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UBC Theses and Dissertations

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 imple- mentation 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 prob- lems. In this work, we choose DG methods to realize adaptive algorithms. These methods yield stable and robust discretization schemes for convection- dominated 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 effi- cient 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++ implementa- tions 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 es- ii Abstract timator are observed to converge exponentially fast in the number of degrees of freedom. iii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii Statement of Co-Authorship . . . . . . . . . . . . . . . . . . . . . xiii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Motivation and objectives . . . . . . . . . . . . . . . . . . . . 1 1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.1 Discontinuous Galerkin methods . . . . . . . . . . . . 5 1.2.2 A-posteriori error estimates for convection-diffusion problems . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.1 Model problem . . . . . . . . . . . . . . . . . . . . . . 7 1.3.2 Setting for error estimation . . . . . . . . . . . . . . . 7 1.3.3 Implementation . . . . . . . . . . . . . . . . . . . . . 8 1.3.4 Robust a-posteriori error estimation . . . . . . . . . . 9 1.3.5 Averaging operator . . . . . . . . . . . . . . . . . . . 10 1.3.6 Diffusion problems in three dimensions . . . . . . . . 13 1.3.7 Anisotropically refined meshes . . . . . . . . . . . . . 14 1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2 An h-version a-posteriori error estimator . . . . . . . . . . 24 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.2 Interior penalty discretization . . . . . . . . . . . . . . . . . 26 iv Table of Contents 2.2.1 Model problem . . . . . . . . . . . . . . . . . . . . . . 26 2.2.2 Discretization . . . . . . . . . . . . . . . . . . . . . . 27 2.3 Robust a-posteriori error estimation . . . . . . . . . . . . . . 28 2.3.1 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3.2 A robust a-posteriori error estimator . . . . . . . . . 30 2.3.3 Reliability and efficiency . . . . . . . . . . . . . . . . 31 2.3.4 A robust estimator for reaction-diffusion problems . . 32 2.4 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.4.1 Auxiliary forms and their properties . . . . . . . . . . 34 2.4.2 Approximation operators . . . . . . . . . . . . . . . . 37 2.4.3 Proof of Theorem 2.3.2 . . . . . . . . . . . . . . . . . 37 2.4.4 Proof of Theorem 2.3.3 . . . . . . . . . . . . . . . . . 41 2.5 Numerical experiments . . . . . . . . . . . . . . . . . . . . . 46 2.5.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . 46 2.5.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . 47 2.5.3 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . 49 2.5.4 Example 4 . . . . . . . . . . . . . . . . . . . . . . . . 50 2.5.5 Example 5 . . . . . . . . . . . . . . . . . . . . . . . . 52 2.5.6 Example 6 . . . . . . . . . . . . . . . . . . . . . . . . 53 2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.7 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3 An hp-version a-posteriori error estimator . . . . . . . . . 62 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.2 Interior penalty discretization . . . . . . . . . . . . . . . . . 64 3.2.1 Model problem . . . . . . . . . . . . . . . . . . . . . . 64 3.2.2 Discretization . . . . . . . . . . . . . . . . . . . . . . 65 3.3 Robust a-posteriori error estimates . . . . . . . . . . . . . . . 66 3.3.1 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.3.2 A robust a-posteriori error estimate . . . . . . . . . . 67 3.4 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.4.1 Stability and auxiliary forms . . . . . . . . . . . . . . 69 3.4.2 Auxiliary meshes . . . . . . . . . . . . . . . . . . . . 70 3.4.3 Averaging operator . . . . . . . . . . . . . . . . . . . 72 3.4.4 Proof of Theorem 3.3.1 . . . . . . . . . . . . . . . . . 72 3.4.5 Proof of Theorem 3.3.3 . . . . . . . . . . . . . . . . . 77 3.5 Proof of Theorem 3.4.4 . . . . . . . . . . . . . . . . . . . . . 81 3.5.1 Polynomial basis functions . . . . . . . . . . . . . . . 81 3.5.2 Extension operators . . . . . . . . . . . . . . . . . . . 83 3.5.3 Decomposition of functions in Sp(T ) . . . . . . . . . 84 v Table of Contents 3.5.4 Proof of Theorem 3.4.4 . . . . . . . . . . . . . . . . . 86 3.5.5 Proof of Lemma 3.5.3 . . . . . . . . . . . . . . . . . . 88 3.5.6 Proof of Lemma 3.5.4 . . . . . . . . . . . . . . . . . . 92 3.6 Numerical experiments . . . . . . . . . . . . . . . . . . . . . 93 3.6.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . 94 3.6.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . 96 3.6.3 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . 99 3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.8 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4 Diffusion problems in three dimensions . . . . . . . . . . . 106 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.2 Discontinuous Galerkin discretization of a diffusion problem 108 4.2.1 Meshes and traces . . . . . . . . . . . . . . . . . . . . 108 4.2.2 Finite element spaces . . . . . . . . . . . . . . . . . . 109 4.2.3 Interior penalty discretization . . . . . . . . . . . . . 110 4.3 Energy norm a-posteriori error estimates . . . . . . . . . . . 111 4.3.1 Energy norm and residuals . . . . . . . . . . . . . . . 111 4.3.2 Reliability . . . . . . . . . . . . . . . . . . . . . . . . 112 4.3.3 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.4 Proof of Theorem 4.3.1 . . . . . . . . . . . . . . . . . . . . . 114 4.4.1 Edges and nodes . . . . . . . . . . . . . . . . . . . . . 114 4.4.2 Auxiliary meshes . . . . . . . . . . . . . . . . . . . . 114 4.4.3 Averaging operator . . . . . . . . . . . . . . . . . . . 116 4.4.4 Proof of Theorem 4.3.1 . . . . . . . . . . . . . . . . . 116 4.5 Proof of Theorem 4.3.3 . . . . . . . . . . . . . . . . . . . . . 121 4.5.1 Inverse estimates . . . . . . . . . . . . . . . . . . . . 121 4.5.2 Polynomial extension over faces . . . . . . . . . . . . 122 4.5.3 Proof of Theorem 4.3.3 . . . . . . . . . . . . . . . . . 124 4.6 Proof of Theorem 4.4.1 . . . . . . . . . . . . . . . . . . . . . 127 4.6.1 Polynomial basis functions . . . . . . . . . . . . . . . 128 4.6.2 Edge extension operators . . . . . . . . . . . . . . . . 131 4.6.3 Face extension operators . . . . . . . . . . . . . . . . 132 4.6.4 Decomposition of functions in Sp(T ) . . . . . . . . . 134 4.6.5 Interior part . . . . . . . . . . . . . . . . . . . . . . . 138 4.6.6 Proof of Theorem 4.4.1 . . . . . . . . . . . . . . . . . 138 4.6.7 Proof of Proposition 4.6.4 . . . . . . . . . . . . . . . . 140 4.7 Numerical experiments . . . . . . . . . . . . . . . . . . . . . 148 4.7.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . 149 4.7.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . 151 vi Table of Contents 4.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 4.9 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 5 Anisotropic meshes . . . . . . . . . . . . . . . . . . . . . . . . 158 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 5.2 Interior penalty discretization . . . . . . . . . . . . . . . . . 159 5.2.1 Model problem . . . . . . . . . . . . . . . . . . . . . . 160 5.2.2 Discretization . . . . . . . . . . . . . . . . . . . . . . 160 5.2.3 Mesh sizes . . . . . . . . . . . . . . . . . . . . . . . . 161 5.2.4 Polynomial degrees . . . . . . . . . . . . . . . . . . . 162 5.2.5 Bilinear form . . . . . . . . . . . . . . . . . . . . . . . 163 5.3 A-posteriori error estimates . . . . . . . . . . . . . . . . . . . 164 5.3.1 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . 164 5.3.2 An a-posteriori error estimate . . . . . . . . . . . . . 165 5.4 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 5.4.1 Stability and auxiliary forms . . . . . . . . . . . . . . 167 5.4.2 Auxiliary meshes . . . . . . . . . . . . . . . . . . . . 168 5.4.3 Averaging operator . . . . . . . . . . . . . . . . . . . 170 5.4.4 Proof of Theorem 5.3.2 . . . . . . . . . . . . . . . . . 170 5.4.5 Proof of Theorem 5.3.4 . . . . . . . . . . . . . . . . . 175 5.5 Proof of Theorem 5.4.4 . . . . . . . . . . . . . . . . . . . . . 180 5.5.1 Polynomial basis functions . . . . . . . . . . . . . . . 180 5.5.2 Extension operators . . . . . . . . . . . . . . . . . . . 182 5.5.3 Decomposition of functions in Sp(T ) . . . . . . . . . 184 5.5.4 Proof of Theorem 5.4.4 . . . . . . . . . . . . . . . . . 186 5.5.5 Proof of Lemma 5.5.3 . . . . . . . . . . . . . . . . . . 188 5.5.6 Proof of Lemma 5.5.4 . . . . . . . . . . . . . . . . . . 192 5.6 Numerical experiments . . . . . . . . . . . . . . . . . . . . . 193 5.6.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . 194 5.6.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . 196 5.6.3 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . 200 5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 5.8 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 6 Conclusions and future work . . . . . . . . . . . . . . . . . . . 207 6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 6.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 6.3 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 vii List of Figures 1.1 The piecewise linear solutions for the convection-diffusion equa- tion (1.1.1) on a uniform mesh. . . . . . . . . . . . . . . . . . 2 1.2 The piecewise linear DG approximation for the convection- diffusion equation (1.1.1) with ε = 10−4, h = 0.01 and Pe = 100. 3 1.3 Adaptively generated h-version meshes after 7 refinement steps. 9 1.4 Adaptively generated hp-version meshes after 7 refinement steps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.5 Comparison of h-adaptive and hp-adaptive DG methods. . . . 12 1.6 The construction of the auxiliary mesh T̃ from T . . . . . . . 12 1.7 Adaptively generated hp-version mesh after 7 refinement steps. 13 1.8 Anisotropically generated hp-meshes after 7 refinement steps for ε = 2 · 10−4. . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.9 Convergence behavior for ε = 2 · 10−4. . . . . . . . . . . . . . 15 2.1 Example 1: Convergence behavior for ε = 1, 10−2, 10−4 and p = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.2 Example 1: Convergence behavior for ε = 1, 10−2, 10−4 and p = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.3 Example 1: Adaptively generated meshes after 7 refinement steps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.4 Example 2: Convergence behavior for ε = 10−2, 10−3 and p = 1, 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.5 Example 2: Adaptively generated meshes after 7 refinement steps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.6 Example 3: Convergence behavior for ε = 10−2, 10−4 and p = 1, 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.7 Example 3: Adaptively generated meshes after 7 refinement steps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.8 Example 4: Convergence behavior with ε = 10−2, 10−4 and p = 1 (left), 2 (right). . . . . . . . . . . . . . . . . . . . . . . 54 viii List of Figures 2.9 Example 4: The adaptively generated meshes after 7 refine- ment steps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.10 Example 5: Convergence behavior for ε = 10−2, 10−3 and p = 1 (left), 2 (right). . . . . . . . . . . . . . . . . . . . . . . 55 2.11 Example 5: Adaptively generated meshes after 7 refinement steps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.12 Example 6: Convergence behavior for ε = 10−2, 10−4 and p = 1, 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.13 Example 6: Adaptively generated meshes after 7 refinement steps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.1 The construction of the auxiliary mesh T̃ from T . . . . . . . 70 3.2 Reference element with variable edge polynomial degrees: p = 5, p Ê1 = 2, p Ê2 = 3, p Ê3 = 4, p Ê4 = 1. . . . . . . . . . . . . . 82 3.3 Left: Partition of K into K1 and K2. Right: Element K and K̃ ∈ R(K). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 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 re- finement 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 4.1 Reference element K̂ with the numbering of faces, edges and vertices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.2 Case 2: The elemental edge E ∈ E(K) has a hanging node located in its midpoint. . . . . . . . . . . . . . . . . . . . . . 132 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. . . 132 4.4 Case 2: Partition of K associated with the partition of face F . 133 4.5 Case 3: Partition of K associated with the partition of face F . 133 4.6 The element K is refined into 8 elements K̃ ∈ R(K). . . . . . 135 ix List of Figures 4.7 Example 1. (a) Comparison of the actual and estimated en- ergy norm of the error with respect to the (third root of the) number of degrees of freedom with hp-adaptive mesh refine- ment; (b) Effectivity indices; (c) Comparison of the actual error with h- and hp-adaptive mesh refinement. . . . . . . . . 149 4.8 Example 1. Finite element mesh after 8 adaptive refinements, with 440 elements and 100578 degrees of freedom: (a) hp- mesh; (b) Three-slice of the hp-mesh. . . . . . . . . . . . . . . 150 4.9 Example 2. (a) Comparison of the actual and estimated en- ergy norm of the error with respect to the (fourth root of the) number of degrees of freedom with hp-adaptive mesh refine- ment; (b) Effectivity indices; (c) Comparison of the actual error with h- and hp-adaptive mesh refinement. . . . . . . . . 151 4.10 Example 2. Finite element mesh after 7 adaptive refinements, with 686 elements and 197670 degrees of freedom: (a) hp- mesh; (b) Three-slice of the hp-mesh. . . . . . . . . . . . . . . 153 5.1 Mapping of the element K. . . . . . . . . . . . . . . . . . . . 160 5.2 The construction of the auxiliary mesh T̃ from T . . . . . . . 169 5.3 Reference element with variable edge polynomial degrees: p K̂ = (5, 4), p Ê1 = 2, p Ê2 = 3, p Ê3 = 4, p Ê4 = 1. . . . . . . . . 181 5.4 Partition of K into K1, · · · ,KN . . . . . . . . . . . . . . . . . 183 5.5 K̃ ∈ R(K) at the corner of K. . . . . . . . . . . . . . . . . . 185 5.6 K̃ ∈ R(K) locates inside K. . . . . . . . . . . . . . . . . . . . 185 5.7 K̃ ∈ R(K) shares an edge of K. . . . . . . . . . . . . . . . . . 185 5.8 Example 1: Convergence behavior for ε = 10−3. . . . . . . . . 195 5.9 Example 1: Convergence behavior for ε = 2 · 10−4. . . . . . . 196 5.10 Example 1: Adaptively generated meshes after 9 and 15 re- finement steps for ε = 10−3. . . . . . . . . . . . . . . . . . . . 197 5.11 Example 1: Adaptively generated meshes after 9 and 15 re- finement steps for ε = 2 · 10−4. . . . . . . . . . . . . . . . . . 198 5.12 Example 2: Convergence behavior for ε = 10−3. . . . . . . . . 199 5.13 Example 2: Convergence behavior for ε = 5 · 10−6. . . . . . . 199 5.14 Example 2: Adaptively generated meshes after 9 and 15 re- finement steps for ε = 10−3. . . . . . . . . . . . . . . . . . . . 200 5.15 Example 2: Adaptively generated meshes after 9 and 15 re- finement steps for ε = 5 · 10−6. . . . . . . . . . . . . . . . . . 201 5.16 Example 3: Convergence behavior for ε = 10−3. . . . . . . . . 202 5.17 Example 3: Adaptively generated meshes after 9 and 15 re- finement steps for ε = 10−3. . . . . . . . . . . . . . . . . . . . 203 x Acknowledgements First and foremost I express my sincerest gratitude to my supervisor, Pro- fessor 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 two- dimensional isotropically and anisotropically refined meshes, respectively. The numerical examples in these three chapters were all implemented by my- self. 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 approx- imation of such problems is of significant importance in science and engi- neering. 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 convection- dominated problems; see e.g., [52, 54, 56]. To illustrate this, we consider the simple one-dimensional convection-diffusion problem: −εu′′(x) + u′(x) = 1, in (0, 1), u(0) = u(1) = 0, (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, oscil- lations 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 ele- ment 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. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 x u h piecewise linear FEM, uniform mesh with h= 0.01   FEM solution (a) ε = 1, h = 10−2, Pe = 10−2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x u h piecewise linear FEM, uniform mesh with h= 0.01   FEM solution (b) ε = 10−4, h = 10−2, Pe = 102 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 10 20 30 40 50 60 x u h piecewise linear FEM, uniform mesh with h= 0.01   FEM solution (c) ε = 10−6, h = 10−2, Pe = 104 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x u h piecewise linear FEM, uniform mesh with h= 10−6   FEM solution (d) ε = 10−6, h = 10−6, Pe = 1 Figure 1.1: The piecewise linear solutions for the convection-diffusion equa- tion (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 di- rections. 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 dramat- ically 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 convection- diffusion problem (1.1.1) with ε = 10−4 and h = 0.01. Clearly, the DG ap- proximation does not show spurious oscillations in contrast to Figure 1.1(b), except in the last two elements near x = 1. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 x u h piecewise linear DGM, uniform mesh with h=0.01   analytical solution DG solution (a) DG approximation 0.8 0.85 0.9 0.95 1 0.85 0.9 0.95 1 1.05 x u h piecewise linear DGM, uniform mesh with h=0.01   analytical solution DG solution (b) DG approximation near x = 1 Figure 1.2: The piecewise linear DG approximation for the convection- diffusion equation (1.1.1) with ε = 10−4, h = 0.01 and Pe = 100. A stable numerical method, however, is generally not enough to ap- proximate convection-dominated problems effectively. This is because the solutions to such problems may have layers of small width where their gra- dients 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 knowl- edge 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 meth- ods to resolve layers and other solution singularities. These methods are designed to control and minimize the discretization errors by locally adapt- ing the mesh sizes and polynomial degrees according to the features of the analytical solution. The main focus of our work is on hp-adaptive discon- tinuous Galerkin methods. In hp-adaptive algorithms, a combination of h- and 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 expo- nential 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 elemen- tal 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 desir- able 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 poly- nomial 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 prob- lems, 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 in- troduction 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 prob- lems. 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 interest- ing 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 re- fined meshes and variable approximation degrees, since refinement or dere- finement 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 develop- ment 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 esti- mation 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 robust- ness 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 meth- ods. These methods are based on employing a fixed, usually low polynomial degree. As a consequence, adaptive h-version methods yield at most alge- braic 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 re- finement, 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 meth- ods for elliptic problems, see e.g. [24, 25, 35, 36, 40, 50], as well as for hy- perbolic 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 esti- mators 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: ‖u− uh‖ = ‖u− uh‖E + |u− uh|O. (1.3.2) 7 1.3. Main results Here, ‖ · ‖E 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]: Locality : η2 = ∑ K η2K , (1.3.3) Reliability : ‖u− uh‖ ≤ CR η, (1.3.4) Efficiency : ‖u− uh‖ ≥ CE η. (1.3.5) Locality (1.3.3) ensures that η can be written as a sum of elemental es- timators ηK . This requirement allows us to identify elements with large local errors. Reliability (1.3.4) means that the estimator always overesti- mates 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 discretiza- tion 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 de- sirable 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 perfor- mance of our estimators in adaptive refinement strategies. The implemen- tations 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 discretiza- tions 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 smooth- ness of the analytical solution. Here, we employ the hp-adaptive strategy developed in [39], where the local regularity of the analytical solution is esti- mated 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 er- ror 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 con- stants 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 hp- adaptive 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 avail- ability of an averaging operator. It allows us to split the error u − uh into two parts, a conforming part and a remainder. The conforming contribu- 10 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 approx- imation 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 ‖∇h(uh − Ihuh)‖2L2(Ω) ≤ C1 ∑ E p2Eh −1 E ‖[[uh]]‖2L2(E), (1.3.7) ‖uh − Ihuh‖2L2(Ω) ≤ C2 ∑ E p−2E hE‖[[uh]]‖2L2(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 H1-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 H1-seminorm estimate (1.3.7) were shown on regular meshes and for a fixed polynomial degree. The new con- tribution 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 102 103 104 10−6 10−5 10−4 10−3 10−2 10−1 100 N   hp−Error Indicator hp−Energy Error h−Error Indicator h−Energy Error (a) ε = 10−3 104 105 106 107 108 10−5 10−4 10−3 10−2 10−1 100 101 N   hp−Error Indicator hp−Energy Error h−Error Indicator h−Energy Error (b) ε = 2 · 10−4 Figure 1.5: Comparison of h-adaptive and hp-adaptive DG methods. continuous function Ihuh is piecewise polynomial on an auxiliary mesh T̃ , 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 T̃ 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 T̃ 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 (The- orem 4.4.1), these results are then extended to 1-irregularly refined hexahe- dral meshes in three dimensions. The same construction can be used on anisotropic meshes. This is de- tailed 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. The- orem 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 fun- damental 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 algo- rithm 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 cor- ner, 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 ‖u− uh‖ ≤ CMT η, where C is a constant independent of the parameters of interest, η the estimator andMT 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, sinceMT 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. 