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On approximate inertial manifolds for the Navier-Stokes equations using finite elements Walsh, Owen 1994

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ON APPROXIMATE INERTIAL MANIFOLDS FOR THE NAVIER-STOKES EQUATIONS USING FINITE ELEMENTS By Owen Walsh B.Sc.(Mathernatics) University of Victoria, 1986 M.Sc.(IVlathernatics) University of British Columbia, 1988  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  in THE FACULTY OF GRADUATE STUDIES i)EPARTMENT OF MATHEMATICS  We  accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBiA  April 1994  ©  Owen Walsh, 1994  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department  or  by  his  or  her  representatives.  It  is  understood  that  copying  or  publication of this thesis for financial gain shall not be allowed without my written permission.  (Signature)  Department of  mcthc.’  The University of British Columbia Vancouver, Canada  Date  DE.6 (2/88)  abstract  The nonlinearity in the Navier-Stokes equations couples the large and small scales of motion in turbulent flow. The nonlinear Galerkin method (NGM) consists of inserting into the equation for the large scale motion the small scale motion as determined by an “approximate inertial manifold”. Despite the conceptual appeal of this idea, its theoreti cal justification has been recently thrown into question. However, its actual performance as a computational method has remained largely untested. Temam and collaborators have reported a 50% speed p in their spectral code for spatially periodic flow but their experiments have been recently criticized. In any case, spatially periodic computations are of little practical use. The aim of this thesis has been to test the NGM in the more practical context of the finite element method. Using finite elements, there is ambiguity and difficulty because the coarse grid has no natural supplementary space.  We analyze a family of supplementary spaces and  it is found that the quality of the asymptotic error estimates depends on the choice. Choosing the space by the L -projection, we prove that the resulting approximation is 2 “asymptotically good”. These results extend and improve upon recent error estimates of Marion and collaborators. For any other choice, the estimates are weaker and if suspect  —  --  as we  they are optimal it seems possible that the NGM may actually decrease the  accuracy of calculations. We also analyzed a variant of the NGM that we call “microscale linearization” (MSL). We prove that the MSL is “asymptotically good” for any member of this family of supplementary spaces. Turning to calculations, choosing the supplementary space by the Ritz projection, we implemented the NGM by modifying a 2-D Navier Stokes code of Turek; it performed very poorly. We implemented a variant of the MSL. ii  It performed better, but still not as well as the original code.  We sought a further  understanding of these results by considering the 1-D Burgers equation. In conclusion, we find no numerical evidence that these methods are better than the standard finite element method. In fact, unless the coarse mesh is itself very fine, all versions performed poorly.  ill  Table of Contents  abstract  11  Table of Contents  iv  List of Figures  vi  List of Tables  VI”  acknowledgement  ix  glossary  x  1  Introduction  1  2  The Navier-Stokes equations  8  2.1  The NGM and MSL using finite elements  11  2.2  Results of the error analysis  15  2.3  the Numerical schemes  22  2.4  2.5  2.3.1  The finite elements spaces  22  2.3.2  Turek’s solver  27  2.3.3  The implementation of the NCM and MSL  29 34  Numerical results 2.4.1  Flow around a cylinder  34  2.4.2  Diffuser calculations  48  Proof of Theorem 2.1  58 iv  3  4  2.5.1  Preliminaries  2.5.2  Organization  66  2.5.3  Proof of the Theorem  69  .  58  Burgers’ equation  86  3.1  The NGM  87  3.2  The MSL  89  3.3  Results of the error analysis of the NGM and MSL  90  3.4  Proof of Theorem 3.1  95  3.4.1  Prelirniiaries  95  3.4.2  Proof of the Theorem  99  3.5  Numerical results  110  3.6  Implementation  127  Conclusion  130  A Proof of Theorem 2.2  132  B Proof of weighted error estimates for the Navier-Stokes equations  141  C Error estimates for Burgers’ equation  151  Bibliography  158  V  List of Figures  2.1  Refining strategies  2.2  Basis functions of Jh  2.3  Particle tracing of a flow around a cylinder, Re  2.4  Flow past a cylinder, grids  35  2.5  Cylinder, SM solution, grid level 3  37  2.6  Cylinder, SM solution, grid level 4  38  2.7  Cylinder, SM solution, grid level 5  39  2.8  Cylinder, NOM solution grid levels 3-4  2.9  Cylinder, NGM solution grid levels 3-4, “high modes”  11  =  100  34  2.10 Cylinder, MSL solution grid levels 3-4 2.11 Cylinder, NGM grid levels 4-5 2.12 Cylinder, NGM solution grid levels 4-5, “high modes” 2.13 Cylinder, NGM solution grid levels 4-5, “low modes” 2.14 Cylinder, MSL solution grid levels 4-5  .  2.15 Cylinder, MSL solution grid levels 4-5, “high modes” 2.16 Diffuser, grids 2.17 Diffuser, SM at grid level 5 2.18 Diffuser, SM at grid level 6 2.19 Diffuser, Test 2, NGM 2.20 Diffuser, Test 2, NGM, “high modes”  .  .  .  .  .  .  2.21 Diffuser, Test 2, MSL 2.22 Diffuser, Test 2, MSL, “high modes” vi  55  2.23 Diffuser, Test 3, NGM  •  56  2.24 Diffuser, Test 3, NGM, “high modes”  •  56  2.25 Diffuser, Test 3, MSL  •  57  2.26 Diffuser, Test 3, MSL, “high modes” 3.27 “Moving shock”; A series of snapshots of the exact solution 3.28 “Moving shock”: SM, h  =  3.29 “Moving shock”: SM, h 3.30 “Moving shock”: NGM, H  =  ,  h  =  3.31 “Moving shock”: MSL, H  =  ,  h  =  3.32 “Moving shock”: MSL, H  =  ,  ii  =  3.33 “Moving shock”: MSL, H  =  ,  Ii  =  3.34 Test 3: n  =  sin t sin rx, comparison of L 2 errors  vii  57 114  List of Tables  3.1  Testi, u  =  sin(7rx): errors from SM  112  3.2  Testl, u  =  sin(irx): errors from NGM, p + q  112  3.3  Testi, u  =  sin(7rx): errors from NGM, p  112  3.4  Test 1, u  =  sin(lO7rx): errors from SM  113  3.5  Test 1, u  =  sin(lO7rx): errors from NGM, p + q  113  3.6  Test 1, u  =  sin(lO7rx): errors from NGM, p  113  3.7  “Moving shock”; standard method ju  3.8  “Moving shock”: NGM  3.9  “Moving shock”: MSL u  Lu  3.10 “Moving shock”: SM  —  —  3.11 “Moving shock”: NGM !u  lLu  3.13 “Moving shock”: SM u 3.14 “Moving shock”: NGM 3.15 “Moving shock”: IVISL  —  Lu Lu  p  —  116  2k  p  —  qW, H  —  qW, H  —  =  =  2k 2k  a —  —  qI!  —  p  —  3.20 Higher order convergence of 3.21 Higher order convergence of 3.22 Higher order convergence of  .  .  117 117  118  .3  119  .3  119  .3  119  =  .3, H  =  Lu !Lu a  118  q  =  values at t  -errors at t 2 3.19 “Moving shock”: L  .  118  3.17 “Moving shock”: 7h values at t  -rn  =  116  2k  =  117  p  3.16 “Moving shock”; L -errors at t 2  3.18 “Moving shock”:  H  q, H  —  116  —  ajj  —  3.12 “Moving shock”: MSL  q!I,  —  .  —  p  —  —  —  Px  —  viii  =  2h  125  q —  122  qj  qjI  125 125  acknowledgement  I would like to thank my thesis supervisor Dr. John Heywood. I thank him for introducing to this fascinating area of study, and I am grateful that he so freely shared his ideas and afforded me the possibility of undertaking this work. I am indebted to Dr. R. Rannacher and Dr. S. Turek from the university of Heidelberg. Without their help and the time I spent in Heidelberg this thesis would not have been possible. A special thanks to Stefan and Monica for their generous hospitality. Finally, I would like to thank my parents and Karen. They were always there to help and encourage me.  ix  glossary  Chapter 2 u  the fluid velocity, 8  0  the fluid pressure, 8  f  an external force, 8  ii  the kinematic viscosity, 8 a domain in R 2 or R , 8 3  0 u  the initial velocity field, 8  C°°()  the space of smooth vector fields in 2  C°(2)  the space of smooth vector fields with compact support in  (f) 2 L  the space of vector fields with finite L -norm in 12 2  H(12)  the completion C°(12) in the Dirichiet-norm  (12) ’ 2 W  the completion  J(12)  the divergence free space, 8  A  the Stokes’ operator, 8  a  the it_eigenfunction of the Stokes’ operator, 8  (., .)  the L -inner product 2  (V., V.)  the Dirichiet inner product  V(12)  the space spanned by the first n-eigenfunctions, 8  W’(12)  the complement space of V, 8  u  the spectral Calerkin approximation, 8  pfl, qfl  the spectral NCM approximation, 9  Cc0(12)  in  2 IID (z<  or the spectral MSL approximation, 9 x  2)  h, H  constants, 1 > H> h> 0  Gh  a grid, h is some measure on the size of the elements  Xh  a “velocity approximating” finite element space  Lh  a “pressure approximating” finite element space  Jh  the space of discretely divergence free finite elements, 10  a,  the standard finite element approximation, 9  n(u,v,w)  a discrete version of (u.Vv,w)  a(u, v)  a discrete version of (Vu, Vv)  b(q,u)  a discrete version of (q,V.u) constants > 0  W’  the “high mode” finite element space, 11 the “low” mode projection, 12 the “high” mode projection, 12  PH,  qh, 7t  the finite element NGM approximation, 13 or the finite element MSL approximation, 14  r(t)  the weight function, r(t)  to  a time (fixed), to > 0  u(t)  the weight function, a  k  the discrete time step  0  a constant depending on the time discretization, 28  =  =  min(t, 1)  min(t  —  to, 1)  interpolation operator onto Xh, 58 jh  interpolation operator, onto Lh, 58  Ph  -projection onto Xh, 58 2 L  Rh  -projection onto Xh, 58 1 H  rh  projection operator onto J , 59 1  xi  Ah  the discrete Laplacian, 63  I Ih  the L -norm 3  116  the L -norrn 6 the  112 e,C  -norrn ’ 2 the V 16,66 =Fe,i=Qe  s,fr  66  ê,  67  ,t  67  Chapter 3 solution to Burgers’ equation, 86  u  the initial condition for Burgers’ equation, 86 the interval [0, 1]  f  an external force, 86  v  the “viscosity” in Burgers’ equation, 86  Gh  a gridding of 1  Xh  a finite element space the finite element approximation to u, 86  c3  Wh  the high mode space, 87  F’  the “low” mode projection, 87 the “high” mode projection, 87  h 147  the space spanned by the induced basis, 88  Ph, qh  the finite element NCI\’I approximation, 88 or the finite element MSL approximation, 89  XII  to  a time (fixed), to > 0  u(t)  the weight function, a  Ph  -projection onto Xh, 95 2 L  Rh  H’-projection onto Xh, 95  e  e=u—ü,99  s  101  =  rnin(t  —  to, 1)  ê=t—s,102  =lje è=p+q—s,104  k  the discrete time step size  0  constant describing the one-step theta scheme  xlii  Chapter 1  Introduction  A turbulent fluid flow contains a large range of scales of motion. The largest scales describe the main flow itself and the large vortices in it. Smaller scales describe increas ingly tiny “turbulent” vortices. An inertial manifold, as conceptualized by Foias, Sell and Temam [11], determines the small scale of motion as a function of the large scale of motion. In a sense, an inertial manifold is a physical “law” which connects the motion of the small scales to that of the large scales. The velocity and pressure in a viscous in compressible fluid are believed to satisfy a system of partial differential equations called the (viscous incompressible) Navier-Stokes equations. In the special case of the twodimensional, spatially periodic equations, Kwak [18] has recently proved the existence of an inertial manifold. Otherwise, it is not known if an inertial manifold exists. Despite this, there has been a great interest in the development of computational methods loosely based on the concept of an inertial manifold. Inertial manifolds have limitations as computational tools. As we have remarked, it is not known if one exists; even if one does there is no guarantee that it would have a closed form or be of small dimension. To work around these limitations, approximate in ertial manifolds along with a related computational method called the nonlinear Galerkin method (NOM) were proposed in [9]. These ideas have received widespread attention (see [19], [11], [10], [20], [6], [8], [17] and the many references therein). Many of these papers aim to give a theoretical justification to approximate inertial manifolds and the nonlin ear Galerkin method. In this vein, a recent result in a paper of Dedulvier, Marion and  1  Titi [8] appeared to demonstrate a strong theoretical basis for the NGM. In the context of spectral eigenfunctions and Dirichiet boundary conditions, for approximations calcu lated using the NGM (see equation (2.3)), they proved an order of convergence which was higher than that possible for approximations calculated using the standard spectral Galerkin method (see equation (2.2)). This higher order convergence was cited as a theo retical justification for the NGM and ultimately for approximate inertial manifolds. This has been thrown into question in a recent paper of Heywood and Rannacher [16]. They do not dispute the higher order convergence; it is true. They dispute any connection to the physics of turbulence. On the computational side, Jauberteau, Rosier and Temam [17] have reported a 50% speed-up in their spectral code for spatially periodic flow using the NGM. However, their experiments have been recently criticized in a new paper of Garcia-Archilla and Frutos [7]. Thus, the NGM has generated disagreement over both its theoretical justification and its practicality as a computational tool. We feel that as a computational tool the NGM has not been truly tested since spatially periodic calcu lations are of little practical importance. The major goal of this thesis is to study the nonlinear Galerkin method for the Navier-Stokes equations using finite elements. We looked at it from both the theoretical and computational sides, and we find there is trouble from each side. The trouble, in each, case stems from the fact that the method forces us to ignore significant terms involving the time derivative. We felt there was no compelling reason to omit these terms, so we developed another method called Microscale Linearization, or MSL. The MSL, like the NGM, separates the “high” and “low” modes of the solution  solution  —  we refer to “modes” rather than “scales of motion” when discussing the  but with MSL, there is no longer any connection to an approximate inertial  manifold. The NGM and MSL require function spaces for the “low” and “high” modes of a solution. In the spectral case, the low mode space is spanned by the first n eigenfunctions 2  of the Stokes operator while the “high” mode space is spanned by the rest. With finite elements, the “low” mode space, XH, is spanned by finite element functions associated with a coarse grid while the “high” mode space (or supplementary space),  is  spanned by finite element functions associated with a more refined grid. A difficulty, not present when using spectral eigenfunctions, is defining this supplementary space; it is not clear that any one is better than another. As a consequence, we study a class of possible choices. Our main theoretical result is Theorem 2.1. Here, we considered the difference be tween the MSL approximation, with the main part calculated on a coarse grid and a “correction” calculated on a fine grid, and the standard approximation, where all is cal culated on the fine grid. We prove this difference satisfies higher order estimates than the difference between the exact solution and this standard approximation. Its consequence, Corollary 2.1, is (paraphrased): • MSL approximations to the Navier-Stokes equations, with a main part calculated on a coarse grid and a correction calculated on a fine grid, are as accurate as the standard approximation, where all is calculated on the fine grid. This result does not depend on the choice of The theoretical results for the NGM were  mixed.  We considered the difference be  tween the NGM approximation, with the main part calculated on a coarse grid and a “correction” calculated on a fine grid, and the standard approximation, where all is calculated on the fine grid. On the positive side, • for one special case of the NGM, a case very recently studied by Ammi and Marion in [1], we were able to prove higher order estimates, similar to those of Theorem 2.1. These are proved as Corollary 2.2. In this case, XH and W’ are orthogonal with respect to the L -inner product. 2 3  The estimates of Corollary 2.2 improve the results of [1] since we prove higher order convergence for the L -norm of the difference. 2  With L -orthogonality, several terms 2  involving the time derivative are identically zero. This is the mechanism that leads to the higher order in these estimates. On the negative side, • If W’ is chosen in another way, using a different orthogonality, we are unable to prove these higher order estimates. The best general results for the NGM is the weaker Theorem 2.2. In the general case, the time derivative terms mentioned above are not zero and they are the cause of the problems. Theorem 2.2 is not strong enough to imply a result similar to Corollary 2.1. In fact, as we remark in Section 3.3, these estimates are not strong enough to guarantee that the “correction” calculated in the NGM will not, in fact, damage the solution. Computationally, we considered the NCM and MSL for the two-dimensional Navier Stokes equations. The implementation was difficult. We require a supplemental space that results in a “high” mode equation which is solvable and moreover, solvable in an efficient way. We suspect that implementational problems are endemic to the NCM (and MSL) for the Navier-Stokes equations using finite elements. They were present in the case we considered. Using discretely divergence-free finite elements we implemented both a NGM and a MSL. In our numerical tests (described in Section 2.4), we found that: • Our version of the NGM performed very poorly, much worse than the standard method. • Our version of MSL performed better than the NCM. However, compared to the standard method the results were disappointing and we saw no advantage in using our version of MSL. 4  An illustrative test was our calculations of flow around a cylinder at Reynolds’ number 100.  (See Figures 2.3, 2.5,  .,  2.15).  