] ds, e . E e E∈E ? (K)  191  5.5. Proof of Theorem 5.4.4 This proves the first inequality in Lemma 5.5.3. Moreover, for any function e ∈ Te , we have the following inverse inequality from [26, v ∈ Spe (Te ), K Theorem 4.76] and a simple scaling argument: 1  1 2 1 −2 k∇vkL2 (K) + (h2K )−2 ) 2 kvkL2 (K) e . pK e . e pK e ((hK )  Together with (5.5.30), we then obtain X Z −2 2 k∇(vKe − ϑ)k2L2 (K) p2E h⊥ . E hmin,E [[v]] ds, e e E∈E ? (K)  E  (5.5.31)  e ∈ Te , (5.5.32) K  which shows the second assertion in Lemma 5.5.3.  5.5.6  Proof of Lemma 5.5.4  Fix an element K ∈ T and let E be an elemental edge in E(K). We define E as follows: If E ∈ E (T ), we set the function WK B  E nod WK = LE pK ,K (vK − vK )|E ,  0 e with the extension operator LE pK ,K (·) in (5.5.3). If E ∈ EI (T ), let K in T be 0 the element such that E is also an elemental edge of K , that is, E ∈ E(K 0 ). Denote by K 0 the element which has the lower edge polynomial degree of the elements K and K 0 , i.e., K 0 = K if pE,K ≤ pE,K 0 and K = K 0 otherwise. E by We define WK  E nod WK = LE pK ,K (vK 0 − vK 0 )|E .  with LE pK ,K (·) in (5.5.3). In the case where E contains N hanging nodes (N ≤ M ), E is partitioned into E = E0 ∪ · · · ∪ EN with E0 , · · · , EN ∈ EI (T ). There exist N + 1 elements K0 , · · · , KN ∈ T such that Ei ∈ E(Ki ), i = 0, · · · , N . Denote by K i the element that has the lower edge polynomial E by degree of K and Ki . We now define WK  E nod nod WK = LE pK ,K (vK 0 − vK 0 )|E0 , · · · , (vK N − vK N )|EN ,  with LE pK ,K (·, . . . , ·) in (5.5.4). P E, Then we define ϑedge elementwise by setting ϑedge |K = E∈E(K) WK E defined above. Clearly, the function ϑedge belongs to S c (T e ). Next, with WK e p we prove the approximation properties of Lemma 5.5.4. By Lemma 5.5.2  192  5.6. Numerical experiments (with a scaling argument), we have X X X kv edge −ϑedge k2L2 (K) e = e Te K∈  .  X  X  K∈T K∈R(K) e  K∈T E∈E(K)  .  X  X  kv edge − ϑedge k2L2 (K) e   nod E 2 kLE pK ,K (vK − vK )|E − WK kL2 (K)  −2 ⊥ nod E 2 (p⊥ E,K ) hE,K k(vK − vK )|E − WK |E kL2 (E)  K∈T E∈E(K)  .  X  X Z  −2 ⊥ 2 nod 2 (p⊥ ]] ) ds. E ) hE ([[v]] + [[v  K∈T E∈E(K) E  This completes the proof of the first assertion of Lemma 5.5.4; the second one follows again from the first one by using the inverse inequality in (5.5.31).  5.6  Numerical experiments  In this section, we reimplement the three numerical examples tested in Chapter 3, but use now η in (5.3.4) as an error indicator in an anisotropic refinement strategy. Our implementation of the DG method (5.2.10) is again based on the Deal.II finite element library [4, 5]. The non-symmetric sparse linear systems of equations are solved by using the UMFPACK package [8, 9]. Due to restrictions in the Deal.II library, our implementation cannot handle anisotropic polynomial degrees and we only show results for isotropic elemental degrees. The hp-adaptive algorithms employed here are similar to the ones in Section 3.6. In all the examples, the hp-adaptive meshes are constructed by first marking the elements K for refinement and derefinement according to the size of the local error indicator ηK in (5.3.3), with refinement and derefinement fractions set to 25% and 10%, respectively. Once an element has been flagged for refinement, we employ the smoothness estimation algorithm developed in [15], see also Section 3.6 for details, to assess the local regularity of the analytical solution. If, by our criterion, the solution on an element K is locally smooth enough, we increase the polynomial degree pK on K isotropically by one; otherwise, we refine the element K anisotropically by using an anisotropic refinement strategy available in the Deal.II library. To be specific, we evaluate the mean value of [[uhp ]], ξ1 and ξ2 , over the edges  193  5.6. Numerical experiments E1 , · · · , E4 ∈ E(K) of an element K (numbered as in