100 150 200 250 300 350 400 450 500 550 600 10−5 10−4 10−3 10−2 10−1 100 101 N1/2   Indicator (anisotropic) Error (anisotropic) Indicator (isotropic) Error (isotropic) 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 hp- version 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. 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Energy norm a-posteriori error estimation for hp-adaptive discontinuous Galerkin methods for elliptic problems in three dimensions. Math. Models Meth- ods Appl. Sci., 2010. accepted for publication. [70] L. Zhu and D. Schötzau. A robust a-posteriori error estimate for hp- adaptive DG methods for convection-diffusion equations. IMA J. Nu- mer. 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 convec- tion-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 convection- diffusion 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 1A version of this chapter has been published. Schötzau, D. and Zhu, L. (2009) A robust a-posteriori error estimator for discontinuous Galerkin methods for convection- diffusion 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 es- timator for discontinuous Galerkin discretizations of convection-diffusion problems. The estimator yields upper and lower bounds of the error mea- sured 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 us- ing standard techniques, while the remainder can be controlled using the stabilizing jump terms. The same techniques were used in the related pa- pers [16, 17, 18] on energy norm error estimation for discontinuous Galerkin discretizations of saddle point problems. The error measure used in our anal- ysis includes a non-local norm similar to the one in [32]. Our numerical ex- amples 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 discon- tinuous Galerkin methods for convection-diffusion equations and reaction- diffusion 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 up- wind 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 numer- ical 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 discontin- uous 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 Sec- tion 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 Interior penalty discretization 2.2.1 Model problem We consider the convection-diffusion model problem:{ −ε∆u+ a(x) · ∇u+ b(x)u = f(x) in Ω, u = 0 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 2 ∇ · a(x) + b(x) ≥ β, x ∈ Ω. (2.2.2) Finally, we suppose that there is a second constant c? ≥ 0 such that ‖ − ∇ · a+ b‖L∞(Ω) ≤ c?β. (2.2.3) The weak form of (2.2.1) is to find u ∈ H10 (Ω) such that A(u, v) := ∫ Ω ( ε∇u · ∇v + a · ∇uv + buv) dx = ∫ Ω fv dx for all v ∈ H10 (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 A(u, v) = ∫ Ω ( ε∇u · ∇v − au · ∇v + (b−∇ · a)uv) dx. 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 parallel- ograms. 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 Ke. For a piecewise smooth function v, we denote by v|E its trace on E taken from inside K, and by ve|E the one taken inside Ke. The average and jump of v across the edge E are then defined as {{v}} = 1 2 (v|E + ve|E), [[v]] = v|E nK + ve|E nKe . Similarly, if q is piecewise smooth vector field, its average and (normal) jump across E are given by {{q}} = 1 2 ( q|E + qe|E ) , [[q]] = q|E · nK + qe|E · nKe . 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 Ah(uh, v) = ∫ Ω fv dx (2.2.4) for all v ∈ Vh, with the bilinear Ah given by Ah(u, v) = ∑ K∈T ∫ K (ε∇u · ∇v + a · ∇uv + buv) dx − ∑ E∈E(Th) ∫ E {{ε∇u}} · [[v]] ds− ∑ E∈E(Th) ∫ E {{ε∇v}} · [[u]] ds + ∑ E∈E(T ) εγ hE ∫ E [[u]] · [[v]] ds− ∑ K∈T ∫ ∂Kin∩Γin a · nK uv ds + ∑ K∈T ∫ ∂Kin\Γ 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 Ah(u, v) = ∑ K∈T ∫ K ( ε∇u · ∇v − au · ∇v + (b−∇ · a)uv ) dx − ∑ E∈E(Th) ∫ E {{ε∇u}} · [[v]] ds− ∑ E∈E(Th) ∫ E {{ε∇v}} · [[u]] ds + ∑ E∈E(T ) εγ hE ∫ E [[u]] · [[v]] ds+ ∑ K∈T ∫ ∂Kout∩Γout a · nK uv ds + ∑ K∈T ∫ ∂Kout\Γ a · nKu(v − ve) ds. 2.3 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 ‖u ‖2E,T = ∑ K∈T ( ε‖∇u‖2L2(K) + β‖u‖2L2(K) ) + ∑ E∈E(T ) γε hE ‖[[u]]‖2L2(E). (2.3.1) 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 |q|? = sup v∈H10 (Ω)\{0} ∫ Ω q · ∇v dx ‖ v ‖E,T . 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 ∈ H10 (Ω) solves∫ Ω ∇ϕ · ∇v dx = ∫ Ω q · ∇v dx ∀ v ∈ H10 (Ω), and q 0 = q −∇ϕ is divergence-free in the sense that∫ Ω q 0 · ∇v dx = 0 ∀ v ∈ H10 (Ω). This decomposition is unique and orthogonal in L2(Ω)2. Thus, we observe that |q|? = 0 if and only if q = q0. Furthermore, if we introduce the norm ‖ϕ‖? = sup v∈H10 (Ω)\{0} ∫ Ω ∇ϕ · ∇v dx ‖ v ‖E,T , we have that |q|? = ‖ϕ‖?. We now define |u |2O,T = |au|2? + ∑ E∈E(T ) ( βhE + hE ε ) ‖[[u]]‖2L2(E). (2.3.2) The semi-norm |au|2? and the jump terms hEε−1‖[[u]]‖2L2(E) will be used to bound the convective derivative, analogously to [32]. Here we note that hEε −1 is the local mesh Péclet number, see also the discussion in Re- mark 2.3.5. Finally, the jump terms βhE‖[[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 ρK = min{hKε− 12 , β− 12 }, ρE = min{hEε− 12 , β− 12 }. In the case β = 0, we set ρK = ε − 1 2hK and ρE = ε − 1 2hE . Let now uh be the discontinuous Galerkin approximation obtained by (2.2.4). Moreover, let fh, ah, and bh denote piecewise polynomial approx- imations in Vh to the right-hand side and the coefficient functions, respec- tively. For each element K ∈ T , we introduce a local error indicator ηK which is given by the sum of three terms η2K = η 2 RK + η2EK + η 2 JK . The first term ηRK is the interior residual defined by η2RK = ρ 2 K‖fh + ε∆uh − ah · ∇uh − bhuh‖2L2(K). The second term η2EK is the edge residual given by η2EK = 1 2 ∑ E∈∂K\Γ ε− 1 2 ρE‖[[ε∇uh]]‖2L2(E). The last term ηJK measures the jumps of the approximate solution uh and is defined by η2JK = 1 2 ∑ E∈∂K\Γ ( γε hE + βhE + hE ε ) ‖[[uh]]‖2L2(E), + ∑ E∈∂K∩Γ ( γε hE + βhE + hE ε ) ‖[[uh]]‖2L2(E). We also introduce a data approximation term by Θ2K = ρ 2 K ( ‖f − fh‖2L2(K) + ‖(a− ah) · ∇uh‖2L2(K) + ‖(b− bh)uh‖2L2(K) ) . We then define the a-posteriori error estimator η = ( ∑ K∈Th η2K ) 1 2 . (2.3.3) The data approximation error is given by Θ = ( ∑ K∈T Θ2K ) 1 2 . (2.3.4) 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 ap- proximation 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 ap- proximation 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 ‖u− uh ‖E,T + |u− uh |O,T . η + Θ. Our next theorem presents a lower bound for the error and shows the effi- ciency of the error estimator η. Theorem 2.3.3 Let u be the solution of (2.2.1) and uh ∈ Vh its DG ap- proximation 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 . ‖u− uh ‖E,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 num- ber 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 ‖u− uh ‖E,T and even of higher-order, once the local mesh Péclet num- ber is sufficiently small. Heuristically, this can be explained as follows. We expect the error ‖u− uh ‖E,T to converge with the optimal order O(N− p 2 ), where N is the number of degrees of freedom; cf. [4, 20]. We then have the bound |a(u− uh)|? . 1√ ε ‖u− uh‖L2(Ω). (2.3.5) 31 2.3. Robust a-posteriori error estimation If we now also assume that the L2-error ‖u − uh‖L2(Ω) converges with the optimal rate O(N− p2− 12 ), we obtain |a(u− uh)|? . N − 1 2 ε √ εN− p 2 . The fraction N − 12 ε 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 ( ∑ E∈T hE ε ‖[[u− uh]]‖2L2(E) ) 1 2 . N − 1 2 ε √ εN− p 2 , ( ∑ E∈T βhE‖[[u− uh]]‖2L2(E) ) 1 2 . √ βN− p 2 − 1 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, u = 0 on Γ, (2.3.6) 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 ‖b‖L∞(Ω) ≤ c?β. For reaction-diffusion equations, the energy norm is now defined by ‖u ‖2E,T = ∑ K∈T ( ε‖∇u‖2L2(K) + β‖u‖2L2(K) ) + ∑ E∈E(T ) ( γε hE + βhE)‖[[u]]‖2L2(E). The local error indicator ηK on every element K ∈ T becomes η2K = η 2 RK + η2EK + η 2 JK , 32 2.3. Robust a-posteriori error estimation where η2RK = ρ 2 K‖fh + ε∆uh − bhuh‖2L2(K), η2EK = 1 2 ∑ E∈∂K\Γ ε− 1 2 ρE‖[[ε∇huh]]‖2L2(E), η2JK = 1 2 ∑ E∈∂K\Γ ( γε hE + βhE)‖[[uh]]‖2L2(E) + ∑ E∈∂K∩Γ ( γε hE + βhE)‖[[uh]]‖2L2(E), where ρK and ρE are defined by ρK = hKε − 1 2 , ρE = hEε 1 2 . Furthermore, since the convection term disappears in the data approxima- tion term, we have, Θ2K = ρ 2 K ( ‖f − fh‖2L2(K) + ‖(b− bh)uh‖2L2(K) ) , with uh being the discontinuous Galerkin approximation. We then employ the same notations as before and set η = ( ∑ K∈Th η2K ) 1 2 , (2.3.7) as well as Θ = ( ∑ 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 ‖u− uh ‖E,T . η + Θ, as well as the lower bound η . ‖u− uh ‖E,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 H10 (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 Dh(u, v) = ∑ K∈T ∫ K ( ε∇u · ∇v + (b−∇ · a)uv ) dx, (2.4.1) Oh(u, v) = − ∑ K∈T ∫ K au · ∇v dx+ ∑ K∈Th ∫ ∂Kout∩Γout a · nK uv ds + ∑ K∈T ∫ ∂Kout\Γ a · nKu(v − ve) ds, (2.4.2) Kh(u, v) = − ∑ E∈E(Th) ∫ E {{ε∇u}} · [[v]]ds− ∑ E∈E(Th) ∫ E {{ε∇v}} · [[u]]ds, (2.4.3) Jh(u, v) = ∑ E∈E(T ) εγ hE ∫ E [[u]] · [[v]] ds. (2.4.4) Then, we set Ãh(u, v) = Dh(u, v) + Jh(u, v) +Oh(u, v). This form is well-defined for all u, v ∈ Vh +H10 (Ω). Obviously, we have Ãh(u, v) = A(u, v), (2.4.5) for all u, v ∈ H10 (Ω). Furthermore, Ah(u, v) = Ãh(u, v) +Kh(u, v), (2.4.6) 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 ∈ H10 (Ω), we have Ãh(u, u) ≥ ‖u ‖2E,T . Furthermore, the auxiliary forms are continuous. Lemma 2.4.2 There holds |Dh(u, v)| . ‖u ‖E,T ‖ v ‖E,T , u, v ∈ Vh +H10 (Ω), |Jh(u, v)| . ‖u ‖E,T ‖ v ‖E,T , u, v ∈ Vh +H10 (Ω), |Oh(u, v)| . |au|?‖ v ‖E,T , u ∈ Vh +H10 (Ω), v ∈ H10 (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 ∈ H10 (Ω) ∩ Vh we have Kh(u, v) . γ− 1 2 ( ∑ E∈E(Th) εγ hE ‖[[u]]‖2L2(E) ) 1 2 ‖ v ‖E,T . Proof : Since v ∈ H10 (Ω) ∩ Vh, we have Kh(u, v) = − ∑ E∈E(Th) ∫ E {{ε∇v}} · [[u]] ds. Using the Cauchy-Schwarz inequality, the inverse estimate, ‖v‖L2(∂K) . h− 1 2 K ‖v‖L2(K), v ∈ Sp(K), and the shape-regularity of the mesh, we obtain Kh(u, v) . ∑ E∈E(T ) ∫ E |ε∇v||[[u]]| ds . γ− 12 ( ∑ E∈E(Th) εhE ∫ E |∇v|2 ds) 12 ( ∑ E∈E(T ) εγ hE ∫ E |[[u]]|2 ds) 12 . γ− 12 ( ∑ K∈T ε‖∇v‖2L2(K) ) 1 2 ( ∑ E∈E(Th) εγ hE ‖[[u]]‖2L2(E) ) 1 2 . 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 inf u∈H10 (Ω)\{0} sup v∈H10 (Ω)\{0} Ãh(u, v) (‖u ‖E,T + |au|?)‖ v ‖E,T ≥ C > 0. Proof : Let u ∈ H10 (Ω) and θ ∈ (0, 1). Then there exists wθ ∈ H10 (Ω) such that ‖wθ ‖E,T = 1, Oh(u,wθ) = − ∫ Ω au · ∇wθ dx ≥ θ|au|?. From the continuity properties in Lemma 2.4.2, we obtain Ãh(u,wθ) = Dh(u,wθ) + Jh(u,wθ) +Oh(u,wθ) ≥ θ|au|? − C1‖u ‖E,T ‖wθ ‖E,T = θ|au|? − C1‖u ‖E,T , for a constant C1 > 0. Let us then define vθ = u+ ‖u ‖E,T 1 + C1 wθ. Obviously, ‖ vθ ‖E,T ≤ (1 + 1 1 + C1 )‖u ‖E,T . By Lemma 2.4.1, A(u, u) ≥ ‖u ‖2E,T , so that sup v∈H10 (Ω)\{0} Ãh(u, v) ‖ v ‖E,T ≥ Ãh(u, vθ) ‖ vθ ‖E,T ≥ ‖u ‖ 2 E,T + (1 + C1) −1‖u ‖E,T (θ|au|? − C1‖u ‖E,T ) (1 + 11+C1 )‖u ‖E,T = 1 2 + C1 (‖u ‖E,T + θ|au|?). Since θ ∈ (0, 1) and u ∈ H10 (Ω) are arbitrary, we obtain the inf-sup condi- tion. 2 36 2.4. Proofs 2.4.2 Approximation operators Let V ch be the conforming subspace of Vh given by V ch = Vh ∩H10 (Ω). We denote by Ah : Vh → V ch the approximation operator defined in [22, Theorem 2.2 and Theorem 2.3]; see also [19, Proposition 5.4] for an exten- sion to the hp-version of the discontinuous Galerkin method. The following approximation result holds. Lemma 2.4.5 For any v ∈ Vh, we have∑ K∈T ‖v −Ahv‖2L2(K) . ∑ E∈E(T ) ∫ E hE |[[v]]|2 ds, ∑ K∈T ‖∇(v −Ahv)‖2L2(K) . ∑ E∈E(T ) ∫ E h−1E |[[v]]|2 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 : H 1 0 (Ω)→ {ϕ ∈ C(Ω) : ϕ|K ∈ S1(K), ∀K ∈ T , ϕ = 0 on Γ}, that satisfies ‖ Ihv ‖E,T . ‖ v ‖E,T and( ∑ K∈T ρ−2K ‖v − Ihv‖2L2(K) ) 1 2 . ‖ v ‖E,T , ( ∑ E∈E(T ) ε 1 2 ρ−1E ‖v − Ihv‖2L2(E) ) 1 2 . ‖ v ‖E,T , for any v ∈ H10 (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 = u c h + u r h, 37 2.4. Proofs where uch = Ahuh ∈ V ch , 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 ‖u− uh ‖E,T + |u− uh |O,T ≤ ‖u− uch ‖E,T + |u− uch |O,T + ‖urh ‖E,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 ‖urh ‖E,T + |urh |O,T . η. Proof : Since [[urh]] = [[uh]], we have ‖urh ‖2E,T + |urh |2O,T = ∑ K∈T ( ε‖∇urh‖2L2(K) + β‖urh‖2L2(K) ) + |aurh|2? + ∑ E∈E (T ) ( γε hE + βhE + hE ε ) ‖[[uh]]‖2L2(E) . ∑ K∈T (ε‖∇urh‖2L2(K) + β‖urh‖2L2(K)) + |aurh|2? + ∑ K∈T η2JK . Hence, only the volume terms and the expression involving the | · |? semi- norm need to be bounded further. Lemma 2.4.5 yields ε ∑ K∈T ‖∇urh‖2L2(K) . γ−1 ∑ E∈E(T ) εγ hE ‖[[uh]]‖2L2(E) . γ−1 ∑ K∈T η2JK , β ∑ K∈T ‖urh‖2L2(K) . ∑ E∈E(T ) βhE‖[[uh]]‖2L2(E) . ∑ K∈T η2JK . To estimate |aurh|?, we apply Lemma 2.4.5 once more, and obtain |aurh|2? . 1 ε ‖urh‖2L2(Ω) . ∑ E∈E(T ) hE ε ‖[[uh]]‖2L2(E) . ∑ K∈T η2JK . This finishes the proof. 2 Lemma 2.4.8 For any v ∈ H10 (Ω), we have∫ Ω f(v − Ihv) dx− Ãh(uh, v − Ihv) . (η + Θ) ‖ v ‖E,T . Here, Ih is the interpolant introduced in Lemma 2.4.6. 38 2.4. Proofs Proof : Set T = ∫ Ω f(v − Ihv) dx− Ãh(uh, v − Ihv). Integration by parts immediately yields T = ∑ K∈T ∫ K (f + ε∆uh − a · ∇uh − buh)(v − Ihv) dx − ∑ K∈T ∫ ∂K ε∇uh · nK(v − Ihv) ds + ∑ K∈T ∫ ∂Kin\Γ a · nK(ueh − uh)(v − Ihv) ds = T1 + T2 + T3. In the term T1, we first add and subtract the data approximations terms. This gives T1 = ∑ K∈T ∫ K (fh + ε∆uh − ah · ∇uh − bhuh)(v − Ihv) dx + ∑ K∈T ∫ K ( (f − fh)− (a− ah) · ∇uh − (b− bh)uh ) (v − Ihv) dx. Using the Cauchy-Schwarz inequality and Lemma 2.4.6 yields T1 . ( ∑ K∈Th η2RK ) 1 2 ( ∑ K∈T ρ−2K ‖v − Ihv‖2L2(K) ) 1 2 + ( ∑ K∈Th Θ2K ) 1 2 ( ∑ K∈T ρ−2K ‖v − Ihv‖2L2(K) ) 1 2 . ( ∑ K∈T (η2RK + Θ 2 K) ) 1 2 ‖ v ‖E,T . Next, if we rewrite the term T2 in terms of the jumps of ε∇uh, we obtain T2 = − ∑ E∈EI(T ) ∫ E [[ε∇uh]](v − Ihv) ds. 39 2.4. Proofs Hence, the Cauchy-Schwarz inequality and Lemma 2.4.6 yield T2 . ( ∑ E∈EI(Th) ε− 1 2 ρE‖[[ε∇uh]]‖2L2(E) ) 1 2 ( ∑ E∈E(T ) ε 1 2 ρ−1E ‖v − Ihv‖2L2(E) ) 1 2 . ( ∑ K∈T η2EK ) 1 2 ‖ v ‖E,T . In order to bound T3, we use the Cauchy-Schwarz inequality, Lemma 2.4.6, and the fact that ρE ≤ hKε− 12 : T3 . ( ∑ E∈E(T ) ε− 1 2 ρE‖[[uh]]‖2L2(E) ) 1 2 ( ∑ E∈Eh(T ) ε 1 2 ρ−1E ‖v − Ihv‖2L2(E) ) 1 2 . ( ∑ K∈T η2JK ) 1 2 ‖ v ‖E,T . This finishes the proof. 2 Lemma 2.4.9 There holds: ‖u− uch ‖E,T + |u− uch |O,T . η + Θ. Proof : Note that |u− uch |O,T = |a(u − uch)|?. Then the inf-sup condition yields: ‖u− uch ‖E,T + |a(u− uch)|? . sup v∈H10 (Ω)\{0} Ãh(u− uch, v) ‖ v ‖E,T . (2.4.8) The properties (2.4.5) and (2.4.6) allow us to conclude that, for any v ∈ H10 (Ω), Ãh(u− uch, v) = ∫ Ω fv dx− Ãh(uch, v) = ∫ Ω fv dx−Dh(uch, v)− Jh(uch, v)−Oh(uch, v) = ∫ Ω fv dx− Ãh(uh, v) +Dh(urh, v) + Jh(urh, v) +Oh(urh, v). From the discontinuous Galerkin method in (2.2.4), we have∫ Ω fIhv dx = Ah(uh, Ihv) = Ãh(uh, Ihv) +Kh(uh, Ihv), 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 = ∫ Ω f(v − Ihv) dx− Ãh(uh, v − Ihv), T2 = Dh(u r h, v) + Jh(u r h, v) +Oh(u r h, v), T3 = Kh(uh, Ihv). The estimate in Lemma 2.4.8 yields T1 . (η + Θ) ‖ v ‖E,T . Similarly, the continuity result in Lemma 2.4.2 and the approximation prop- erties in Lemma 2.4.7 give T2 . (‖urh ‖E,T + |aurh|?)‖ v ‖E,T ≤ η‖ v ‖E,T . Finally, we use Lemma 2.4.3 to bound the term T3. We obtain T3 . γ− 1 2 (∑ K∈T η2JK ) 1 2 ‖ Ihv ‖E,T . γ− 1 2 (∑ K∈T η2JK ) 1 2 ‖ v ‖E,T . Here, we have also used the ‖ · ‖E,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 H10 (K), while ψE is in H 1 0 (wE). We have ‖ψK‖L∞(K) = 1, ‖ψE‖L∞(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 ‖ · ‖E,D the local energy norm ‖u ‖2E,D = ∑ K∈D ( ε‖∇u‖2L2(K) + β‖u‖2L2(K) ) . We also set ‖u‖2L2(D) = ∑ K∈D ‖u‖2L2(K), and define ∫ D f(x) dx = ∑ K∈D ∫ K f(x) dx. The following result holds; cf. [32, Lemma 3.6]. Lemma 2.4.10 We have ‖v‖2L2(K) . (v, ψKv)K , (2.4.10) ‖ψKv ‖E,K . ρ−1K ‖v‖L2(K), (2.4.11) ‖σ‖2L2(E) . (σ, ψEσ)E , (2.4.12) ‖ψEσ‖L2(wE) . ε 1 4 ρ 1 2 E‖σ‖L2(E), (2.4.13) ‖ψEσ ‖E,wE . ε 1 4 ρ − 1 2 E ‖σ‖L2(E), (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( ∑ K∈T η2JK ) 1 2 . ‖u− uh ‖E,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( ∑ K∈T η2RK ) 1 2 . ‖u− uh ‖E,T + |u− uh |O,T + Θ. 42 2.4. Proofs Proof : Let K be an element in T . We define R|K = (fh + ε∆uh − ah · ∇uh − bhuh)|K , and set W |K = ρ2KRψK . By inequality (2.4.10) in Lemma 2.4.10,∑ K∈T η2RK = ∑ K∈T ρ2K‖R‖2L2(K) . ∑ K∈T (R, ρ2KψKR)K = ∑ K∈T (R,W )K = ∑ K∈T (fh + ε∆uh − ah · ∇uh − bhuh,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,∑ K∈T η2RK . ∑ K∈T ( ε(∇(uh − u),∇W )K − (a(u− uh),∇W )K ) + ∑ K∈T ((b−∇ · a)(u− uh),W )K + ∑ 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∑ K∈T η2RK . (‖u− uh ‖E,T + |u− uh |O,T + Θ) × ( ∑ K∈Th ‖W ‖2E,K + ρ−2K ‖W‖2L2(K) ) 1 2 . By (2.4.11) and (2.4.9), we have the following estimates ‖W ‖2E,K . ρ2K‖R‖2L2(K), ρ−2K ‖W‖2L2(K) . ρ2K‖R‖2L2(K). This yields∑ K∈T η2RK . ( ‖u− uh ‖E,T + |u− uh |O,T + Θ )( ∑ K∈T η2RK ) 1 2 , which shows the assertion. 2 43 2.4. Proofs Lemma 2.4.12 There holds( ∑ K∈T η2EK ) 1 2 . ‖u− uh ‖E,T + |u− uh |O,T + Θ. Proof : We set τ = ∑ E∈EI(T ) ε− 1 2 ρE [[ε∇uh]]ψE . By (2.4.12) in Lemma 2.4.10 and the fact that [[ε∇u]] = 0 on interior edges, we obtain∑ K∈T η2EK . ∑ E∈EI(T ) ([[ε∇uh]], τ)E = ∑ E∈EI(T ) ([[ε∇(uh − u)]], τ)E . After integration by parts over each of the two elements of wE , we have∑ E∈EI(T ) ([[ε∇(uh−u)]], τ)E = ∑ E∈EI(T ) ∫ wE (ε(∆uh−∆u)τ+ε(∇uh−∇u)·∇τ)dx. Using the differential equation and approximating the data, we obtain∑ K∈T η2EK . ∑ E∈EI(T ) ∫ wE (fh + ε∆uh − ah · ∇uh − buh)τ dx + ∑ E∈EI(T ) ∫ wE ( (a · ∇(uh − u) + b(uh − u))τ + ε(∇uh −∇u) · ∇τ ) dx + ∑ E∈EI(T ) ∫ wE ( (f − fh) + (ah − a) · ∇uh + (bh − b)uh ) τ dx Integration by parts over wE of the convection term a · ∇(uh − u) yields,∑ K∈T η2EK . T1 + T2 + T3 + T4 + T5, where T1 = ∑ E∈EI(T ) ∫ wE (fh + ε∆uh − ah · ∇uh − buh)τ dx, T2 = ∑ E∈EI(T ) ∫ wE ( (−∇ · a+ b)(uh − u)τ + ε(∇uh −∇u) · ∇τ ) dx, T3 = − ∑ E∈EI(T ) ∫ wE a(uh − u) · ∇τ dx, 44 2.4. Proofs T4 = ∑ E∈EI(T ) ∫ E a · [[uh]]τ ds, T5 = ∑ 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 T1 . (‖u− uh ‖E,T + |u− uh |O,T + Θ)( ∑ E∈EI(T ) ρ−2E ‖τ‖2L2(wE) ) 1 2 . By inequality (2.4.13) in Lemma 2.4.10 we obtain,( ∑ E∈EI(T ) ρ−2E ‖τ‖2L2(wE) ) 1 2 . ( ∑ K∈T η2EK ) 1 2 , so that T1 . (‖u− uh ‖E,T + |u− uh |O,T + Θ)( ∑ K∈T η2EK ) 1 2 . Using the shape regularity of the mesh and inequality (2.4.14), the term T2 can be bounded by T2 . ‖u− uh ‖E,T ( ∑ E∈EI(T ) ‖ τ ‖2E,wE ) 1 2 . ‖u− uh ‖E,T ( ∑ K∈T η2EK ) 1 2 . For the term T3, we use the previous estimate and obtain T3 . |u− uh |O,T ( ∑ E∈EI(T ) ‖ τ ‖2E,wE ) 1 2 . |u− uh |O,T ( ∑ K∈Th η2EK ) 1 2 . Since the support of ψE intersects the support of at most two other ψE ’s and since ‖ψE‖L∞(E) = 1, we have T4 . ( ∑ E∈EI(T ) ε− 1 2 ρE‖[[uh]]‖2L2(E) ) 1 2 ( ∑ E∈EI(T ) ε 1 2 ρ−1E ‖τ‖2L2(E) ) 1 2 . Then, due to ρE ≤ hEε− 12 , we obtain T4 . ( ∑ E∈EI(T ) hE ε ‖[[uh]]‖2L2(E) ) 1 2 ( ∑ K∈Th η2EK ) 1 2 . |u− uh |O,T ( ∑ K∈T η2EK ) 1 2 . 45 2.5. Numerical experiments Finally, the data error term T5 can be bounded by T5 . Θ ( ∑ K∈Th ρ−2K ‖τ‖2L2(K) ) 1 2 . Θ ( ∑ K∈Th η2EK ) 1 2 . This finishes the proof. 2 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 pre- sented below, we construct adaptively refined mesh sequences by marking elements for refinement or derefinement according to the size of the local in- dicators 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 el- ements. 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) = (ex−1ε − 1 e− 1 ε − 1 + x− 1 )(e y−1ε − 1 e− 1 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 ‖u− uh ‖E,T and 46 2.5. Numerical experiments the value of the estimator η against N− 1 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 en- ergy error, in agreement with Theorem 2.3.2. Asymptotically, the curves are straight lines, indicating convergence of the optimal order O(N− 12 ). 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 curve ”DERR”, we also calculate the error ε− 1 2 ‖u − uh‖L2(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 are approximated well enough. The same behavior is ob- served for the error ∑ E∈E(T ) ( βhE + hEε −1) ‖[[uh]]‖2L2(E), which is computed in the curve ”JERR”; see also Remark 2.3.5. In the second row of subfig- ures 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 101 102 103 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 N1/2   EST ERR DERR JERR (a) ε = 1 101 102 103 10−5 10−4 10−3 10−2 10−1 100 101 N1/2   EST ERR DERR JERR (b) ε = 10−2 102 103 104 10−4 10−3 10−2 10−1 100 101 N1/2   EST ERR DERR JERR (c) ε = 10−4 101 102 103 2 3 4 5 6 7 8 N1/2   ratio (d) ε = 1 101 102 103 2 3 4 5 6 7 8 N1/2   ratio (e) ε = 10−2 102 103 104 2 3 4 5 6 7 8 N1/2   ratio (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 u(x, y) = erf( x√ 2ε )(1− y2). Here, erf(x) is the error function defined by erf(x) = 2√ pi ∫ x 0 e −t2dt. 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 cor- respondingly. 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 ap- proximations. In the asymptotic regime, we observe linear and quadratic convergence rates of the optimal order O(N− p2 ) 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 103 104 105 106 10−11 10−10 10−9 10−8 10−7 10−6 10−5 10−4 10−3 N   EST ERR DERR JERR (a) ε = 1 103 104 105 106 10−6 10−5 10−4 10−3 10−2 10−1 100 101 N   EST ERR DERR JERR (b) ε = 10−2 104 105 106 107 108 10−5 10−4 10−3 10−2 10−1 100 101 N   EST ERR DERR JERR (c) ε = 10−4 103 104 105 106 6 7 8 9 10 11 12 N   ratio (d) ε = 1 103 104 105 106 6 7 8 9 10 11 12 N   ratio (e) ε = 10−2 104 105 106 107 108 6 7 8 9 10 11 12 N   ratio (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−1ε 1− e− 2ε . 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 convergence rates of the optimal order O(N− p2 ) 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 pi6 , cos pi6 )> and b = f = 0, 50 2.5. Numerical experiments 102 103 10−5 10−4 10−3 10−2 10−1 100 N1/2   EST ERR DERR JERR (a) ε = 10−2, p = 1 102 103 10−5 10−4 10−3 10−2 10−1 100 N1/2   EST ERR DERR JERR (b) ε = 10−3, p = 1 102 103 1 2 3 4 5 6 7 8 N1/2   ε=10−2 ε=10−3 (c) p = 1 103 104 105 106 107 10−7 10−6 10−5 10−4 10−3 10−2 10−1 N   EST ERR DERR JERR (d) ε = 10−2, p = 2 104 105 106 107 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 N   EST ERR DERR JERR (e) ε = 10−3, p = 2 103 104 105 106 107 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 N   ε=10−2 ε=10−3 (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 on x = −1 and y = 1, u = tanh( 1− y ε ) on x = 1, u = 1 2 ( tanh( x ε ) + 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 and p = 2 against N− p 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. of order one, the error estimator converges with the optimal rate O(N− p2 ). 