In this test, for all mesh sizes we considered,  the NGM was unstable and blew up. MSL did not have stability problems. However, MSL approximations, calculated on a coarse and a fine grid, were no better than standard approximations calculated on the equivalent coarse grid, despite the fact that when using MSL most of the computational time was spent calculating on the fine grid. Motivated to supplement our Navier-Stokes results, in Chapter 3, we studied Burgers’ equation in one-dimension. A particular goal was to supplement our numerical results and Burgers’ equation is a simpler problem free of numerical difficulties. Let Xff be the “low” mode finite element space. For the same reasons as before, we considered a class of possible choices for the “high” mode space, 1’V’. Our study of this mirrors that of the Navier-Stokes equations. We considered the NGM; of particular interest was one proposed by Marion and Temam in [20]. In parallel we considered the MSL. Our theoretical results were similar to those for Navier-Stokes equations. We considered the difference between MSL approximations and the standard approximation calculated on the associated fine grid. We prove, in Theorem 3.1, this difference satisfies higher order estimates than the difference between the exact solution and this standard approximation. Its consequence, Corollary 3.1 states (paraphrased): • MSL approximations of the solution to Burgers’ equation, with a main part calcu lated on a coarse grid and a correction calculated on a fine grid, are asymptotically as accurate as the standard finite element method in which everything is calculated on the fine grid. This result is true independent of  c3 147 h  The same was not true for the NCM. • For a special case of the NOM, we can prove versions of these higher order estimates. -inner 2 In this particular case, XH and T’V,’ are orthogonal with respect to the L 5  product. • The best general result for the NGM was the weaker Theorem 3.2. Our numerical results suggest the weaker estimates in this theorem are optimal. Though we characterize Theorem 3.2 as weak, it does imply a positive order of conver gence for NGM approximations. For the particular Marion/Temam NGM, this result is stronger than the original one proved in [20] in which approximations were shown to converge to the correct solution in a very weak sense; no order of convergence was proven. However, the estimates of Theorem 3.2 are not strong enough to guarantee that the “correction” obtained in the NOM will improve the calculation. In fact our numerical results indicate that the “correction” can even harm the approximation (see below). In Section 3.5, we tested the NGM and the MSL numerically. The particular version of the NGM tested was the Marion/Temam one. We tested them in a wide variety of situations. (For instance, we tested cases where the fine grid was obtained from several refinements of the coarse grid, instead of just a single one). In summary, • the NGM performed very poorly. In our “moving shock” test, the second test of Section 3.5, the L -norm of the error associated with the NGM approximation was, 2 for all grid sizes tested, more than 120 times larger than the error associated with the standard method of approximation calculated on the equivalent fine grid. This is more than 30 times larger than the error associated with the standard method of approximation when calculated on the equivalent coarse grid! In other words, the NGM “correction” damaged the calculations. The results are summarized in Tables 3.16, 3.17 and 3.18. The effect of these large errors on the approximations is visible; NGM approximations have unnatural “wiggles”. Figure 3.30 is a series of snapshots, at fixed times, of one such approximation.  6  • The results for the NGM did not improve if we refined the fine grid while keeping the same coarse grid (see Table 3.19). • As our theoretical results predicted, the MSL worked well. If the grid size was small enough, the MSL approximation, which is calculated on a coarse and a fine grid, is as good as the standard approximation in which everything is calculated on the fine grid. • However, we found no advantage to using the MSL. The requirement that the grid size be “small enough” limits its use. We found that if the coarse grid was fine enough so that a standard approximation calculated on this grid was pretty well able to fully resolve the solution, then the MSL approximation was good. If the coarse grid was out of this range the MSL approximation was not so good.  7  Chapter 2  The Navier-Stokes equations  The initial boundary value problem for the Navier-Stokes in a bounded domain n  =  2, 3, with a given external force f  velocity, u  =  =  u(x,t), and the pressure, 0  C R,  f(x, t) and kinematic viscosity v is to find the =  0(x,t), satisfying V•u=O  u+u•VuvAu—V0+f, uj=O, u(x,O)=u . 0  (2.1)  J  Let, J()  e  =  () 2 L  =  n  0,  0 in the weak sense}  =  and let P be the L -projection onto J. The Stokes operator, A : W 2 (!) fl H(f) ’ 2 J(Q), uniquely maps u to Au  —  P A u. Let & be the ithleigenfunction of the Stokes  =  operator. it is well know that there is a countable number of these eigenfunctions dense -inner product, 2 in J and orthogonal in both the L  (., .),  and the Dirichlet inner product,  (V.,V.). Let V  =  V()  =  span  {a1,..  .  ,a’},  and  W  =  W(Q)  =  span  .}  Let Pv be the L -projection onto V. The standard spectral Galerkin approximation of 2 the velocity u, is the solution u’  V of the ordinary differential equations  (u,) + v(Vu’,V) + (u.Vu,)  8  =  (f,y) ,V € V,  (2.2)  satisfying u(O,.)  =  Pvu(O,  The spectral “nonlinear” Calerkin approximation of the  .).  velocity is the solution (pTh, q”) E V x W of .Vp + pVq + •Vp Th Th q ,) Th (p,) + v(Vp,V) + (p  =  (f,) ,V E V  v(Vq,V) + (p ,) .Vp Th  =  (f,) ,V E W,  satisfying p(O,.)  J  (23)  .).  Pvu(O,  Remark 2.1 (Orthogonality) Because the eigenfunctions are orthogonal in both the L 2 and Dirichiet inner products: (VpV) (q,  )  =  (p’,  =  )  =  0, for all  W’ and (Vq’,Vç)  e V’. Hence, in (2.3,) the only linear term omitted is (q,  0, for all  The spectral MSL approximation of the velocity is the solution (p”, qfl) (p + Th .Vp •Vq T p h+ (p,) + v(Vp,V) + Th (q,  )  + v(Vq, V) + (p.Vp,  Pvu(0,) and q(0,.)  satisfying p(O,.)  =  Pwu(0,  )  ).  (Vi’, W) of  =  (f,) ,V E , Th V  =  (f,  )  =  ,V E W,  J  (24)  .).  Remark 2.2 (Spectral MSL) We have presented a spectral version of the MSL to high light that no linear terms are omitted; in particular, no terms involving time derivatives have been dropped. Consider finite element approximations of the Navier-Stokes equations. Suppose  is subdivided into a grid Gh with associated finite element spaces: Xi,, a velocity ap proximating space, and Li,, a pressure approximating space. We assume Xh and Li, are “good” spaces possessing some necessary approximability and stability properties. Let n(u,v,w) denote a discrete version of (u.Vv,w), b(q,u) a discrete version of (q,V.u) and a(u,v) a discrete version of (Vu,Vv). The standard finite element approximation of (2.1) is the solution  a  (Ut, h) + v a(U,  b(qh,U)  =  0,  =  Üh  mi,)  e +  Xh,  =  ‘in, E Lh of  n(u, U, )  —  b(, mi,)  =  (f,  h),  Vi, E Xh,  J  Vqh ELi,  9  (2 5)  satisfying u(O,.)  =  ao. Let  Jh C Xh be the space of discretely divergence-free finite  elements: Jh  =  {h  e  Xh  b(qh,  h)  =  0, Vqh E Lh}.  An alternative formulation of (2.5) is to seek solutions (at, h)  b(, bh)  + v a(ã, =  h)  +  n(u, a, )  (ut, h) + v a(ü, I’h) +  =  (f,  h)  n(a, a, ‘)  10  a  =  Üh E J,,  =  lrh  e  Lh of  Vh E Jh, —  (f, h)  (2.6) Vh  e  Xh/Jh.  J  2.1  The NGM and MSL using finite elements  Suppose we have a domain Q with a given coarse grid, GH. A standard way of producing a finer grid, GH/2, is to systematically refine GH. One refining strategy, used by the software package FEAT (see [3]), is illustrated in Figure 2.1. This process can be repeated  Figure 2.1: refining strategies to obtain afine grid, Gh. A consequence is the cascade (or hierarchy) of grids GH, GH/ , 2  This hierarchy of grids has several uses; one is in the construction of multigrid solvers. For the NGM and MSL, these grids can be used to define the modes of a solution using finite elements.  Basically, the low modes are modes which can be approximated by  finite element functions associated with a coarse grid while the high modes are modes which can be approximated in some supplementary finite element space associated with a more refined grid. More precisely, let XH be a “standard” velocity approximating space associated with the coarse grid, GH, and let Xh be a “standard” velocity approximating space associated with the fine grid Gh. Assume H  < h  and  C Xh C H. We  define the low modes as finite element functions in XH. A major difficulty is faced in defining the high modes. Unlike the spectral case, there does not seem to be a natural supplementary space to choose. There are other possibilities, but our idea is to consider a class of spaces, W  defined for c,  =  {  Xh  > 0 (not both zero), where  (, ) + 11  a(, ) =0, V E XH}.  (2.7)  It is clear that Xh  =  XH + W’ for any a,  is the operator which maps u a (P’u, H) + Also, let  : Xh  —4  0. The a, /9-projection P’ : H  /9  —*  XH  H to the unique element P’u E XH which satisfies  /9 a(’u, H)  =  WZ’ where  a (u, =  I  H)  +  /9 a(u, H) VH  For notational convenience we  —  sometimes suppress the superscripts a, /9 and refer to Wh, PH, and Remark 2.3 (Special cases of P’) If a  =  0,  /9  E XH.  =  QH.  1 then PH is the Ritz projection and  XH arid V7h are orthogonal with respect to the Dirichlet inner product. If a  =  1,  /9  =  0  then PH is the L -projection and XH and W 2 1 are orthogonal with respect to the L -inner 2 product. Remark 2.4 (The spaces V’) In the finite element case, different choices of a,/3 lead  to different spaces  This is a major difference from the spectral case where, as  we know, the spectral eigenfunctions are mutually orthogonal in both the Dirichiet and -inner products. 