Figure 5.3) by R R P P i=1,3 Ei |[[uhp ]]| ds i=2,4 Ei |[[uhp ]]| ds P P ξ1 = , ξ2 = . i=1,3 hEi i=2,4 hEi  If the average jump in one direction is larger than the average of the jumps in the other direction by a certain factor c, i.e., if ξi > cξ3−i , i = 1, 2, the element K is refined only along that particular direction i by connecting the middle points of the edges Ei and Ei+2 ; otherwise the element K is refined isotropically into four elements by bisecting the elemental edges of K. In the following tests, we take c = 2. In our implementation, we only allow one hanging node on each elemental edge. In all our tests, we set the stabilization parameter to γ = 10. The approximate right-hand side fhp is taken as the L2 -projection of f onto Sp (T ). Moreover, since the flow field a is either constant or linear, we simply choose ahp = a in the residual ηRK . We numerically reproduce solutions that are analytic over the computational domain, although they have steep gradients along boundary and internal layers. In all our examples, we observe p-refinement to be dominating once the local mesh size is sufficiently resolved. Based on the a-priori error analysis for p-version methods in [26], we 1 thus plot all computed quantities against N 2 in a semi-logarithmic scale, with N = dim(Sp (T )).  5.6.1  Example 1  In this example, we take Ω = (0, 1)2 , choose a = (1, 1)> and select the right-hand side f so that the analytical solution to the convection-diffusion problem (5.2.1) is given by u(x1 , x2 ) =   e x1ε−1 − 1 1  e− ε   e x2ε−1 − 1  + x1 − 1 + x2 − 1 . 1 −1 e− ε − 1  The solution is smooth, but has boundary layers at x1 = 1 and x2 = 1; their widths are both of order O(ε). This problem is well-suited to test whether the indicator η is able to pick up the steep gradients near these boundaries using anisotropic refinement. We begin this test with a uniform mesh of 8×8 elements and the uniform polynomial degree pK = 1. In Figure 5.8(a), we show the performance of our hp-adaptive algorithm with anisotropic mesh refinement for ε = 10−3 . In the curves labeled “Error Indicator” and “Energy Error”, we see that the indicator η always overestimates the true energy error k u − uhp kE,T , 194  5.6. Numerical experiments in agreement with Theorem 5.3.2. Additionally, the convergence lines using hp-refinement are (roughly) straight on a linear-log scale, which indicates that exponential convergence is attained for this problem. In the 1 curve “L2 Error”, we calculate the error ε− 2 ku − uh kL2 (Ω) , which is an upper bound for |a(u − uh )|? . We see that this error is at least of the same order as the energy error. The same behavior is observed for the error P 1/2 ⊥ p2 h−2 −1 h⊥ p⊥ p−2 2 (εh + ε shown in ”Jump )k[[u ]]k 2 h E∈E(T ) E E min,E E E min,K L (E) Error”. Finally, in the curve labeled ”Theta”, we calculate an approximation to the data error Θ in (5.3.4) by using a Gauss-Legendre quadrature rule of higher order on each element. This example is tested in Chapter 3 with the same settings. In Figure 5.8(b), we compare the true energy error and the error estimator for both an isotropic and anisotropic refinement algorithm. Here, the anisotropic refinement result is identical as shown in Figure 5.8(a), while the isotropic refinement result is exactly the same as shown in Figure 3.7. The superiority of using anisotropic refinement is clearly visible although we observe exponential convergence for all quantities. In Figure 5.8(c), we plot the ratio of the indicator and the true energy error for the anisotropic refinement 1 method. It stays around 4, uniformly in N 2 . (a)  1  (b)  1  (c)  10  10  Error Indicator Energy Error L2 Error Jump Error Theta  0  10  −1  10  10 Indicator (anisotropic) Error (anisotropic) Indicator (isotropic) Error (isotropic)  0  10  ratio (anisotropic) 9 8  −1  10  7  −2  10  6  −2  10  −3  10  