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 − y2))> and b = f = 0. The boundary conditions are: u = tanh( 1− y ε ) on x = 1, u = 0 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 lay- ers near the corners. 52 2.5. Numerical experiments 101 102 103 10−5 10−4 10−3 10−2 10−1 100 101 N1/2   EST ERR DERR JERR (a) ε = 10−2, p = 1 102 103 104 10−4 10−3 10−2 10−1 100 101 N1/2   EST ERR DERR JERR (b) ε = 10−4, p = 1 101 102 103 104 1 2 3 4 5 6 7 8 N1/2   ε=10−2 ε=10−4 (c) p = 1 103 104 105 106 107 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 101 N   EST ERR DERR JERR (d) ε = 10−2, p = 2 104 105 106 107 108 10−5 10−4 10−3 10−2 10−1 100 101 N   EST ERR DERR JERR (e) ε = 10−4, p = 2 102 103 104 105 106 107 108 3 4 5 6 7 8 9 10 11 N   ε=10−2 ε=10−4 (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. Asymptoti- cally, 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 u(x, y) = ( 1− e x/ √ ε + e(1−x)/ √ ε 1 + e1/ √ ε )( 1− e y/ √ ε + e(1−y)/ √ 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 convection- diffusion 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. 101 102 103 104 10−2 10−1 100 101 102 N1/2   ε=10−2 ε=10−4 103 104 105 106 107 108 10−4 10−3 10−2 10−1 100 101 102 N   ε=10−2 ε=10−4 h min=1.10e−02 h min= 5.52e−03 h min=1.73e−04 h min=3.45e−04 h min=8.63e−05 h min=2.21e−02 h min=2.21e−02 h min=1.10e−2 h min=5.52e−3 h min=1.73e−04 h min=3.45e−04 h min=8.63e−05 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. 101 102 103 104 10−2 10−1 100 101 N1/2   ε=10−2 ε=10−3 103 104 105 106 107 10−4 10−3 10−2 10−1 100 101 N   ε=10−2 ε=10−3 h min=2.21e−2 h min=1.10e−2 h min=5.52e−3 h min=2.76e−3 h min=1.38e−3 h min=6.91e−4 h min=2.21e−2 h min=1.10e−2 h min=5.52e−3 h min=2.76e−3 h min=1.38e−3 h min=6.91e−4 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. O(N− p2 ). 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 asso- ciated 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 re- sults 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. 101 102 103 10−3 10−2 10−1 100 N1/2   EST ERR (a) ε = 10−2, p = 1 101 102 103 104 10−3 10−2 10−1 N1/2   EST ERR (b) ε = 10−4, p = 1 101 102 103 104 1 2 3 4 5 6 7 8 N1/2   ε=10−2 ε=10−4 (c) p = 1 103 104 105 106 107 10−5 10−4 10−3 10−2 10−1 N   EST ERR (d) ε = 10−2, p = 2 104 105 106 107 10−6 10−5 10−4 10−3 10−2 10−1 N   EST ERR (e) ε = 10−4, p = 2 103 104 105 106 107 4 5 6 7 8 9 10 11 N   ε=10−2 ε=10−4 (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 Fi- nite Element Analysis. 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A-posteriori error estimators for convection-diffusion equa- tions. 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 dif- fer 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 discontin- uous 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. 2A 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 convection- diffusion equations. IMA Journal of Numerical Analysis. 62 3.1. Introduction Discontinuous Galerkin methods are naturally suited for realizing hp- adaptivity. 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 meth- ods 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 el- liptic 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 de- grees. 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 contri- bution 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 averag- ing operator. In [19], an optimal H1-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 H1-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 consist- ing 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 de- cide 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 convection- diffusion 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) in Ω, u = 0 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 ‖a‖L∞(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 ∈ H10 (Ω) such that A(u, v) = ∫ Ω ( ε∇u · ∇v+ a · ∇uv) dx = ∫ Ω fv dx ∀ v ∈ H10 (Ω). (3.2.3) Under assumption (3.2.2), the variational problem (3.2.3) is uniquely solv- able. 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. Each element K ∈ T is the image of the reference square K̂ = (−1, 1)2 under an affine elemental mapping FK : K̂ → K. As usual, we denote 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 byNI(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 ′ of two elements K,K ′ ∈ 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 ′ ∈ T , we have %−1 ≤ pK/pK′ ≤ %. For E ∈ E(T ), we introduce the edge polynomial degree pE by pE = { max{pK , pK′}, E = ∂K ∩ ∂K ′ ∈ EI(T ), pK , E = ∂K ∩ Γ ∈ EB(T ). (3.2.4) 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 Sp(T ) = { v ∈ L2(Ω) : v|K ◦ FK ∈ QpK (K̂), K ∈ T }, (3.2.5) 65 3.3. Robust a-posteriori error estimates with Qp(K̂) denoting the set of all polynomials on the reference square K̂ of degree less or equal than p in each variable. We now consider the following discontinuous Galerkin method for the ap- proximation of the convection-diffusion problem (3.2.1): Find uhp ∈ Sp(T ) such that Ahp(uhp, v) = ∫ Ω fv dx (3.2.6) for all v ∈ Sp(T ), with the bilinear form Ahp given by Ahp(u, v) = ∑ K∈T ∫ K (ε∇u · ∇v + a · ∇uv) dx − ∑ E∈E(T ) ∫ E {{ε∇u}} · [[v]] ds− ∑ E∈E(T ) ∫ E {{ε∇v}} · [[u]] ds + ∑ E∈E(T ) εγp2E hE ∫ E [[u]] · [[v]] ds− ∑ K∈T ∫ ∂Kin∩Γin a · nK uv ds + ∑ K∈T ∫ ∂Kin\Γ a · nK(ue − u)v ds. 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: ‖ v ‖2E,T = ∑ K∈T ε‖∇v‖2L2(K) + ejumpp,T (v)2, ejumpp,T (v) 2 = ∑ E∈E(T ) γεp2E hE ‖[[v]]‖2L2(E). (3.3.1) Next, we define |q|? = sup v∈H10 (Ω)\{0} ∫ Ω q · ∇v dx ‖ v ‖E,T ∀ q ∈ L 2(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, ojumpp,T (v) 2 = ∑ E∈E(T ) hE εpE ‖[[v]]‖2L2(E). (3.3.2) Notice that hEε −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 approxi- mations 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 η2K = η 2 RK + η2EK + η 2 JK . (3.3.3) The first term ηRK is the interior residual defined by η2RK = ε −1p−2K h 2 K‖fhp + ε∆uhp − ahp · ∇uhp‖2L2(K). The second term ηEK is the edge residual given by η2EK = 1 2 ∑ E∈∂K\Γ ε−1p−1E hE‖[[ε∇uhp]]‖2L2(E). 67 3.3. Robust a-posteriori error estimates The last residual ηJK measures the jumps of the approximate solution uhp: η2JK = 1 2 ∑ E∈∂K\Γ [ γ2εp3E hE + hE εpE ] ‖[[uhp]]‖2L2(E) + ∑ E∈∂K∩Γ [ γ2εp3E hE + hE εpE ] ‖[[uhp]]‖2L2(E). We also introduce the local data approximation term Θ2K = ε −1p−2K h 2 K ( ‖f − fhp‖2L2(K) + ‖(a− ahp) · ∇uhp‖2L2(K) ) . We then introduce the global error estimator and data approximation error η2 = ∑ K∈T η2K , Θ 2 = ∑ K∈T Θ2K . (3.3.4) 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 ‖u− uhp ‖E,T + |u− uhp |O,T . η + Θ. Remark 3.3.2 The power γ2p3E in ηJK is slightly suboptimal with respect to the one used in the jump terms of the energy norm (3.3.1). This sub- optimality is due to the possible presence of hanging nodes in T . Indeed, for conforming meshes, the conforming hp-version Clément interpolant con- structed in [26] can be employed in our proof; see also [19]. As a conse- quence, Theorem 3.3.1 holds true with the following version of ηJK : η2JK = 1 2 ∑ E∈∂K\Γ [ γεp2E hE + hE εpE ] ‖[[uhp]]‖2L2(E) + ∑ E∈∂K∩Γ [ γεp2E hE + hE εpE ] ‖[[uhp]]‖2L2(E). 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 neces- sary 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 η . |p|δ+1‖u− uhp ‖E,T + |p|2δ+1|u− uhp |O,T + |p|2δ+ 1 2Θ. 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 estima- tor 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∈H10 (Ω)\{0} sup v∈H10 (Ω)\{0} A(u, v) (‖u ‖E,T + |au|?) ‖ v ‖E,T ≥ 1 3 . Next, we split the discontinuous Galerkin form Ahp into two parts, see Chapter 2, and define Ãhp(u, v) = ∑ K∈T ∫ K (ε∇u · ∇v + a · ∇uv) dx+ ∑ E∈E(T ) εγp2E hE ∫ E [[u]] · [[v]] ds − ∑ K∈T ∫ ∂Kin∩Γin a · nK uv ds+ ∑ K∈T ∫ ∂Kin\Γ a · nK(ue − u)v ds, Khp(u, v) = − ∑ E∈E(T ) ∫ E {{ε∇u}} · [[v]] ds− ∑ 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 A(u, v) = Ãhp(u, v), u, v ∈ H10 (Ω), (3.4.1) Ahp(u, v) = Ãhp(u, v) +Khp(u, v), u, v ∈ Sp(T ). (3.4.2) 3.4.2 Auxiliary meshes We shall make use of an auxiliary 1-irregular mesh T̃ 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 T̃ 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(T̃ ) the set of vertices in N (T̃ ) and EA(T̃ ) the set of edges in E(T̃ ) which are inside an element K of T , respectively. Moreover, we write R(K) for the elements in T̃ 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 T̃ from T . Next, we introduce the following auxiliary discontinuous Galerkin finite element space on the mesh T̃ : Sp̃(T̃ ) = { v ∈ L2(Ω) : v|K̃ ◦ FK̃ ∈ QpK̃ (K̂), K̃ ∈ T̃ }, where the auxiliary polynomial degree vector p̃ is defined by p K̃ = pK , for K̃ ∈ R(K). Thus, we clearly have the inclusion Sp(T ) ⊆ Sp̃(T̃ ). 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 T̃ are given by ‖ v ‖2 E,T̃ = ∑ K̃∈T̃ ε‖∇v‖2 L2(K̃) + ejump p̃,T̃ (v) 2, | v |2 O,T̃ = |av| 2 ? + ojumpp̃,T̃ (v) 2, (3.4.3) where the auxiliary edge polynomial degrees p Ẽ for the jump terms over T̃ are defined as in (3.2.4), using the auxiliary degrees p K̃ . Obviously, we have ‖ v ‖E,T = ‖ v ‖E,T̃ , | v |O,T = | v |O,T̃ , for all v ∈ H10 (Ω). Furthermore, the following result holds. Lemma 3.4.2 Let v ∈ Sp̃(T̃ ) + H10 (Ω) be such that [[v]]|E = [[w]]|E for all E ∈ E(T̃ ), for a function w ∈ Sp(T ) +H10 (Ω). Then we have ejumpp,T (w) . ejumpp̃,T̃ (v) . ejumpp,T (w), ojumpp,T (w) . ojumpp̃,T̃ (v) . ojumpp,T (w). Proof : Since w ∈ Sp(T ) + H10 (Ω), we have that [[v]]|E = [[w]]|E = 0 over newly created edges in EA(T̃ ). To look at the jumps over refined edges, let E ∈ ER(T ). We have E = E1 ∪ E2 with E1 and E2 in E(T̃ ) and hE1 = hE2 = hE/2. Thus hE1 εpE1 ‖[[v]]‖2L2(E1) + hE2 εpE2 ‖[[v]]‖2L2(E2) = 1 2 hE εpE ‖[[w]]‖2L2(E). We conclude that ojump p̃,T̃ (v) 2 = ∑ E∈E(T )∩E(T̃ ) hE εpE ‖[[w]]‖2L2(E) + ∑ E∈ER(T ) 1 2 hE εpE ‖[[w]]‖2L2(E). 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 ) +H10 (Ω), we have the bounds ‖ v ‖E,T . ‖ v ‖E,T̃ , | v |O,T . | v |O,T̃ . 71 3.4. Proofs Proof : Clearly, we have∑ K∈T ε‖∇v‖2L2(K) = ∑ K̃∈T̃ ε‖∇v‖2 L2(K̃) for v ∈ Sp(T ) +H10 (Ω). Applying Lemma 3.4.2 with w = v yields ejumpp,T (v) . ejumpp̃,T̃ (v), ojumpp,T (v) . ojumpp̃,T̃ (v). The assertion now follows readily. 2 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 Scp̃(T̃ ) be the conforming subspace of Sp̃(T̃ ) given by Scp̃(T̃ ) = Sp̃(T̃ ) ∩H10 (Ω). Theorem 3.4.4 (Averaging operator) There is operator Ihp : Sp(T )→ Scp̃(T̃ ) that satisfies∑ K̃∈T̃ ‖∇(v − Ihpv)‖2L2(K̃) . ∑ E∈E(T ) p2Eh −1 E ‖[[v]]‖2L2(E), (3.4.4)∑ K̃∈T̃ ‖v − Ihpv‖2L2(K̃) . ∑ E∈E(T ) p−2E hE‖[[v]]‖2L2(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 con- structed 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 esti- mates 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 = u c hp + u r hp, (3.4.6) 72 3.4. Proofs where uchp = Ihpuhp ∈ Scp̃(T̃ ) ⊂ H10 (Ω), with Ihp the approximation operator from Theorem 3.4.4. The remainder is then given by urhp = uhp − uchp = uhp − Ihpuhp ∈ Sp̃(T̃ ). By Lemma 3.4.3 and the triangle inequality, we obtain ‖u− uhp ‖E,T + |u− uhp |O,T . ‖u− uhp ‖E,T̃ + |u− uhp |O,T̃ . ‖u− uchp ‖E,T̃ + |u− uchp |O,T̃ + ‖urhp ‖E,T̃ + |urhp |O,T̃ = ‖u− uchp ‖E,T + |u− uchp |O,T + ‖urhp ‖E,T̃ + |urhp |O,T̃ . (3.4.7) 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 ‖urhp ‖E,T̃ + |urhp |O,T̃ . η. Proof : Since [[urhp]]|E = [[uhp]]|E for all E ∈ E(T̃ ) and uhp ∈ Sp(T ), the definition of the jump residual ηJK and Lemma 3.4.2 yield ‖urhp ‖2E,T̃ + |u r hp |2O,T̃ = ∑ K̃∈T̃ ε‖∇urhp‖2L2(K̃) + |au r hp|2? + ejumpp̃,T̃ (urhp)2 + ojumpp̃,T̃ (urhp)2 . ∑ K̃∈T̃ ε‖∇urhp‖2L2(K̃) + |au r hp|2? + ∑ K∈T η2JK . Hence, only the volume terms and |aurhp|? need to be bounded further. Since pE ≥ 1, Theorem 3.4.4 yields ε ∑ K̃∈T̃ ‖∇urhp‖2L2(K̃) . γ −1 ∑ E∈E(T ) εγp2E hE ‖[[uhp]]‖2L2(E) . γ−1 ∑ K∈T η2JK . To estimate |aurhp|?, we again use Theorem 3.4.4 and the fact that pE ≥ 1, |aurhp|2? . 1 ε ∑ K̃∈T̃ ( ‖a‖2 L∞(K̃)‖u r hp‖2L2(K̃) ) . ∑ E∈E(T ) hE εp2E ‖[[uhp]]‖2L2(E) . ∑ K∈T η2JK . 73 3.4. Proofs This finishes the proof. 2 Next, we recall the following standard hp-version approximation result from [23, Lemma 3.7]: For any v ∈ H10 (Ω), there exists a function vhp ∈ Sp(T ) such that p2K h2K ‖v − vhp‖2L2(K) + ‖∇(v − vhp)‖2L2(K) + pK hK ‖v − vhp‖2L2(∂K) . ‖∇v‖2L2(K), (3.4.8) for any K ∈ T . Lemma 3.4.7 For any v ∈ H10 (Ω), we have∫ Ω f(v − vhp) dx− Ãhp(uhp, v − vhp) +Khp(uhp, vhp) . (η + Θ) ‖ v ‖E,T . 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∫ Ω f(v−vhp) dx− Ãhp(uhp, v−vhp)+Khp(uhp, vhp) = T1 +T2 +T3 +T4 +T5, where T1 = ∑ K∈T ∫ K (f + ε∆uhp − a · ∇uhp)(v − vhp) dx, T2 = − ∑ E∈EI(T ) ∫ E [[ε∇uhp]]{{v − vhp}} ds, T3 = − ∑ E∈E(T ) ∫ E {{ε∇vhp}} · [[uhp]] ds, T4 = ∑ K∈T ∫ ∂Kin\Γ a · nK(uhp − uehp)(v − vhp) ds + ∑ K∈T ∫ ∂Kin∩Γin a · nKuhp(v − vhp) ds, T5 = − ∑ E∈E(T ) εγp2E hE ∫ 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 . ( ∑ K∈T (η2RK + Θ 2 K) ) 1 2 ‖ v ‖E,T . Similarly, by the Cauchy-Schwarz inequality and (3.4.8), we have T2 . ( ∑ E∈EI(T ) ε−1p−1E hE‖[[ε∇uhp]]‖2L2(E) ) 1 2 ( ∑ E∈EI(T ) εpEh −1 E ‖v − vhp‖2L2(E) ) 1 2 . ( ∑ K∈T η2EK ) 1 2 ‖ v ‖E,T . To estimate T3, we employ the Cauchy-Schwarz inequality, the hp-version trace inequality and the H1-stability of vhp from (3.4.8). This results in T3 . ( ∑ E∈E(T ) εp2Eh −1 E ‖[[uhp]]‖2L2(E) ) 1 2 ( ∑ K∈T εp−2K hK‖∇vhp‖2L2(∂K) ) 1 2 . ( ∑ K∈T η2JK ) 1 2 ( ∑ K∈T ε‖∇vhp‖2L2(K) ) 1 2 . ( ∑ K∈T η2JK ) 1 2 ‖ v ‖E,T . For T4, we apply again the Cauchy-Schwarz inequality and (3.4.8) to get T4 . ( ∑ E∈E(T ) ε−1p−1E hE‖[[uhp]]‖2L2(E) ) 1 2 ( ∑ E∈E(T ) εpEh −1 E ‖v − vhp‖2L2(E) ) 1 2 . ( ∑ K∈T η2JK ) 1 2 ‖ v ‖E,T . Finally, we have T5 . ( ∑ E∈E(T ) εγ2p3Eh −1 E ‖[[uhp]]‖2L2(E) ) 1 2 ( ∑ E∈E(T ) εpEh −1 E ‖v − vhp‖2L2(E) ) 1 2 . ( ∑ K∈T η2JK ) 1 2 ‖ v ‖E,T . The above estimates for the terms T1 through T5 imply the assertion. 2 Lemma 3.4.8 There holds: ‖u− uchp ‖E,T + |u− uchp |O,T . η + Θ. 75 3.4. Proofs Proof : Since u− uchp ∈ H10 (Ω), we have |u− uchp |O,T = |a(u− uchp)|?. Then the inf-sup condition in Lemma 3.4.1 yields: ‖u− uchp ‖E,T + |u− uchp |O,T . sup v∈H10 (Ω)\{0} A(u− uchp, v) ‖ v ‖E,T . (3.4.9) To bound (3.4.9), let v ∈ H10 (Ω). Then, property (3.4.1) shows that A(u− uchp, v) = ∫ Ω fv dx−Ahp(uchp, v) = ∫ Ω fv dx− Ãhp(uchp, v). By employing the fact that v ∈ H10 (Ω) and integrating by parts the convec- tion term, one can readily see that Ãhp(u c hp, v) = Ãhp(uhp, v) +R, with R = ∑ K̃∈T̃ ∫ K̃ (−ε∇urhp + aurhp) · ∇v dx. Furthermore, from the DG method in (3.2.6) and property (3.4.2), we have∫ Ω fvhp dx = Ãhp(uhp, vhp) +Khp(uhp, vhp), where vhp ∈ Sp(T ) is the hp-version interpolant of v introduced in (3.4.8). Combining the above results yields A(u− uchp, v) = ∫ Ω f(v − vhp) dx− Ãhp(uhp, v − vhp) +Khp(uhp, vhp)−R. The estimate in Lemma 3.4.7 now shows that |A(u− uchp, v)| . (η + Θ) ‖ v ‖E,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| . ( ‖urhp ‖E,T̃ + |urhp |O,T̃ ) ‖ v ‖E,T . η‖ v ‖E,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 inequal- ity (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 K̂ = (−1, 1)2, we define the weight function Ψ K̂ (x̂) = dist(x̂, ∂K̂). For an arbitrary element K ∈ T , we set ΨK = cKΨK̂ ◦ F−1K , where cK is a scaling factor chosen such that ∫ K(ΨK − 1) dx = 0. Similarly, on the reference interval Î = (−1, 1), we define the weight function Ψ Î (x̂) = 1− x̂2. For an interior edge E, we let ΨE = cEΨÎ ◦ F−1E , where FE is the affine transformation that maps Î 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 ( ∑ K∈T η2RK ) 1 2 . |p| ‖u− uhp ‖E,T + |p| |a(u− uhp)|? + |p|δ+ 1 2Θ. Proof : For any element K ∈ T , we set vK = ε −1(fhp + ε∆uhp − ahp · ∇uhp)|KΨαK , where α ∈ (12 , 1]. Applying the inverse inequality from [27, Theorem 2.5], we obtain ‖fhp + ε∆uhp − ahp · ∇uhp‖L2(K) . pαK‖(fhp + ε∆uhp − ahp · ∇uhp)Ψα/2K ‖L2(K) = εpαK‖vKΨ−α/2K ‖L2(K). This leads to∑ K∈T η2RK . S 2 with S2 = ∑ K∈T p2α−2K h 2 Kε‖vKΨ−α/2K ‖2L2(K). 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 , S2 = ∑ K∈T p2α−2K h 2 K ∫ K (fhp + ε∆uhp − ahp · ∇uhp)vK dx = ∑ K∈T p2α−2K h 2 K ∫ K (ε∇(u− uhp)− a(u− uhp)) · ∇vK dx + ∑ K∈T p2α−2K h 2 K ∫ K (((fhp − f) + (a− ahp) · ∇uhp)ε− 1 2Ψ α 2 K)(ε 1 2 vKΨ −α 2 K )dx. 77 3.4. Proofs Here, we have also used that vK |∂K = 0. From the proof of [27, Lemma 3.4], we have ‖∇vK‖L2(K) . h−1K p2−αK ‖vKΨ−α/2K ‖L2(K). By the Cauchy-Schwarz inequality, the definition of the dual norm and the data approximation error Θ, we obtain S2 . S (|p|‖u− uhp ‖E,T + |p||a(u− uhp)|? + |p|αΘ) . Therefore, ( ∑ K∈T η2RK ) 1/2 . |p| ‖u− uhp ‖E,T + |p| |a(u− uhp)|? + |p|αΘ. 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 }, w̃E = { K̃ ∈ T ∪T̃ : E ∈ E(K̃) }. For simplicity, we also use the notation wE and w̃E to denote the domain formed by the elements in wE and in w̃E , respectively. Lemma 3.4.10 Under the assumptions of Theorem 3.3.3, there holds ( ∑ K∈T η2EK ) 1 2 . |p|δ+1‖u− uhp ‖E,T + |p|2δ+1|u− uhp |O,T + |p|2δ+ 1 2 Θ. 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 ∈ H10 (wE) with ψE |E = τE , ψE |∂wE = 0 and ‖ψE‖L2(wE) . h1/2E p−1E ‖τEΨ−α/2E ‖L2(E), (3.4.12) ‖∇ψE‖L2(wE) . h−1/2E pE‖τEΨ−α/2E ‖L2(E). (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. In this case, wE is concave, and there exists an element K̃1 ∈ T̃ , such that 78 3.4. Proofs K̃1 ( K1 and K̃1 ∩K2 = E. Thus w̃E = K̃1 ∪K2 ( wE . By Lemma 2.6 of [27] we can find a function ψ̃E ∈ H10 (w̃E) with ψ̃E |E = τE , ψ̃E |∂w̃E = 0 and ‖ψ̃E‖L2(w̃E) . h1/2E p−1E ‖τEΨ−α/2E ‖L2(E), ‖∇ψ̃E‖L2(w̃E) . h−1/2E pE‖τEΨ−α/2E ‖L2(E). Now define the function ψE on wE by ψE = ψ̃E on w̃E , and by zero oth- erwise. Thus, we have ψE ∈ H10 (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 ‖[[∇uhp]]‖L2(E) . pαE ‖[[∇uhp]]Ψ α 2 E‖L2(E) = pαE ‖τEΨ −α 2 E ‖L2(E). (3.4.14) Therefore,∑ K∈T η2EK . S 2 with S2 = ∑ E∈EI(T ) p2α−1E hEε‖τEΨ −α 2 E ‖2L2(E). Since [[ε∇u]] = 0 on interior edges, integration by parts over wE yields∫ E [[ε∇(uhp − u)]]τE ds = ∫ wE ε(∆uhp −∆u)ψE + ε(∇uhp −∇u) · ∇ψE dx, where ∆uhp and ∇uhp are understood piecewise. Using the differential equa- tion, approximating the data and integrating by parts the convective term, we readily obtain S2 = T1 + T2 + T3 + T4 + T5, with T1 = ∑ E∈EI(T ) ∫ wE p2α−1E hE (fhp + ε∆uhp − ahp · ∇uhp)ψE dx, T2 = ∑ E∈EI(T ) ∫ wE p2α−1E hE (ε∇uhp − ε∇u) · ∇ψE dx, T3 = ∑ E∈EI(T ) ∫ wE p2α−1E hE a(u− uhp) · ∇ψE dx, T4 = ∑ E∈EI(T ) ∫ E p2α−1E hE a · [[uhp]]τE ds, T5 = ∑ E∈EI(T ) ∫ wE p2α−1E 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 T1 . S |p|α− 12 (|p| ‖u− uhp ‖E,T + |p| |a(u− uhp)|? + |p|αΘ). Similarly, we obtain T2 . S |p| 12+α‖u− uhp ‖E,T , as well as T3 . S |p| 12+α|a(u− uhp)|?. To bound T4, we first notice that ‖ΨE‖L∞(E) = cE ≡ 32 . By (3.4.14) and the definition of semi-norm | · |O,T , we conclude that T4 . ∑ E∈EI(T ) ε− 1 2 h 1 2 E p − 1 2 E ‖[[uhp]]‖L2(E) ε 1 2 h 1 2 E p 3α− 1 2 E ‖τEΨ −α 2 E ‖L2(E) . S |p|2α|u− uhp |O,T . Finally, the data error term T5 can be bounded by T5 . ∑ E∈EI(T ) ε− 1 2 p−1E hE‖(f − fhp) + (ahp − a) · ∇uhp‖L2(wE) × ε 12 p2αE ‖ψE‖L2(wE) . S |p|α− 12Θ. Combining the above bounds for T1 through T5, we obtain S2 . S ( |p|α+ 12 ‖u− uhp ‖E,T + |p|2α|u− uhp |O,T + |p|2α− 1 2Θ ) . Thus,(∑ K∈T η2EK ) 1 2 . |p|α+ 12 ‖u− uhp ‖E,T + |p|2α|u− uhp |O,T + |p|2α− 1 2Θ. 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 ( ∑ K∈T η2JK ) 1/2 . |p| 12 ‖u− uhp ‖E,T + |u− uhp |O,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 Î = (−1, 1) be the reference interval. We denote by Ẑp = { ẑp0 , · · · , ẑpp } the Gauss-Lobatto nodes of order p ≥ 1 on Î. Recall that ẑp0 = −1 and ẑpp = 1. We denote by Ẑpint = { ẑp1 , · · · , ẑpp−1 } the interior Gauss-Lobatto nodes of order p on Î. Let now E ∈ E(K) be an edge of an element K. The nodes in Ẑp can be affinely mapped onto E and we denote by Zp(E) = { zE,p0 , · · · , zE,pp } the Gauss-Lobatto nodes of order p on E. The points zE,p0 and z E,p p coincide with the two end points of E. The set Zpint(E) = { zE,p1 , · · · , zE,pp−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 P intp (E) = { q ∈ Pp(E) : q(zE,p0 ) = q(zE,pp ) = 0 }, Pnodp (E) = { q ∈ Pp(E) : q(z) = 0, z ∈ Zpint(E) }. By construction, we have Pp(E) = P intp (E)⊕ Pnodp (E). For an element K and p ≥ 1, we now define basis functions for polyno- mials 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. We first consider the reference element K̂ = (−1, 1)2. We denote its four edges by Ê1, . . . , Ê4 and its four vertices by ν̂1, . . . , ν̂4, numbered as in Figure 3.2. Let {ϕ̂pi }0≤i≤p be the Lagrange basis functions associated with the nodes Ẑp. We denote by {ẑpi,j = (ẑpi , ẑpj )}1≤i,j≤p the interior tensor- product Gauss-Lobatto nodes on K̂. The interior basis functions are then given by Φ̂int,pi,j (x̂1, x̂2) = ϕ̂ p i (x̂1) ϕ̂ p j (x̂2), 1 ≤ i, j ≤ p− 1. 81 3.5. Proof of Theorem 3.4.4 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Ê1 Ê4 Ê3 Ê2 ν̂1 ν̂3ν̂4 ν̂2 Figure 3.2: Reference element with variable edge polynomial degrees: p = 5, p Ê1 = 2, p Ê2 = 3, p Ê3 = 4, p Ê4 = 1. Next, we consider as an example the edge Ê1 in Figure 3.2 with edge de- gree p Ê1 . The edge basis functions for Ê1 are Φ̂ Ê1,pÊ1 i (x̂1, x̂2) = ϕ̂ p Ê1 i (x̂1) ϕ̂ p 0(x̂2), i = 1, · · · , pÊ1 − 1. Note that Φ̂ Ê1,pÊ1 i vanishes on Ê2, Ê3 and Ê4. The other edge basis functions are defined analogously. Finally, we consider the vertex ν̂1, which is shared by Ê1 and Ê4; see Figure 3.2. We then introduce the associated vertex basis function Φ̂ν̂1 K̂ (x̂1, x̂2) = ϕ̂ p Ê1 0 (x̂1) ϕ̂ p Ê4 0 (x̂2). It vanishes on Ê2 and Ê3. The vertex basis functions associated with the other vertices of K̂ are defined analogously. This completes the definition of the shape functions on the reference element K̂. For an arbitrary parallelogram K, shape functions Φ on K can be defined from the analogous ones on K̂ by using the pull-back map Φ = Φ̂ ◦ F−1K , giving rise to shape functions ΦνK , Φ E,pE i and Φ int,p i,j on K. Therefore, a polynomial v of the form (3.5.1) can be expanded into v(x) = ∑ ν∈N (K) v(ν) ΦνK(x) + ∑ E∈E(K) pE−1∑ i=1 v(zE,pEi ) Φ E,pE i (x) + ∑ 1≤i, j≤p−1 cij Φ int,p i,j (x), 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 v̂ ∈ Qp(Îd), d = 1, 2, that vanishes at the interior Gauss-Lobatto nodes on Îd, there holds ‖v̂‖2 L2(Îd) . p−2‖v̂‖2 L2(∂Îd) . (ii) If the vertex ν̂ of the reference element K̂ is shared by two edges Ên and Êm, the associated vertex basis function Φ̂ ν̂ K̂ can be bounded by ‖Φ̂ν̂ K̂ ‖ L2(K̂) . p−1 Ên p−1 Êm . 3.5.2 Extension operators Next, we define extension operators over edges. Let Ê ∈ E(K̂) be an ele- mental edge of the reference element K̂. We define L̂Êp by L̂Êp : P intp (Ê) −→ Qp(K̂), q̂(x) 7−→ p−1∑ i=1 q̂(ẑÊ,pi )Φ̂ Ê,p i (x). (3.5.2) By construction, L̂Êp (q̂) = q̂ on Ê, and L Ê p (q̂) vanishes in all the interior tensor-product Gauss-Lobatto nodes {ẑpi,j}1≤i,j≤p−1 of K̂ and on the other three edges of K̂. From [10, Lemma 3.1], we have the following inequality. Lemma 3.5.2 The linear extension operator L̂Êp introduced in (3.5.2) sat- isfies ‖L̂Êp (q̂)‖L2(K̂) . p−1‖q̂‖L2(Ê). 