2 L  12  The NGM Given spaces XH and W’ for the low and high modes of the velocity and Lh for the pressure, we obtain a NGM by mimicking the spectral version (2.3). Let t > to > 0. A NGM approximation to (2.1) is a solution (p,q,ir)  e  x W’ x Lh of  (ps, b)+va(p+q, ‘,b)+ri(p, p, ‘/)-l-n(p, q,b)+n(q, p,’4’)=(f, va(p+q, )+n(p, p, b)  =  (f, b)+b(7r,b), V W’  (qh,V(p+q)) satisfying p(.,to)  =  +b(r,b),Vb EXH, (2.8)  =  (2.9) (2.10)  O,VqhE Lh,  ). Equations (2.8) and (2.9) can be rewritten more com 0 Fu(.,t  pactly as  (Pt, b)+va(p+q, &)+n(p, p, &)+n(p, q,b)+n(q, p,b)=(f, b)+b(7r,b),  e  for all b  (2.11)  Alternatively, (2.8), (2.9), (2.10), can be rewritten in terms of the  Xh.  discretely divergence-free functions:  (pt, P)+va(p+q, )+n(p, p, )+n(p, q,)+n(q, p,) b(7r, ) for all ‘çb  (pt,  e  =  (f,  ),  &)+va(p+q, )+n(p, p, )+n(p, q,b)+n(q, p,P)  Xh/Jh.  13  VEJh,  —  (2.12)  (f, ), (2.13)  The MSL method As mentioned in the introduction, MSL is like the NOM except that all terms involving time derivatives of the solution are retained. Notice in (2.8), (2.9) the terms (qt,i/), (p,’) and (qt,’’) are not present. Let t > to > 0. The MSL approximation to (2.1) is the solution (p, q, ir) E XH x W’ x Lh of (pt-I-qt, b)+va(p+q, ‘b)+n(p, p, ‘)+n(p, q,’)+n(q, p,’b)  (pt+qt, )+va(p+q, )+n(p, p,  =  (qh,V(p+q)) satisfying po  =  0 FHU(,to), q  =  =  (f, /)+b(ir,’z/’), V’zb E XFJ, (2.14)  (f, )+b(,) V eW’, O,Vqhe Lh,  (2.15) (2.16)  Hu(,to). The velocity equations (2.14), (2.15), may  be rewritten as (pt+q, ‘)+va(p+q, ‘)+n(p, p, ‘)+n(p, q,P)+n(q, p,Pb)=(f, )+b(r,’), (2.17) for all L’  e  Xh. Alternatively, the discretely divergence-free formulation is  (pt+qt, ç)+va(p+q, )+n(p, p, ç)+n(p, q,)+n(q, p,i)=(f, y), Vç e Jh, (2.18)  b(7r, &) for all  ‘  =  (pt+qt, &)+va(p+q, ‘)+n(p, p, b)+n(p, q,b)+n(q, p,Pb)—(f, ), (2.19)  E Xh/Jh.  14  2.2  Results of the error analysis  In this section we state the results of our error analysis and consider some of the conse quences. The proof of our main result, Theorem 2.1, is in Section 2.5. Let u, 9 be the solution to the Navier-Stokes equations (2.1) and let  a,  be the  approximation calculated using the standard finite element method: (Üt,)  with u(.,0)  =  + va(ü,) + n(ã,ã,) + b(,)  =  (f,),  V E Xh,  b(qh,a)  =  0,  Vqh E Lh,  uo.  Let p  =  Pf a, and let  =  J  a. We suppose that the spaces 3 Qj  (25)  Xh, Lh  are “nice” approximating spaces satisfying some necessary approxirnability and stability properties. These properties are discussed in more detail in Subsection 2.5.1. We suppose that u, 0 satisfy some standard regularity properties for 0 < t < T < cc. Let sm to highlight that this constant is linked to the “standard method”  —  sm  —  be a positive  constant independent on ii but depending on u, T, t1 and some properties related to the pair Xh, Lh. Let r(t)  =  min(t, 1). We suppose  Ilu all —  +  hllvu  —  VaR +  rll0  —  . 2 <srnh  (2.20)  For a proof of this estimate we refer the reader to [15]. We also suppose, some “weighted error” estimates hold for the u  —  a  and  Utt  —  a  (see Subsection 2.5.1). In fact we prove  the first of these “weighted estimates” in Appendix B. For the NGM and MSL we suppose we have a coarse grid GH and a fine grid Gh with  o  <  h < H < 1 and further, we suppose the associated “velocity approximating spaces”  satisfy XH C Xh C H.  15  Error estimates for the MSL method Consider the MSL approximation (p, q, ir) = (pH,  qh, 7Th)  E XHxWxLh, the solution to  (pt+qt, )+va(p+q, )+n(p, p, )+n(p, q,P) + n(q, p, P)=(f, ) + b(ir, ), for all b E Xh, b(qh, p + q) q(.,to)  =  =  ) 0 0, for all q E Lh, satisfying p(, t  /ã(.,to). Let us now introduce e  =  a  —  P”°ã(, to) and  (p + q), the difference between the  finite element approximation obtained on the fine grid Gh and the MSL approximation obtained using the coarse mesh GH and the fine mesh Gh, and the difference We also define the weight function a(t)  =  rnin(t  —  —  7r.  to, 1).  Theorem 2.1 (Error estimates for the MSL method) There exists a constant Cmsi, in dependent of h but dependent on u, T,  and some properties related to the pair Xh, Lh,  such that  IVeW jell uejj  , 2 H  (2.21)  msiH,  (2.22)  , 3 H  (2.23)  , 2 jmsiH  (2.24)  rnsi  <  msi  lCH for all t 0 <t < T.  We have used the notation Cmsi to highlight its connection to MSL. The proof of Theorem 2.1 is constructed using a series of Lemmas and Corollaries in Subsection 2.5.3. In fact (2.21) is proved as Lemma 2.7, (2.22) as Corollary 2.4, (2.24)  in  Lemma 2.9 and (2.23)  as Corollary 2.5. Estimates (2.21), (2.23) and (2.24) are a full order higher than the basic estimate (2.20). A simple consequence of Theorem 2.1 is Corollary 2.1. Consider the full velocity 16  error, u  —  +  (pH  q),  and the full pressure error, 0  imations and consider the corresponding errors u  —  —  7rh,  Uh  associated with MSL approx  and 0  —  h  associated with the  “standard” finite element approximation calculated on the fine grid. Corollary 2.1 proves that the absolute size of the errors u  —  PH  —  q  and u  —  2 Üh, measured in either the L  or the Dirichiet norm, are, for all intents and purposes, the same if H is small enough. Similarly for 0  —  7rh  Corollary 2.1 Let  and 0  —  measured in the L -norm. 2  h  and Jmsi be as previously defined. Then,  IlVu  —  VPH  a2u  —  —  pH  Vqh —  qhW  , 2 1sm h + Jmsl H  (2.25)  2 + ms1 H , 3 sm h  (2.26)  h+  —  for all to  t  <  , 2 H  (2.27)  T. We stress that Csm is the best possible constant of estimate (2.20,).  Suppose we have a coarse grid and a fine grid which is the nhhrefinement of it, that is H = 2h. In this case, the approximation obtained using the MSL method (calculated partly on the coarse grid, partly on the fine grid) is as accurate as the standard finite element approximation (where all is calculated on the fine grid) in the following sense: given any e  >  0 there exist h such that if h Vu  —  VPH —  —  PH  J2W0  <  h, then  Vqhj —  —  (sm + e)h,  (2.28)  q,  <  , 2 (Jsm + e)h  (2.29)  lrhW  <  sm + e)h.  (2.30)  Proof: The proofs of (2.25), (2.26), (2.27) are straightforward consequences of the esti mates in Theorem 2.1, the estimate (2.20) and the triangle inequality. Remark 2.5 As a consequence of (2.22), the weighting factor u in (2.26) can be re moved at the cost of one half a power of H on the right hand side. 17  When H  2”h, (2.25) is equivalent to  =  Vu  and if h < h  =  h  —  VPH  —  —  Vqh  then Vu  =  h+  sm  —  VPH  msI  ”h , 2 (“ + e)h. The proofs of  Vqh  —  msl  (2.29) and (2.30) are similar. For the estimate (2.29) use Remark 2.5. D Error estimates for the NGM Let  a,  and  ,  e1 be as before. Consider the NGM approximation (p. q, r) = (pH, qh, 7rh)E  XH x Xh x W, the solution to (Pt, P)+va(p + q, )+n(p, p, )+n(p, q,  for all b e  =  a  —  p  e —  Xh,  b(qh, p + q)  q and  (  =  0  —  =  ) + n(q, p, P)=(f, ) + b(, ),  0, for all q E Lh, satisfying p(., to)  =  PHu(., to). Let  r where p, q and r are now the NOM solution above. In the  NGM, the choice of supplementary space,  plays a crucial role in the error analysis.  The best general result corresponding to Theorem 2.1 is the much weaker Theorem 2.2. On the positive side, there is a specific case where things work out nicely. If W is chosen with respect to the L -inner product (a 2  =  1, 41  =  0), the resulting nonlinear  Galerkin method is one recently proposed by Ammi and Marion in [1]. For this scheme, as a simple consequence of Theorem 2.1, we can prove the higher order estimates. Corollary 2.2 (‘Error estimates for a special NGM,.) Consider the nonlinear Galerkin method proposed by Ammi and Marion in [1] where Wh and XH are orthogonal with respect to the L -inner product. Approximations calculated using this scheme satisfy: 2 VeW + ue+He  ‘S  <  , 2 12 H C  (2.31)  , 3 uH  (2.32)  for all t 0 < t < T. Here  l2  —  -orthogonality” 2 NGM with “L  —  the subscript 12 highlights the connection to the special is a constant independent of h but dependent on u, T,  and some properties related to the pair Xh, Lh. It is clear that because of these estimates we can make a conclusion similar to Corollary 2.1. In [1], they proved (2.31). We have extended their results and proved the higher order estimates for the L -norm in (2.32). This Corollary is proved at the end of Subsection 2 2.5.3. The key point is that, in this special case, (Pt, QH)  =  0, V  (q, PH)  X.  This is not true without the L -orthogonality and in any other case, because these terms 2 are non-zero, we can only prove the weaker Theorem 2.2. Theorem 2.2 (Error estimate for the NGM,) There exists a constant  ngm, 1 t  independent  of h but dependent on u, T, S2 and some properties related to the pair Xh, Lh, such that  aIeW  + HeW +  HVejj  +  uHIICII  (2.33)  , 2 ngm H  for all t 0 <t < T.  The proof of this theorem is in Appendix A. We suspect that estimate (2.33) is of optimal order. If it is, and if we take H (as we did in Corollary 2.1), and consider IPH —qhI  JU_ ÜhW  + Üh  PH  U  —  —qhU  PH  —  sm  q,  2 h  =  2h  then  +ngm  2 H  =  CsrnH2 +ngmH . 2  This suggests that: • the size of  lu  —  PH  —  qhll depends on the approxirnability properties of the coarse  grid. Fixing the coarse grid but taking more refined fine grids (larger n), we ex pect  Ilu  —  PH  —  qhll  . In other words we see no reason to expect the 2 Cngm H  approximation to improve. 19  • Consider the standard approximation calculated on the coarse grid, —  aH!I  sm  . It is not clear which is smaller, sm H 2 H 2 or  It is possible that u  —  PH  + q,W  >  u  —  UH.  ÜH.  H2  We know,  . 2 + ngmH  In other words, the “correction” q  may damage the approximation. Our numerical experiments seem to indicate that this happens. A further remark From Theorem 2.1 we obtain some further insight about the following question: • in studying the NO or MSL methods, should one focus attention on p or p + q as the main approximation to the solution u? Supposing the higher order estimates in Theorem 2.1 hold, the following is a heuristic argument which shows why p + q is the main approximation. For simplicity consider MSL approximations. Consider the relative sizes of u  since PHq  —  p and u  0. Now, using Lemma 2.1 and estimate (2.23), l1H(u  -  P  Cu  -  -  p  -  Using Lemma 2.1, Mu  —  PHUII  2 OH  and hence, for small enough H, Mu  -  pM  On the other hand,  20  OH2.  . 3 q <CH  —  p  —  q.  and IIFh(u  —  p  —  q)  CWu  <  —  p  —  , 3 q <CH  Oh.  U—PhUjj Thus, for small enough H, -  p  -  II  Clearly, since h < H, for H small enough, ju where H  =  —  . 2 Oh  l1  lu  >  —  p  —  qll.  