5 −4  10  −3  10  4  −5  10  3  −4  10 −6  10  2 −5  10  −7  10  1  −8  10  −6  40  60  80 N1/2  100  120  10  0  100  200 N1/2  300  400  0  40  60  80 N1/2  100  120  Figure 5.8: Example 1: Convergence behavior for ε = 10−3 . In Figure 5.9, we show the same plots for ε = 2 · 10−4 . Qualitatively, we observe the same behavior as before. Together with Figure 5.8(c), we see that the ratio of the indicator and the true energy error oscillates around 4 for both ε = 10−3 and ε = 2 · 10−4 , in agreement with Theorems 5.3.2 and 5.3.4. That shows the alignment measure M(v, T ) is of moderate size for this problem and our estimate in Theorems 5.3.2 and 5.3.4 is numerically observed to be robust. 195  5.6. Numerical experiments (a)  1  (b)  1  (c)  10  10  10 Indicator (anisotropic) Error (anisotropic) Indicator (isotropic) Error (isotropic)  0  10  0  10 −1  10  −2  8 7  −1  10  ratio (anisotropic) 9  10  6  −3  10  −2  10  5  −4  10  4 −5  −3  10  10  3 −6  10  −7  10  −8  10  50  Error Indicator Energy Error 2 L Error Jump Error Theta 100  2  −4  10  1 −5  150 N1/2  200  10  100  200  300  400 N1/2  500  600  700  0 50  100  150  200  N1/2  Figure 5.9: Example 1: Convergence behavior for ε = 2 · 10−4 . In Figure 5.10 and Figure 5.11, we show the meshes and polynomial degree distribution after 9 and 15 hp-adaptive refinement steps for ε = 10−3 and ε = 2 · 10−4 , respectively. We see strong anisotropic mesh refinement into the layers. Once they are sufficiently resolved, p-refinement is starting to dominate in the layers.  5.6.2  Example 2  Next, we consider an example with an internal layer and with variable coefficients. In the domain Ω = (−1, 1)2 , we take a(x1 , x2 ) = (−x1 , x2 )> . We choose f and the inhomogeneous Dirichlet boundary conditions such that the solution to (5.2.1) is given by Z x x1 2 2 2 u(x1 , x2 ) = erf( √ )(1 − x2 ), with erf(x) = √ e−t dt. π 0 2ε For small values of ε, the solution u has an internal layer around x1 = 0, √ whose width is of order O( ε). We use standard DG terms to incorporate the inhomogeneous boundary conditions, and modify the error indicator η correspondingly. For details we refer to [13]. We begin this test with a uniform mesh of 8 × 8 elements and the uniform polynomial degree pK = 2. In Figure 5.12 and Figure 5.13, the numerical results for this example are shown for the values ε = 10−3 and ε = 5 · 10−6 , respectively. We plot the same quantities as in Example 1. For ε = 10−3 , we observe exponential convergence rates for both the energy error and the indicator. The curves associated with the convection and approximation errors are of the same order as the energy error. If we now decrease the value of ε to ε = 5·10−6 , the jump related to the convection term depicted in ”Jump Error” is dominating 196  5.6. Numerical experiments  (a) 9 hp-adaptive refinements  (b) Enlarged mesh refinement around the upper-right corner after 9 refinement steps  (c) 15 hp-adaptive refinements  (d) Enlarged mesh refinement around the upper-right corner after 15 refinement steps  Figure 5.10: Example 1: Adaptively generated meshes after 9 and 15 refinement steps for ε = 10−3 .  197  5.6. Numerical experiments  (a) 9 hp-adaptive refinements  (b) Enlarged mesh refinement around the upper-right corner after 9 refinement steps  (c) 15 hp-adaptive refinements  (d) Enlarged mesh refinement around the upper-right corner after 15 refinement steps  Figure 5.11: Example 1: Adaptively generated meshes after 9 and 15 refinement steps for ε = 2 · 10−4 .  