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 operator LEp,K(q) : P intp (E)→ Qp(K) by LEp,K(q) = [L Ê p (q ◦ FK)] ◦ F−1K , q ∈ P intp (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 ∈ P intp (E1) and q2 ∈ P intp (E2), we define the extension operator LEp,K(q1, q2) by LEp,K(q1, q2) = L E1 p,K1 (q1) + L E2 p,K2 (q2), (3.5.4) 83 3.5. Proof of Theorem 3.4.4 with LE1p,K1 and L E2 p,K2 given in (3.5.3). By definition, the extensions LEp,K(q) and L E p,K(q1, q2) are continuous in K and satisfy LEp,K(q)|E = q, LEp,K(q1, q2)|E1 = q1 and LEp,K(q1, q2)|E2 = q2. Moreover, LEp,K(q) and L E p,K(q1, q2) both vanish on the other edges of E(K). ν̃2 ν3 K̃ ν̃4 ν̃3 ν̃1ν1 ν4 ν2 E4 E3 Ẽ4 Ẽ2 Ẽ1 Ẽ3 E2 E1 E1 E2 E K2 K1 Figure 3.3: Left: Partition of K into K1 and K2. Right: Element K and K̃ ∈ R(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(T̃ ), we set pE = min{ pK̃ : K̃ ∈ w̃E }. (3.5.5) Notice that an elemental edge E in E(K̃), K̃ ∈ T̃ , belongs to E(T ) ∪ E(T̃ ). Hence, for any K̃ ∈ T̃ , equation (3.5.5) defines the elemental edge polyno- 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 ∪ T̃ . Let v ∈ Sp(T ). Firstly, we introduce a (nodal) interpolant vnod ∈ Sp̃(T̃ ). For each element K ∈ T and K̃ ∈ R(K), we will construct the restriction vnod K̃ of vnod to K̃ such that vnod K̃ ∈ QpK (K̃), vnodK̃ |E ∈ PpE (E), E ∈ E(K̃), (3.5.6) with pE given in (3.5.5). To define v nod K̃ , we distinguish two cases. Case 1: If R(K) = {K} (i.e., if K is unrefined), the interpolant vnod K̃ = vnodK is simply defined by vnodK (x) = ∑ ν∈N (K) vK(ν) Φ ν K(x). (3.5.7) 84 3.5. Proof of Theorem 3.4.4 Case 2: If R(K) consists of four newly created elements, we define vnod K̃ on each element K̃ ∈ R(K) separately. To do so, fix K̃ ∈ R(K). Without loss of generality, we may consider the situation shown in Figure 3.3 (right), where {Ei}4i=1 and {νi}4i=1 denote the edges and vertices of K, {Ẽi}4i=1 and {ν̃i}4i=1 the ones of K̃. Notice that here we have ν̃2 = ν2 and ν̃4 ∈ NA(T̃ ). Furthermore, Ẽ3 and Ẽ4 are in EA(T̃ ) and pẼ3 = pẼ4 = pK̃ = pK . Let us now define the value of vnod K̃ at the edge and vertex nodes of K̃. At the interior nodes of Ẽ3 and Ẽ4, we set vnod K̃ (z) = vK(z), z ∈ Z p Ẽ3 int (Ẽ3) ∪ Z p Ẽ4 int (Ẽ4). (3.5.8) Similarly, we set vnod K̃ (ν) = vK(ν) for the vertices ν = ν̃2 and ν = ν̃4. It remains to define the values of vnod K̃ on the nodes of the edges Ẽ1 and Ẽ2, as well as on ν̃1 and ν̃3. We only consider ν̃1 and Ẽ1 (the construction for ν̃3 and Ẽ2 is completely analogous). If ν̃1 ∈ N (T ) (i.e., ν̃1 is a hanging node in T ), then we define vnod K̃ (z) = 0, z ∈ ZpẼ1int (Ẽ1), vnodK̃ (ν̃1) = vK(ν̃1). (3.5.9) On the other hand, if ν̃1 /∈ N (T ), then E1 ∈ E(T ) and ν1 ∈ N (T ). In this case, we interpolate the values of the nodal interpolant over the long edge E1 at the Gauss-Lobatto nodes on Ẽ1. That is, we set vnod K̃ (z) = vK(ν1) Φ ν1 K (z)+vK(ν2) Φ ν2 K (z), z ∈ Z p Ẽ1 int (Ẽ1)∪{ν̃1}. (3.5.10) With the nodal values of vnod K̃ constructed (3.5.8)–(3.5.10), we have vnod K̃ (x) = ∑ ν∈N (K̃) vnod K̃ (ν) Φν K̃ (x) + ∑ E∈E(K̃) pE−1∑ i=1 ( vnod K̃ (z E,pE i )Φ E,pE i (x) ) . This finishes the construction of the interpolant of vnod. Notice that vnod ∈ Sp̃(T̃ ) is continuous over edges E ∈ EA(T̃ ) and satisfies vK(ν)− vnodK (ν) = 0, ν ∈ N (T ) located on ∂K, (3.5.11) as well as vnod K̃ |E ∈ PnodpE (E), E ∈ E(T ), K̃ ∈ w̃E . (3.5.12) 85 3.5. Proof of Theorem 3.4.4 Secondly, we construct a function vedge ∈ Sp̃(T̃ ) related to the edge degrees of freedom. To do so, fix an element K ∈ T . For any edge E ∈ E(K), we define vEK by vEK = { LEpK ,K((vK − vnodK )|E), E ∈ E(T ), LEpK ,K((vK − vnodK )|E1 , (vK − vnodK )|E2), E = E1 ∪ E2, E1,2 ∈ E(T ), with LEpK ,K(·) in (3.5.3) and LEpK ,K(·, ·) in (3.5.4), respectively. We then define vedge elementwise as vedgeK (x) = ∑ E∈E(K) vEK(x). Thirdly, we construct a function vint ∈ Sp̃(T̃ ) simply by setting elemen- twise vintK = vK − vnodK − vedgeK , K ∈ T . Notice that vintK belongs to H 1 0 (K). Hence, we have v int ∈ Scp̃(T̃ ). In conclusion, any function v ∈ Sp(T ) can be decomposed into three parts: v = vnod + vedge + vint, (3.5.13) with vnod, vedge and vint in Sp̃(T̃ ) 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 = vnod + vedge + vint, according to (3.5.13). We will define the averaging operator Ihpv in three parts: Ihpv = ϑ nod + ϑedge + ϑint, (3.5.14) with ϑnod, ϑedge, ϑint ∈ Scp̃(T̃ ). Since vint ∈ Scp̃(T̃ ), we simply take ϑint = vint. 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 ∈ Scp̃(T̃ ) that satisfies ∑ K̃∈T̃ ‖vnod − ϑnod‖2 L2(K̃) . ∑ E∈E(T ) ∫ E p−2E hE [[v nod]]2ds, ∑ K̃∈T̃ ‖∇(vnod − ϑnod)‖2 L2(K̃) . ∑ E∈E(T ) ∫ E p2Eh −1 E [[v nod]]2ds. 86 3.5. Proof of Theorem 3.4.4 Lemma 3.5.4 There is a conforming approximation ϑedge ∈ Scp̃(T̃ ) that satisfies ∑ K̃∈T̃ ‖vedge − ϑedge‖2 L2(K̃) . ∑ E∈E(T ) ∫ E p−2E hE([[v]] 2 + [[vnod]]2)ds, ∑ K̃∈T̃ ‖∇(vedge − ϑedge)‖2 L2(K̃) . ∑ E∈E(T ) ∫ E p2Eh −1 E ([[v]] 2 + [[vnod]]2)ds. By the triangle inequality and Lemmas 3.5.3 and 3.5.4, we then obtain∑ K̃∈T̃ ‖v − Ihpv‖2L2(K̃) . ∑ E∈E(T ) ∫ E p−2E hE ( [[v]]2 + [[vnod]]2 ) ds, ∑ K̃∈T̃ ‖∇(v − Ihpv)‖2L2(K̃) . ∑ E∈E(T ) ∫ E p2Eh −1 E ( [[v]]2 + [[vnod]]2 ) ds. Theorem 3.4.4 now follows if we show that ‖[[vnod]]‖2L2(E) . ‖[[v]]‖2L2(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 vnod, the jump over E satisfies [[vnod]](νi) = [[v]](νi), i = 1, 2. Since [[vnod]] vanishes on all the interior Gauss-Lobatto nodes on E, item (i) in Lemma 3.5.1 and a scaling argument yield ‖[[vnod]]‖2L2(E) . p−2E hE([[vnod]](ν1)2 + [[vnod]](ν2)2) = p−2E hE([[v]](ν1) 2 + [[v]](ν2) 2) . p−2E hE‖[[v]]‖2L∞(E). From [33, Theorem 3.92], we further have the inverse estimate ‖[[v]]‖2L∞(E) . p2Eh−1E ‖[[v]]‖2L2(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 vnod ∈ Sp̃(T̃ ) be the nodal interpolant in the decomposition (3.5.13). We now shall construct the conforming approximation ϑnod in Scp̃(T̃ ). For simplicity, we shall omit the superscript ”nod” and, in the sequel, write v for vnod and ϑ for ϑnod. For a node ν, we introduce the sets: w̃(ν) = { K̃ ∈ T̃ : ν ∈ N (K̃) }, wE(ν) = {E ∈ E(T ) : ν ∈ E }. Fix K ∈ T and K̃ ∈ R(K). We proceed by distinguishing the same two cases as in Subsection 3.5.3. Case 1: If R(K) = {K}, we have K = K̃. Then any elemental edge Ẽ ∈ E(K̃) belongs to E(T ) and we have v K̃ | Ẽ ∈ Pnodp Ẽ (Ẽ). For any Gauss- Lobatto node ν located on ∂K̃, we define the value of ϑ(ν) by ϑ(ν) =  |w̃(ν)|−1 ∑ K̃∈w̃(ν) v K̃ (ν), ν ∈ NI(T ), 0, otherwise. (3.5.16) Here, |w̃(ν)| denotes the cardinality of the set w̃(ν). In the case considered, we have |w̃(ν)| = 4 for ν ∈ NI(T ). Then we define ϑ on K̃ by ϑ(x) = ∑ ν∈N (K̃) ϑ(ν) Φν K̃ (x). (3.5.17) From (3.5.7) and (3.5.17), we have ‖v K̃ − ϑ‖ L2(K̃) . ∑ ν∈N (K̃) |v K̃ (ν)− ϑ(ν)| ‖Φν K̃ ‖ L2(K̃) . (3.5.18) Analogously to [10, Page 1125], we have |v K̃ (ν)− ϑ(ν)| . ∑ E∈wE(ν) pEh −1/2 E ‖[[v]]‖L2(E). (3.5.19) Hence, by combining (3.5.18), (3.5.19) and item (ii) in Lemma 3.5.1 (with scaling), we obtain ‖v K̃ − ϑ‖2 L2(K̃) . ∑ E∈{wE(ν)} ν∈N (K̃) ∫ E p−2E hE [[v]] 2ds. (3.5.20) 88 3.5. Proof of Theorem 3.4.4 Case 2: If R(K) consists of four elements, we define ϑ on each ele- ment K̃ ∈ R(K) separately, analogously to the construction of the nodal interpolant in Subsection 3.5.3. Without loss of generality, we may again consider the case illustrated in Figure 3.3 (right). Since Ẽ3, Ẽ4 ∈ EA(T̃ ), the function v is continuous over Ẽ3 and Ẽ4. The values of ϑ on the edge nodes z ∈ ZpK̃int (Ẽ3) ∪ Z p K̃ int (Ẽ4), and the vertex ν̃4 are defined by ϑ(z) = v K̃ (z), z ∈ ZpK̃int (Ẽ3) ∪ Z p K̃ int (Ẽ4), ϑ(ν̃4) = vK̃(ν̃4). (3.5.21) We further define the value of ϑ on the vertex ν̃2 by (3.5.16). It remains to define the values of ϑ for the nodes on the edges Ẽ1 and Ẽ2, as well as for ν̃1 and ν̃3. We only consider ν̃1 and Ẽ1 (the construction for ν̃3 and Ẽ2 is completely analogous). If ν̃1 ∈ N (T ), then ν̃1 is a hanging node of T and Ẽ1 ∈ E(T ). Thus, vK̃ |Ẽ1 ∈ PnodpẼ1 (Ẽ1). For any z ∈ Z p Ẽ1 int (Ẽ1) ∪ {ν̃1}, the value of ϑ(z) is taken as in (3.5.16). On the other hand, if ν̃1 /∈ N (T ), then E1 ∈ E(T ) and vK |E1 ∈ PnodpE1 (E1). We define ϑ(ν1) again by (3.5.16). Recall that ϑ(ν2) = ϑ(ν̃2) has already been defined. Then we set ϑ(z) = ϑ(ν1)Φ ν1 K (z) + ϑ(ν2)Φ ν2 K (z), z ∈ Z p Ẽ1 int (Ẽ1) ∪ {ν̃1}. (3.5.22) Now we construct ϑ on K̃ by setting ϑ(x) = ∑ ν∈N (K̃) ϑ(ν)Φν K̃ (x) + ∑ Ẽ∈E(K̃) p Ẽ −1∑ i=1 ( ϑ(z Ẽ,p Ẽ i )Φ Ẽ,p Ẽ i (x) ) . This completes the construction of ϑ. Clearly, ϑ ∈ Scp̃(T̃ ). We shall now derive an estimate analogous to (3.5.20). To do so, we bound the difference between v K̃ and ϑ on K̃ as follows: ‖v K̃ − ϑ‖ L2(K̃) . ∑ ν∈N (K̃) ‖(v K̃ (ν)− ϑ(ν))Φν K̃ ‖ L2(K̃) + ∑ Ẽ∈E(K̃) ‖ς Ẽ ‖ L2(K̃) . ∑ ν∈N (K̃) ‖(v K̃ (ν)− ϑ(ν))Φν K̃ ‖ L2(K̃) + p−1 K̃ h 1/2 K̃ ∑ Ẽ∈E(K̃) ‖ς Ẽ ‖ L2(Ẽ) = T1 + T2, (3.5.23) with ς Ẽ (x) = ∑p Ẽ −1 i=1 (( v K̃ (z Ẽ,p Ẽ i ) − ϑ(z Ẽ,p Ẽ i ) ) Φ Ẽ,p Ẽ i (x) ) . For the second 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 ς Ẽ (x) vanishes at all the interior tensor- product Gauss-Lobatto nodes in K̃ and on the edges of K̃ that are different from Ẽ. Let us first bound the term T1 in (3.5.23). If the node ν ∈ NA(T̃ ), then, by (3.5.21), ( v K̃ (ν)− ϑ(ν))Φν K̃ (x) = 0, x ∈ K̃. (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 ‖(v K̃ (ν)− ϑ(ν))Φν K̃ ‖ L2(K̃) . ∑ E∈wE(ν) p−1E h 1 2 E‖[[v]]‖L2(E). (3.5.25) Finally, if ν /∈ N (T ) ∪NA(T̃ ), 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 |v K̃ (ν)− ϑ(ν)| ≤ |(vK(ν1)− ϑ(ν1))|+ |(vK(ν2)− ϑ(ν2))|. Thus, as before, we obtain ‖(v K̃ (ν)− ϑ(ν))Φν K̃ ‖ L2(K̃) . ∑ E∈wE(ν1)∪wE(ν2) p−1E h 1 2 E‖[[v]]‖L2(E). (3.5.26) To combine the results in (3.5.24)-(3.5.26), we define the set N ?(K̃) as follows. We start from N (K̃) and first remove all the vertices belonging to NA(T̃ ). Then, any vertex ν̃ ∈ N (K̃) with ν̃ /∈ N (T ) ∪ NA(T̃ ) is replaced by the vertex ν ∈ N (K) which is on the same elemental edge of K as ν̃. For example, in the case shown in Figure 3.3 (right), we have N ?(K̃) = { ν1, ν̃2, ν̃3 } if ν̃1 /∈ N (T ) and ν̃3 ∈ N (T ). We also set E?(K̃) = {E ∈ wE(ν) : ν ∈ N ?(K̃) }. In conclusion, the term T1 is bounded by T1 . ∑ E∈E?(K̃) p−1E h 1 2 E‖[[v]]‖L2(E). (3.5.27) Next, let us estimate the term T2 in (3.5.23). If Ẽ ∈ E(T ) or Ẽ ∈ EA(T̃ ), by the constructions of v and ϑ, we clearly have ‖ς Ẽ ‖ L2(Ẽ) = 0. Otherwise, 90 3.5. Proof of Theorem 3.4.4 one of the two end points of Ẽ, say ν̃1, is a newly created node in T̃ and the other one, ν̃2, belongs to N (T ). Thus, we have p−1 K̃ h 1 2 K̃ ‖ς Ẽ ‖ L2(Ẽ) ≤ p−1 K̃ h 1 2 K̃ 2∑ i=1 ‖(v K̃ (ν̃i)− ϑ(ν̃i) ) Φν̃i K̃ ‖ L2(Ẽ) + p−1 K̃ h 1 2 K̃ ‖v K̃ − ϑ‖ L2(Ẽ) = T21 + T22. Then there exists an elemental edge E ∈ E(K) such that ν̃1 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 T21 . ∑ E∈wE(ν1)∪wE(ν2) p−1E h 1 2 E‖[[v]]‖L2(E). In view of (3.5.22), we proceed as in (3.5.19) and obtain T22 . p−1 K̃ h 1 2 K̃ ‖ 2∑ i=1 (( vK(νi)− ϑ(νi) ) ΦνiK ) ‖L2(E) . ∑ E∈wE(ν1)∪wE(ν2) p−1E h 1 2 E‖[[v]]‖L2(E). Combining the above results shows that T2 . ∑ E∈E?(K̃) p−1E h 1 2 E‖[[v]]‖L2(E). (3.5.28) The bounds for T1 and T2 in (3.5.27) and (3.5.28) yield ‖v K̃ − ϑ‖2 L2(K̃) . ∑ E∈E?(K̃) ∫ E p−2E hE [[v]] 2ds. (3.5.29) This finishes the discussion of Case 2. Thus, by the key estimates in (3.5.20) and (3.5.29), we have ‖v K̃ − ϑ‖2 L2(K̃) . ∑ E∈E?(K̃) ∫ E p−2E hE [[v]] 2ds, K̃ ∈ T̃ . (3.5.30) 91 3.5. Proof of Theorem 3.4.4 This proves the first inequality in Lemma 3.5.3. Moreover, by the inverse inequality ‖∇v‖ L2(K̃) . p2 K̃ h−1 K̃ ‖v‖ L2(K̃) , v ∈ Sp̃(T̃ ), K̃ ∈ T̃ , (3.5.31) see [33], we obtain from (3.5.30) ‖∇(v K̃ − ϑ)‖2 L2(K̃) . ∑ E∈E?(K̃) ∫ E p2Eh −1 E [[v]] 2ds, K̃ ∈ T̃ , (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 the function WEK as follows: If E ∈ EB(T ), we set WEK = L E pK ,K ( (vK − vnodK )|E ) , with the extension operator LEpK ,K(·) in (3.5.3). If E ∈ EI(T ), let K ′ in T̃ be the element such that E is also an elemental edge of K ′, that is, E ∈ E(K ′). Denote by K ′ the element which has the lower polynomial degree of the elements K and K ′, i.e., K ′ = K if pK ≤ pK′ and K = K ′ otherwise. We define WEK by WEK = L E pK ,K ( (vK′ − vnodK′ )|E ) . with LEpK ,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 ′,K ′′ ∈ T such that E1 ∈ E(K ′) and E2 ∈ E(K ′′). Denote by K ′ the element that has the lower polynomial degree of K and K ′, and by K ′′ the element that has the lower degree of K and K ′′. We now define WEK by WEK = L E pK ,K ( (vK′ − vnodK′ )|E1 , (vK′′ − vnodK′′ )|E2 ) , with LEpK ,K(·, ·) in (3.5.4). Then we define ϑedge elementwise by setting ϑedge|K = ∑ E∈E(K)W E K , with WEK defined above. Clearly, the function ϑ edgebelongs to Scp̃(T̃ ). Next, 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∑ K̃∈T̃ ‖vedge − ϑedge‖2 L2(K̃) = ∑ K∈T ∑ K̃∈R(K) ‖vedge − ϑedge‖2 L2(K̃) . ∑ K∈T ∑ E∈E(K) ‖LEpK ,K ( (vK − vnodK )|E )−WEK‖2L2(K) . ∑ K∈T ∑ E∈E(K) p−2K hK‖(vK − vnodK )|E −WEK |E‖2L2(E) . ∑ K∈T ∑ E∈E(K) ∫ E p−2E hE([[v]] 2 + [[vnod]]2) ds. 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 refine- ment 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 algo- rithm 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 lx = 6 2 ∑pK i=0 imi − pK ∑pK i=0mi (pK + 1)((pK + 1)2 − 1) , ly = 6 2 ∑pK i=0 ini − pK ∑pK i=0 ni (pK + 1)((pK + 1)2 − 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; other- wise, 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 per- formed 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 L 2-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 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 [33], we thus plot all computed quantities against N 1 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) = (ex1−1ε − 1 e− 1 ε − 1 + x1 − 1 )(ex2−1ε − 1 e− 1 ε − 1 + x2 − 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 uni- form 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 over- estimates the true energy error ‖u− uhp ‖E,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 ε− 1 2 ‖u − uh‖L2(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 behav- ior is observed for the error (∑ E∈E(T )(ε −1p−1E hE‖[[uh]]‖2L2(E)) )1/2 shown 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 Θ 2 K = ε−1p−2K h 2 K‖f − fhp‖2L2(K) on each element K. The data approximation er- ror 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 esti- mate 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 indica- tor 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 Re- mark 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, uniformly in N 1 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 distribu- tion 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 100 150 200 250 300 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 N1/2 (a)   102 103 104 10−6 10−5 10−4 10−3 10−2 10−1 100 N (b)   100 150 200 250 300 10−6 10−5 10−4 10−3 10−2 10−1 100 N1/2 (c)   100 150 200 250 300 1 2 3 4 5 6 7 8 9 10 N1/2 (d)   Error Indicator Energy Error L2  Error Jump Error Theta hp−Error Indicator hp−Energy Error h−Error Indicator h−Energy Error Error Indicator (p3) Energy Error (p3) Error Indicator (p2) Energy Error (p2) ratio (p3) ratio(p2) 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 co- efficients. 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 u(x1, x2) = erf( x1√ 2ε )(1− x22), with erf(x) = 2√ pi ∫ x 0 e−t 2 dt. 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 200 300 400 500 600 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 N1/2 (a)   Error Indicator Energy Error L2  Error Jump Error Theta 104 105 106 107 108 10−5 10−4 10−3 10−2 10−1 100 101 N (b)   hp−Error Indicator hp−Energy Error h−Error Indicator h−Energy Error 200 300 400 500 600 10−5 10−4 10−3 10−2 10−1 100 101 N1/2 (c)   Error Indicator (p3) Energy Error (p3) Error Indicator (p2) Energy Error (p2) 200 300 400 500 600 1 2 3 4 5 6 7 8 9 10 11 N1/2 (d)   ratio (p3) ratio(p2) 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. 50 60 70 80 90 100 110 120 10−7 10−6 10−5 10−4 10−3 10−2 10−1 N1/2 (a)   Error Indicator Energy Error L2  Error Jump Error Theta 50 60 70 80 90 100 110 120 1 2 3 4 5 6 7 8 9 10 11 12 N1/2 (b)   ratio 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 distri- butions 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 210 220 230 240 250 260 270 280 290 300 10−5 10−4 10−3 10−2 10−1 100 N1/2 (a)   200 210 220 230 240 250 260 270 280 290 300 0 5 10 15 20 25 30 35 40 N1/2 (b)   Error Indicator Energy Error L2  Error Jump Error Theta ratio 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 pi6 , cos pi6 ), 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 = 1 2 ( tanh( x1 ε ) + 1 ) 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 h- adaptive methods, where the h-adaptive algorithm is still based on employ- ing 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 refine- ment steps. 3.7 Conclusions We have derived a robust a-posteriori error estimate for hp-adaptive discon- tinuous 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 esti- mate 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 100 150 200 250 300 350 400 10−4 10−3 10−2 10−1 100 101 N1/2 (a)   hp−Error Indicator 102 103 104 10−4 10−3 10−2 10−1 100 101 N1/2 (b)   hp−Error Indicator h−Error Indicator 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 avail- ability of an averaging operator as in Theorem 3.4.4. The difficulties in extending our results to three-dimensional problems are only related to the technicalities of the construction and analysis of such an operator on three- dimensional irregular meshes. These are discussed in the following chapters. Finally, we remark that our analysis in this chapter only holds for isotrop- ically refined meshes. In view of the powerful hp-version approximation results on anisotropic meshes, see [33], it would be desirable to allow for anisotropic h- and p- refinement as well, similarly to the approach for func- tional error estimation in [17]. This is studied in Chapter 5. 101 3.8. Bibliography 3.8 Bibliography [1] M. Ainsworth and J.T. Oden. A-posteriori Error Estimation in Fi- nite 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 anal- ysis of non-conforming finite elements with face penalty for advection- diffusion equations. 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Robust a-posteriori error estimates for stationary convection-diffusion equations. SIAM J. Numer. Anal., 43:1766–1782, 2005. [37] M. Vohraĺık. A-posteriori error estimates for lowest-order mixed fi- nite 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) in Ω ⊂ R3, u = 0 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 ∈ H10 (Ω) such that A(u, v) ≡ ∫ Ω ∇u · ∇v dx = ∫ Ω fv dx ∀ v ∈ H10 (Ω). (4.1.2) DG methods are ideally suited for realizing hp-adaptivity for second- order 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 3A 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 prob- lems 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 de- rived 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 in- terest 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 reen- trant 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 anal- ysis 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 hp- version 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. Each element K ∈ T is the image of the cube K̂ = (−1, 1)3 under an affine elemental mapping TK : K̂ → K. As usual, we denote by hK the 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 ′ for two neighboring elements K,K ′ ∈ 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=1Fi, 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 Ke 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 ve|F the one taken from inside Ke. The average and jump of v across the face F are then defined as {{v}} = 1 2 (v|F + ve|F ), [[v]] = v|F nK + ve|F nKe . Here, nK and nKe denote the unit outward normal vectors on the boundary of elements K and Ke, respectively. Similarly, if q is piecewise smooth vector field, its average and (normal) jump across F are given by {{q}} = 1 2 ( q|F + qe|F ) , [[q]] = q|F · nK + qe|F · nKe . 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 Qp(K) = { v : K → R : v ◦ TK ∈ Qp(K̂) }, (4.2.1) with Qp(K̂) denoting the set of tensor product polynomials on the reference element K̂ of degree less than or equal to p in each coordinate direction on K̂. In addition, if F ∈ F(K) is a face of K and F̂ the corresponding one on the reference element K̂, we define Qp(F ) = { v : F → R : v ◦ TK |F ∈ Qp(F̂ ) }, (4.2.2) where Qp(F̂ ) denotes the set of tensor product polynomials on F̂ of degree less than or equal to p in each coordinate direction on F̂ . 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′ ≤ % (4.2.3) for any pair of neighboring elements K,K ′ ∈ T . For a mesh face F ∈ F(T ), we introduce the face polynomial degree pF by pF = { max{pK , pK′}, if F = ∂K ∩ ∂K ′ ∈ FI(T ), pK , if F = ∂K ∩ Γ ∈ FB(T ). (4.2.4) 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 }. (4.2.5) 4.2.3 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 Ahp(uhp, v) = ∫ Ω fv dx ∀ v ∈ Sp(T ). (4.2.6) The bilinear form Ahp(u, v) is given by Ahp(u, v) = ∑ K∈T ∫ K ∇u · ∇v dx− ∑ F∈F(T ) ∫ F ( {{∇u}} · [[v]] + {{∇v}} · [[u]] ) ds + ∑ F∈F(T ) γp2F hF ∫ 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 param- eter 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): ‖u ‖2E,T = ∑ K∈T ‖∇u‖2L2(K) + ∑ F∈F(T ) γp2F hF ‖[[u]]‖2L2(F ). (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 ele- ment K ∈ T , we introduce the following local error indicator ηK which is given by the sum of the three terms η2K = η 2 RK + η2FK + η 2 JK . (4.3.2) The first term ηRK is the interior residual defined by η2RK = p −2 K h 2 K‖fhp + ∆uhp‖2L2(K). The second term ηFK is the face residual given by η2FK = 1 2 ∑ F∈FI(K) p−1F hF ‖[[∇uhp]]‖2L2(F ). The last residual ηJK measures the jumps of the approximate solution uhp and is defined as η2JK = 1 2 ∑ F∈FI(K) γ2p3F hF ‖[[uhp]]‖2L2(F ) + ∑ F∈FB(K) γ2p3F hF ‖[[uhp]]‖2L2(F ). We also introduce the local data approximation term Θ2K = p −2 K h 2 K‖f − fhp‖2L2(K). (4.3.3) Summing up the local error indicators, we introduce the global a-posteriori error estimator η = (∑ K∈T η2K ) 1 2 . (4.3.4) 111 4.3. Energy norm a-posteriori error estimates Similarly, we define the global data approximation term Θ = (∑ K∈T Θ2K ) 1 2 . (4.3.5) 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 ‖u− uhp ‖E,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 convection- diffusion equations. Crucial in our proof, however, is the use of a three- dimensional 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 p2Fh −1 F on each face, while the corresponding weight in the jump residual ηJK is of the different order p 3 Fh −1 F . This suboptimality with respect to the powers of pF is due to the possible presence of hanging nodes in the un- derlying mesh T . Indeed, on meshes without irregular nodes, Theorem 4.3.1 holds true with the following (optimal) jump residual: η̂2JK = 1 2 ∑ F∈FI(K) γ2p2F hF ‖[[uhp]]‖2L2(F ) + ∑ F∈FB(K) γ2p2F hF ‖[[uhp]]‖2L2(F ); see also Remark 4.4.3 below. The associated estimator η̂ is then given by η̂2 = ∑ K∈T η̂2K with η̂ 2 K = η 2 RK + η2FK + η̂ 2 JK . (4.3.6) Our numerical experiments in Section 4.7 indicate that the indicators η and η̂ 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 ′ ∈ T : ∂K ′ ∩ ∂K ∈ F(T )}. (4.3.7) The local energy norm over wK is defined by ‖u ‖2E,wK = ∑ K′∈wK ‖∇u‖2L2(K′) + ∑ F∈F(K) γp2F hF ‖[[u]]‖2L2(F ). (4.3.8) Similarly, we set ΘwK =  ∑ K′∈wK Θ2K′ 1/2 . (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 estima- tors ηK be defined by (4.3.2) and the local data approximation error ΘK by (4.3.3). Then, for any δ ∈ (0, 12), we have the local upper bound ηK . pδ+1K ‖u− uhp ‖E,wK + p 2δ+ 1 2 K 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 ∈ H1/2(Γ). In this case, the local error indicators ηK have to be modified by redefining the jump estimators ηJK as η2JK = 1 2 ∑ F∈FI(K) γ2p3F hF ‖[[uhp]]‖2L2(F ) + ∑ F∈FB(K) γ2p3F hF ‖uhp − ghp‖2L2(F ), 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 ) = ⋃ K∈T N (K). 