In the particular case  2h, we expect  lu ll —  For instance if n  =  1,  2 CH  we expect  lu  while  lu  p  4llu  —  21  —  —  p  —  —  ll qll•  OH2.  the Numerical schemes  2.3  Much of the numerical work was a collaborative work with S. Turek.  For the two-  dimensional Navier-Stokes equations, Turek has designed and implemented an excellent finite element solver (see [30], [28], [29], [27], [23]). We have taken his basic code and modified it to allow the possibility of solving using either a nonlinear Galerkin or a Microscale Linearization method. To start we give a brief overview of the finite element spaces involved and some of the basic ideas of Turek’s solver. After this, we describe the implementation of the NGM and MSL that we used. 2.3.1  The finite elements spaces  The velocity approximating space is the nonconforming finite element space Xh spanned by functions which are, in each coordinate, “rotated bilinear” on each quadrilateral of the grid with degrees of freedom at the midpoints. If the grid is composed of triangles then Xh is spanned by functions which are, in each coordinate, linear on each triangle of the grid with degrees of freedom at the midpoints. In both cases, the pressure space, Lh, is the space spanned by functions which are piecewise constant on each quadrilateral (or triangle). The triangular case has been studied in [13] while the quadrilateral case has been studied in [23]. The beauty of these nonconforming spaces is they allow a natural construction of a basis, with local support, for the discretely divergence-free space  Jh.  This allows one to calculate using these basis functions and as a consequence to calculate using the formulation (2.6) where the velocity is calculated separately from the pressure. Let us consider these spaces in more detail. Let Gh be a regular decomposition of 1 into either triangles or quadrilaterals. We write Gh  =  UT where T is an individual triangle or quadrilateral. The parameter h > 0  is a measure on the size of the elements of Gh. The family of grids, {Gh}, is assumed 22  to satisfy a uniform shape and size condition and for the case of quadrilaterals each T must be convex. The common edge between adjacent T, T is denoted by midpoint of this edge is boundary edges by F  We let uGh  and the  UFj he the set of all edges and denote the  =  (0G fl 811). Since the spaces are nonconforming, we have to  =  work with bilinear forms that are defined elementwise:  f  ah(uh,vh) TEGh  VUh  : Vvhdx, (2.34)  q.f Vuhdx. T;EGh  Remark 2.6 It is convenient to work with a slightly different bh for one of the cases  of quadrilateral elements. T’Ve discuss this particular case at the end of this section (see (2.35)). Let: Lh  Xh Jh  = {qh  I  q =  e  const, VT  Gh,  j qhdx  =  Sh x Sh,  =  =  (11) 2 E L  {ui  Xh  I  bh(h,uh)  =  0, VA,  e  Lh},  where Sh is carefully defined below for both triangular and quadrilateral elements. Suppose we have a grid of triangles. Consider T E Gh. Let P (T) 1  =  span {1,x,y}.  On T, the degrees of freedom are determined by the nodal functionals {F-  where Fp(v)  =  v(mp). Any  v  P E 8T}  (T) is uniquely determined by the values of the Fi-’(v). 1 E P  Let, (T), VT E Gh, s.t. v is continuous at all edges, 1 vIT E P Sh  =  V  (11) 2 E L  in the sense that Fr(v)T and Fp(v)  =  0, VP  e P. 23  =  Fr,(v)I, VP  c 8Gh,  Pjj,  Remark 2.7 If we choose asfunctionals Fr(v)  FL’  =  , in this triangular case, 7 j v()d  we obtain the same space S,. Suppose we have a grid composed of convex quadrilaterals.  In this situation we  consider the parametric version of Xh since this is what we use in practice. Parametric means the functions in the finite element space are defined with respect to mappings to the reference element. In the case of triangles the parametric and non-parametric forms are equivalent (see Remark 2.8). Let T be the reference square, T Let Qi ?/)T  =  =  [—1, 112.  span {1, x, y, xy}. Since T is convex, there exists a unique invertible mapping  E Qi x Qi mapping (T)  =  {qo’  t  to T. Let (T) be the rotated bilinear elements  Iqe  2 span{1,x,y,x  Remark 2.8 For triangles define  —  y2}}  (T) in the obvious way. The invertible mapping,  ‘/T,  from the reference triangle to any triangle is an affine mapping and hence po’ E P (T) 1 for each p E P,(T). Hence, P,(T)  =  (T) which implies that the parametric and non1 P  parametric cases are the same. On a given quadrilateral T, we consider two types of functionals: F(v)  =  1 j v() d IL  or  F(v)  =  v(mp).  Either choice is unisolvent in Q,(T) but will lead to different spaces, S, rn E =  v E L () 2  Q,(T), VT  =  A function v,  e  Jh  =  3 C äGh, F,(v)I, VF  0, VP E F.  Let us consider the space Jh C Xh. functions.  1 or 2:  Gh, s.t. v is continuous at all edges, F,  in the sense that F(v)IT and F (v) 1  =  In particular, we are interested in its basis  satisfies bh(h,vh) 24  =  0, for all  \h  E Lh.  Consider  the quadrilateral case where Xh  x S (defined using the midpoint functionals).  =  Consider a general quadrilateral T E Gh with vertices a , a 3 , a 1 , a 2 , e 2 , edges e 4 , e 1 , e 3 , 4 , rn 2 midpoints m , m 3 , rn 1 , tangential vectors t 4 , t 1 , t 2 , t 3 , and unit normal vectors n 4 , 1  , n 2 n , n 3 . In this particular case, it is convenient to work with a bilinear form bh(.,.) 4  1 e 1 a slightly different from the one defined in (2.34) (see Remark 2.6). In this case b\h, Vh) is, for each element T E Gh, defined via  17i1 Vh(772)  =  (2.35)  fl.  The following helps to understand the connection between this definition and the first one.  f  T  Let  j(rn)  =  1,  i,j  =  Sh, j  çj =  6 j ,  V.vdx=Tf .  .  .  ,  4, be the nodal basis functions on the quadrilateral T, that is,  1,••. ,4. The functions  i  ,4, are linearly independent  =  and satisfy: =  AIT 7j (?n) t nj .  =  0.  3 y  Thus, a first group of basis functions are the tangential basis functions, v corresponding to the edges (or midpoints) of T. 25  =  qjt,  A second group of basis functions,  corresponding to the vertices of T, are given by =  —  V/i  HYl  i  =  1,... ,4,  j  =  (i+ 2)  4 + 1.  These are the streamfunction (or rotational,) basis functions. It can be seen that the coefficient of the basis functions associated with the vertex a, is an approximation to the value of the streamfunction at this vertex. Physically, the basis function v corresponds to a flow parallel to the side e while v corresponds to a flow around the vertex a.  1 e streamfunction-type  tangential-type  Figure 2.2: basis functions of Jh The construction of the basis for Jh in the other two cases is very similar.  26  2.3.2  Turek’s solver  To introduce the solver, it is best to first consider the stationary Navier-Stokes equations: V•u=O,  uVu=vAu—VO+f,  ulan=O.  The finite element approximation is the solution u 11 E J, va(ui,  h)  b(lrh, I’h)  =  +  (f,  n(uh, Uh, çh)  , 1 va(u  h)  ir  Vh E  E Lh  of (2.36)  Jh,  ) + n(uh, Uh, ‘) + (f, v’),  Vh E Xh/Jh.  (2.37)  We first solve for the velocity using (2.36). Having the velocity, the discrete pressure can be calculated using a marching process from element to element without solving a linear system of equations (see [13]). Consider the velocity equation more carefully. In fact, there are several possible variants of the velocity equation (2.36), correspond ing to one of several possible discrete versions, n(u,v,’), of (u.Vv,w): is centrally discretized  •  f  n(ui, Vh, Wh)  OV/ Uh,i  Wh,j  dx.  IT  • n(.,.,.)  •  .,  )  is discretized using a higher order Sarnarskij upwinding proposed in [25].  be the ith_basis functions in the basis of Jh. Let U,. be the vector with components  Let U  n(.,  is discretized using a first order upwinding proposed in [21].  =  (u,  )  and let F be the vector with components F  matrix with components A  = a(,  ,)  =  (f, j. Let Ah be the  and let N,.(U,.) be the matrix, associated with  the nonlinear term, with components N,.(U,.) 3  =  n(u,  ,  to finding Uh such that: vAU + N/ (Uh)Uh 1 27  =  Fh.  ).  Then, (2.36) is equivalent  This nonlinear algebraic equation is solved using a fixed-point defect correction method where the (n + 1)t_iterate, U’, is obtained from the  u  = u+w[vAh+h(  u)r (Fh 1  —  flth  by solving  vAhU  —  N(U)U).  Here, w is a damping parameter which can be set beforehand, generally close to 1, or calculated using some sort of optimization scheme. The matrix ]cTh(UJ) is built using one of the upwinded discretizations, not the central one. This is done to take advantage of the nice algebraic properties of such matrices (see [21], [26]) which allow the use of a fast multigrid solver (see [27]). This multigrid solver is used to invert the matrix uAh+NhWL) in each fixed point iteration. The generalization to the time-dependent problem is not difficult.  For temporal  discretization, Turek’s solver allows the possibility of using either the fractional-step theta scheme (see [12], [4]) or one-step theta schemes. The properties of such schemes are discussed in [22]. Consider one-step theta schemes and let k he the discrete time step. In this discussion, let m U ( .) = U(., mk). We note that 0 =  ,  0 and 6 = 1 correspond to  0  the Crank-Nicholson, Forward Euler and Backward Euler schemes respectively. Applying a one-step theta scheme to the discretized nonstationary equations (2.6), suppressing the subscript h, we obtain the algebraic system (I -1- kOvA + kON(Um+l))Um+l = G , m  where G m = (I + (6  —  1)k(vA + N(Um)))Urn+OkFrn+l + (1  —  . Suppressing the m 6)kF  time step superscripts, this is simply (I + kOvA + kON(U))U = G.  