198  5.6. Numerical experiments the estimator η. (Recall that the error plotted in the curve ”L2 Error” is only an upper bound for the error |a(u − uh )|? and can overestimate η.) Nevertheless, exponential convergence rates are observed for all quantities. This illustrates the fact that the estimator η is not robust in estimating the energy error alone; the inclusion of the dual norm in the error measure is essential. This is further reflected in the plots at the right-hand sides of Figures 5.12 and 5.13 where we show the ratio of the indicator and the energy error. While for ε = 10−3 the values are between 5 and 9, they clearly increase for ε = 5 · 10−6 . Initially, they also strongly oscillate. Again, this is due to the fact that we do not include the dual norm in the error measure. Eventually, the ratio stays bounded around 15. This also indicates that the alignment measure is also of moderate size for this example. (a)  −1  (b)  −1  10  (c)  10 Error Indicator Energy Error L2 Error Jump Error Theta  −2  10  −3  10 Indicator (anisotropic) Error (anisotropic) Indicator (isotropic) Error (isotropic)  −2  10  8 7  −3  10  9  10  6 −4  −4  10  10  5 4  −5  −5  10  10  3  −6  2  −6  10  10  1 −7  10  ratio (anisotropic)  −7  30  40  50 N1/2  60  10  70  40  60  80 N1/2  100  120  140  0 30  40  50 N1/2  Figure 5.12: Example 2: Convergence behavior for ε =  (a)  0  (b)  0  10  −1  10  70  (c)  10 Error Indicator Energy Error L2 Error Jump Error Theta  60  10−3 .  55 ratio (anisotropic) 50  −1  10  45  −2  40  10  −2  10  35 −3  10  30  −3  10  25  −4  10  20  −4  10 −5  10  15 Indicator (anisotropic) Error (anisotropic) Indicator (isotropic) Error (isotropic)  −5  10  −6  10  −7  10  70  10 5  −6  75  80  85  90 N1/2  95  100  105  10  50  100  150  200 N1/2  250  300  70  75  80  85  90  95  100  105  N1/2  Figure 5.13: Example 2: Convergence behavior for ε = 5 · 10−6 . Figure 5.14 and Figure 5.15 shows the hp-adaptive meshes and poly199  5.6. Numerical experiments nomial degree distributions after 9 refinement and 15 refinement steps, for ε = 10−3 and ε = 5 · 10−6 , respectively. We observe that the mesh refinement is mainly along the x1 -direction. We also see that the mesh refinement √ stops once the local mesh size along the x1 -direction is of order O( ε) and p-refinement starts to take over in the vicinity of the layer, similarly to the phenomenon observed in Section 3.6 for isotropic refinement.  (a) 9 hp-adaptive refinements  (b) 15 hp-adaptive refinements  Figure 5.14: Example 2: Adaptively generated meshes after 9 and 15 refinement steps for ε = 10−3 .  5.6.3  Example 3  Finally, we consider a problem with convection not aligned with the mesh. We take Ω = (−1, 1)2 , a = (− sin π6 , cos π6 ), f = 0 and consider the boundary conditions u = 0 on x1 = −1 and x2 = 1, as well as u = tanh(  1 − x2 ) ε  on x1 = 1,  u=   x1 1 tanh( ) + 1 2 ε  on x2 = −1.  The boundary data is almost discontinuous near the point (0, −1) √ and √ causes u to have an internal layer of width O( ε) along the line x2 + 3x1 = −1, with values u = 0 to the left and u = 1 to the right, as well as a boundary layer along the outflow boundary. There is no exact solution available to this problem. We test this problem with ε = 10−3 and start the algorithm for pK = 1 on a uniform mesh of 8 × 8 elements. In Figure 5.16, we plot the value of the error indicator η. Asymptotically, we observe exponential convergence for the indicator.  200  5.6. Numerical experiments  (a) 9 hp-adaptive refinements  (b) Enlarged mesh refinement around the internal layer after 9 refinement steps  (c) 15 hp-adaptive refinements  (d) Enlarged mesh refinement around the internal layer after 15 refinement steps  Figure 5.15: Example 2: Adaptively generated meshes after 9 and 15 refinement steps for ε = 5 · 10−6 .  201  5.7. Conclusions  1  Error Indicator(Anisotropic)  10  0  10  −1  10  −2  10  −3  10  −4  10  40  60  80  100  120  140 N1/2  160  180  200  220  Figure 5.16: Example 3: Convergence behavior for ε = 10−3 .  