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 ′ is a line segment given by the intersection of two faces F, F ′ 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 T̃ of affine hexahedra. We construct the auxiliary mesh T̃ 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 T̃ 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 T̃ anymore. In the following, we shall write R(K) for the elements in T̃ 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 T̃ . 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 T̃ are denoted by N (T̃ ), E(T̃ ) and F(T̃ ), 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 (T̃ ), E(T̃ ) and F(T̃ ): NA(T̃ ) = { ν̃ ∈ N (T̃ ) : ∃K ∈ T such that ν̃ is inside K }, EA(T̃ ) = { Ẽ ∈ E(T̃ ) : ∃K ∈ T such that Ẽ is inside K }, FA(T̃ ) = { F̃ ∈ F(T̃ ) : ∃K ∈ T such that F̃ is inside K }. We then introduce the following auxiliary DG finite element space on the mesh T̃ : Sp̃(T̃ ) = { v ∈ L2(Ω) : v|K̃ ◦ TK̃ ∈ QpK̃ (K̂), K̃ ∈ T̃ }, where the auxiliary polynomial degree vector p̃ is defined by p K̃ = pK for K̃ ∈ R(K) and T K̃ is the affine mapping from K̂ onto K̃. We clearly have the following inclusion: Sp(T ) ⊆ Sp̃(T̃ ). (4.4.1) In analogy with (4.3.1), the energy norm associated with T̃ is defined by ‖u ‖2 E,T̃ = ∑ K̃∈T̃ ‖∇u‖2 L2(K̃) + ∑ F̃∈F(T̃ ) γp2 F̃ h F̃ ‖[[u]]‖2 L2(F̃ ) , (4.4.2) where the auxiliary face polynomial degrees p F̃ for the jump terms over T̃ are defined as in (4.2.4), but using the auxiliary degrees p K̃ . 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 op- erators 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 Scp̃(T̃ ) be the conforming subspace of Sp̃(T̃ ) given by Scp̃(T̃ ) = Sp̃(T̃ ) ∩H10 (Ω). (4.4.3) Theorem 4.4.1 There exists an averaging operator Ihp : Sp(T ) → Scp̃(T̃ ) that satisfies∑ K̃∈T̃ ‖∇(v − Ihpv)‖2L2(K̃) . ∑ F∈F(T ) p2Fh −1 F ‖[[v]]‖2L2(F ), (4.4.4) ∑ K̃∈T̃ ‖v − Ihpv‖2L2(K̃) . ∑ F∈F(T ) p−2F hF ‖[[v]]‖2L2(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 H1-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 two- dimensional 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 T̃ 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 = u c hp + u r hp, 116 4.4. Proof of Theorem 4.3.1 where uchp = Ihpuhp ∈ Scp̃(T̃ ) ⊂ H10 (Ω) is defined using the averaging opera- tor Ihp in Theorem 4.4.1. The remainder u r hp is given by u r hp = uhp − uchp ∈ Sp̃(T̃ ). Analogously to Lemma 3.4.3, one can show that ‖u− uhp ‖E,T . ‖u− uhp ‖E,T̃ . Therefore, by the triangle inequality, ‖u− uhp ‖E,T . ‖u− uchp ‖E,T̃ + ‖urhp ‖E,T̃ . Finally, since u− uchp ∈ H10 (Ω), we have ‖u− uchp ‖E,T = ‖u− uchp ‖E,T̃ . As the starting point of our proof, we thus obtain the following inequality: ‖u− uhp ‖E,T . ‖u− uchp ‖E,T + ‖urhp ‖E,T̃ . (4.4.6) We first show that ‖urhp ‖E,T̃ 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 ‖urhp ‖E,T̃ . η. Proof : Recall from (4.4.2) that ‖urhp ‖2E,T̃ = ∑ K̃∈T̃ ‖∇urhp‖2L2(K̃) + ∑ F̃∈F(T̃ ) γp2 F̃ h F̃ ‖[[urhp]]‖2L2(F̃ ). Since uhp ∈ Sp(T ) and [[urhp]]|F = [[uhp]]|F for all F ∈ F(T̃ ), an argument similar to 3.4.2 allows us to bound the jump terms by ∑ F̃∈F(T̃ ) γp2 F̃ h F̃ ‖[[urhp]]‖2L2(F̃ ) . γ −1 ∑ F∈F(T ) γ2p2F hF ‖[[uhp]]‖2L2(F ) . γ−1 ∑ K∈T η2JK , 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∑ K̃∈T̃ ‖∇urhp‖2L2(K̃) . γ −2 ∑ F∈F(T ) γ2p2F hF ‖[[uhp]]‖2L2(F ) . γ−2 ∑ K∈T η2JK . This completes the proof. 2 117 4.4. Proof of Theorem 4.3.1 To bound ‖u− uchp ‖E,T in (4.4.6), we shall make use of the following two auxiliary forms: Dhp(u, v) = ∑ K∈T ∫ K ∇u · ∇v dx+ ∑ F∈F(T ) γp2F hF ∫ F [[u]] · [[v]] ds, Khp(u, v) = − ∑ F∈F(T ) ∫ F {{∇u}} · [[v]] ds− ∑ F∈F(T ) ∫ F {{∇v}} · [[u]] ds. The form Dhp(u, v) is well-defined for u, v ∈ Sp(T ) + H10 (Ω), whereas the term Khp(u, v) is only well-defined for discrete functions u, v ∈ Sp(T ). Fur- thermore, we have A(u, v) = Dhp(u, v) ∀u, v ∈ H10 (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 ∈ H10 (Ω), there exists a function vhp ∈ Sp(T ) such that p2Kh −2 K ‖v − vhp‖2L2(K) . ‖∇v‖2L2(K), ‖∇(v − vhp)‖2L2(K) . ‖∇v‖2L2(K), pKh −1 K ‖v − vhp‖2L2(∂K) . ‖∇v‖2L2(K), (4.4.9) for any element K ∈ T . Next, we prove the following auxiliary estimate. Lemma 4.4.5 For any v ∈ H10 (Ω), we have∫ Ω f(v − vhp) dx−Dhp(uhp, v − vhp) +Khp(uhp, vhp) . (η + Θ) ‖ v ‖E,T . Here, vhp ∈ Sp(T ) is the hp-version approximation of v defined in (4.4.9). Proof : For notational convenience, let us set T = ∫ Ω f(v − vhp) dx−Dhp(uhp, v − vhp) +Khp(uhp, vhp). 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 T = ∑ K∈T ∫ K (f + ∆uhp)(v − vhp) dx− ∑ F∈F(T ) γp2F hF ∫ F [[uhp]] · [[v − vhp]] ds − ∑ F∈FI(T ) ∫ F [[∇uhp]]{{v − vhp}} ds− ∑ F∈F(T ) ∫ F {{∇vhp}} · [[uhp]] ds ≡ T1 + T2 + T3 + T4. To bound term T1, we first add and subtract the approximation fhp to f : T1 = ∑ K∈T ∫ K (fhp + ∆uhp)(v − vhp) dx+ ∑ K∈T ∫ K (f − fhp)(v − vhp) dx. Using the approximation properties (4.4.9) and the Cauchy-Schwarz inequal- ity shows that T1 . ( ∑ K∈T ( η2RK + Θ 2 K ) ) 12( ∑ K∈T p2Kh −2 K ‖v − vhp‖2L2(K) ) 1 2 . ( ∑ K∈T ( η2RK + Θ 2 K ) ) 12 ‖ v ‖E,T . For term T2, we again exploit the Cauchy-Schwarz inequality to conclude that T2 ≤ ( ∑ F∈F(T ) γ2p3Fh −1 F ‖[[uhp]]‖2L2(F ) ) 1 2 ( ∑ F∈F(T ) pFh −1 F ‖[[v − vhp]]‖2L2(F ) ) 1 2 . 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 de- grees, we get the bound T2 . ( ∑ K∈T η2JK ) 1 2 ‖ v ‖E,T . Similarly, term T3 can be bounded by T3 ≤ ( ∑ F∈FI(T ) p−1F hF ‖[[∇uhp]]‖2L2(F ) ) 1 2 ( ∑ F∈FI(T ) pFh −1 F ‖{{v − vhp}}‖2L2(F ) ) 1 2 . ( ∑ K∈T η2FK ) 1 2 ‖ v ‖E,T . 119 4.4. Proof of Theorem 4.3.1 Finally, for term T4, we use the Cauchy-Schwarz inequality, the shape- regularity of the meshes, and the bounded variation property (4.2.3) of the polynomial degrees, to obtain T4 . γ−1 ( ∑ F∈F(T ) γ2p2Fh −1 F ‖[[uhp]]‖2L2(F ) ) 1 2 ( ∑ K∈T p−2K hK‖∇vhp‖2L2(∂K) ) 1 2 . From the standard hp-version inverse trace inequality, see [32], we conclude that T4 . γ−1 ( ∑ K∈T η2JK ) 1 2 ( ∑ K∈T ‖∇vhp‖2L2(K) ) 1 2 . From the approximation properties in (4.4.9) it follows that∑ K∈T ‖∇vhp‖2L2(K) . ∑ K∈T ‖∇(v − vhp)‖2L2(K) + ∑ K∈T ‖∇v‖2L2(K) . ‖ v ‖2E,T . Hence, T4 . γ−1 ( ∑ K∈T η2JK ) 1 2 ‖ v ‖E,T . The above bounds for terms T1, T2, T3, and T4 now imply the assertion. 2 We are now ready to bound ‖u− uchp ‖E,T in (4.4.6). Lemma 4.4.6 Under the foregoing assumptions, the following upper bound holds ‖u− uchp ‖E,T . η + Θ. Proof : Since u− uchp ∈ H10 (Ω), we have that ‖u− uchp ‖E,T = A(u− uchp, v) ‖ v ‖E,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), A(u− uchp, v) = ∫ Ω fv dx−A(uchp, v) = ∫ Ω fv dx−Dhp(uchp, v). One can now readily see that Dhp(u c hp, v) = Dhp(uhp, v) +R, 120 4.5. Proof of Theorem 4.3.3 with R = − ∑ K̃∈T̃ ∫ K̃ ∇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∫ Ω fvhp dx = Dhp(uhp, vhp) +Khp(uhp, vhp), where vhp ∈ Sp(T ) is the hp-version approximation of v in (4.4.9). Combin- ing these results, we thus arrive at A(u− uchp, v) = ∫ Ω f(v − vhp) dx−Dhp(uhp, v − vhp) +Khp(uhp, vhp)−R, The estimate in Lemma 4.4.5 now yields |A(u− uchp, v)| . (η + Θ) ‖ v ‖E,T + |R|. (4.4.11) It remains to bound |R|; from Lemma 4.4.4 and the Cauchy-Schwarz in- equality, we readily obtain |R| . ‖urhp ‖E,T̃ ‖ v ‖E,T . η‖ v ‖E,T . (4.4.12) The desired result now follows from (4.4.10), (4.4.11) and (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 Îd = (−1, 1)d 1 ≤ d ≤ 3, be the unit hypercube in d dimensions. Analogously to (4.2.1)– (4.2.2), we write Qp(Îd) for the space of the polynomials on Îd of degree less or equal than p in each variable. Lemma 4.5.1 For 1 ≤ d ≤ 3, let Ψ̂d(x) be the bubble function on Îd given by Ψ̂d(x) = Π d i=1(1− x2i ). (4.5.1) 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(Îd), p ∈ N, we have the inverse estimates∫ Îd Ψ̂d|∇φ|2 dx ≤ C1p2 ∫ Îd |φ|2 dx, (4.5.2)∫ Îd Ψ̂αd |φ|2 dx ≤ C2p2(β−α) ∫ Îd Ψ̂βd |φ|2 dx, (4.5.3)∫ Îd Ψ̂2δd |∇φ|2 dx ≤ C3p2(2−δ) ∫ Îd Ψ̂δd|φ|2 dx. (4.5.4) In addition, if φ = 0 on ∂Îd, then∫ Îd |∇φ|2 dx ≤ C1p2 ∫ Îd (Ψ̂d) −1|φ|2 dx. (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 F̂ ∈ F(K̂) be an elemental face of the reference element K̂. Starting from Ψ̂2 in (4.5.1) and using an affine transformation from Î 2 onto F̂ , we can readily define a face bubble function Ψ̂ F̂ on F̂ . Lemma 4.5.2 For α ∈ (1/2, 1], there exists Cα > 0 such that for any polynomial φ ∈ Qp(F̂ ) of degree p ≥ 1 and any δ ∈ (0, 1] there is an extension v̂ F̂ ∈ H1(K̂) satisfying v̂ F̂ = φ · Ψ̂α F̂ on F̂ and v̂ F̂ = 0 on ∂K̂ \ F̂ , as well as ‖v̂ F̂ ‖2 L2(K̂) ≤ Cαδ‖φΨ̂α/2 F̂ ‖2 L2(F̂ ) , (4.5.6) ‖∇v F̂ ‖2 L2(K̂) ≤ Cα(δp2(2−α) + δ−1)‖φΨ̂α/2 F̂ ‖2 L2(F̂ ) . (4.5.7) Proof : Without loss of generality, we may assume that the face F̂ is given by F̂ = (−1, 1)× (−1, 1)× {−1}. 122 4.5. Proof of Theorem 4.3.3 In this case, we have Ψ̂ F̂ (x1, x2,−1) = Ψ̂2(x1, x2). We now define the ex- tension v F̂ by v F̂ (x1, x2, x3) = 2 −2αφ(x1, x2)ΨF̂ (x1, x2) α(1− x3)αe− 1+x3 2δ . Obviously, we have v̂ F̂ | F̂ = φ · Ψα F̂ and v F̂ | ∂K̂\F̂ = 0. To prove (4.5.6), we note that Ψ2α F̂ ≤ Ψα F̂ on F̂ . Therefore, a direct computation shows that ‖v F̂ ‖2 L2(K̂) = 4−2α‖φΨα F̂ ‖2 L2(F̂ ) ∫ Î (1−x3)2αe−(1+x3)/δ dx3 ≤ Cαδ‖φΨ α 2 F̂ ‖2 L2(F̂ ) . To show (4.5.7), we proceed as follows: ‖∂x1vF̂ ‖2L2(K̂) ≤ Cα (‖∂x1φΨαF̂ ‖2L2(F̂ ) + ‖φΨα−1F̂ ‖2L2(F̂ ))δ ≤ Cαδp2(2−α)‖φΨα/2 F̂ ‖2 L2(F̂ ) , where we used Lemma 4.5.1 twice (for d = 2). The estimate for ∂x2vF̂ can be derived analogously. For ∂x3vF̂ , we have ‖∂x3vF̂ ‖2L2(K̂) ≤ Cα‖φΨ α F̂ ‖2 L2(F̂ ) [∫ 1 −1 ( 1− x3 2 )2(α−1)e− x3+1 δ dx3 + 1 4δ2 ∫ 1 −1 ( 1− x3 2 )2αe− x3+1 δ dx3 ] ≤ Cαp−2α‖φΨα/2 F̂ ‖2 L2(F̂ ) [δ−1+2α + δ−1], where we estimated the integral ∫ 1 −1( 1−x3 2 ) 2(α−1)e− 2(x3+1) δ dx3 as follows: ∫ 1 −1 ( 1− x3 2 )2(α−1)e− x3+1 δ dx3 = ∫ 1−2δ −1 ( 1− x3 2 )2(α−1)e−(x3+1)/δ dx3 + ∫ 1 1−2δ ( 1− x3 2 )2(α−1)e−(x3+1)/δ dx3 ≤ δ2(α−1) ∫ 1−2δ −1 e−(x3+1)/δ dx3 + 2 2α− 1 [ e 2δ−2 δ δ2α−1 + δ2α−1 δ ∫ 1 1−2δ e−(x3+1)/δdx3 ] ≤ δ2(α−1)δ + 2 2α− 1δ 2α−1 + 2 2α− 1(e 2 − 1)δ2α−1 ≤ Cαδ−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. For an element K ∈ T , we set ΨK = Ψ̂3 ◦T−1K , where Ψ̂3 is the reference bubble defined in (4.5.1). For an elemental face F ∈ F(K) corresponding to F̂ ∈ F(K̂), we similarly define ΨF = Ψ̂F̂ ◦ (TK |F )−1, where Ψ̂F̂ is the face 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 ηRK . pK‖∇(u− uhp)‖L2(K) + p δ+ 1 2 K ΘK . Proof : For any element K ∈ T , we set vK = (fhp + ∆uhp)|KΨαK , where α ∈ (1/2, 1]. Applying the inverse inequality in (4.5.3) with a simple scaling argument, we obtain ‖fhp + ∆uhp‖L2(K) . pαK‖(fhp + ∆uhp)Ψα/2K ‖L2(K) = pαK‖vKΨ−α/2K ‖L2(K). This leads to η2RK . S 2 K with S 2 K = p 2α−2 K h 2 K‖vKΨ−α/2K ‖2L2(K). (4.5.8) Since the exact solution satisfies (f + ∆u)|K = 0, we obtain S2K = p 2α−2 K h 2 K ∫ K (fhp + ∆uhp)vK dx = p2α−2K h 2 K ∫ K (∆(uhp − u) + (fhp − f)) vK dx = p2α−2K h 2 K( ∫ K ∇(u− uhp) · ∇vK dx+ ∫ K (fhp − f)Ψα/2K (vKΨ−α/2K ) dx). 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 ‖∇vK‖L2(K) . h−1K p2−αK ‖vKΨ−α/2K ‖L2(K). 124 4.5. Proof of Theorem 4.3.3 By the Cauchy-Schwarz inequality and the definition of the data approxi- mation term ΘK , we conclude that S2K . ( pK‖∇(u− uhp)‖L2(K) + pαKΘK ) ( p2α−2K h 2 K‖vKΨ−α/2K ‖2L2(K) ) 1 2 . ( pK‖∇(u− uhp)‖L2(K) + pαKΘK ) SK . Therefore, by this inequality and (4.5.8), we have ηRK . pK‖∇(u− uhp)‖L2(K) + pαKΘK . Choosing δ = α− 1/2 finishes the proof. 2 For a mesh face F ∈ F(T ), we define wF = {K1,K2 ∈ T : F = ∂K1 ∩ ∂K2 }, w̃F = { K̃ ∈ T ∪ T̃ : F ∈ F(K̃) }. (4.5.9) For simplicity, we also use the notation wF and w̃F to denote the domain formed by the elements in wF and in w̃F , respectively. Lemma 4.5.4 Under the assumptions of Theorem 4.3.3, there holds ηFK . pδ+1K ‖u− uhp ‖E,wK + p 2δ+ 1 2 K 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 ∈ H10 (wF ) with vF |F = τF , vF |∂wF = 0 such that ‖vF ‖L2(wF ) . h1/2F p−1F ‖τFΨ−α/2F ‖L2(F ), (4.5.10) ‖∇vF ‖L2(wF ) . h−1/2F pF ‖τFΨ−α/2F ‖L2(F ). (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 an element K̃1 ∈ T̃ , such that K̃1 ( K1 and F ∈ F(K̃1) ∩ F(K2). Thus, w̃F = K̃1 ∪ K2 ( wF . As before, by Lemma 4.5.2 (with δ = 1/p2F ) and a scaling argument, we can then find a function ṽF ∈ H10 (w̃F ) with ṽF |F = τF , ṽF |∂w̃F = 0 and ‖ṽF ‖L2(w̃F ) . h1/2F p−1F ‖τFΨ−α/2F ‖L2(F ), ‖∇ṽF ‖L2(w̃F ) . h−1/2F pF ‖τFΨ−α/2F ‖L2(F ). 125 4.5. Proof of Theorem 4.3.3 We then define the function vF on wF by vF = { ṽF on w̃F , 0 otherwise. Thus, vF ∈ H10 (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 ‖[[∇uhp]]‖L2(F ) . pαF ‖[[∇uhp]]Ψ α 2 F ‖L2(F ) = pαF ‖τFΨ −α 2 F ‖L2(F ). Therefore, η2FK . S 2 K with S 2 K = ∑ F∈FI(K) p2α−1F hF ‖τFΨ −α 2 F ‖2L2(F ). (4.5.12) Since [[∇u]] = 0 on interior edges, integration by parts over wF yields ‖τFΨ−α/2F ‖2L2(F ) = ∫ F [[∇(uhp − u)]]τF ds = ∑ K∈wF ∫ K (∆uhp −∆u)vF + (∇uhp −∇u) · ∇vF dx. Using the differential equation and approximating the data, we obtain S2K = ∑ F∈FI(K) p2α−1F hF ∑ K∈wF ∫ K (fhp + ∆uhp)ψF dx + ∑ F∈FI(K) p2α−1F hF ∑ K∈wF ∫ K (∇uhp −∇u) · ∇ψF dx + ∑ F∈FI(K) p2α−1F hF ∑ K∈wF ∫ K (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 T1 . p α− 1 2 K (pK‖u− uhp ‖E,wK + pαKΘwK )( ∑ F∈FI(K) p2α−1F hF ‖τFΨ −α 2 F ‖L2(F )) 1 2 . pα− 1 2 K (pK‖u− uhp ‖E,wK + pαKΘwK ) SK . 126 4.6. Proof of Theorem 4.4.1 Similarly, the term T2 can be bounded by T2 . ‖u− uhp ‖E,wK  ∑ F∈FI(K) hF p 2α−1 F ‖∇ψF ‖L2(wF )  . ‖u− uhp ‖E,wK  ∑ F∈FI(K) p 1 2 +α F h 1 2 F p α− 1 2 F ‖τFΨ −α 2 F ‖L2(F )  . p 1 2 +α K ‖u− uhp ‖E,wKSK . Finally, we estimate the data error term T3 as follows: T3 . ∑ F∈FI(K) p−1F hF ‖f − fhp‖L2(wF ) p2αF ‖ψF ‖L2(wF ) . ΘwK  ∑ F∈FI(K) p2α−1F hF p 2α−1 F ‖τFΨ −α 2 F ‖2L2(F )  12 . pα− 1 2 K ΘwKSK . Combining the above bounds for T1 through T3, we obtain SK . p α+ 1 2 K ‖u− uhp ‖E,wK + p 2α− 1 2 K ΘwK . By (4.5.12), we conclude that ηFK . p α+ 1 2 K ‖u− uhp ‖E,wK + p 2α− 1 2 K 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 ηJK . p 1 2 K‖u− uhp ‖E,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 Î = (−1, 1) be the reference interval. We denote by Ẑp(Î) = { ẑp0 , · · · , ẑpp } the Gauss-Lobatto nodes of order p ≥ 1 on Î. Recall that ẑp0 = −1 and ẑpp = 1. We denote by Ẑpint(Î) = { ẑp1 , · · · , ẑpp−1 } the interior Gauss-Lobatto nodes of order p on Î. Now let E ∈ E(K) be an elemental edge of K ∈ T . The nodes in Ẑp can be affinely mapped onto E and we denote by Zp(E) = { zE,p0 , · · · , zE,pp } the Gauss-Lobatto nodes of order p on E. The points zE,p0 and z E,p p coincide with the two end points of E. The set Zpint(E) = { zE,p1 , · · · , zE,pp−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 P intp (E) = { q ∈ Pp(E) : q(zE,p0 ) = q(zE,pp ) = 0 }, Pnodp (E) = { q ∈ Pp(E) : q(z) = 0, z ∈ Zpint(E) }. By construction, we have Pp(E) = P intp (E)⊕ Pnodp (E). On the reference square Î2 = (−1, 1)2, we define the tensor-product Gauss-Lobatto nodes of order p by Ẑp(Î2) = { ẑpi,j = (ẑpi , ẑpj ) }0≤i,j≤p. These nodes can be affinely mapped onto an elemental face F ∈ F(K) of K ∈ T and we define Zp(F ) = { zF,pi,j }0≤i,j≤p to be the Gauss-Lobatto nodes of order p on F . Furthermore, we write Zpint(F ) = { zF,pi,j }1≤i,j≤p−1 for the interior Gauss-Lobatto points on F . We also define QintpF (F ) = { q ∈ QpF (F ) : q = 0 on ∂F}. Similarly, we define the interior Gauss-Lobatto nodes of order p on the reference element K̂ by Ẑpint(K̂) = {ẑpi,j,k = (ẑpi , ẑpj , ẑpk)}1≤i,j,k≤p−1. For an element K ∈ T and a polynomial degree p ≥ 1, we denote its interior Gauss- Lobatto points by Zpint(K) = { zK,pi,j,k }1≤i,j,k≤p−1. Here, the points zK,pi,j,k are the affine mappings of ẑpi,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 ν̂4 ν̂7 ν̂3 ν̂6 ν̂8 ν̂5 ν̂1 ν̂2 (a) Numbering of nodes Ê5 Ê1 Ê2Ê4 Ê6 Ê8 Ê9 Ê7 Ê10 Ê11 Ê3 Ê12 (b) Numbering of edges F̂6 x̂1 x̂2x̂3 F̂1 F̂5 F̂3 F̂2 F̂4 (c) Numbering of faces Figure 4.1: Reference element K̂ with the numbering of faces, edges and vertices. As usual, we shall divide the basis functions into interior, face, edge and vertex basis functions. We first consider the reference element K = K̂ = (−1, 1)3. We denote its faces by F̂1, . . . F̂6, its edges by Ê1, . . . , Ê12 and its vertices by ν̂1, . . . , ν̂8, numbered as in Figure 4.1. Let {ϕ̂pi }0≤i≤p be the Lagrange basis functions associated with the Gauss-Lobatto nodes Ẑp(Î) on Î. The interior basis functions are then Φ̂int,pi,j,k (x̂1, x̂2, x̂3) = ϕ̂ p i (x̂1) ϕ̂ p j (x̂2) ϕ̂ p k(x̂3), 1 ≤ i, j, k ≤ p− 1. Next, we define the face basis functions exemplary for the face F̂1 in Fig- ure 4.1 with face polynomial degree p F̂1 . They are given by Φ̂ F̂1,pF̂1 i,j (x̂1, x̂2, x̂3) = ϕ̂ p F̂1 i (x̂1) ϕ̂ p 0(x̂2) ϕ̂ p F̂1 j (x̂3), 1 ≤ i, j ≤ pF̂1 − 1. Note that Φ̂ F̂1,pF̂1 i,j vanishes on F̂2 through F̂6. The other face basis functions are then defined analogously. To define the edge basis functions, we consider exemplary the edge Ê1 in Figure 4.1 with edge degree pÊ1 . The edge basis functions for Ê1 are Φ̂ Ê1,pÊ1 i (x̂1, x̂2, x̂3) = ϕ̂ p Ê1 i (x̂1) ϕ̂ p F̂5 0 (x̂2) ϕ̂ p F̂1 0 (x̂3), i = 1, . . . , pÊ1 − 1. Note that Φ̂ Ê1,pÊ1 i vanishes on all the other edges and on the faces F̂2, F̂3, F̂4 and F̂6. Moreover, it vanishes on the interior nodes {ẑ F̂1,pF̂1 i,j } p F̂1 −1 i,j=1 and 129 4.6. Proof of Theorem 4.4.1 {ẑF̂5,pF̂5i,j } p F̂5 −1 i,j=1 of the faces F̂1 and F̂5, respectively. The other edge basis functions are then defined analogously. Finally, we consider the vertex ν̂1, which is shared by the edges Ê1, Ê4 and Ê5; see Figure 4.1. The associated vertex basis function is then defined by Φ̂ν̂1 K̂ (x̂1, x̂2, x̂3) = ϕ̂ p Ê1 0 (x̂1) ϕ̂ p Ê4 0 (x̂2) ϕ̂ p Ê5 0 (x̂3). The vertex basis functions associated with the other vertices of K̂ can be defined analogously. This completes the definition of the shape functions on the reference element K̂. For an arbitrary element K, the basis functions Φ on K can be defined from the analogous ones on K̂ by the pull-back map TK : Φ(x1, x2, x3) = Φ̂ ◦ T−1K (x1, x2, x3), giving rise to shape functions ΦνK , ΦE,pEi , ΦF,pFi,j and Φint,pi,j,k on K. Therefore, a polynomial v ∈ Qp(K) satisfying (4.6.1) can be expanded in the following form: v(x) = ∑ ν∈N (K) v(ν) ΦνK(x) + ∑ E∈E(K) pE−1∑ i=1 v(zE,pEi ) Φ E,pE i (x) + ∑ F∈F(K) pF−1∑ i,j=1 cFi,jΦ F,pF i,j (x) + ∑ 1≤i,j,k≤p−1 ci,j,k Φ int,p i,j,k (x), with coefficients cFi,j and ci,j,k. In the sequel, we will make use of the following two estimates for poly- nomials, 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-Loba- tto nodes of K, then there holds ‖v‖2L2(K) . hKp−2K ‖v‖2L2(∂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 ‖ΦνK‖L2(K) . h3/2K p−1Ei p−1Ej p−1Ek . (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 Φ ν K can be bounded by ‖ΦνK‖L2(F ) . hKp−1Ei p−1Ej . 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 LEp by LEp,K : P intp (E) −→ Qp(K), q(x) 7−→ p−1∑ i=1 q(zE,pi )Φ E,p i (x). (4.6.2) 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 ∈ P intp (E1) and q2 ∈ P intp (E2), we then define the extension operator LEp,K(q1, q2) by LEp,K(q1, q2) = L E1 p,K1 (q1) + L E2 p,K2 (q2), (4.6.3) with LE1p,K1(·) and LE2p,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=1Ki into four elements, as illustrated in Figure 4.3. If E is shared by K1 and K2 and if q ∈ P intp (E), the extension LEp,K(q) is then defined by LEp,K(q) = L E p,K1(q) + L E p,K2(q), (4.6.5) with LEp,K1 and L E p,K2 given in (4.6.2) and extended by zero to the other two elements. By construction, the extension operators LEp,K(q) in (4.6.2), (4.6.5) and LEp,K(q1, q2) in (4.6.3) are continuous on K and satisfy LEp,K(q)|E = q, LEp,K(q1, q2)|E1 = q1, LEp,K(q1, q2)|E2 = q2. Moreover, LEp,K(q) and L E p,K(q1, q2) vanish at the interior Gauss-Lobatto nodes in Zpint(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                                                      u E2 E1 E K2 K1 Figure 4.2: Case 2: The elemen- tal edge E ∈ E(K) has a hanging node located in its midpoint.                                      u                                       E4 E2 E1 E3 K1 K2 K3 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. Lemma 4.6.2 The linear edge extension operators LEp introduced above sat- isfy ‖LEp,K(q)‖L2(K) . p−2hK‖q‖L2(E), E ∈ E(K), ‖LEp,K(q)‖L2(K) . p−2hK‖q‖L2(E), E ∈ EF (K), F ∈ F(K), ‖LEp,K(q1, q2)‖L2(K) . p−2hK 2∑ i=1 ‖qi‖L2(Ei), E ∈ E1 ∪ E2, E1, E2 ∈ E(T ). 4.6.3 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(T̃ ) or F ∈ FN (T )), we define LFp,K by LFp,K : Qintp (F ) −→ Qp(K), q(x) 7−→ p−1∑ i,j=1 q(zF,pi,j )Φ F,p i,j (x). (4.6.6) Second, if F has a hanging node in its midpoint (i.e., F /∈ F(T )), we write F as F = ∪4i=1Fi, for four faces Fi ∈ F(T ). We then partition K into four parallelepipeds, K = ∪4i=1Ki, as illustrated in Figure 4.4. For polyno- mials qi ∈ Qintp (Fi), i = 1, . . . , 4, we define the operator LFp,K(q1, q2, q3, q4) 132 4.6. Proof of Theorem 4.4.1 by LFp,K(q1, q2, q3, q4) = 4∑ i=1 LFip,Ki(qi), (4.6.7) with LEip,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(T̃ ) and again partition K into four parallelepipeds, K = ∪4i=1Ki, 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∑ i=1 LFip,Ki(q|Fi), (4.6.8) where, for 1 ≤ i ≤ 4, LFip,Ki(q|Fi) = p−1∑ k,l=1 q(zFi,pk,l )Φ Fi,p k,l + ∑ E∈E(Fi) p−1∑ k=1 q(zE,pk )Φ E,p k + q(νc)Φ νc Ki .                                                      u                       F1 F2 F4 F3 K1 K2 K4 K3 Figure 4.4: Case 2: Partition of K associated with the partition of face F .                                                      u                       F1 F2 F4 F3 K1 K2 K4 K3 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 L F p,K(q1, q2, q3, q4) both vanish in the interior Gauss- Lobatto nodes in Zpint(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 ‖LFp,K(q)‖L2(K) . p−1h1/2K ‖q‖L2(F ), F ∈ F(T ) ∩ F(T̃ ) or F ∈ FN (T ), ‖LFp,K(q)‖L2(K) . p−1h1/2K ‖q‖L2(F ), F ∈ FH(T ), ‖LFp,K(q1, . . . , q4)‖L2(K) . p−1h1/2K 4∑ i=1 ‖qi‖L2(Fi), F = F1 ∪ F2 ∪ F3 ∪ F4, and F1, . . . , F4 ∈ F(T ). 