This nonlinear algebraic system is solved using a fixed-point defect correction method where the (n + 1)st_iterate is obtained from the U’ = U tm + w {i + kvA + kêN(U)]’ (G 28  by solving —  (I + kOvA)U  —  k0N(U)Uj.  2.3.3  The implementation of the NGM and MSL  As we have seen, things that were natural for spectral eigenfunctions are not true for finite element functions. For instance it is not possible to choose the finite element spaces XH and Wh in such a way that they are mutually orthogonal with respect to both the L 2 and Dirichiet inner products. This is why in our analysis we considered a whole class of possibilities. When implementing a NGM (or a MSL) further obstacles are faced. We now have the added problem of choosing XH and Wh in such a way that the resulting NOM is numerically solvable. Most of the problems are associated with the “high mode” equation and in general we expect this equation to be difficult to solve. For this reason, we did not attempt to implement the NGM of Ammi and Marion (where Xh and Wh are orthogonal in the L -norm). In general, we suspect this difficulty is endemic to the NOM 2 (and MSL) when using finite elements and this may be reason enough not to consider using the method. Nevertheless, in our chosen setting of discretely divergence-free finite elements, we did implement and test some schemes. We chose discretely divergence-free finite elements for  two major reasons.  The first  was our initial feeling that a variational formulation that separates the velocity from the pressure was somehow “closer” to the spectral eigenfunction equations (in which the velocity and pressure are separated). The second reason, perhaps the most important, was that we saw a possibility of implementing a NGM (and MSL)  simplifications  —  making some  as a modification of Turek’s basic code. This was especially appealing  because the implementation from ground zero of a Navier-Stokes solver is a very large task.  29  Implementing the NGM Consider the velocity approximation. Let GH be the coarse grid with associated XH, JH  and let Gh be the fine grid with associated Xh,  Jh.  In all our calculations, Gh is  one refinement of GH. The refining process is done by joining opposing midpoints (see Figure 2.1). When working with the discretely divergence-free basis functions, we work from a formulation where the velocity is separated from the pressure. Thus, consider the velocity formulations (2.12) which can be rewritten as (Pt,  P ço)+va(p+q,p)+n(p, p, Ph)+n(P, q,o )+n(q, P,P’h)=(f, Sh), (2.38) 11 Qco)=(f, Qh), (2.39)  iía(p+q, Qh)+n(p, p, for all cp E  Jh.  (2.38) is the “low mode” equation and (2.39) the “high mode” or  supplementary equation. We make the following simplification. We suppose PJh One consequence of this is that (2.38) is approximated by seeking p (ps,  H)  + va(p,  H)  +  n(p,  p,  h  e  a(H, H)  +  Another consequence is that for  We consider only the case where c  H)  +  n(p, q, H)  Jh, Qh  =  0 and  =  such that  H)  =  where  ?I)H  is the solution to  VH E  a(, H)  JH  + n(q, p,  I’H  —  e  (f,  H)  JH.  1 (the case where the spaces are  orthogonal with respect to the Dirichlet inner product) because then (2.38) approximated by finding the solutions p, q  =  —  ,  where (p,  ,  )  JH  ,  x  (2.39) is Jh X  satisfy (Pt, H)  for all  + va(p+q, H) +  n(p,  p,  H)  +  n(p, q, H)  + n(q, P,  H)  =  (f,  H)  (2.40)  ‘H  va(p+h, h)+n(p, p, Sh) =  =  (f,  va(h,p),  30  (2.41)  °h) VsZ’HEJH.  (2.42)  Remark 2.9 We consider only the case where the “high” and “low” mode spaces are  orthogonal with respect to the Dirichlet inner product because in this case we have this method of solving the high mode equation as the difference of two Stokes problems. This does not work for any other cases. For the temporal discretization we use one step-theta schemes with a slight modifi cation. The two nonlinear terms involving q are discretized explicitly. This is done in order to be able to use a robust fixed-point solver. We let k be the discrete time step and again use the notation where pfl p(., nk) (and similarly qfl, f , 2 Applying this =  discretization, we obtain seek the solution p , 1  our  fully discrete NCM solver. At a fixed time I  =  (n + l)k, we  t’ to  (pP’, ci)+kÔv a(p’, coH)+k n(p’, p’, 1 va(p  .).  •.  =  gfl(i),  H)  ,j+n(p’,  =  =  (fn+l,  ç)  va(’,ff),  VH E JH, (2.43) VSOh  J,  (2.44)  VçoJJ-j. (2.45)  where gfl(i)  =  (p,H)+k(f,H)+k(O —1) (va(p,yH)+n(p,p,yH))  —k ((pfl, qfl, y)+n(q, , Th SH)) p Remark 2.10 As initial conditions we often use po  =  ü(.,to), where ü(.,to) E JH is the  solution, at time t 0 o, calculated using the standard method (2.5), and q  0.  The MSL method  To work with the discretely divergence-free basis functions, we work from the formulation (2.18), (2.19). The velocity equation (2.18) can be rewritten as (pt+qt, P)+va(p+q, i)+n(p, p, )+n(p, q,P)+n(q, p,I)=(f,  Pp),  (p + q, )+va(p+q,)+n(p, p, )=(f, ),  31  e  for all p  Jh. Discretize this system using a one-step theta scheme with an explicit  discretization of the two nonlinear terms involving q. The result is , 1 (p’ + q  , pfl+l 1 P)+ kÔa(p7’ +q’, Pço)+ kOn(p  , 1 (p’+q  )+k 1 +p a(q ,  co)+kên(p1  ,  P(p)  g’Qp), (2.46)  ii(ç),  =  pr’, )  (2.47)  where gfl()  =  (p + q, ) + k(f,  -  )+ n(p, p, Pp)]  i){va(pj+q,  P)  (p +  + k(f,c) + k(ê  +  -  k [n(q, p, P)+n(p, q, co)}  and =  We now fix a  =  1, /3  =  Jh,  1) [va(p+q)+n(pH,pH,)].  kOz-’ and we assume P4’Jh  approximation, at a fixed time t q E  —  =  (n + 1)k, is the solution PH E  JH,  q  =  q  —  q, where  clEf E JEf to  (P’,H)  + kOa(p ,H) + k0fl(p’,p 1 ,H) 1  ,H) 1 (  =  , qo 0 u  + k8va(’,H) =  = ( ,H) 1  gfl(),  VH E JR, (2.48)  1(h), V/h  E Jh,  (2.49)  + kua(1,ii), V  JH,  (2.50)  =  with Po  The fully discrete MSL  Jj-.  . 0 u  Remark 2.11 We often use as initial conditions Po  =  ã(.,to)  e  JH,  qo  =  0 where  a  is  obtained from a calculations performed using the standard finite element method (2.5.  The algebraic system (2.43), (2.44), (2.45), from the NGM, and the algebraic system (2.48), (2.49), (2.50), from the MSL niethod, are solved in a similar manner. To avoid confusion omit the superscript associated with the time step. 32  Let PH be the vector  of coefficients of p with respect to the standard basis of Jjj. Let GH the vector with components G = g(i). The nonlinear “low mode” velocity equations (2.43) (or (2.48)) is solved using a fixed-point defect correction method. Given Gif and an initial guess F°, the (m + l)st iterate prn+l  =  ]3+1  +  is obtained from the rnth iterate P# by solving  [ + Nh(P)]  (GH  -  P 11 vA  -  N(P)P).  Having obtained the solution P (when the fixed-point iteration converges) the linear supplementary problems (2.44), (2.45) (or (2.49), (2.50)) can be easily solved.  2.4  Numerical results  We test the NG and MSL methods by comparing the qualitative properties of solutions computed using these methods with the qualitative properties of those calculated using Turek’s code, which we often refer to as the “standard method”. 2.4.1  Flow around a cylinder  We consider flow around a cylinder with a Reynolds’ number of 100. A Reynolds’ number of 100 corresponds (approximately) to towing a 1 cm. diameter cylinder through water at a rate of 1 cm/sec. In this situation, the long time solution to the Navier-Stokes equations is known to be the stable time periodic von Kármán vortex street. In his book on fluid motion, Van Dyck [31] has experimental pictures of flows in this Reynolds’ number range. Below is a particle tracing visualization of a calculation we made.  :  •  •.,,  •  ••—..  .  •  •  :  Figure 2.3: particle tracing of a flow around a cylinder, Re  =  100  The computational domain is a rectangular box of length 32, height 16 with a circle of diameter 1 centered at the point (7,8). In the test calculations we prescribe: 34  at inflow, a parabolic velocity profile with maximum velocity one, • at the walls, a velocity of zero, • at the outflow, a “do nothing” boundary condition. This “do nothing” boundary conditions is a Neumann-like boundary condition which is naturally defined by the variational problem. Let n be the normal vector at the outflow boundary. Let u boundary, p  —  =  ii—  u n and let uT = u u. It is shown in [14], that at the outflow 7 8u = c, = 0, where c is a constant. This boundary condition has —  an been analyzed and carefully tested in [14]. In all of our calculations we used the Crank Nicholson scheme (the one-step theta scheme with U  =  ).  Figure 2.4 pictures the coarse  grid and we used several refinements of it. The grids we calculated with are: Grid level 3 =  1984 elements, 6160 unknowns, Grid level 4  level 5  =  =  7936 elements, 24224 unknowns, Grid  31744 elements, 96064 unknowns.  Figure 2.4: cylinder grids, levels 1,  •..,  4  Results for the standard method We calculated the standard method for 200 seconds (800 time steps) in three cases: Level 3  No von Krmn vortex street resulted. The solution had small oscillations  -  getting smaller with time. There is too much artificial viscosity from a combination of too coarse a grid and upwinding. The streamlines are depicted in Figure 2.5. Level 4  -  A good von Kármán vortex street was obtained. This was certainly a great im  provement from the Level 3 calculation. The streamlines of the results are depicted in Figure 2.6. Level 5  -  Compared to the level 4 calculation, the result was a fuller vortex street.  The oscillations are larger, better defined with shorter periods. The streamlines are shown in Figure 2.7 and Figure 2.3 is a particle tracing visualization of the calculation.  36  ==  I ..  Figure 2.5: SM solution, grid level 3, t =0, 20, 37  ,  180  0  -  5( w  Figure 2.6: SM solution, grid level 4, t 38  =  0, 20,  ,  180  a  a  a  a  a  1W  o Figure 2.7: SM solution, grid level 5, t 39  0, 20,  ,  180  Results for the NGM and MSL We propose two tests: Test 1  -.  