Since the internal layer is neither along the x1 -direction nor the x2 direction, the mesh refinement around the internal layer ends up being isotropic, while anisotropic element is employed by our algorithm in the boundary layer. This is clearly visible in Figure 5.17, where we show the adaptive meshes after 9 refinement and 15 refinement steps. Since the solution is almost constant away from the layers, p-refinement is again concentrated along the layers.  5.7  Conclusions  We have derived an a-posteriori error estimate for hp-adaptive discontinuous Galerkin methods for convection-diffusion equations on anisotropically refined rectangular meshes. The constant in the upper bound depends on a so-called alignment measure. We have applied our estimate as an error indicator for error estimation in an hp-adaptive refinement algorithm. Our numerical results indicate that the indicator is capable of anisotropically refining the mesh at interior and boundary layers. We observe that, as soon as a reasonable resolution of the layer is achieved, the alignment measure is of moderate size and the ratio of the error estimator and the true energy error is independent of the diffusion parameter ε and the mesh size. This means that, after a few refinement steps, our estimator is robust in practice. Moreover, once the local mesh size is anisotropically of the same order as the width of the boundary or interior layer, both the energy error and the error indicator are observed to converge exponentially. Compared to the isotropic algorithm presented in Chapter 3, anisotropic adaptive algorithms use far less degrees of freedom to achieve an approximation with the same 202  5.7. Conclusions order of accuracy.  (a) 9 hp-adaptive refinements  (b) Enlarged mesh refinement around upperleft corner after 9 refinement steps  (c) 15 hp-adaptive refinements  (d) Enlarged mesh refinement around upperleft corner after 15 refinement steps  Figure 5.17: Example 3: Adaptively generated meshes after 9 and 15 refinement steps for ε = 10−3 .  203  5.8. Bibliography  5.8  Bibliography  [1] T. Apel. Anisotropic Finite Elements: Local Estimates and Application. Advances in Numerical Mathematcis. Teubner, Stuttgart, 1999. [2] T. Apel, S. Grosman, P. Jimack, and A. Meyer. A new methodology for anisotropic mesh refinement based upon error gradients. Appl. Numer. Math., 50:329–341, 2004. [3] T. Apel and G. Lube. Anisotropic mesh refinement in stabilized Galerkin methods. Numer. Math., 74:261–282, 1996. [4] W. Bangerth, R. Hartmann, and G. Kanschat. Differential Equations Analysis Library, Technical http://www.dealii.org.  deal.II Reference.  [5] W. Bangerth, R. Hartmann, and G. Kanschat. deal.II — a general purpose object oriented finite element library. ACM Trans. Math. Software, 33:24:1–24:27, 2007. [6] E. Burman and A. Ern. Continuous interior penalty hp-finite element methods for advection and advection-diffusion equations. Math. Comp., 76:1119–1140, 2007. [7] E. Creusé and S. Nicaise. Anisotropic a-posteriori error estimation for the mixed discontinuous Galerkin approximation of the Stokes problem. Numer. Meth. PDEs., 22:449–483, 2006. [8] T. A. Davis. Algorithm 832: UMFPACK, an unsymmetric-pattern multifrontal method. ACM Transactions on Mathematical Software, 30:196–199, 2004. [9] T. A. Davis. A column pre-ordering strategy for the unsymmetricpattern multifrontal method. ACM Transactions on Mathematical Software, 30:165–195, 2004. [10] L. Formaggia and S. Perotto. Anisotropic error estimates for elliptic problems. Numer. Math., 94:67–92, 2003. [11] E.H. Georgoulis, E. Hall, and P. Houston. Discontinuous Galerkin methods on hp-anisotropic meshes I: A priori error analysis. Int. J. Comput. Sci. Math., 1:221–244, 2007.  