4.6.4 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(T̃ ) and a face F ∈ F(T )∪F(T̃ ), we set pE = min{ pK̃ : K̃ ∈ T ∪ T̃ , E ∈ E(K̃) }, pF = min{ pK̃ : K̃ ∈ T ∪ T̃ , F ∈ F(K̃) }. (4.6.9) Let v ∈ Sp(T ). We denote by vK the restriction of v to an element K ∈ T ∪ T̃ . We decompose v into a nodal, edge, face and interior part, respectively: v = vnod + vedge + vface + vint, (4.6.10) with vnod, vedge, vface and vint in Sp̃(T̃ ) introduced below. Nodal part First, we construct the nodal part vnod ∈ Sp̃(T̃ ) in (4.6.10). For each element K ∈ T and K̃ ∈ R(K), we will construct the restriction vnod K̃ of vnod to K̃ such that vnod K̃ ∈ Qp K̃ (K̃) (note that pK = pK̃) and vnod K̃ |E ∈ PpE (E), E ∈ E(K̃), vnodK̃ |F ∈ PpF (F ), F ∈ F(K̃), with pE and pF given in (4.6.9). To define v nod K̃ , we distinguish the following two cases. Case 1: If R(K) = {K} (i.e., if K is unrefined), the interpolant vnod K̃ = vnodK is simply defined by vnodK (x) = ∑ ν∈N (K) vK(ν) Φ ν K(x). (4.6.11) 134 4.6. Proof of Theorem 4.4.1 Case 2: If R(K) consists of eight newly created elements, we define vnod K̃ on each element K̃ ∈ R(K) separately. To do so, fix K̃ ∈ R(K). Without loss of generality, we may consider the situation shown in Figure 4.6, where we denote by ν̃i, Ẽj and F̃k the vertices, edges and faces of K̃, respectively, 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 ν̃8 ∈ NA(T̃ ), F̃3, F̃4, F̃6 ∈ FA(T̃ ), as well as Ẽ8, Ẽ11, Ẽ12 ∈ EA(T̃ ). Hence, the polynomial degrees are given by p F̃i = p Ẽj = p K̃ = pK , i ∈ {3, 4, 6}, j ∈ {8, 11, 12}. Let us now define the value of vnod K̃ at the nodes located on ∂K̃. At the interior nodes shared by F̃i and Ẽj for i ∈ {3, 4, 6} and j ∈ {8, 11, 12}, we set vnod K̃ (z) = vK(z), z ∈ {Z p F̃i int (F̃i)}i∈{3,4,6} ∪ {Z p Ẽj int (Ẽj)}j∈{8,11,12}. (4.6.12) Similarly, we set vnod K̃ (ν) = vK(ν) for the vertices ν = ν̃2 and ν = ν̃8. ν1 = ν2 ν5 ν6 E1 E5 E6 E9                                                                       ν̃1 ν̃2Ẽ1 F̃1 ν̃5 ν̃6 Ẽ5 Ẽ6 Ẽ9                     Figure 4.6: The element K is refined into 8 elements K̃ ∈ R(K). It remains to define vnod K̃ on the nodes located on the faces F̃1, F̃2 and F̃5 (excluding the vertex ν̃2). We only consider F̃1 (the construction for F̃2 and F̃5 is completely analogous); see Figure 4.6. If F̃1 ∈ F(T ), then we have ν̃1, ν̃5, ν̃6 ∈ N (T ). The four edges Ẽi ∈ E(F̃1) for i ∈ {1, 5, 6, 9} belong to 135 4.6. Proof of Theorem 4.4.1 E(T ). For i ∈ {1, 5, 6, 9} and j ∈ {1, 5, 6}, we define vnod K̃ (z) = 0, z ∈ ZpF̃1int (F̃1) ∪ {Z p Ẽi int (Ẽi)}Ẽi∈E(F̃1), (4.6.13) vnod K̃ (ν̃j) = vK(ν̃j). (4.6.14) Otherwise, if F̃1 /∈ 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 pF̃1 = pF1 . In this case, we interpolate the values of the nodal interpolant over the face F1 at the Gauss-Lobatto nodes on F̃1. That is, we define vnod K̃ (z) = ∑ ν∈N (F1) vK(ν) Φ ν K(z), (4.6.15) for all z ∈ {ZpẼint (Ẽ)}Ẽ∈E(F̃1) ∪ Z p F̃1 int (F̃1) ∪ {ν̃i}i∈{1,5,6}. Second, if F1 ∈ FH(T ), then ν̃5 /∈ N (T ), but ν̃1 and ν̃6 may or may not belong to N (T ). We define the value of vnod K̃ at the nodes located on F̃1 for this case as follows. First, noticing that pẼ5 = pẼ9 = pF̃1 = pF1 and ν̃2 ∈ N (T ), we set vnod K̃ (z) = 0, z ∈ ZpF1int (F̃1) ∪ Z pF1 int (Ẽ5) ∪ Z pF1 int (Ẽ9) ∪ {ν̃5}, (4.6.16) Next, we define the values of vnod K̃ on the nodes of the edges Ẽ1 and Ẽ6, as well as on the nodes ν̃1 and ν̃6. We only consider ν̃1 and Ẽ1 (the construction for ν̃6 and Ẽ6 is completely analogous). If ν̃1 ∈ N (T ) (i.e., ν̃1 is a hanging node in T ), then we define vnod K̃ (z) = 0, z ∈ ZpẼ1int (Ẽ1), vnodK̃ (ν̃1) = vK(ν̃1). (4.6.17) If ν̃1 /∈ N (T ), then we have E1 ∈ E(T ) and ν1 ∈ N (T ). In this case, p Ẽ1 = pE1 , and we interpolate the values of the nodal interpolant over the long edge E1 at the Gauss-Lobatto nodes on Ẽ1. That is, we set vnod K̃ (z) = vK(ν1) Φ ν1 K (z)+vK(ν2) Φ ν2 K (z), z ∈ Z p Ẽ1 int (Ẽ1)∪{ν̃1}. (4.6.18) 136 4.6. Proof of Theorem 4.4.1 With the nodal values of vnod K̃ constructed in (4.6.12)-(4.6.18), we have vnod K̃ (x) = ∑ ν∈N (K̃) vnod K̃ (ν) Φν K̃ (x) + ∑ E∈E(K̃) pE−1∑ i=1 ( vnod K̃ (z E,pE i )Φ E,pE i (x) ) + ∑ F∈F(K̃) pF−1∑ i,j=1 ( vnod K̃ (z F,pF i,j )Φ F,pF i,j (x) ) . This finishes the construction of the interpolant vnod. Notice that vnod ∈ Sp̃(T̃ ); it is continuous over faces F ∈ FA(T̃ ) and over edges inside faces F ∈ F(T ). Moreover, it satisfies vK(ν)− vnodK (ν) = 0, ν ∈ N (T ) located on ∂K, and vnod K̃ |E ∈ PnodpE (E), E ∈ E(T ), K̃ ∈ w̃E , with w̃E defined by w̃E = { K̃ ∈ T ∪ T̃ : E ∈ E(K̃) }, ∀E ∈ E(T ). Edge part Second, we construct the edge function vedge ∈ Sp̃(T̃ ) in the decomposi- tion (4.6.10). To do so, fix an element K ∈ T . For an edge E on ∂K, we define vEK by vEK =  LEpK ,K((vK − vnodK )|E), E ∈ E(K) ∩ E(T ), LEpK ,K((vK − vnodK )|E), E ∈ EF (K), F ∈ F(K), LEpK ,K((vK − vnodK )|E1 , (vK − vnodK )|E2), E = E1 ∪ E2, E1,2 ∈ E(T ), with LEpK ,K(·) defined for Case 1 in (4.6.2) or for Case 3 in (4.6.5), and LEpK ,K(·, ·) for Case 2 in (4.6.3), respectively. We then define vedge on each element as: vedgeK (x) = ∑ E∈E(K) vEK(x) + ∑ F∈F(K) ∑ E∈EF (K) vEK(x). 137 4.6. Proof of Theorem 4.4.1 Face part Third, we construct the face function vface ∈ Sp̃(T̃ ) in (4.6.10). Fix an element K ∈ T and let F be an elemental face in F(K). If F ∈ F(T ), we define vFK by vFK = { LFpK ,K((vK − vnodK − v edge K )|F ), F /∈ FH(T ), LFpK ,K((vK − vnodK − v edge K )|F ), F ∈ FH(T ), with LFpK ,K(·) defined for Case 1 in (4.6.6) and for Case 3 in (4.6.8). Other- wise, there exists four faces Fi ∈ F(T ), i = 1, . . . , 4, such that F = ∪4i=1{Fi}. We define vFK by vFK = L F pK ,K ((vK − vnodK − vedgeK )|F1 , . . . , (vK − vnodK − vedgeK )|F4), with LFpK ,K(·, ·, ·, ·) defined for Case 2 in (4.6.7). We then define vface ele- mentwise as vfaceK (x) = ∑ F∈F(K) vFK(x). 4.6.5 Interior part Finally, the interior function vint ∈ Sp̃(T̃ ) in (4.6.10) is simply obtained by setting on each element vintK = vK − vnodK − vedgeK − vfaceK , K ∈ T . Notice that vintK belongs to H 1 0 (K). Hence, we have v int ∈ Scp̃(T̃ ). 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 = vnod + vedge + vface + vint, according to (4.6.10). We shall define the averaging operator Ihpv in four parts: Ihpv = ϑ nod + ϑedge + ϑface + ϑint, (4.6.19) with ϑnod, ϑedge, ϑface, ϑint ∈ Scp̃(T̃ ). Since vint ∈ Scp̃(T̃ ), we simply take ϑint = vint. 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 ap- proximation ϑnod ∈ Scp̃(T̃ ) that satisfies:∑ K̃∈T̃ ‖vnod − ϑnod‖2 L2(K̃) . ∑ F∈F(T ) hF p2F ∫ F [[vnod]]2 ds, ∑ K̃∈T̃ ‖∇(vnod − ϑnod)‖2 L2(K̃) . ∑ F∈F(T ) p2F hF ∫ F [[vnod]]2 ds. (4.6.20) (ii) Edge approximation: There is a conforming approximation ϑedge ∈ Scp̃(T̃ ) that satisfies:∑ K̃∈T̃ ‖vedge − ϑedge‖2 L2(K̃) . ∑ F∈F(T ) hF p2F ∫ F ([[v]]2 + [[vnod]]2) ds, ∑ K̃∈T̃ ‖∇(vedge − ϑedge)‖2 L2(K̃) . ∑ F∈F(T ) p2F hF ∫ F ([[v]]2 + [[vnod]]2) ds. (4.6.21) (iii) Face approximation: There is a conforming approximation ϑface ∈ Scp̃(T̃ ) that satisfies:∑ K̃∈T̃ ‖vface − ϑface‖2 L2(K̃) . ∑ F∈F(T ) hF p2F ∫ F ([[v]]2 + [[vnod]]2) ds, ∑ K̃∈T̃ ‖∇(vface − ϑface)‖2 L2(K̃) . ∑ F∈F(T ) p2F hF ∫ F ([[v]]2 + [[vnod]]2) ds. (4.6.22) By the triangle inequality and Proposition 4.6.4, we then obtain∑ K̃∈T̃ ‖v − Ihpv‖2L2(K̃) . ∑ F∈F(T ) p−2F hF (‖[[v]]‖2L2(F ) + ‖[[vnod]]‖2L2(F )),∑ K̃∈T̃ ‖∇(v − Ihpv)‖2L2(K̃) . ∑ F∈F(T ) p2Fh −1 F (‖[[v]]‖2L2(F ) + ‖[[vnod]]‖2L2(F )). Hence, Theorem 4.4.1 follows if we show that ‖[[vnod]]‖2L2(F ) . ‖[[v]]‖2L2(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 vnod, the jump over F satisfies [[vnod]](ν) = [[v]](ν), ν ∈ NT (F ). If F ∈ F(T ) ∩ F(T̃ ) 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 ‖[[vnod]]‖L2(F ) . ∑ ν∈N (F ) |[[vnod(ν)]]|‖ΦνK‖L2(F ) . p−2F hF max ν∈NT (F ) |[[vnod]](ν)|, 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 F̃i ∈ F(T̃ ), 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, pro- ceeding as in the corresponding proof of Lemma 3.5.3, we obtain from (4.2.3) and the construction of vnod that ‖[[vnod]]‖L2(F ) = 4∑ i=1 ‖[[vnod]]‖ L2(F̃i) . p−2F hF max ν∈NT (F ) |[[vnod]](ν)|. Thus, for any face F ∈ F(T ), we have ‖[[vnod]]‖L2(F ) . p−2F hF max ν∈NT (F ) |[[vnod]](ν)| = p−2F hF max ν∈NT (F ) |[[v]](ν)|. Without loss of generality, we suppose that |[[vnod]](ν)| reaches its maxi- mum 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 max ν∈NT (F ) ‖[[v]](ν)‖ = ‖[[v]](ν1)‖ . pEh−1/2E ‖[[v]]‖L2(E) . p2Fh−1F ‖[[v]]‖L2(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 vnod ∈ Sp̃(T̃ ) be the nodal part of v ∈ Sp(T ) in the decomposi- tion (4.6.10). We shall now construct the conforming approximation ϑnod in Scp̃(T̃ ). For simplicity, we shall omit the superscript “nod” and, in the sequel, write v for vnod and ϑ for ϑnod. We introduce the sets: w̃(ν) = { K̃ ∈ T̃ : ν ∈ N (K̃) }, wF (ν) = {F ∈ F(T ) : ν ∈ F }. Fix K ∈ T and K̃ ∈ R(K). We proceed by distinguishing the same two cases as in Subsection 4.6.4. Case 1: If R(K) = {K}, we have K = K̃. Then, any elemental face F̃ ∈ F(K̃) belongs to F(T ) and we have v K̃ | F̃ ∈ Qp F̃ (F̃ ). Moreover, any elemental edge Ẽ ∈ E(K̃) belongs to E(T ) and v K̃ | Ẽ ∈ Pnodp Ẽ (Ẽ). For any Gauss-Lobatto node ν located on ∂K̃, we define the value of ϑ(ν) by ϑ(ν) =  |w̃(ν)|−1 ∑ K̃∈w̃(ν) v K̃ (ν), ν ∈ NI(T ), 0, otherwise. (4.6.24) Here, |w̃(ν)| denotes the cardinality of the set w̃(ν). Note that we have |w̃(ν)| = 8 for ν ∈ NI(T ). Then we define ϑ on K̃ by: ϑ(x) = ∑ ν∈N (K̃) ϑ(ν) Φν K̃ (x). (4.6.25) From (4.6.11) and (4.6.25), we have ‖v K̃ − ϑ‖ L2(K̃) . ∑ ν∈N (K̃) |v K̃ (ν)− ϑ(ν)| ‖Φν K̃ ‖ L2(K̃) . (4.6.26) Analogously to [11, Pages 1125-1126], we conclude that |v K̃ (ν)− ϑ(ν)| . ∑ F∈wF (ν) p2Fh −1 F ‖[[v]]‖L2(F ). (4.6.27) 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 ‖v K̃ − ϑ‖ L2(K̃) . ∑ F∈{wF (ν)} ν∈N (K̃) p−1F h 1/2 F ‖[[v]]‖L2(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 ele- ment K̃ ∈ R(K) separately, analogously to the construction of the nodal interpolant in Subsection 4.6.4. Without loss of generality, we may again consider the case illustrated in Figure 4.6. Since the faces F̃3, F̃4, F̃6 belong to FA(T̃ ), the function v is continuous over them. The values of ϑ on the face nodes z ∈ {ZpK̃int (F̃i)}i∈{3,4,6} ∪ {Z p K̃ int (Ẽj)}j∈{8,11,12} and the vertex ν̃8 are defined by ϑ(ν̃8) = vK̃(ν̃8) and ϑ(z) = v K̃ (z), z ∈ {ZpK̃int (F̃i)}i∈{3,4,6} ∪ {Z p K̃ int (Ẽj)}j∈{8,11,12}. (4.6.29) We further define the value of ϑ on the vertex ν̃2 by (4.6.24). It remains to define the values of v K̃ on the nodes located on the faces F̃1, F̃2 and F̃5, excluding the vertex ν̃2. We only consider F̃1 (the construction for F̃2 and F̃5 is completely analogous); see Figure 4.6. If F̃1 ∈ F(T ), then for any Gauss-Lobatto node on F̃1, z ∈ Z p F̃1 int (F̃1)∪{ZpEint (E)}E∈E(F̃1)∪{ν1}∪ {ν5} ∪ {ν6}, the value of ϑ(z) is taken as in (4.6.24). Otherwise, if F̃1 /∈ 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 F̃1. That is, we set ϑ(z) = ∑ ν∈N (F1) ϑ(ν) ΦνK(z), (4.6.30) for z ∈ {ZpẼint (Ẽ)}Ẽ∈E(F̃1) ∪ Z p F̃1 int (F̃1) ∪ {ν̃i}i∈{1,5,6}. Second, if F1 ∈ FH(T ), then ν̃5 /∈ N (T ), but ν̃1 and ν̃6 may or may not belong to N (T ). We first define ϑ(z) = 0, z ∈ ZpF1int (F̃1) ∪ Z pF1 int (Ẽ5) ∪ Z pF1 int (Ẽ9) ∪ {ν̃5}, (4.6.31) Next, we define the values of ϑ on the nodes of the edges Ẽ1 and Ẽ6, as well as on ν̃1 and ν̃6. We only consider ν̃1 and Ẽ1 (the definition for ν̃6 and Ẽ6 is completely analogous). If ν̃1 ∈ N (T ) (i.e., ν̃1 is a hanging node of T ), then we define ϑ(z) for z ∈ ZpẼ1int (Ẽ1) ∪ {ν̃1} by (4.6.24). If ν̃1 /∈ N (T ), then E1 ∈ E(T ) and ν1 ∈ N (T ). We define ϑ(ν1) again by (4.6.24). Recall that ϑ(ν2) = ϑ(ν̃2) has already been defined. Then, for the nodes on Ẽ1, we set ϑ(z) = ϑ(ν1)Φ ν1 K (z) + ϑ(ν2)Φ ν2 K (z), z ∈ Z p Ẽ1 int (Ẽ1) ∪ {ν̃1}. (4.6.32) 142 4.6. Proof of Theorem 4.4.1 Now we construct ϑ on K̃ by setting ϑ(x) = ∑ ν∈N (K̃) ϑ(ν) Φν K̃ (x) + ∑ E∈E(K̃) pE−1∑ i=1 ( ϑ(z E,pE i )Φ E,pE i (x) ) + ∑ F∈F(K̃) pF−1∑ i,j=1 ( ϑ(z F,pF i,j )Φ F,pF i,j (x) ) . (4.6.33) This completes the construction of ϑ. It can be readily seen that ϑ ∈ Scp̃(T̃ ). We shall now derive an estimate analogous to (4.6.28) for Case 2. To do so, we estimate the difference between v K̃ and ϑ on K̃ as follows: ‖v K̃ − ϑ‖ L2(K̃) . ∑ ν̃∈N (K̃) ‖ςν̃‖L2(K̃) + ∑ Ẽ∈E(K̃) ‖ς Ẽ ‖ L2(K̃) + ∑ F̃∈F(K̃) ‖ς F̃ ‖ L2(K̃) , (4.6.34) with ςν̃(x) = ( v K̃ (ν̃)− ϑ(ν̃))Φν̃ K̃ (x), ς Ẽ (x) = p Ẽ −1∑ i=1 (( v K̃ (z Ẽ,p Ẽ i )− ϑ(z Ẽ,p Ẽ i ) ) Φ Ẽ,p Ẽ i (x) ) , ς F̃ (x) = p F̃ −1∑ i,j=1 (( v K̃ (z F̃ ,p F̃ i,j )− ϑ(z F̃ ,p F̃ i,j ) ) Φ F̃ ,p F̃ i,j (x) ) . Proceeding as in the two-dimensional proof in Lemma 3.5.3, we obtain the following estimates. First, we have that ‖ςν̃‖L2(K̃) = 0 for ν̃ ∈ NA(T̃ ) and ‖ςν̃‖L2(K̃) . ∑ F∈wF (ν̃) p−1F h 1/2 F ‖[[v]]‖L2(F ), ν̃ ∈ N (T ). Second, for ν̃ /∈ N (T ), we have ‖ςν̃‖L2(K̃) .  ∑ F∈{wF (ν)}ν∈∂E p−1F h 1 2 F ‖[[v]]‖L2(F ), ∃E ∈ E(K), ν̃ is inside E,∑ F∈{wF (ν)}ν∈N (F?) p−1F h 1 2 F ‖[[v]]‖L2(F ), ∃F ? ∈ F(K), ν̃ inside F ?. Similarly, for ς Ẽ in (4.6.34), we have that ς Ẽ = 0 if Ẽ ∈ EA(T̃ ) or if Ẽ ∈ EF ?(K) for a face F ? ∈ FH(T )∩F(K). Moreover, if Ẽ ∈ EF ?(K) for a face 143 4.6. Proof of Theorem 4.4.1 F ? ∈ FN (T ) ∩ F(K), we have ‖ς Ẽ ‖ L2(K̃) . ∑ F∈{wF (ν)}ν∈N (F?) p−1F h 1/2 F ‖[[v]]‖L2(F ). For the situation when there exists an edge E ∈ E(T ) such that Ẽ ⊆ E, we have ‖ς Ẽ ‖ L2(K̃) . ∑ F∈{wF (ν)}ν∈∂E p−1F h 1/2 F ‖[[v]]‖L2(F ). Now we only need to bound ‖ς F̃ ‖ L2(K̃) in (4.6.34) for any face F̃ ∈ F(K̃). If F̃ ∈ F(T ) or F̃ ∈ FA(T̃ ), by the construction of v and ϑ, we have ‖ς F̃ ‖ L2(K̃) = 0. Otherwise, there exist a face F ∈ F(K) such that F ∈ FR(T ) and F̃ 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 F̃ discussed being F1 and F̃1, respectively. If F1 ∈ FH(T ), then ‖ςF̃1‖L2(K̃) = 0. Otherwise, F1 ∈ FN (T ). Since ςF̃1 vanishes at all the interior tensor-product Gauss-Lobatto nodes in K̃ and on the faces of ZpK̃int (K̃) that are different from F̃1, we obtain from Lemma 4.6.1(i) and the construction of v and ϑ that ‖ς F̃1 ‖ L2(K̃) . p−1 K̃ h 1/2 K̃ ‖ς F̃1 ‖ L2(F̃1) . p−1 K̃ h 1/2 K̃ (‖v K̃ − ϑ‖ L2(F̃1) + ∑ Ẽi∈E(F̃1) ‖ς Ẽi ‖ L2(F̃1) + ∑ ν̃j∈N (F̃1) ‖ςν̃j‖L2(F̃1) ) . p−1K h 1 2 K‖vK̃ − ϑ‖L2(F1) + p−1K h 1 2 K( ∑ Ẽi∈E(F̃1) ‖ς Ẽi ‖ L2(F̃1) + ∑ ν̃j∈N (F̃1) ‖ςν̃j‖L2(F̃1)) ≡ T1 + T2. Using (4.6.27), Lemma 4.6.1(iii) and (4.2.3), we get T1 . p−1K h 1/2 K ∑ ν∈N (F1) ‖ςν‖L2(F1) . p−1K h 1/2 K ∑ ν∈N (F1) (|vK(ν)− ϑ(ν)| ‖ΦνK‖L2(F1)) . ∑ F∈{wF (ν)}ν∈N (F1) p−1F h 1/2 F ‖[[v]]‖L2(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 T2 . ∑ F∈{wF (ν)}ν∈N (F1) p−1F h 1/2 F ‖[[v]]‖L2(F ). Hence, ς F̃ in (4.6.34) can be bounded by ‖ς F̃ ‖ L2(K̃) . ∑ F∈{wF (ν)}ν∈N (F1) p−1F h 1/2 F ‖[[v]]‖L2(F ). To combine the bounds for ςν̃ , ςẼ and ςF̃ , we define the set N ?(K̃) as follows. We start from N (K̃) and first remove all the vertices belonging to NA(T̃ ). Then, any vertex ν̃ ∈ N (K̃) with ν̃ /∈ N (T ) ∪ NA(T̃ ) is replaced by the vertex ν ∈ N (K) which lies on the same elemental edge of K as ν̃; see Section 3.5.5. We also set F?(K̃) = {F ∈ wF (ν) : ν ∈ N ?(K̃) }. Thus, we have ‖v K̃ − ϑ‖ L2(K̃) . ∑ F∈F?(K̃) p−1F h 1/2 F ‖[[v]]‖L2(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 ‖v K̃ − ϑ‖ L2(K̃) . ∑ F∈F?(K̃) p−1F h 1/2 F ‖[[v]]‖L2(F ), K̃ ∈ T̃ . (4.6.36) This proves the first inequality in (4.6.20). Moreover, by the inverse inequal- ity, ‖∇v‖ L2(K̃) . p2 K̃ h−1 K̃ ‖v‖ L2(K̃) , v ∈ Sp̃(T̃ ), K̃ ∈ T̃ , (4.6.37) see [12], we obtain from (4.6.36) and (4.2.3) ‖∇(v K̃ − ϑ)‖ L2(K̃) . ∑ F∈F?(K̃) pFh −1/2 F ‖[[v]]‖L2(F ), K̃ ∈ T̃ , (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 define the function WEK as follows: if E ∈ EB(T ), we set WEK = L E pK ,K ( (vK − vnodK )|E ) , with the extension operator LEpK ,K(·) defined in (4.6.2). If E ∈ EI(T ), let K ′ ∈ wE be the element which has the lowest polyno- mial degree in the set wE defined in (4.6.39); see Section 3.5.6. We define WEK by WEK = L E pK ,K ( (vK′ − vnodK′ )|E ) , with LEpK ,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 ′ ∈ wE1 and K ′′ ∈ wE2 the elements in T which have the lowest polynomial degree in the set wE1 and wE2 , respectively; see Section 3.5.6. We now define W E K by WEK = L E pK ,K ( (vK′ − vnodK′ )|E1 , (vK′′ − vnodK′′ )|E2 ) , with LEpK ,K(·, ·) in (4.6.3). Next, for an edge E ∈ EF (K), F ∈ F(K), the function WEK is given analogously. Let K ′ ∈ wE be the element which has the lowest polynomial degree in the set wE . We define W E K by WEK = L E pK ,K ( (vK′ − vnodK′ )|E ) , with LEpK ,K(·) given in (4.6.5). Then we define ϑedge elementwise by setting ϑedge|K = ∑ E∈E(K) WEK + ∑ F∈F(K) ∑ E∈EF (K) WEK , with WEK defined above. Clearly, the function ϑ edgebelongs to Scp̃(T̃ ). By employing Lemma 4.6.2 and proceeding as in Section 3.5.6, the approxima- tion 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 the function WFK as follows: if F ∈ FB(T ), we set WFK = L F pK ,K ( (vK − vnodK − vedgeK )|F ) , with LFpK ,K(·) defined in (4.6.6). If F ∈ FI(T ), let K ′ in T be the neighbor- ing element such that F ∈ F(K)∩F(K ′). Denote by K ′ the element which has the lower polynomial degree of the elements K and K ′. We define WFK by WFK = L F pK ,K ( (vK′ − vnodK′ − vedgeK′ )|F ) , 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=1Fi with Fi ∈ F(T ), i = 1, . . . , 4, cf. Figure 4.4. There exist four elements Ki ∈ T such that Fi ∈ F(Ki). Denote by Ki the element that has the lower polynomial degree of K and Ki, i = 1, . . . , 4. We now define W F K by WFK = L F pK ,K ( (vK1 − vnodK1 − v edge K1 )|F1 , . . . , (vK4 − vnodK4 − v edge K4 )|F4 ) , with LFpK ,K(·, ·, ·, ·) defined in (4.6.7). Next, we prove the face approximation property (4.6.22). By the lo- cal bounded variation of p (4.2.3), Lemma 4.6.3 and the polynomial trace inequality (see [32]), we have∑ K̃∈T̃ ‖vface − ϑface‖2 L2(K̃) = ∑ K∈T ∑ K̃∈R(K) ‖vface − ϑface‖2 L2(K̃) . ∑ K∈T ∑ F∈F(K) ‖LFpK ,K ( (vK − vnodK − vedgeK )|F )−WFK‖2L2(K) . ∑ K∈T ∑ F∈F(K) p−2K hK‖(vK − vnodK − vedgeK )|F −WFK |F ‖2L2(F ) . ∑ K∈T ∑ F∈F(K) p−2F hF (‖[[v]]‖2L2(F ) + ‖[[vnod]]‖2L2(F ) + ‖[[vedge]]‖2L2(F )) . ∑ K∈T ∑ F∈F(K) p−2F hF (‖[[v]]‖2L2(F ) + ‖[[vnod]]‖2L2(F )) + ∑ K̃∈T̃ ‖[[vedge]]‖2 L2(K̃) . 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 exploit- ing 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 dere- finement 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 ad- justed 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 hp- adaptively 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 factor of p 1/2 F ) in the face polynomial order pF , with the indicator η̂ 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 10 15 20 25 30 35 40 45 50 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100   Error Estimator (p2) True Error (p2) Error Estimator (p3) True Error (p3) (DOF)1/3 0 2 4 6 8 10 0 1 2 3 4 5 6 7 8 9 10   p2 p3 E ff ec ti vi ty Mesh Number 20 40 60 80 100 120 140 160 10−7 10−6 10−5 10−4 10−3 10−2 10−1   h−Refinement hp−Refinement ‖u − u h p ‖ E ,T (DOF)1/3 (a) (b) (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 de- grees 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 η̂, 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 se- lect f and an appropriate inhomogeneous boundary condition, so that the analytical solution to (4.1.1) is given by u(x1, x2, x3) = sin(pix1) cos(pix2) cos(pix3). 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 indi- cator η stated in Theorem 4.3.1 (denoted by p3 in the figure) and η̂ outlined in Remark 4.3.2 (denoted by p2). Here, we observe that the two error in- dicators perform in a very similar manner: in each case the error bound over-estimates the true error by a (reasonably) consistent factor. From Fig- ure 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 ef- fectivity 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) Three- slice 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 frac- tion 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 re- finement 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 10 12 14 16 18 20 22 10−2 10−1 100 101   Error Estimator (p2) True Error (p2) Error Estimator (p3) True Error (p3) (DOF)1/4 0 1 2 3 4 5 6 7 8 9 0 5 10 15 20 25   p2 p3 E ff ec ti vi ty Mesh Number 10 15 20 25 30 10−1  h−Refinement hp−Refinement ‖u − u h p ‖ E ,T (DOF)1/4 (a) (b) (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 de- grees 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) = (x 2 1 + x 2 2 + x 2 3) 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 ∈ H1(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 η̂ 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 el- ement 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 singu- larity 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 per- form 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 η̂ in (4.3.6). Indeed, from Figure 4.9(b) we observe that the effec- tivity 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 Theo- rem 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 ‖u− uhp‖E,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 indica- tors 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 cor- ner 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, isotro- pically 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 three- dimensional 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 discretiza- tions 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) Three- slice 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. 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Schwab. hp-FEM for fluid flow simulation. In T. Barth and H. De- conink, editors, High-Order Methods for Computational Physics, vol- ume 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 hp- adaptive DG methods for convection-diffusion equations. IMA J. Nu- mer. 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 condi- tions. 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 incorpo- rate 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 sin- gularities 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 el- emental polynomial degrees. 