Calculate using grid level 3 as coarse grid and grid level 4 as fine grid. The  interest in this test is whether either method somehow “bridges the gap” between the no vortex street solution we obtained with the standard method at grid level 3 and the fairly well developed vortex street solution we obtained with the standard method at grid level 4  .  As initial condition use the final solution of the standard  method level 3 calculation (a symmetric solution). Test 2  -  Calculate using grid level 4 as coarse grid and grid level 5 as fine grid. Here, the  main interest is whether either method produces a fuller von IKármán vortex street like the solution to the standard method at grid level 5. The initial condition is the final solution of the standard method grid level 4 calculation (vortex street is developed). The results of these tests: Test 1, NOM The results are shown in Figure 2.4.1. The solution developped unnatural -  “wiggles” and eventually, at time t  .7 (70 time steps), the solver no longer  converged. Test 1, MSL  The results are shown in Figure 2.10. The so]ution was stable and we  -  calculated for 20 seconds (2000 time steps). At the end of this time there was no vortex street. There were very small oscillations but they were decreasing with time. Test 2, NOM  -  The results are in Figure 2.11.  “wiggles” and by time t  =  The solution developped unnatural  1.65 (165 time steps) the solver no longer converged. 40  In Figure 2.12, we have included some streamlines of the “high modes” q and in Figure 2.13, we have included some stream lines of the “low modes” p. Test 2, MSL  -  The results are in Figure 2.14. We calculated until t  =  30 (3000 time  steps). The vortex street continued unimpeded. However, it retained the charac teristics of the original vortex street. It did not become “fuller” or oscillate with a shorter period. In Figure 2.15, we have included some streamlines of the high modes q.  41  I  CD  aq  CD  .Ij  Figure 2.10: Test 1; MSL grid levels 3-4, t  43  =  0, 5, 10, 15, 20  a -w  0 Figure 2.11: Test 2; NGM grid levels 4-5, t  44  =  0, 1, 1.5, 1.62, 1.63, 1.64  Figure 2.12: Test 2; NOM grid levels 4-5, “high modes” q, L  =  1, 1.5, 1.64  Figure 2.13: Test 2; NOM grid levels 4-5, “low modes” p, times near blow up  45  \%  r  o e  w a  r  Figure 2.14: Test 2; MSL grid levels 4-5, t  46  =  0, 5, 10, 15, 20, 25, 30  I  CD  2.4.2  Diffuser calculations  We consider flow into a diffuser, that is jet flow into a widening pipe. The intake is of width 1 and the outflow is of width 5 while the pipe is of length 32. In figure 2.16, the domain, coarse grid along with several refinements are shown. In the actual calculations, we used: • Grid level 5  =  8704 elements, 26721 unknowns,  • Grid level 6  =  34816 elements, 105665 unknowns.  IIIIIillh1tllhilI  ‘‘‘  I11I 41 1t 1  I I Ill lU’ IIiLWIiI IllS  Figure 2.16: diffuser grids, levels 1, 2, 4, 5 In the test calculations we prescribed: • at inflow, a parabolic velocity profile with maximum velocity one, • at the walls, a velocity of zero, at the outflow, as before, the natural “do nothing” boundary conditions, • a kinematic viscosity v  =  This corresponds to a Reynolds’ number of 5000. As time discretization, in all calcula tions we used the Crank-Nicholson scheme. 48  The results for the standard method Starting from Stokes’ flow, we calculated the solution at grid levels 5 and 6. Level 5  -  We calculated until t  =  550 (about 2200 time steps). Streamlines at various  times are depicted in Figure 2.17. Level 6  -  We also calculated until t  550. Streamlines at various times are depicted in  Figure 2.18. Compare the two results. For instance, compare the streamlines at t  =  550 in Figures  2.17 and 2.18. The shape of the large scale structures is fairly similar though the larger vortices are much “stronger” in the level 6 solution. In addition, in the level 6 solution, there is more fine structure (smaller vortices). In Figures 2.17 and 2.18 compare the streamlines at times t  =  50, 100, 150, 200, 250. In this time frame, both flows bifurcate  (in earnest) from a symmetric to a non-symmetric flow. Clearly, the level 6 calculation has stronger large vortices and more fine structure. Remark 2.12 We were unable to consider a grid level 7 calculation. This grid involves >  400000 unknowns, and the memory required for this calculation is beyond the capacity  of our workstations.  49  c  —  II  C  CD Th  cD  CI1  CD  <  CD  —  CD  CD  LID  CD  0  01  0 0,  I  0  o  0 0  Results for the NGM and MSL We considered several tests: Test 1  -  At a Reynolds’ number of 5000, we calculated using as initial conditions the  standard method level 5 solution from various times. Test 2  -  As a result of the tests above, we lowered the Reynolds’ to 1500 by increasing  the viscosity. At this Reynolds’ number, we calculated using as initial data the standard method level 5 solution at t  =  100 (with q  =  0 initially). At this time  the solution is still fairly symmetric and our interest was to see what happened through the “bifurcation” to the non-symmetric solution. Test 3  -  Again at Reynolds’ number 1500, we calculated used as starting data the level  5 standard method solution at t  =  400 (with q  =  0 initially). At this time the  solution is not symmetric but seems to have a fairly stable large scale structure with, however, some interesting nonsteady smaller structure on top of it. The results of these tests were in all cases pretty had. We remark that by “blow-up”, we mean that the solution developped oscillations and eventually the solver no longer converged. Test 1  -  At this Reynolds’ number of 5000 both the NG and MSL methods performed  very poorly and both blew up. We have not included any results of these tests. Test 2, NOM  -  The solution was unstable and blew up by time t  =  .11 (11 time steps).  The results are in Figures 2.19, 2.20. Test 2, MSL  -  The solution was also unstable and blew up by time t  =  .89 (89 time  steps). The results are in Figures 2.21, 2.22. Test 3, NGM  -  The solution was unstable and blew up by time t  The results are in Figures 2.23, 2.24. 52  =  .11 (11 time steps).  Test 3, MSL  -  The solution was also unstable and blew up by time t  steps). The results are in Figures 2.25, 2.26.  53  =  .56 (56 time  cc  II  II  cc  -  0  Eji  b  b  C,)  ct  Li  n H-1  /,/({  HI  /11,4  IL\  IL  SD  II  .Ij  0  *11  I  I \JIIIIII s._ I iitrn I  I Si iIIIIIIIIIi  I  —‘)1IiII I/J)I I flJ LA  IIIIIlIIII !1IJ1III  ri \c/ ftJf  Hi  ll1  /  \V/(  I\’S I  I hi ISI  )III  II CID  It l(((NlI WT/)1i C  -  S -  2.5 2.5.1  Proof of Theorem 2.1 Preliminaries  We let Gh  =  C R, n  T be a finite decomposition of mesh size h, 0 < h < 1, of the domain 2,3, into closed subsets T. Let C, i  1,... , 8, be positive constants  =  independent of h. For a given grid, Gh, we associate finite element spaces Xh C H, a velocity approximating space, and Lh C L, where L denotes an element in L 2 with mean zero, a pressure approximating space. We make some standard assumptions. The space Xh is assumed to satisfy the inverse inequality: <C hH h VbhW 1  (2.51)  for all Jh in Xh. For each u E H(1l) fl W (l) and q ’ 2  1 there exist H  ihU  Xh,  jq E Lh such that  —  VihuW  hluW G , 2  jq jhqIL2/R —  <  G3hIIqH1/R.  (2.52)  The pair Xh, Lh satisfy the classical inf-sup condition for mixed finite element methods: inf  b(qh, h)  sup  qhLh  qhU WVi’hH  >  4  >  0.  (2.53)  —  Let Ph be the L -projection onto X 2 11 and let Rh be the Ritz  projection onto  Xh. We also  assume (though some of the following can be shown to be consequences of or previous assumptions):  —  PhuU+ju -  —  Rhuj + h(W\ u 7  PhUj  + lu  -  RhuU) 7 \  <  WuW (2.54) h 5 C , 2  RhujI + h(ll\7Phull + lVRhull)  <  h11 Vu!!, (2.55) 6 C  —  VPhu+IlVu  —  !Rhu!I + !Phu!l  58  07!Iull.  (2.56)  Let J  =  {EH  : V•  =  o},  we assume for every v E J ’ there exists an 2 1 Fl W  approximation rhv E Jh such that: Mv  rv +  —  hWy  —  (2.57)  rhvM . vW 2 <Csh  In the “two-grid” situation which arises in the NOM and MSL, we assume that GH is a coarse grid and Gh is a fine grid with 0 < h < H < 1. Let XH and Xh be the associated “velocity approximating spaces”. We assume XH C X, C H.  e  Let F, be the aI3-projection where for each u  Pu E X is the unique  element such that ) + /3(Vu,Vh) 1 a(Pu,  V  o(u,h) + /3(Vu,Vh),  =  Lemma 2.1 Let ,/3 > 0, not both zero and let u  X.  H nW . There exists a constant ’ 2  C independent of h, c, /3 such that  Ilu In fact C Proof:  =  —  PuM + hWVu  —  VPuII  UuII Ch . 2  5 is the best possible constant of estimate (2.54’). 5 where C C  To simplify the notation we drop the superscripts , Phu, h)  —  Letting  (2.58)  /‘h =  u  allu  —  -  Fhu  —  Phu,  —  -  —  U  2 FhUW  —  VPhu, V)  0,  1 Vb,  (2.59)  X.  Phu) in (2.59), we obtain  /3Vu  2 + PhM  = a(u  (u  + /3(Vu  /3. By definition,  -  —  2 VPhuU  =  Phu) + /3 (Vu  + u  -  —  VFhu,  2 + /3MVu Phj  -  Vu  —  VPhu)  VPVu  -  VPhUM.  Hence,  [u  -  2 FhuU  -  Mu  -  phuM2] +  /3HVu  -  2 VPujj  59  /3Vu  -  uMIjVu 1 Vi  -  VPhuII  and since  lu  —  lu  Phull  -projection minimizes the L 2 -norm) 2 Phull (because the L  we obtain /WVu  -  2 VPhull  Cancelling terms and using (2.54)  IlVu  Vu  -  VFhUll.  -  implies  Vhull <IlVu  -  VPhuIIWVu  -  The L -estimate is proved similarly. Choosing 2  . 2 <GhIjujI  VPhull  /‘h = U  —  (2.60)  Fhu (u Rhu) as test function —  —  in (2.59), implies +  2 FhUII  -  (u  —  PhU,  Ilu  -  u  lvu —  -  2 VPhuj  Rhu) + 9 (Vu  PhuIilIu  -  RhuIl +  —  VPhu,  llVu  -  Vu  —  VRiu)  Vhull2 +  çIIVu  -  . 