204  BIBLIOGRAPHY [12] E.H. Georgoulis, E. Hall, and P. Houston. Discontinuous Galerkin methods on hp-anisotropic meshes II: A-posteriori error analysis and adaptivity. Appl. Numer. Math., 59:2179–2194, 2009. [13] P. Houston, D. Schötzau, and T. P. Wihler. Energy norm a-posteriori error estimation of hp-adaptive discontinuous Galerkin methods for elliptic problems. Math. Models Methods Appl. Sci., 17:33–62, 2007. [14] P. Houston, C. Schwab, and E. Süli. Stabilized hp-finite element methods for first-order hyperbolic problems. SIAM J. Numer. Anal., 37:1618–1643, 2000. [15] P. Houston and E. Süli. A note on the design of hp–adaptive finite element methods for elliptic partial differential equations. Comput. Methods Appl. Mech. Engrg., 194:229–243, 2005. [16] O. A. Karakashian and F. Pascal. A-posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems. SIAM J. Numer. Anal., 41:2374–2399, 2003. [17] G. Kunert. An a-posteriori residual error estimator for the finite element method on anisotropic tetrahedral meshes. Numer. Math., 86:471–490, 2000. [18] G. Kunert. A local problem error estimator for anisotropic tetrahedral finite element meshes. SIAM J. Numer. Anal., 39:668–689, 2001. [19] J.M. Melenk and B.I. Wohlmuth. On residual-based a-posteriori error estimation in hp-FEM. Adv. Comp. Math., 15:311–331, 2001. [20] J.J.H. Miller, E. O’Riordan, and G.I. Shishkin. Fitted Numerical Methods for Singular Perturbation Problems. World Scientific, 1996. [21] M. Picasso. An anisotropic error indicator based on Zienkiewicz-Zhu error estimator: Application to elliptic and parabolic problems. SIAM J. Sci. Comput., 24:1328–1355, 2003. [22] H.-G. Roos, M. Stynes, and L. Tobiska. Robust Numerical Methods for Singularly Perturbed Differential Equations, volume 24 of Springer Series in Computational Mathematics. Springer–Verlag, 2008. [23] D. Schötzau, C. Schwab, and T. Wihler. hp-DGFEM for second-order elliptic problems in polyhedra I: Stability and quasioptimality on geometric meshes. submitted, 2010. 205  BIBLIOGRAPHY [24] D. Schötzau, C. Schwab, and T. Wihler. hp-DGFEM for second-order elliptic problems in polyhedra II: Exponential convergence. submitted, 2010. [25] D. Schötzau and L. Zhu. A robust a-posteriori error estimator for discontinuous Galerkin methods for convection-diffusion equations. Appl. Numer. Math., 59:2236–2255, 2009. [26] C. Schwab. p- and hp-Finite Element Methods: Theory and Application to Solid and Fluid Mechanics. Oxford University Press, Oxford, 1998. [27] C. Schwab. hp-FEM for fluid flow simulation. In T. Barth and H. Deconink, editors, High-Order Methods for Computational Physics, volume 9 of Lect. Notes Comput. Sci. Engrg., pages 325–438. Springer– Verlag, 1999. [28] C. Schwab and M. Suri. The p and hp versions of the finite element method for problems with boundary layers. Math. Comp., 65:1403– 1429, 1996. [29] L. Zhu, S. Giani, P. Houston, and D. Schötzau. Energy norm a-posteriori error estimation for hp-adaptive discontinuous Galerkin methods for elliptic problems in three dimensions. Math. Models Methods Appl. Sci., 2010. accepted for publication. [30] L. Zhu and D. Schötzau. A robust a-posteriori error estimate for hpadaptive DG methods for convection-diffusion equations. IMA J. Numer. Anal., 2009. accepted for publication.  206  Chapter 6  Conclusions and future work 6.1  Conclusions  In this thesis, we establish and numerically test robust a-posteriori error estimators for h- and hp-adaptive interior penalty discontinuous Galerkin (DG) methods for stationary convection-diffusion equations in two and three dimensions. In Chapter 2, a robust a-posteriori error estimator is first derived for hadaptive DG methods. The estimator yields global upper and lower bounds of the error measured in terms of the energy norm and a semi-norm associated with the convective term in the equation. The constants in the upper and lower bounds are independent of the magnitude of the Péclet number of the problem; in this sense the estimator is fully robust for convectiondominated convection-diffusion problems. At the end of this chapter, we present a series of numerical examples where we use the error estimator as an error indicator in an