4A 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 esti- mation for hp-adaptive DG methods for convection-diffusion equations on anisotropically refined rectangular meshes with anisotropic polynomial de- gree 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 asso- ciated with the convective term. The constant in the lower bound is inde- pendent 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 convection- diffusion 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 c1x ≤ y ≤ c2x, 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 on Γ. (5.2.1) Here, Ω is a bounded Lipschitz polyhedral domain 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. (5.2.2) Without loss of generality, we shall assume that ‖a‖L∞(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 ∈ H10 (Ω) such that A(u, v) = ∫ Ω ( ε∇u · ∇v+ a · ∇uv) dx = ∫ Ω fv dx ∀ v ∈ H10 (Ω). (5.2.3) Under assumption (5.2.2), the variational problem (5.2.3) is uniquely solv- able. 5.2.2 Discretization Let T be a subdivision of the polygonal domain Ω ⊂ R2 into disjoint el- ements, which for simplicity we assume to be rectangles. Each element K ∈ T is the image of the reference square under an affine elemental map- ping FK : K̂ → K; see Figure 5.1. - 6 (−1,−1) (1, 1) K̂ x̂1 x̂2 R FK A AAK   *   A AA v1K v2K x1 x2 K h2K h1K 6 - 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 ′ of two elements K,K ′ ∈ 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 h 2 K , respectively: h1K = length(v 1 K), h 2 K = length(v 2 K). Further, we set hmin,K = min{h1K , h2K}, hmax,K = max{h1K , h2K}. We then define the matrix MK = [v 1 K , v 2 K ]. (5.2.4) Note that MK is orthogonal and M>KMK = [ (h1K) 2 (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: h⊥E,K = h 3−i K , if E is parallel to v i K , i = 1, 2. Moreover, for any E ∈ EI(T ), we assume that h⊥E,K ∼ h⊥E,K′ , E = K ∩K ′, K,K ′ ∈ T . (5.2.5) Note that this assumption does not bound the aspect ratios of elements. For any edges E,E′ ∈ E(K) and E∩E′ 6= ∅, hE/hE′ 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 = h i K , i = 1, 2. For any edge E ∈ E(T ), we further set h⊥E = { min{h⊥E,K , h⊥E,K′}, E ∈ EI(T ), E = ∂K ∩ ∂K ′, h⊥E,K , E ∈ EB(T ), E = ∂K ∩ Γ. In our analysis, we allow for irregularly refined meshes T , where each ele- mental 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 Î = (−1, 1) be the reference interval. Then we take M points, −1 < ŷ1 < ŷ2 < · · · < ŷM < 1, on Î and set Ĥ = { ŷ1, · · · , ŷM }. Let now E ∈ E(K) be an edge of an element K. The nodes of Ĥ can be affinely mapped onto E and we denote by H(E) = { yE1 , · · · , yEM } the possible loca- tions of the hanging nodes on E. In other words, any hanging node on E belongs to H(E). Setting ŷ0 = −1 and ŷM+1 = 1, we further assume there exists a positive constant κ ≤ 1 independent of the mesh T , such that |ŷi+1 − ŷi| ≥ 2κ for i = 0, 1, · · · ,M . Under this assumption, if the edge E′ ∈ E(T ) is also a part of the elemental edge E (i.e., E′ ⊂ E), we have κ ≤ hE′/hE ≤ 1. (5.2.6) Next, we define hmin,E by setting hmin,E = { min{hmin,K , hmin,K′}, E ∈ EI(T ), E = ∂K ∩ ∂K ′, hmin,K , 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 ∼ h⊥E,K , hmin,E ∼ hmin,K . (5.2.7) 5.2.4 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 polyno- mial degrees pK = ( p 1 K , p 2 K ) with p 1 K , p 2 K ≥ 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 = p i K , if E is parallel to v i K , i = 1, 2, p⊥E,K = p 3−i K , if E is parallel to v i K , i = 1, 2. We define a polynomial degree vector p = {pK : K ∈ T } and set |p| = max K∈T pmax,K . Moreover, we assume that p is of bounded local variation. That is, for any pair of neighboring elements K,K ′ ∈ T sharing an edge E ∈ T , we have p⊥E,K ∼ p⊥E,K′ , pE,K ∼ pE,K′ . (5.2.8) For an edge E ∈ E(T ), we introduce two sets of edge polynomial de- grees pE and p ⊥ E by: pE = { max{pE,K , pE,K′}, E ∈ EI(T ), E = ∂K ∩ ∂K ′, pE,K , E ∈ EB(T ), E = ∂K ∩ Γ, p⊥E = { max{p⊥E,K , p⊥E,K′}, E ∈ EI(T ), E = ∂K ∩ ∂K ′, p⊥E,K , E ∈ EB(T ), E = ∂K ∩ Γ. (5.2.9) 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 Sp(T ) = { v ∈ L2(Ω) : v|K ◦ FK ∈ QpK (K̂), K ∈ T }, withQpK (K̂) denoting the set of all polynomials on the reference square K̂ of degree less or equal than p1K and p 2 K in the two space directions, respectively. We now consider the following discontinuous Galerkin method for the ap- proximation of the convection-diffusion problem (5.2.1): Find uhp ∈ Sp(T ) such that Ahp(uhp, v) = ∫ Ω fv dx (5.2.10) 163 5.3. A-posteriori error estimates for all v ∈ Sp(T ), with the bilinear form Ahp given by Ahp(u, v) = ∑ K∈T ∫ K (ε∇u · ∇v + a · ∇uv) dx − ∑ E∈E(T ) ∫ E {{ε∇u}} · [[v]] ds− ∑ E∈E(T ) ∫ E {{ε∇v}} · [[u]] ds + ∑ E∈E(T ) ∫ E εγ(p⊥E) 2 h⊥E [[u]] · [[v]] ds− ∑ K∈T ∫ ∂Kin∩Γin a · nK uv ds + ∑ 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 con- stant 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: ‖ v ‖2E,T = ∑ K∈T ε‖∇v‖2L2(K) + ejumpp,T (v)2, ejumpp,T (v) 2 = ∑ E∈E(T ) εγ(p⊥E) 2 h⊥E ‖[[v]]‖2L2(E). (5.3.1) Next, we define |q|? = sup v∈H10 (Ω)\{0} ∫ Ω q · ∇v dx ‖ v ‖E,T ∀ q ∈ L 2(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, ojumpp,T (v) 2 = ∑ K∈T ∑ E∈E(K) (εh⊥E,Kp2E,K h2min,K + h⊥E,Kp ⊥ E,K εp2min,K )‖[[v]]‖2L2(E). (5.3.2) Notice that the definition of ojumpp,T (v) is different from the one in Chap- ter 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 approx- imations 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 η2K = η 2 RK + η2EK + η 2 JK . (5.3.3) The first term ηRK is the interior residual defined by η2RK = ε −1p−2min,Kh 2 min,K‖fhp + ε∆uhp − ahp · ∇uhp‖2L2(K). The second term ηEK is the edge residual given by η2EK = 1 2 ∑ E∈∂K\Γ h2min,Kp ⊥ E,K εp2min,Kh ⊥ E,K ‖[[ε∇uhp]]‖2. The last residual ηJK measures the jumps of the approximate solution uhp: η2JK = 1 2 ∑ E∈∂K\Γ ∫ E ( εγ2(p⊥E,K) 5 h⊥E,Kp 2 min,K + εh⊥E,Kp 2 E,K h2min,K + h⊥E,Kp ⊥ E,K εp2min,K ) [[uhp]] 2 ds + ∑ E∈∂K∩Γ ∫ E ( εγ2(p⊥E,K) 5 h⊥E,Kp 2 min,K + εh⊥E,Kp 2 E,K h2min,K + h⊥E,Kp ⊥ E,K εp2min,K ) [[uhp]] 2 ds. 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 Θ2K = ε −1p−2min,Kh 2 min,K ( ‖f − fhp‖2L2(K) + ‖(a− ahp) · ∇uhp‖2L2(K) ) . We then introduce the global error estimator and data approximation error η2 = ∑ K∈T η2K , Θ 2 = ∑ K∈T Θ2K . (5.3.4) In the following, we use the symbols . and & to denote bounds that are valid up to positive constants independently of the mesh sizes, the polyno- mial 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 ∈ H1(Ω) be an arbitrary non-constant function and T be a family of triangulations of Ω. The align- ment measure M(v, T ) is then defined by M(v, T ) = ( ∑ K∈T h −2 min,K‖MK∇v‖2L2(K))1/2 ‖∇v‖L2(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 ‖u− uhp ‖E,T + |u− uhp |O,T . M(v, T )(η + Θ). Here, v ∈ H10 (Ω) is the test function such that the inf-sup condition (5.4.10) holds true. Remark 5.3.3 The alignment measureM(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, 12) we have the bound η . |p|δ+ 32 ‖u− uhp ‖E,T + |p|2δ+ 3 2 |u− uhp |O,T + |p|2δ+1Θ. 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 subop- timal 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∈H10 (Ω)\{0} sup v∈H10 (Ω)\{0} A(u, v) (‖u ‖E,T + |au|?) ‖ v ‖E,T ≥ 1 3 . 167 5.4. Proofs Next, we split the discontinuous Galerkin form Ahp into two parts, see Chapter 2, and define Ãhp(u, v) = ∑ K∈T ∫ K (ε∇u · ∇v + a · ∇uv) dx+ ∑ E∈E(T ) ∫ E εγ(p⊥E) 2 h⊥E [[u]] · [[v]]ds − ∑ K∈T ∫ ∂Kin∩Γin a · nK uv ds+ ∑ K∈T ∫ ∂Kin\Γ a · nK(ue − u)v ds, Khp(u, v) = − ∑ E∈E(T ) ∫ E {{ε∇u}} · [[v]] ds− ∑ 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 A(u, v) = Ãhp(u, v), u, v ∈ H10 (Ω), (5.4.1) Ahp(u, v) = Ãhp(u, v) +Khp(u, v), u, v ∈ Sp(T ). (5.4.2) 5.4.2 Auxiliary meshes We shall make use of an auxiliary irregular mesh T̃ 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 yEi , i ∈ {1, · · · ,M}. In this case, we refine K by connecting yEi and yE ′ i along the curve FK(F −1 K (y E i ), F −1 K (y E′ i )) with E ′ 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 T̃ 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(T̃ ) the set of vertices in N (T̃ ) and EA(T̃ ) the set of edges in E(T̃ ) which are inside an element K of T , respectively. Moreover, we write R(K) for the elements in T̃ 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 T̃ : Sp̃(T̃ ) = { v ∈ L2(Ω) : v|K̃ ◦ FK̃ ∈ QpK̃ (K̂), K̃ ∈ T̃ }, 168 5.4. Proofs s s s s s s s s s s s ss s ss ss s ss s s s s s =⇒ s s s s s s s s s s s ss s ss ss s ss s s s s s c c c c cc c c Figure 5.2: The construction of the auxiliary mesh T̃ from T . where the auxiliary polynomial degree vector p̃ is defined by p K̃ = pK , for K̃ ∈ R(K). Thus, we clearly have the inclusion Sp(T ) ⊆ Sp̃(T̃ ). In complete analogy to (5.3.1) and (5.3.2), the energy and convective norms associated with the auxiliary mesh T̃ are given by ‖ v ‖2 E,T̃ = ∑ K̃∈T̃ ε‖∇v‖2 L2(K̃) + ejump p̃,T̃ (v) 2, | v |2 O,T̃ = |av| 2 ? + ojumpp̃,T̃ (v) 2, (5.4.3) where the auxiliary edge polynomial degrees for the jump terms over T̃ are defined as in (5.2.9), using the auxiliary degrees p K̃ . Obviously, we have ‖ v ‖E,T = ‖ v ‖E,T̃ , | v |O,T = | v |O,T̃ , for all v ∈ H10 (Ω). Furthermore, analogously to Lemmas 3.4.2 and 3.4.3, one can show the following results. Lemma 5.4.2 Let v ∈ Sp̃(T̃ ) + H10 (Ω) be such that [[v]]|E = [[w]]|E for all E ∈ E(T̃ ), for a function w ∈ Sp(T ) +H10 (Ω). Then we have ejumpp,T (w) . ejumpp̃,T̃ (v) . ejumpp,T (w), ojumpp,T (w) . ojumpp̃,T̃ (v) . ojumpp,T (w). Lemma 5.4.3 For v ∈ Sp(T ) +H10 (Ω), we have the bounds ‖ v ‖E,T . ‖ v ‖E,T̃ , | v |O,T . | v |O,T̃ . 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 con- tinuous ones, analogously to the one used in Chapters 3 and 4. To define this operator, we let Scp̃(T̃ ) be the conforming subspace of Sp̃(T̃ ) given by Scp̃(T̃ ) = Sp̃(T̃ ) ∩H10 (Ω). We then have the following approximation result. Theorem 5.4.4 (Averaging operator) There is operator Ihp : Sp(T )→ Scp̃(T̃ ) that satisfies∑ K̃∈T̃ ‖v − Ihpv‖2L2(K̃) . ∑ E∈E(T ) ∫ E (p⊥E) −2h⊥E [[v]] 2 ds, (5.4.4) ∑ K̃∈T̃ ‖∇(v − Ihpv)‖2L2(K̃) . ∑ E∈E(T ) ∫ E p2Eh ⊥ Eh −2 min,E [[v]] 2 ds. (5.4.5) 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 H1-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 = u c hp + u r hp, (5.4.6) where uchp = Ihpuhp ∈ Scp̃(T̃ ) ⊂ H10 (Ω), with Ihp the approximation operator from Theorem 5.4.4. The remainder is then given by urhp = uhp − uchp = uhp − Ihpuhp ∈ Sp̃(T̃ ). By Lemma 5.4.3 and the triangle inequality, we obtain ‖u− uhp ‖E,T + |u− uhp |O,T . ‖u− uhp ‖E,T̃ + |u− uhp |O,T̃ . ‖u− uchp ‖E,T̃ + |u− uchp |O,T̃ + ‖urhp ‖E,T̃ + |urhp |O,T̃ = ‖u− uchp ‖E,T + |u− uchp |O,T + ‖urhp ‖E,T̃ + |urhp |O,T̃ . (5.4.7) 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 ‖urhp ‖E,T̃ + |urhp |O,T̃ . η. Proof : Since [[urhp]]|E = [[uhp]]|E for all E ∈ E(T̃ ) and uhp ∈ Sp(T ), the definition of the jump residual ηJK and Lemma 5.4.2 yield ‖urhp ‖2E,T̃ + |u r hp |2O,T̃ = ∑ K̃∈T̃ ε‖∇urhp‖2L2(K̃) + |au r hp|2? + ejumpp̃,T̃ (urhp)2 + ojumpp̃,T̃ (urhp)2 . ∑ K̃∈T̃ ε‖∇urhp‖2L2(K̃) + |au r hp|2? + ∑ K∈T η2JK . Hence, only the volume terms and |aurhp|? need to be bounded further. The- orem 5.4.4, the assumptions (5.2.7) and (5.2.8) yield ε ∑ K̃∈T̃ ‖∇urhp‖2L2(K̃) . ∑ E∈E(T ) ∫ E εp2Eh ⊥ Eh −2 min,E [[uhp]] 2 ds . ∑ K∈T η2JK . 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), |aurhp|2? . 1 ε ∑ K̃∈T̃ ( ‖a‖2 L∞(K̃)‖u r hp‖2L2(K̃) ) . ∑ E∈E(T ) h⊥E ε(p⊥E)2 ‖[[uhp]]‖2L2(E) . ∑ K∈T η2JK . This finishes the proof. 2 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 ∈ H10 (Ω), there exists a function vhp ∈ Sp(T ) such that p2min,K‖v − vhp‖2L2(K) . ‖MK∇v‖2L2(K), ‖MK∇(v − vhp)‖2L2(K) . ‖MK∇v‖2L2(K),∑ E∈E(K) h⊥E,Kp 2 min,K p⊥E,K ‖v − vhp‖2L2(E) . ‖MK∇v‖2L2(K), (5.4.8) 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, ‖v − vhp‖2L2(E) . 1 h⊥E,K ‖MK∇(v − vhp)‖L2(K)‖v − vhp‖L2(K) + 1 h⊥E,K ‖v − vhp‖2L2(K), the Cauchy-Schwarz inequality, the fact that p⊥E,K ≥ 1 and the results in the first and the second inequalities, we obtain ‖v − vhp‖2L2(E) . 1 h⊥E,Kp ⊥ E,K ‖MK∇(v − vhp)‖2L2(K) + p⊥E,K h⊥E,K ‖v − vhp‖2L2(K) . 1 h⊥E,Kp ⊥ E,K ‖MK∇v‖2L2(K) + p⊥E,K h⊥E,Kp 2 min,K ‖MK∇v‖2L2(K) . p⊥E,K h⊥E,Kp 2 min,K ‖MK∇v‖2L2(K), which shows the third inequality. 2 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 ∈ H10 (Ω), we have∑ K∈T p2min,K h2min,K ‖v − vhp‖2L2(K) .M(v, T )2‖∇v‖2L2(Ω), ∑ K∈T ∑ E∈∂K h⊥E,Kp 2 min,K p⊥E,Kh 2 min,K ‖v − vhp‖2L2(E) .M(v, T )2‖∇v‖2L2(Ω). (5.4.9) We then have the following result. Lemma 5.4.9 For any v ∈ H10 (Ω), we have∫ Ω f(v−vhp)dx−Ãhp(uhp, v−vhp)+Khp(uhp, vhp) .M(v, T ) (η + Θ) ‖ v ‖E,T . 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∫ Ω f(v−vhp) dx− Ãhp(uhp, v−vhp)+Khp(uhp, vhp) = T1 +T2 +T3 +T4 +T5, where T1 = ∑ K∈T ∫ K (f + ε∆uhp − a · ∇uhp)(v − vhp) dx, T2 = − ∑ E∈EI(T ) ∫ E [[ε∇uhp]]{{v − vhp}} ds, T3 = − ∑ E∈E(T ) ∫ E {{ε∇vhp}} · [[uhp]] ds, T4 = ∑ K∈T ∫ ∂Kin\Γ a · nK(uhp − uehp)(v − vhp) ds + ∑ K∈T ∫ ∂Kin∩Γin a · nKuhp(v − vhp) ds, T5 = − ∑ E∈E(T ) ∫ E εγ(p⊥E) 2 h⊥E [[uhp]] · [[v − vhp]] ds. 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 T1 .M(v, T ) ( ∑ K∈T (η2RK + Θ 2 K) ) 1 2 ‖ v ‖E,T . Similarly, by the Cauchy-Schwarz inequality and (5.4.9), we have T2 . ( ∑ K∈T ∑ E∈∂K\Γ h2min,Kp ⊥ E,K εp2min,Kh ⊥ E,K ‖[[ε∇uhp]]‖2L2(E) ) 1 2 × ( ∑ K∈T ∑ E∈∂K\Γ εp2min,Kh ⊥ E,K h2min,Kp ⊥ E,K ‖v − vhp‖2L2(E) ) 1 2 .M(v, T )( ∑ K∈T η2EK ) 1 2 ‖ v ‖E,T . 173 5.4. Proofs To estimate T3, we employ the Cauchy-Schwarz inequality, the trace in- equality in [6, Lemma 3.1] for anisotropic polynomial degrees. This results in T3 . ( ∑ E∈E(T ) ∫ E ε(p⊥E) 2 h⊥E [[uhp]] 2 ds ) 1 2 ( ∑ K∈T ∑ E∈E(K) ∫ E εh⊥E (p⊥E)2 |∇vhp|2 ds ) 1 2 . ( ∑ K∈T η2JK ) 1 2 ( ∑ K∈T ε‖∇vhp‖2L2(K) ) 1 2 . ( ∑ K∈T η2JK ) 1 2 ‖ v ‖E,T . For T4, we apply again the Cauchy-Schwarz inequality and (5.4.9) to get T4 . ( ∑ K∈T ∑ E∈E(K) h2min,Kp ⊥ E,K εp2min,Kh ⊥ E,K ‖[[uhp]]‖2L2(E) ) 1 2 × ( ∑ K∈T ∑ E∈E(K) εp2min,Kh ⊥ E,K h2min,Kp ⊥ E,K ‖v − vhp‖2L2(E) ) 1 2 .M(v, T )( ∑ K∈T η2JK ) 1 2 ‖ v ‖E,T . Finally, we have T5 . ( ∑ K∈T ∑ E∈E(K) εγ2h2min,K(p ⊥ E,K) 5 p2min,K(h ⊥ E,K) 3 ‖[[uhp]]‖2L2(E) ) 1 2 × ( ∑ K∈T ∑ E∈E(K) εp2min,Kh ⊥ E,K h2min,Kp ⊥ E,K ‖v − vhp‖2L2(E) ) 1 2 .M(v, T )( ∑ K∈T η2JK ) 1 2 ‖ v ‖E,T . The above estimates for the terms T1 through T5 imply the assertion. 2 Lemma 5.4.10 There holds: ‖u− uchp ‖E,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 ∈ H10 (Ω), we have |u− uchp |O,T = |a(u− uchp)|?. Then the inf-sup condition in Lemma 5.4.1 yields: ‖u− uchp ‖E,T + |u− uchp |O,T . sup v∈H10 (Ω)\{0} A(u− uchp, v) ‖ v ‖E,T . (5.4.10) 174 5.4. Proofs To bound (5.4.10), let v ∈ H10 (Ω). Then, property (5.4.1) shows that A(u− uchp, v) = ∫ Ω fv dx−Ahp(uchp, v) = ∫ Ω fv dx− Ãhp(uchp, v). By employing the fact that v ∈ H10 (Ω) and integrating by parts the convec- tion term, one can readily see that Ãhp(u c hp, v) = Ãhp(uhp, v) +R, with R = ∑ K̃∈T̃ ∫ K̃ (−ε∇urhp + aurhp) · ∇v dx. Furthermore, from the DG method in (5.2.10) and property (5.4.2), we have∫ Ω fvhp dx = Ãhp(uhp, vhp) +Khp(uhp, vhp), where vhp ∈ Sp(T ) is the hp-version interpolant of v in Lemma 5.4.7. Com- bining the above results yields A(u− uchp, v) = ∫ Ω f(v − vhp) dx− Ãhp(uhp, v − vhp) +Khp(uhp, vhp)−R. The estimate in Lemma 5.4.9 now shows that |A(u− uchp, v)| .M(v, T ) (η + Θ) ‖ v ‖E,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 |R| . ( ‖urhp ‖E,T̃ + |urhp |O,T̃ ) ‖ v ‖E,T . η‖ v ‖E,T . (5.4.12) Equations (5.4.10)–(5.4.12) imply the desired result. 2 The proof of Theorem 5.3.2 now immediately follows from the inequal- ity (5.4.7), Lemma 5.4.6 and Lemma 5.4.10. 5.4.5 Proof of Theorem 5.3.4 In the proof of Theorem 5.3.4, we shall make use of the bubble functions, Ψ̂1 and Ψ̂2, constructed in Lemma 4.5.1. For an arbitrary element K ∈ T , we set ΨK = Ψ2 ◦F−1K ; for an interior edge E, we let ΨE = Ψ1 ◦F−1E , 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 , respec- tively. 175 5.4. Proofs Lemma 5.4.11 Under the assumptions of Theorem 5.3.4, there holds ( ∑ K∈T η2RK ) 1 2 . |p| ‖u− uhp ‖E,T + |p| |a(u− uhp)|? + |p|δ+ 1 2Θ. Proof : For any element K ∈ T , we set vK = ε −1(fhp + ε∆uhp − ahp · ∇uhp)|KΨαK , where α ∈ (12 , 1]. Applying the inverse inequality (4.5.3), we obtain ‖fhp + ε∆uhp − ahp · ∇uhp‖L2(K) . (p1Kp2K) α 2 ‖(fhp + ε∆uhp − ahp · ∇uhp)Ψα/2K ‖L2(K) = ε(p1Kp 2 K) α 2 ‖vKΨ−α/2K ‖L2(K). This leads to∑ K∈T η2RK . S 2 with S2 = ∑ K∈T εh2min,K(p 1 Kp 2 K) α/p2min,K‖vKΨ−α/2K ‖2L2(K). 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 , S2 = ∑ K∈T h2min,K(p 1 Kp 2 K) α/p2min,K ∫ K (fhp + ε∆uhp − ahp · ∇uhp)vK dx = ∑ K∈T h2min,K(p 1 Kp 2 K) α/p2min,K (∫ K (ε∇(u− uhp)− a(u− uhp)) · ∇vK dx + ∫ K (((fhp − f) + (a− ahp) · ∇uhp)ε− 1 2Ψ α 2 K)(ε 1 2 vKΨ −α 2 K )dx ) . Here, we have also used that vK |∂K = 0. From the proof of [19, Lemma 3.4], we have ‖∇vK‖2L2(K) . (p1Kp 2 K) 2−α h2min,K ‖vKΨ−α/2K ‖2L2(K). By the Cauchy-Schwarz inequality, the definition of the dual norm and the data approximation error Θ, we obtain S2 . S (|p|‖u− uhp ‖E,T + |p||a(u− uhp)|? + |p|αΘ) . Therefore, ( ∑ K∈T η2RK ) 1/2 . |p| ‖u− uhp ‖E,T + |p| |a(u− uhp)|? + |p|αΘ. 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 }, w̃E = { K̃ ∈ T ∪ T̃ : E ∈ E(K̃) }. (5.4.13) For simplicity, we also use the notation wE and w̃E to denote the domain formed by the elements in wE and in w̃E , respectively. Lemma 5.4.12 Under the assumptions of Theorem 5.3.4, there holds ( ∑ K∈T η2EK ) 1 2 . √ DT (|p|δ+1‖u− uhp ‖E,T +|p|2δ+1|u− uhp |O,T +|p|2δ+ 12Θ), 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 ∈ H10 (wE) with ψE |E = τE , ψE |∂wE = 0 and ‖ψE‖2L2(wE) . h⊥E p−2E ‖τEΨ −α/2 E ‖2L2(E), (5.4.15) ‖∇ψE‖2L2(wE) . h⊥Eh−2min,Ep2E‖τEΨ −α/2 E ‖2L2(E). (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. In this case, wE is concave, and there exists an element K̃1 ∈ T̃ , such that K̃1 ( K1 and K̃1 ∩ K2 = E. Thus w̃E = K̃1 ∪ K2 ( wE . By Lemma 2.6 of [19] we can find a function ψ̃E ∈ H10 (w̃E) with ψ̃E |E = τE , ψ̃E |∂w̃E = 0 and ‖ψE‖2L2(wE) . h⊥E p−2E ‖τEΨ −α/2 E ‖2L2(E), ‖∇ψE‖2L2(wE) . h⊥Eh−2min,Ep2E‖τEΨ −α/2 E ‖2L2(E). Now define the function ψE on wE by ψE = ψ̃E on w̃E , and by zero oth- erwise. Thus, we have ψE ∈ H10 (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 ‖[[∇uhp]]‖L2(E) . ‖pαE [[∇uhp]]Ψ α 2 E‖L2(E) = ‖pαE τEΨ −α 2 E ‖L2(E). (5.4.17) Therefore,∑ K∈T η2EK . S 2 with S2 = ∑ K∈T ∑ E∈∂K\Γ ∫ E ε h2min,Kp ⊥ E,K p2min,Kh ⊥ E,K p2αE τ 2 EΨ −α E ds. Since [[ε∇u]] = 0 on interior edges, integration by parts over wE yields∫ E [[ε∇(uhp − u)]]τE ds = ∫ wE ε(∆uhp −∆u)ψE + ε(∇uhp −∇u) · ∇ψE dx, where ∆uhp and ∇uhp are understood piecewise. Using the differential equa- tion, approximating the data and integrating by parts the convective term, we readily obtain S2 = T1 + T2 + T3 + T4 + T5, with T1 = ∑ K∈T ∑ E∈∂K\Γ ∫ wE h2min,Kp ⊥ E,K p2min,Kh ⊥ E,K p2αE (fhp + ε∆uhp − ahp · ∇uhp)ψE dx, T2 = ∑ K∈T ∑ E∈∂K\Γ ∫ wE h2min,Kp ⊥ E,K p2min,Kh ⊥ E,K p2αE (ε∇uhp − ε∇u) · ∇ψE dx, T3 = ∑ K∈T ∑ E∈∂K\Γ ∫ wE h2min,Kp ⊥ E,K p2min,Kh ⊥ E,K p2αE a(u− uhp) · ∇ψE dx, T4 = ∑ K∈T ∑ E∈∂K\Γ ∫ E h2min,Kp ⊥ E,K p2min,Kh ⊥ E,K p2αE a · [[uhp]]τE ds, T5 = ∑ K∈T ∑ E∈∂K\Γ ∫ wE h2min,Kp ⊥ E,K p2min,Kh ⊥ E,K p2αE ( (f − fhp) + (ahp − a) · ∇uhp ) ψE dx. The Cauchy-Schwarz inequality, Lemma 5.4.11 and inequality (5.4.15) yield T1 . S √ DT |p|α− 12 (|p| ‖u− uhp ‖E,T + |p| |a(u− uhp)|? + |p|αΘ). Similarly, we obtain T2 . S √ DT |p|α+ 12 ‖u− uhp ‖E,T , 178 5.4. Proofs as well as T3 . S √ DT |p|α+ 12 |a(u− uhp)|?. By (5.4.17) and the definition of semi-norm | · |O,T , we conclude that T4 . ( ∑ K∈T ∑ E∈∂K\Γ ∫ E p4αE h2min,Kp ⊥ E,K εp2min,Kh ⊥ E,K [[uhp]] 2 ds ) 1 2 × ( ∑ K∈T ∑ E∈∂K\Γ ∫ E ε h2min,Kp ⊥ E,K p2min,Kh ⊥ E,K p2αE τ 2 EΨ −α E ds ) 1 2 . S |p|2α|u− uhp |O,T . Finally, the data error term T5 can be bounded by T5 . ( ∑ K∈T ∑ E∈∂K\Γ ∫ wE ε h2min,Kp ⊥ E,K p2min,Kh ⊥ E,K p2αE τ 2 EΨ −α E dx ) 1 2 × ( ∑ K∈T ∑ E∈∂K\Γ ∫ wE p⊥E,Kp 2α−1 E pE h2min,K εp2min,K |(f − fhp) + (ahp − a) · ∇uhp|2dx ) 1 2 . S √ DT |p|α− 12Θ. Combining the above bounds for T1 through T5, we obtain S2 . S √ DT ( |p|α+ 12 ‖u− uhp ‖E,T + |p|2α|u− uhp |O,T + |p|2α− 1 2Θ ) . Thus,(∑ K∈T η2EK ) 1 2 . D 1 2 T (|p|α+ 12 ‖u− uhp ‖E,T + |p|2α|u− uhp |O,T + |p|2α− 12Θ). 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 ( ∑ K∈T η2JK ) 1/2 . DT |p| 12 ‖u− uhp ‖E,T + |u− uhp |O,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 Î = (−1, 1) be the reference interval. We denote by Ẑp = { ẑp0 , · · · , ẑpp } the Gauss-Lobatto nodes of order p ≥ 1 on Î. Recall that ẑp0 = −1 and ẑpp = 1. We denote by Ẑpint = { ẑp1 , · · · , ẑpp−1 } the interior Gauss-Lobatto nodes of order p on Î. Let now E ∈ E(K) be an edge of an element K. The nodes in Ẑp can be affinely mapped onto E and we denote by Zp(E) = { zE,p0 , · · · , zE,pp } the Gauss-Lobatto nodes of order p on E. The points zE,p0 and z E,p p coincide with the two end points of E. The set Zpint(E) = { zE,p1 , · · · , zE,pp−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 P intp (E) = { q ∈ Pp(E) : q(zE,p0 ) = q(zE,pp ) = 0 }, Pnodp (E) = { q ∈ Pp(E) : q(z) = 0, z ∈ Zpint(E) }. By construction, we have Pp(E) = P intp (E)⊕ Pnodp (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. We first consider the reference element K̂ = (−1, 1)2. We denote its four edges by Ê1, . . . , Ê4 and its four vertices by ν̂1, . . . , ν̂4, numbered as in Figure 5.3. Let {ϕ̂pi }0≤i≤p be the Lagrange basis functions associated with the nodes Ẑp. We denote by {ẑpK̂i,j = (ẑ p1 K̂ i , ẑ p2 K̂ j )}, 1 ≤ i ≤ p1K̂ − 1, 1 ≤ j ≤ p2 K̂ − 1, the interior tensor-product Gauss-Lobatto nodes on K̂. Note that p Ê1,K̂ = p Ê3,K̂ = p1 K̂ and p Ê2,K̂ = p Ê4,K̂ = p2 K̂ . The interior basis functions are then given by Φ̂ int,p K̂ i,j (x̂1, x̂2) = ϕ̂ p1 K̂ i (x̂1) ϕ̂ p2 K̂ j (x̂2), 1 ≤ i ≤ p1K̂ − 1, 1 ≤ j ≤ p 2 K̂ − 1. 180 5.5. Proof of Theorem 5.4.4 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 ν̂4 ν̂1 Ê3 Ê1 ν̂2 ν̂3 Ê4 Ê2 Figure 5.3: Reference element with variable edge polynomial degrees: p K̂ = (5, 4), p Ê1 = 2, p Ê2 = 3, p Ê3 = 4, p Ê4 = 1. Next, we consider exemplarily the edge Ê1 in Figure 5.3 with edge degree p Ê1 . The edge basis functions for Ê1 are Φ̂ Ê1,pÊ1 i (x̂1, x̂2) = ϕ̂ p Ê1 i (x̂1) ϕ̂ p2K 0 (x̂2), i = 1, · · · , pÊ1 − 1. Note that Φ̂ Ê1,pÊ1 i vanishes on Ê2, Ê3 and Ê4. The other edge basis functions are defined analogously. Finally, we consider the vertex ν̂1, which is shared by Ê1 and Ê4; see Figure 5.3. We then introduce the associated vertex basis function Φ̂ν̂1 K̂ (x̂1, x̂2) = ϕ̂ p Ê1 0 (x̂1) ϕ̂ p Ê4 0 (x̂2). It vanishes on Ê2 and Ê3. The vertex basis functions associated with the other vertices of K̂ are defined analogously. This completes the definition of the shape functions on the reference element K̂. For an arbitrary parallelogram K, shape functions Φ on K can be defined from the analogous ones on K̂ by using the pull-back map Φ = Φ̂ ◦ F−1K , giving rise to shape functions ΦνK , Φ E,pE i and Φ int,pK i,j on K. Therefore, a polynomial v of the form (5.5.1) can be expanded into v(x) = ∑ ν∈N (K) v(ν) ΦνK(x) + ∑ E∈E(K) pE−1∑ i=1 v(zE,pEi ) Φ E,pE i (x) + ∑ 1≤i≤p1K−1 1≤j≤p2K−1 cij Φ int,pK i,j (x), 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: (i) For a function v̂ ∈ Qp(Î) that vanishes at the interior Gauss-Lobatto nodes on Î, there holds ‖v̂‖2 L2(Î) . p−2(v̂(−1)2 + v̂(1)2). (ii) For a function v̂ ∈ Qp K̂ (K̂) that vanishes at the interior Gauss- Lobatto nodes on K̂, there holds ‖v̂‖2 L2(K̂) . 4∑ i=1 (p⊥ Êi,K̂ )−2‖v̂‖2 L2(Êi) . (iii) If the vertex ν̂ of the reference element K̂ is shared by two edges Ên and Êm, the associated vertex basis function Φ̂ ν̂ K̂ can be bounded by ‖Φ̂ν̂ K̂ ‖ L2(K̂) . p−1 Ên p−1 Êm . 5.5.2 Extension operators Next, we define extension operators over edges. Let Ê ∈ E(K̂) be an ele- mental edge of the reference element K̂. We define L̂Êp K̂ by L̂Êp K̂ : P intp Ê,K̂ (Ê) −→ Qp K̂ (K̂), q̂(x) 7−→ p Ê,K̂ −1∑ i=1 q̂(ẑ Ê,p Ê,K̂ i )Φ̂ Ê,p Ê,K̂ i (x). (5.5.2) By construction, L̂Êp K̂ (q̂) = q̂ on Ê, and LÊp K̂ (q̂) vanishes in all the interior tensor-product Gauss-Lobatto nodes {ẑpK̂i,j } of K̂ and on the other three edges of K̂. From [6, Lemma 3.1], we have the following inequality. Lemma 5.5.2 The linear extension operator L̂Êp K̂ introduced in (5.5.2) sat- isfies ‖L̂Êp K̂ (q̂)‖ L2(K̂) . (p⊥ Ê,K̂ )−1‖q̂‖ L2(Ê) . 