adaptive refinement strategy. Our numerical results indicate that the estimate is effective in locating and resolving interior and boundary layers. Once the local mesh Péclet number is of order one, the energy error converges with optimal order. For h-adaptive DG methods, our error indicator is reliable, efficient and, most importantly, robust for the error in the energy norm. In Chapters 3 and 4, this analysis is then extended to hp-adaptive DG methods. Differently from the h-adaptive methods which employ a fixed low polynomial degree and yield algebraic rates of convergence, hp-adaptive algorithms take the p-refinement into account and achieve exponential rates of convergence for piecewise analytic data. In Chapter 3, we derive a robust aposteriori error estimate for two-dimensional convection-diffusion equations while the error is again measured in terms of the energy norm and a dual norm associated with the convection. The constants in the upper and lower bounds are independent of the local mesh sizes, the Péclet number of the problem, although the one in the lower bound weakly depends on the polynomial degrees. In Chapter 4, we propose an a-posteriori error estimate for three-dimensional elliptic equations which gives rise to global upper and lo207  6.1. Conclusions cal lower bounds of the error measured in the natural DG energy norm. The constants in these error bounds are independent of the local mesh sizes and the one in the lower bound weakly depends on the local polynomial degrees as well. The theoretical properties of our estimators (reliability, efficiency, robustness and exponential rates of convergence), are further numerically shown in our numerical experiments. The above error estimates are all developed on 1-irregular isotropically refined meshes. However, adaptive algorithms which refine the mesh isotropically generally lead to an excessive number of degrees of freedom to derive an accurate approximation of the analytical solution which contains anisotropic singularities. In addition, anisotropically and geometrically refined meshes are mandatory to achieve exponential rates of convergence for diffusion problems in generic polyhedral domains. Therefore, it is highly desirable to develop hp-adaptive DG methods on anisotropic elements. This is carried out in Chapter 5 for two-dimensional problems. The approach in this chapter is similar to the one in Chapter 3. The estimate yields global upper and lower bounds of the errors measured in terms of the energy norm and the semi-norm associated with the convection. The constant in the upper bound depends on an alignment of the anisotropy of the mesh. We apply our estimator as an error indicator in an hp-adaptive refinement algorithm. Our numerical results indicate that the indicator is capable of anisotropically refining the mesh at interior and boundary layers aligned with the mesh. We observe that, as soon as a reasonable resolution of the layer is achieved, the alignment measure is of moderate size and the ratio of the error estimate and the true energy error is independent of the diffusion parameter ε and the mesh size. This means that, after a few refinement steps, our estimator is robust in practice. Moreover, once the local mesh size is anisotropically of the same order as the width of the boundary or interior layer, both the energy error and the error indicator are observed to converge exponentially. Compared to the isotropic algorithm, anisotropic adaptive algorithms use less degrees of freedom to achieve an approximation with the same order of accuracy. When setting the convection coefficient in model problems to zero, we can extend our approach from convection-diffusion equations to singularly perturbed reaction-diffusion problems. A robust a-posteriori error estimate for h- and hp-adaptive DG methods for singularly perturbed reaction-diffusion