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 operator LEpK ,K(q) : P intpE,K (E)→ QpK (K) by LEpK ,K(q) = [L Ê pK (q ◦ FK)] ◦ F−1K , q ∈ P intpE,K (E). (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 ∈ P intpE,K (Ei), i = 0, · · · , N , we define the extension operator LEpK ,K(q0, · · · , qN ) by LEpK ,K(q0, · · · , qN ) = N∑ i=0 LEipK ,Ki(qi), (5.5.4) with LEipK ,Ki , i = 0, · · · , N , given in (5.5.3). By definitions, the extensions LEpK ,K(q) and L E pK ,K (q0, · · · , qN ) are continuous in K and satisfy LEpK ,K(q)|E = q, and LEpK ,K(q0, · · · , qN )|Ei = qi, i = 0, · · · , N. Moreover, LEpK ,K(q) and L E pK ,K (q0, · · · , qN ) both vanish on the other edges of E(K). s s s s c c s sK0 KN EN E0 pp pp p pp pp p E 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(T̃ ), we set pE = min{ pE,K̃ : K̃ ∈ w̃E }, (5.5.5) with w̃E defined in (5.4.13). Notice that an elemental edge E in E(K̃), K̃ ∈ T̃ , belongs to E(T ) ∪ E(T̃ ). Hence, for any K̃ ∈ T̃ , equation (5.5.5) defines 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 ∪ T̃ . Let v ∈ Sp(T ). Firstly, we introduce a (nodal) interpolant vnod ∈ Sp̃(T̃ ). For each element K ∈ T and K̃ ∈ R(K), we will construct the restriction vnod K̃ of vnod to K̃ such that vnod K̃ ∈ QpK (K̃), vnodK̃ |E ∈ PpE (E), E ∈ E(K̃), (5.5.6) with pE given in (5.5.5). To define v nod K̃ , we distinguish two cases. Case 1: If R(K) = {K} (i.e., if K is unrefined), the interpolant vnod K̃ = vnodK is simply defined by vnodK (x) = ∑ ν∈N (K) vK(ν) Φ ν K(x). (5.5.7) Case 2: If R(K) consists of multiple newly created elements, we define vnod K̃ on each element K̃ ∈ R(K) separately. To do so, fix K̃ ∈ R(K). Without loss of generality, we may consider the situation that K̃ located in the corner of K; see Figure 5.5 for illustration. The constructions of vnod K̃ for the cases when K̃ locates inside K (see Figure 5.6) or shares an edge of K (see Figure 5.7) are analogous. Here {Ei}4i=1 and {νi}4i=1 denote the edges and vertices of K, {Ẽi}4i=1 and {ν̃i}4i=1 the ones of K̃. Notice that here we have ν̃2 = ν2 and ν̃4 ∈ NA(T̃ ). Furthermore, Ẽ3 and Ẽ4 are in EA(T̃ ) and p Ẽ3 = pE1,K = pE3,K , pẼ4 = pE2,K = pE4,K . Let us now define the value of vnod K̃ at the edge and vertex nodes of K̃. At the interior nodes of Ẽ3 and Ẽ4, we set vnod K̃ (z) = vK(z), z ∈ Z p Ẽ3 int (Ẽ3) ∪ Z p Ẽ4 int (Ẽ4). (5.5.8) Similarly, we set vnod K̃ (ν) = vK(ν) for the vertices ν = ν̃2 and ν = ν̃4. 184 5.5. Proof of Theorem 5.4.4 s s s s c c c c c ν1 ν2 ν4 ν3 E1 E3 E4 E2 ν̃1 ν̃2 ν̃4 ν̃3 K̃ Ẽ1 Ẽ3 Ẽ4 Ẽ2 Figure 5.5: K̃ ∈ R(K) at the corner of K. s s s s c c c c c c c c c c c c ν1 ν2 ν4 ν3 E1 E3 E4 E2 ν̃1 ν̃2 ν̃4 ν̃3 K̃ Ẽ1 Ẽ3 Ẽ4 Ẽ2 Figure 5.6: K̃ ∈ R(K) locates inside K. s s s s c c c c c c c c c c c c ν1 ν2 ν4 ν3 E1 E3 E4 E2 ν̃1 ν̃2 ν̃4 ν̃3 K̃ Ẽ1 Ẽ3 Ẽ4 Ẽ2 Figure 5.7: K̃ ∈ R(K) shares an edge of K. It remains to define the values of vnod K̃ on the nodes of the edges Ẽ1 and Ẽ2, as well as on ν̃1 and ν̃3. We only consider ν̃1 and Ẽ1 (the construction for ν̃3 and Ẽ2 is completely analogous). If ν̃1 ∈ N (T ) (i.e., ν̃1 is a hanging node in T ), then we define vnod K̃ (z) = 0, z ∈ ZpẼ1int (Ẽ1), vnodK̃ (ν̃1) = vK(ν̃1). (5.5.9) On the other hand, if ν̃1 /∈ N (T ), then E1 ∈ E(T ) and ν1 ∈ N (T ). In this case, we interpolate the values of the nodal interpolant over the long edge E1 at the Gauss-Lobatto nodes on Ẽ1. That is, we set vnod K̃ (z) = vK(ν1) Φ ν1 K (z)+vK(ν2) Φ ν2 K (z), z ∈ Z p Ẽ1 int (Ẽ1)∪{ν̃1}. (5.5.10) With the nodal values of vnod K̃ constructed (5.5.8)–(5.5.10), we have vnod K̃ (x) = ∑ ν∈N (K̃) vnod K̃ (ν) Φν K̃ (x) + ∑ E∈E(K̃) pE−1∑ i=1 ( vnod K̃ (z E,pE i )Φ E,pE i (x) ) . This finishes the construction of the interpolant of vnod. Notice that vnod ∈ Sp̃(T̃ ) is continuous over edges E ∈ EA(T̃ ) and satisfies vK(ν)− vnodK (ν) = 0, ν ∈ N (T ) located on ∂K, (5.5.11) as well as vnod K̃ |E ∈ PnodpE (E), E ∈ E(T ), K̃ ∈ w̃E . (5.5.12) 185 5.5. Proof of Theorem 5.4.4 Secondly, we construct a function vedge ∈ Sp̃(T̃ ) related to the edge degrees of freedom. To do so, fix an element K ∈ T . For any edge E ∈ E(K), we define vEK by vEK =  LEpK ,K((vK − vnodK )|E), E ∈ E(T ), LEpK ,K((vK − vnodK )|E0 , · · · , (vK − vnodK )|EN ), E = E0 ∪ · · · ∪ EN , E0, · · · , EN ∈ E(T ), with LEpK ,K(·) in (5.5.3) and LEpK ,K(·, · · · , ·) in (5.5.4), respectively. We then define vedge elementwise as vedgeK (x) = ∑ E∈E(K) vEK(x). Thirdly, we construct a function vint ∈ Sp̃(T̃ ) simply by setting elemen- twise vintK = vK − vnodK − vedgeK , K ∈ T . Notice that vintK belongs to H 1 0 (K). Hence, we have v int ∈ Scp̃(T̃ ). In conclusion, any function v ∈ Sp(T ) can be decomposed into three parts: v = vnod + vedge + vint, (5.5.13) with vnod, vedge and vint in Sp̃(T̃ ) 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 = vnod + vedge + vint, according to (5.5.13). We will define the averaging operator Ihpv in three parts: Ihpv = ϑ nod + ϑedge + ϑint, (5.5.14) with ϑnod, ϑedge, ϑint ∈ Scp̃(T̃ ). Since vint ∈ Scp̃(T̃ ), we simply take ϑint = vint. 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 ∈ Scp̃(T̃ ) that satisfies ∑ K̃∈T̃ ‖vnod − ϑnod‖2 L2(K̃) . ∑ E∈E(T ) ∫ E (p⊥E) −2h⊥E [[v nod]]2 ds, ∑ K̃∈T̃ ‖∇(vnod − ϑnod)‖2 L2(K̃) . ∑ E∈E(T ) ∫ E p2Eh ⊥ Eh −2 min,E [[v nod]]2 ds. Lemma 5.5.4 There is a conforming approximation ϑedge ∈ Scp̃(T̃ ) that satisfies∑ K̃∈T̃ ‖vedge − ϑedge‖2 L2(K̃) . ∑ E∈E(T ) ∫ E (p⊥E) −2h⊥E([[v]] 2 + [[vnod]]2) ds, ∑ K̃∈T̃ ‖∇(vedge − ϑedge)‖2 L2(K̃) . ∑ E∈E(T ) ∫ E p2Eh ⊥ Eh −2 min,E([[v]] 2 + [[vnod]]2) ds. By the triangle inequality and Lemmas 5.5.3 and 5.5.4, we then obtain∑ K̃∈T̃ ‖v − Ihpv‖2L2(K̃) . ∑ E∈E(T ) ∫ E (p⊥E) −2h⊥E ( [[v]]2 + [[vnod]]2 ) ds, ∑ K̃∈T̃ ‖∇(v − Ihpv)‖2L2(K̃) . ∑ E∈E(T ) ∫ E p2Eh ⊥ Eh −2 min,E ( [[v]]2 + [[vnod]]2 ) ds. Theorem 5.4.4 now follows if we show that ‖[[vnod]]‖2L2(E) . ‖[[v]]‖2L2(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 vnod, the jump over E satisfies [[vnod]](νi) = [[v]](νi), i = 1, 2. Since [[vnod]] vanishes on all the interior Gauss-Lobatto nodes on E, item (i) in Lemma 5.5.1 and a scaling argument yield ‖[[vnod]]‖2L2(E) . p−2E hE([[vnod]](ν1)2 + [[vnod]](ν2)2) = p−2E hE([[v]](ν1) 2 + [[v]](ν2) 2) . p−2E hE‖[[v]]‖2L∞(E). From [26, Theorem 3.92], we further have the inverse estimate ‖[[v]]‖2L∞(E) . p2Eh−1E ‖[[v]]‖2L2(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 vnod ∈ Sp̃(T̃ ) be the nodal interpolant in the decomposition (5.5.13). We now shall construct the conforming approximation ϑnod in Scp̃(T̃ ). For simplicity, we shall omit the superscript ”nod” and, in the sequel, write v for vnod and ϑ for ϑnod. For a node ν, we introduce the sets: w̃(ν) = { K̃ ∈ T̃ : ν ∈ N (K̃) }, wE(ν) = {E ∈ E(T ) : ν ∈ E }. Fix K ∈ T and K̃ ∈ R(K). We proceed by distinguishing the same two cases as in Section 5.5.3. Case 1: If R(K) = {K}, we have K = K̃. Then any elemental edge Ẽ ∈ E(K̃) belongs to E(T ) and we have v K̃ | Ẽ ∈ Pnodp Ẽ (Ẽ). For any Gauss- Lobatto node ν located on ∂K̃, we define the value of ϑ(ν) by ϑ(ν) =  |w̃(ν)|−1 ∑ K̃∈w̃(ν) v K̃ (ν), ν ∈ NI(T ), 0, otherwise. (5.5.16) Here, |w̃(ν)| denotes the cardinality of the set w̃(ν). In the case considered, we have |w̃(ν)| = 4 for ν ∈ NI(T ). Then we define ϑ on K̃ by ϑ(x) = ∑ ν∈N (K̃) ϑ(ν) Φν K̃ (x). (5.5.17) From (5.5.7) and (5.5.17), we have ‖v K̃ − ϑ‖ L2(K̃) . ∑ ν∈N (K̃) |v K̃ (ν)− ϑ(ν)| ‖Φν K̃ ‖ L2(K̃) . (5.5.18) Analogously to [6, Page 1125], we have |v K̃ (ν)− ϑ(ν)| . ∑ E∈wE(ν) pEh −1/2 E ‖[[v]]‖L2(E). (5.5.19) Hence, by combining (5.5.18), (5.5.19), item (iii) in Lemma 5.5.1 (with scal- ing), the local bounded variations (5.2.5) and (5.2.8), we obtain ‖v K̃ − ϑ‖2 L2(K̃) . ∑ E∈{wE(ν)} ν∈N (K̃) ∫ E (p⊥E) −2h⊥E [[v]] 2ds. (5.5.20) 188 5.5. Proof of Theorem 5.4.4 Case 2: If R(K) consists of multiple elements, we define ϑ on each ele- ment K̃ ∈ R(K) separately, analogously to the construction of the nodal in- 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 in Figures 5.6 and 5.7 is analogous and omitted here. Since Ẽ3, Ẽ4 ∈ EA(T̃ ), the function v is continuous over Ẽ3 and Ẽ4. The values of ϑ on the edge nodes z ∈ ZpẼ3,K̃int (Ẽ3) ∪ Z p Ẽ4,K̃ int (Ẽ4), and the vertex ν̃4 are defined by ϑ(z) = v K̃ (z), z ∈ ZpẼ3,K̃int (Ẽ3) ∪ Z p Ẽ4,K̃ int (Ẽ4), ϑ(ν̃4) = vK̃(ν̃4). (5.5.21) We further define the value of ϑ on the vertex ν̃2 by (5.5.16). It remains to define the values of ϑ for the nodes on the edges Ẽ1 and Ẽ2, as well as for ν̃1 and ν̃3. We only consider ν̃1 and Ẽ1 (the construction for ν̃3 and Ẽ2 is completely analogous). If ν̃1 ∈ N (T ), then ν̃1 is a hanging node of T and Ẽ1 ∈ E(T ). Thus, vK̃ |Ẽ1 ∈ PnodpẼ1 (Ẽ1). For any z ∈ Z p Ẽ1 int (Ẽ1)∪{ν̃1}, the value of ϑ(z) is taken as in (5.5.16). On the other hand, if ν̃1 /∈ N (T ), then E1 ∈ E(T ) and vK |E1 ∈ PnodpE1 (E1). We define ϑ(ν1) again by (5.5.16). Recall that ϑ(ν2) = ϑ(ν̃2) has already been defined. Then we set ϑ(z) = ϑ(ν1)Φ ν1 K (z) + ϑ(ν2)Φ ν2 K (z), z ∈ Z p Ẽ1 int (Ẽ1) ∪ {ν̃1}. (5.5.22) Now we construct ϑ on K̃ by setting ϑ(x) = ∑ ν∈N (K̃) ϑ(ν)Φν K̃ (x) + ∑ Ẽ∈E(K̃) p Ẽ −1∑ i=1 ( ϑ(z Ẽ,p Ẽ i )Φ Ẽ,p Ẽ i (x) ) . This completes the construction of ϑ. Clearly, ϑ ∈ Scp̃(T̃ ). We shall now derive an estimate analogous to (5.5.20). To do so, we bound the difference between v K̃ and ϑ on K̃ as follows: ‖v K̃ − ϑ‖2 L2(K̃) . ∑ ν∈N (K̃) ‖(v K̃ (ν)− ϑ(ν))Φν K̃ ‖2 L2(K̃) + ∑ Ẽ∈E(K̃) ‖ς Ẽ ‖2 L2(K̃) . ∑ ν∈N (K̃) ‖(v K̃ (ν)− ϑ(ν))Φν K̃ ‖2 L2(K̃) + ∑ Ẽ∈E(K̃) (p⊥ Ẽ,K̃ )−2h⊥ Ẽ,K̃ ‖ς Ẽ ‖2 L2(Ẽ) = T1 + T2, (5.5.23) with ς Ẽ (x) = ∑p Ẽ −1 i=1 (( v K̃ (z Ẽ,p Ẽ i ) − ϑ(z Ẽ,p Ẽ i ) ) Φ Ẽ,p Ẽ i (x) ) . For the second 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 ς Ẽ (x) vanishes at all the interior tensor-product Gauss-Lobatto nodes in K̃ and on the elemental edges of K̃ that are different from Ẽ. Let us first bound the term T1 in (5.5.23). If the node ν ∈ NA(T̃ ), then, by (5.5.21), ( v K̃ (ν)− ϑ(ν))Φν K̃ (x) = 0, x ∈ K̃. (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 ‖(v K̃ (ν)− ϑ(ν))Φν K̃ ‖2 L2(K̃) . ∑ E∈wE(ν) ∫ E (p⊥E) −2h⊥E [[v]] 2 ds. (5.5.25) Finally, if ν /∈ N (T ) ∪ NA(T̃ ), 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 |v K̃ (ν)− ϑ(ν)| ≤ |(vK(ν1)− ϑ(ν1))|+ |(vK(ν2)− ϑ(ν2))|. Thus, as before, we obtain ‖(v K̃ (ν)− ϑ(ν))Φν K̃ ‖2 L2(K̃) . ∑ E∈wE(ν1)∪wE(ν2) ∫ E (p⊥E) −2h⊥E [[v]] 2 ds. (5.5.26) To combine the results in (5.5.24)-(5.5.26), we define the set N ?(K̃) as follows. We start from N (K̃) and first remove all the vertices belonging to NA(T̃ ). Then, any vertex ν̃ ∈ N (K̃) with ν̃ /∈ N (T )∪NA(T̃ ) is replaced by the vertex ν ∈ N (K) which is on the same elemental edge of K as ν̃. For example, in the case shown in Figure 5.5, we have N ?(K̃) = { ν1, ν̃2, ν̃3 } if ν̃1 /∈ N (T ) and ν̃3 ∈ N (T ). We also set E?(K̃) = {E ∈ wE(ν) : ν ∈ N ?(K̃) }. In conclusion, the term T1 is bounded by T1 . ∑ E∈E?(K̃) ∫ E (p⊥E) −2h⊥E [[v]] 2 ds. (5.5.27) Next, let us estimate the term T2 in (5.5.23). If Ẽ ∈ E(T ) or Ẽ ∈ EA(T̃ ), by the constructions of v and ϑ, we clearly have ‖ς Ẽ ‖ L2(Ẽ) = 0. Otherwise, 190 5.5. Proof of Theorem 5.4.4 one of the two end points of Ẽ, say ν̃1, is a newly created node in T̃ and the other one, ν̃2, belongs to N (T ). Thus, we have (p⊥ Ẽ,K̃ )−2h⊥ Ẽ,K̃ ‖ς Ẽ ‖2 L2(Ẽ) ≤ (p⊥ Ẽ,K̃ )−2h⊥ Ẽ,K̃ 2∑ i=1 ‖(v K̃ (ν̃i)− ϑ(ν̃i) ) Φν̃i K̃ ‖2 L2(Ẽ) + (p⊥ Ẽ,K̃ )−2h⊥ Ẽ,K̃ ‖v K̃ − ϑ‖2 L2(Ẽ) := T21 + T22. Then there exists an elemental edge E ∈ E(K) such that ν̃1 is on E. Denote the end points of E by ν1 and ν2. Similarly to (5.5.25) and (5.5.26), we have T21 . ∑ E∈wE(ν1)∪wE(ν2) ∫ E (p⊥E) −2h⊥E [[v]] 2 ds. In view of (5.5.22), we proceed as in (5.5.19) and obtain T22 . (p⊥Ẽ,K̃) −2h⊥ Ẽ,K̃ ‖ 2∑ i=1 (( vK(νi)− ϑ(νi) ) ΦνiK ) ‖2L2(E) . ∑ E∈wE(ν1)∪wE(ν2) ∫ E (p⊥E) −2h⊥E [[v]] 2 ds. 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 T2 . ∑ E∈E?(K̃) ∫ E (p⊥E) −2h⊥E [[v]] 2 ds. (5.5.28) The bounds for T1 and T2 in (5.5.27) and (5.5.28) yield ‖v K̃ − ϑ‖2 L2(K̃) . ∑ E∈E?(K̃) ∫ E (p⊥E) −2h⊥E [[v]] 2 ds. (5.5.29) 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 ‖v K̃ − ϑ‖2 L2(K̃) . ∑ E∈E?(K̃) ∫ E (p⊥E) −2h⊥E [[v]] 2ds, K̃ ∈ T̃ . (5.5.30) 191 5.5. Proof of Theorem 5.4.4 This proves the first inequality in Lemma 5.5.3. Moreover, for any function v ∈ Sp̃(T̃ ), K̃ ∈ T̃ , we have the following inverse inequality from [26, Theorem 4.76] and a simple scaling argument: ‖∇v‖ L2(K̃) . p1 K̃ p2 K̃ ((h1K) −2 + (h2K) −2) 1 2 ‖v‖ L2(K̃) . (5.5.31) Together with (5.5.30), we then obtain ‖∇(v K̃ − ϑ)‖2 L2(K̃) . ∑ E∈E?(K̃) ∫ E p2Eh ⊥ Eh −2 min,E [[v]] 2ds, K̃ ∈ T̃ , (5.5.32) 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 the function WEK as follows: If E ∈ EB(T ), we set WEK = L E pK ,K ( (vK − vnodK )|E ) , with the extension operator LEpK ,K(·) in (5.5.3). If E ∈ EI(T ), let K ′ in T̃ be the element such that E is also an elemental edge of K ′, that is, E ∈ E(K ′). Denote by K ′ the element which has the lower edge polynomial degree of the elements K and K ′, i.e., K ′ = K if pE,K ≤ pE,K′ and K = K ′ otherwise. We define WEK by WEK = L E pK ,K ( (vK′ − vnodK′ )|E ) . with LEpK ,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 Ki the element that has the lower edge polynomial degree of K and Ki. We now define W E K by WEK = L E pK ,K ( (vK0 − vnodK0 )|E0 , · · · , (vKN − v nod KN )|EN ) , with LEpK ,K(·, . . . , ·) in (5.5.4). Then we define ϑedge elementwise by setting ϑedge|K = ∑ E∈E(K)W E K , with WEK defined above. Clearly, the function ϑ edge belongs to Scp̃(T̃ ). Next, 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∑ K̃∈T̃ ‖vedge−ϑedge‖2 L2(K̃) = ∑ K∈T ∑ K̃∈R(K) ‖vedge − ϑedge‖2 L2(K̃) . ∑ K∈T ∑ E∈E(K) ‖LEpK ,K ( (vK − vnodK )|E )−WEK‖2L2(K) . ∑ K∈T ∑ E∈E(K) (p⊥E,K) −2h⊥E,K‖(vK − vnodK )|E −WEK |E‖2L2(E) . ∑ K∈T ∑ E∈E(K) ∫ E (p⊥E) −2h⊥E([[v]] 2 + [[vnod]]2) ds. 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 Chap- ter 3, but use now η in (5.3.4) as an error indicator in an anisotropic re- finement 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 han- dle 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 algo- rithm 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 ξ1 = ∑ i=1,3 ∫ Ei |[[uhp]]| ds∑ i=1,3 hEi , ξ2 = ∑ i=2,4 ∫ Ei |[[uhp]]| ds∑ 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 ap- proximate right-hand side fhp is taken as the L 2-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 gradi- ents along boundary and internal layers. In all our examples, we observe p-refinement to be dominating once the local mesh size is sufficiently re- solved. Based on the a-priori error analysis for p-version methods in [26], we thus plot all computed quantities against N 1 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) = (ex1−1ε − 1 e− 1 ε − 1 + x1 − 1 )(ex2−1ε − 1 e− 1 ε − 1 + x2 − 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 ‖u− uhp ‖E,T , 194 5.6. Numerical experiments in agreement with Theorem 5.3.2. Additionally, the convergence lines us- ing hp-refinement are (roughly) straight on a linear-log scale, which indi- cates that exponential convergence is attained for this problem. In the curve “L2 Error”, we calculate the error ε− 1 2 ‖u − uh‖L2(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(∑ E∈E(T )(εh ⊥ Ep 2 Eh −2 min,E + ε −1h⊥Ep ⊥ Ep −2 min,K)‖[[uh]]‖2L2(E) )1/2 shown in ”Jump Error”. Finally, in the curve labeled ”Theta”, we calculate an approxima- tion 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 Fig- ure 5.8(b), we compare the true energy error and the error estimator for both an isotropic and anisotropic refinement algorithm. Here, the anisotropic re- finement 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 superi- ority of using anisotropic refinement is clearly visible although we observe exponential convergence for all quantities. In Figure 5.8(c), we plot the ra- tio of the indicator and the true energy error for the anisotropic refinement method. It stays around 4, uniformly in N 1 2 . 40 60 80 100 120 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 101 N1/2 (a)   Error Indicator Energy Error L2  Error Jump Error Theta 0 100 200 300 400 10−6 10−5 10−4 10−3 10−2 10−1 100 101 N1/2 (b)   Indicator (anisotropic) Error (anisotropic) Indicator (isotropic) Error (isotropic) 40 60 80 100 120 0 1 2 3 4 5 6 7 8 9 10 N1/2 (c)   ratio (anisotropic) 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 50 100 150 200 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 101 N1/2 (a)   100 200 300 400 500 600 700 10−5 10−4 10−3 10−2 10−1 100 101 N1/2 (b)   50 100 150 200 0 1 2 3 4 5 6 7 8 9 10 N1/2 (c)   Error Indicator Energy Error L2  Error Jump Error Theta Indicator (anisotropic) Error (anisotropic) Indicator (isotropic) Error (isotropic) ratio (anisotropic) 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 co- efficients. 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 u(x1, x2) = erf( x1√ 2ε )(1− x22), with erf(x) = 2√ pi ∫ x 0 e−t 2 dt. 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 refine- ment 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 refine- ment 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. 30 40 50 60 70 10−7 10−6 10−5 10−4 10−3 10−2 10−1 N1/2 (a)   40 60 80 100 120 140 10−7 10−6 10−5 10−4 10−3 10−2 10−1 N1/2 (b)   30 40 50 60 70 0 1 2 3 4 5 6 7 8 9 10 N1/2 (c)   Error Indicator Energy Error L2  Error Jump Error Theta Indicator (anisotropic) Error (anisotropic) Indicator (isotropic) Error (isotropic) ratio (anisotropic) Figure 5.12: Example 2: Convergence behavior for ε = 10−3. 70 75 80 85 90 95 100 105 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 N1/2 (a)   Error Indicator Energy Error L2  Error Jump Error Theta 50 100 150 200 250 300 10−6 10−5 10−4 10−3 10−2 10−1 100 N1/2 (b)   Indicator (anisotropic) Error (anisotropic) Indicator (isotropic) Error (isotropic) 70 75 80 85 90 95 100 105 5 10 15 20 25 30 35 40 45 50 55 N1/2 (c)   ratio (anisotropic) 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 poly- 199 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 refine- ment 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 refine- ment 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 pi6 , cos pi6 ), 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 = 1 2 ( tanh( x1 ε ) + 1 ) 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 bound- ary 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 in- ternal layer after 9 refinement steps (c) 15 hp-adaptive refinements (d) Enlarged mesh refinement around the in- ternal layer after 15 refinement steps Figure 5.15: Example 2: Adaptively generated meshes after 9 and 15 refine- ment steps for ε = 5 · 10−6. 201 5.7. Conclusions 40 60 80 100 120 140 160 180 200 220 10−4 10−3 10−2 10−1 100 101 N1/2   Error Indicator(Anisotropic) 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 solu- tion is almost constant away from the layers, p-refinement is again concen- trated along the layers. 5.7 Conclusions We have derived an a-posteriori error estimate for hp-adaptive discontinu- ous 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 upper- left corner after 9 refinement steps (c) 15 hp-adaptive refinements (d) Enlarged mesh refinement around upper- left corner after 15 refinement steps Figure 5.17: Example 3: Adaptively generated meshes after 9 and 15 refine- ment 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. deal.II Differential Equations Analysis Library, Technical Reference. http://www.dealii.org. [5] W. Bangerth, R. Hartmann, and G. Kanschat. deal.II — a general pur- pose 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 unsymmetric- pattern multifrontal method. ACM Transactions on Mathematical Soft- ware, 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 el- liptic 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 Meth- ods 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 geo- metric 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 dis- continuous 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. De- conink, editors, High-Order Methods for Computational Physics, vol- ume 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 Meth- ods Appl. Sci., 2010. accepted for publication. [30] L. Zhu and D. Schötzau. A robust a-posteriori error estimate for hp- adaptive DG methods for convection-diffusion equations. IMA J. Nu- mer. 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 h- adaptive DG methods. The estimator yields global upper and lower bounds of the error measured in terms of the energy norm and a semi-norm associ- ated 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 convection- dominated 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 al- gorithms take the p-refinement into account and achieve exponential rates of convergence for piecewise analytic data. In Chapter 3, we derive a robust a- posteriori 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 poly- nomial degrees. In Chapter 4, we propose an a-posteriori error estimate for three-dimensional elliptic equations which gives rise to global upper and lo- 207 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 isotrop- ically 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 prob- lems in generic polyhedral domains. Therefore, it is highly desirable to de- velop 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 es- timator as an error indicator in an hp-adaptive refinement algorithm. Our numerical results indicate that the indicator is capable of anisotropically re- fining 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 per- turbed 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 demon- strated 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 con- forming part and a remainder by employing an averaging operator for dis- 208 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 H1-space for discontinuous functions. In Chap- ter 3 and 4, we construct an averaging operator on two-dimensional and three-dimensional isotropically refined meshes respectively. New L2-norm and H1-norm approximation properties for this hp-averaging operator are shown for 1-irregular meshes consisting of parallelograms and variable poly- nomial degrees. In Chapter 5, these averaging operators and their approx- imation 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 dimen- sions, but also for purely elliptic problems due to the presence of anisotropic edge singularities that might arise along boundary edges of the computa- tional domain. While it is known that hp-methods on geometrically and anisotropically refined meshes resolve edge and corner singularities at expo- nential rates of convergence [2, 3, 4], the design of a-posteriori error estima- tors 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 es- timates 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. Vohraĺık. Guaranteed and robust dis- continuous Galerkin a-posteriori error estimates for convection-diffusion- reaction 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 geo- metric 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. De- conink, editors, High-Order Methods for Computational Physics, vol- ume 9 of Lect. Notes Comput. Sci. Engrg., pages 325–438. Springer– Verlag, 1999. 211

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