equations would be easily obtained following the technique explicitly demonstrated in Section 2.3.4 for the h-version of the DG method. In the analysis in this thesis, the error is always decomposed into a conforming part and a remainder by employing an averaging operator for dis208  6.2. Future work continuous functions. The conforming contribution can then be dealt with using standard techniques, while the remainder can be controlled using the stabilizing jump terms. The construction and analysis of such an averaging operator is a key ingredient in our work. With this tool, we can make use of standard techniques on the H 1 -space for discontinuous functions. In Chapter 3 and 4, we construct an averaging operator on two-dimensional and three-dimensional isotropically refined meshes respectively. New L2 -norm and H 1 -norm approximation properties for this hp-averaging operator are shown for 1-irregular meshes consisting of parallelograms and variable polynomial degrees. In Chapter 5, these averaging operators and their approximation properties are then extended to the anisotropically refined meshes with anisotropic polynomial degree distributions; in this approach, multiple hanging nodes on an elemental edge are allowed.  6.2  Future work  A natural extension of this work is the hp-version analysis of the averaging operator on three-dimensional anisotropically refined meshes. Such meshes are mandatory not only for convection-diffusion problems in three dimensions, but also for purely elliptic problems due to the presence of anisotropic edge singularities that might arise along boundary edges of the computational domain. While it is known that hp-methods on geometrically and anisotropically refined meshes resolve edge and corner singularities at exponential rates of convergence [2, 3, 4], the design of a-posteriori error estimators is still an open question. We believe that the techniques in this thesis can be extended to the three-dimensional anisotropic meshes needed in this situation. In [1], a robust a-posteriori error estimator is derived for adaptive DG methods, where the reliability constant in the upper bound is guaranteed to be one. In this approach, the DG solution is post-processed in order to obtain a divergence-conforming flux and require the use of an averaging operator. However, the results in [1] are only valid for the h-version of the DG method. Therefore, our hp-version techniques will probably be instrumental in extending the approach of efficiency in [1] to the context of the hp-version DG method, especially for the analysis of efficiency. In this thesis, we studied the scalar convection-diffusion problem (1.3.1). This has to be seen as a first model on the way to prove robust a-posteriori estimates for more complicated flow problems. In particular, we are interested in extending our results to mixed hp-version DG methods for incompressible 209  6.2. Future work flow with small Reynolds number. A natural next step would be to study the Oseen equations.  210  6.3. Bibliography  6.3  Bibliography  [1] A. Ern, A.F. Stephansen, and M. Vohralı́k. Guaranteed and robust discontinuous Galerkin a-posteriori error estimates for convection-diffusionreaction problems. J. Comput. Appl. Math., 234:114–130, 2010. [2] D. Schötzau, C. Schwab, and T. Wihler. hp-DGFEM for second-order elliptic problems in polyhedra I: Stability and quasioptimality on geometric meshes. submitted, 2010. [3] D. Schötzau, C. Schwab, and T. Wihler. hp-DGFEM for second-order elliptic problems in polyhedra II: Exponential convergence. submitted, 2010. [4] C. Schwab. hp-FEM for fluid flow simulation. In T. Barth and H. Deconink, editors, High-Order Methods for Computational Physics, volume 9 of Lect. Notes Comput. Sci. Engrg., pages 325–438. Springer– Verlag, 1999.  211  

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