2 VRhu  Now lVu—VRhull < lVu—VPhuIl (because the Ritz projection minimizes the Dirichiet norm) and hence, using (2.54),  lu  -  Phull <Ilu  -  Rhujj  . 2 ChUull  Combining estimates (2.60), (2.61) proves the Lemma.  (2.61)  E  Remark 2.13 It is clear that Lemma 2.1 holds in more general circumstances than considered above. For instance Xh could be nonconforming. In this case we must modify the definition of P in terms of an elementwise bilinear form rather than (V., V.). In the course of  proving Lemma 2.1 we have in fact proved:  Corollary 2.3 For all a,  0 (not both zero), the projection P’ 3 satisfies llU-PU  <  llU-RhUj,  llVu—VPull  <  lVu—VPiull.  60  Lemma 2.2 Let u E Hj then  lu  PhUII <GhljVull.  -  (2.62)  In particular, suppose XH C X,. Let F’ be the cq3-projection onto XH and I  —  P’ then, for all finite element functions IIQFfUhM  llUh  =  —  Qj  =  Xh,  Uh  (2.63)  PHUhll <CHllVuhI.  Proof: Using Corollary 2.3 and (2.55), (2.62) follows since lu  PuW  Mu  -  RhulI  ChWVuU.  Clearly, (2.63) is just a special case. Lemma 2.3 Let  Uh  11 X E ,  PH  =  Pu and q  MPHW +  =  MqhM  WVPHW + Vqhl  Qfu then GMuhW,  (2.64)  CWVuhW.  (2.65)  Proof: The proof relies on the inverse inequality (2.51) and estimate (2.62). Firstly, MHhM < MPiu  -  uhM +  MuhM  <ChllVuhll  +  luhil <Clluhl  and  llQffUhj  =  huh  -  PHuhI <ChhlVuh <Chluhi  imply (2.64). Secondly, since  I VP  =  lVPJJUhl < VFuh  < ChllPuh  —  1 (hlPu Cli 5 <  RHuhll —  uhhl  +  -  \7RHuhj +  + IIVUhI  huh  —  RFJuhhl) + bVuhhl  Ch’hlbVuhhl + bVuhhl <Cl Vuihl 6j  IVRHuh  and =  llVuh  -  u 1 Ch  VPuhlI  -  FHUhli  <CliVuhil,  (2.65) is true. 0 We require estimates for standard finite element approximations to the Navier-Stokes equations. Let Üh, t9, be the solution to (2.5). Let  in  —  ü + hi! Vu  —  T(t) =  Vu + riO  rnin(t, 1) then for 0 < t < T, <2  (2.66)  —  For the proof of this estimate we refer the reader to [15]. The constant C is independent of h but dependent on T, u, Q and some properties of the grid. Under these circumstances one may further prove a sequences of weighted error estimates, much like the sequence of weighted estimates proved in [15] for the continuous problem. For our purposes, it is enough to know that, for all T > t  t > 0, there exists constants C (depending on to,  t as well as other quantities) such that  Hut ilutt  —  —  ãtii + hi! Vu  u!l  + hiiVujt  —  —  Vat!!  (2.67)  . 2 Vütt!i <h  (2.68)  Many of the ingredients for proving such an estimates can be found in [15]. However, it is still a fairly involved technical argument to prove these weighted estimates. We prove the first of these in Appendix B. In [1], they use such estimates and in this paper the reference for the proof of these estimates is [2] (a reference we do not have). Lemma 2.4 Let XH C Xh, let Üh, qh  =  = Üh  —  7Th  PH For all T> t  H, but depending on u,  i’  be the solution to (2.5) and let PH to > 0, there exist constants  ,  =  PÜh and  independent of  and T, such that i!qhIl + H!!Vhii iqh,t!i  + HiiVh,tii  Hqh,ttii + H!iVh,!i 62  <  H, 1 C  (2.69)  , H 2  (2.70) (2.71)  Proof: (2.69) follows from (2.66), and Lemma 2.1 since  iqhii  =  Wuh  -  PHuhW  -  uli  +  Mu  +  -  iiP(ah  u)ll  -  . 2 H  In a similar way, (2.70) follows from (2.67) and while (2.71) follows from (2.68). We stress that estimates (2.70) and (2.71) require no weighting factor since the time t 0 is specifically taken away from t  =  0.  ü  We require some further technical results proved in [15]. Let A,, : Xh invertible operator which maps each a(v,,,,b,,)  For any  ‘h/h  Vh  =  e Xh to  X,, be the  Xh where  Ahvh  —(Ahvh,bh),  —*  Xh.  Jh  E Xh,  i  A  b,,ii  (2.72)  <Gh’ijVhii.  This is easily proved since,  M  Ah hi 2  =  -a(Ah,,,,,) <Ch’ii Ah hIIIhii  We also note that, for all t > t , 0 A  ahM  +  U  A,, uh,tii +  1  A  a,ttii  (2.73)  .  This follows since  U  Ah  2 a,,i1  —a(A,,ü,,, Üh  =  —a(Ahüh, ü,,)  =  <  G  (liVu  Ah ã,,lih  -  —  u)  Vail + UVu  —  —  a(Aü,,, u  VRhÜII)  —  Rhã)  CU A  uii  Similarly for other two terms. To handle the (discrete) nonlinear term n(.,  ,  need discrete analogues of several Sobolev inequalities (see pg. 298 [15]). For all ClV’4’,,ii,  i’/-’hii6  63  <.  .),  we will  v’,, E X,,, (2.74)  Ih3 <  6 IV’’hW  GhIVh,  (2.75)  Cjj A,.  (2.76)  4 Ah CIVhI  WhI{ + IhM3  (2.77)  hW2.  Note, as a consequence,  IjVuI + 00 IR’Boo + 3 IpW  + lPW3  (2.78)  ,  (2.79)  + WVW3 < CVh Ah  Concerning the nonlinear term itself, we assume a discretization such that n(u, v, w)  =  —n(u, w, v). We require some estimates for solutions to the “backwards” Stokes’ equations. Lemma 2.5 Suppose zt + Az + Vp  =  f,  z(t)  0,  =  zIan  =  (2.80)  0.  Then  +  j  Vzjds  2 + L(IzI + BziU BVz(to)W 2  +  jVp ) 2 ds  2 Wz(to)W  cj  fW d 2 s  (2.81)  C] 2 HfW d s.  (2.82)  Proof: Multiplying (2.80) by —z and integrating over the domain, we obtain  _IIzII2 + IIVzH 2  . 2 CjlfH  Integrating this from to to I implies (2.81). Multiplying (2.80) by Az, where A is the Stokes projection, and integrating over the domain one can show _IIVzW2 +  2 WAzH 64  =  2 CfH  and, integrating from to to t,  2+j Vz(to)I  ds < Cf fW 2 AzW ds. 2  The full estimate (2.82) is obtained since  IzI2  CIAzf by the Cattabriga-Solonnikov  estimate ([5], [24]) and since  IIztIi  + Vp  =11  A  z  -  65  zI2  + jjW  Organization  2.5.2  This subsection is included to familiarize the reader with some of the different quantities we study in the course of proving Theorem 2.1. Uh  —  (PH  +  q),  In Theorem 2.1 we estimate eh  =  the difference between the standard approximation on the fine grid and  the MSL approximation on a coarse and this fine grid, and  Ch  =  rh 7  —  7Th,  the difference  of the two discrete pressures. For simplicity we drop the subscripts h, H and also let P  =  Pf and QH  obtain the  Q.  =  error equation  Subtracting (2.17) from the velocity equation of (2.5) we  for e,  (et, ) + a(e, ) +  n(p,  +  n(p,  =  —n(,  C:  p, ) ,  For all —  P)  ,  )  to,  t  n(p, p, -  —  )  =  n(,  n(p, q, n(p,  ,  +  Q)  b(qh,e)  with e(.,to)  +  =  —  0,  p, P)  h 11  = QHeh =  qi  —  n(q, p,  b(C,  (2.83)  n(,  p,  Vqj.  e  0. We divide e into its “low mode part”,  “high mode part”,  —  V  e Xh,  Lh,  eH  =  Pi-je  =  + (f,  )  PH  —  PH  and its  q,.  Let (s, *) E Jh x Lh be the solution to (sj, ) + a(s, ) + b(, ) -n(p, q, P) for all &  Xh, satisfying (x, to)  -  =  =  -n(p,  n(q, p, P)  -  Vp, P)  n(p, p,  Pã(x, to) and (x, 0)  = Qü(x,  (s, *) is a solution “half-way-between” the approximations ê=a—s,  C=—*,  =Fê  and  =Qê. 7 i  Similarly, let è=p+q—s,  =7r—, =Pê  and  66  )  i=Qé.  (2.84)  ). Loosely speaking, 0 t  (a, ) and (p + q, ir). Let  Notice e  ê  —  é and  =  . —  We obtain an equation for ê,  (  by subtracting (2.84)  from (2.17), the result is:  )  (et,  with  e(to,.)  =  +n(,  ,  —n(,  ,  =  ) )  —  —  =  p, )  -  + n(p, ,  n(q, p,  n(,  ,  0. An equation for è, (t,  with é(to,.)  + a(ê, ) + b(, )  b)  —  -  P)  n(p, p, b) —  (2.85)  n(p, q,  n(, 5, b), V E Xh,  is obtained by subtracting (2.84) from (2.17):  ) + a(é, ) + b(, b)  =  n(p, p,  -  n(p, fi, )  (2.86)  0.  Splitting the error e,  C  into the parts ê,  others we work directly with e, that can be proven for e, estimates for e,  (,  .  and é,  helps us in some instances; in  We mention the following rule of thumb: any estimate  can be proven for ê,  and é,  and therefore, at times, having  we will conclude the same estimates are true for ê,  (  and ê,  without  subjecting the reader to the details. It will be necessary to use the equations obtained by differentiating (2.83) and (2.85) with respect to time. We need these when we estimate quantities like et,  Ct  and  t,  Ct  (and some others). Differentiating the error equation (2.83) with respect to t we obtain, for t > to, (ett, ‘b)+a(et, b) + b(Ct, ‘ii’) +n(Pt, j5, &) —n(pt, p, ) +n(p,  P)  ) + n(, p, b) n(pt, q, ) + n(p, , ) +n(pt, , ) —fl(qt, , ) n(Pt, , ) , p, ) —n(, , ) n(p, , ) n(, Pt, Q), +n(, p,  —  n(q, Pt,  —  —  =  —  Pt, b) —ri(p, p, ‘II’)  n(qt, p, n(p, q,  —  —  —  —  67  V E X . 1  )  (2.87)  Similarly, differentiating (2.85) with respect to t, we obtain, for all t > t , 0 (tt,  &) +a(ê, b) H- b(C ,) 1 +n(, Pt, b)  Pt, ) + n(, t, &) +n(,  —n(p,  ,  b)  —  —  —  =  [fl(t, —  1, )  n(pt, p,  n(p, Pt,  ) + fl(t, p, Pb)  n(q, p,  P) + n(Pt,  n(p, qt, Pb)]—n(t,  —  —  n(  p, )  —  n(,  68  ,  ,  P)  —  —  b)  n(qt, p, (2.88)  n(pt, q,  &) n(, t, b) —n(Pt, —  p, b), V e  Xh.  ,  )  2.5.3  Proof of the Theorem  In the following, 0 < to < T < cc are fixed. C represents a generic positive constant independent of h, u and to while  represents a generic positive constant independent  of h but possibly depending on u, to and T. (Both C and C may depend on  and  some parameters, uniform in h, associated with the family of grids). GH is a coarse grid and Gh is a fine grid with associated finite element spaces XH and Xh respectively. We assume XH C  C H and 0 < H < h < 1. P  defined projections. a, /3  —  P’ and  Q  =  3 Q/  are the previously  0 are arbitrary (not both zero).  Lemma 2.6 There exists a constant  such that,  2 + vj hlVehI heW ds 2  , 4 <H  (2.89)  for all t 0 <t < T. Proof: Letting b  =  e in (2.83) we obtain  ld  IheW 2 +WVejh 2 +n(p, , )  —  =  —  [n(ji, , e)  n(p, q, )J  —  —  n(p, p, e) + n(q, P )  n(, j, e)  —  n(,  ,  j)  —  —  n(q, p, )  n(, p, i).  Now, n(p, , e)  n(p, , )  n(, i, )  —  —  —  n(p, p, e)  n(p, q, )  n(q, p, )  =  —n(e, , e) + n(, p, e) + n(p, , e),  =  —n(,  e ,) + n(, j3, e) + n(p, e i)  =  —n(, ii, ) + ri(, j, ) + n(p, ii,  =  n(, , ij) + n(, , )  =  —n(ij, , ) + n(’q, p, ) + n(, , )  =  n(’q,p,). 69  —  n(p,  ,  q)  Hence, [n(p,  15, e)  .}  —  =  n(, 15, e) +  n(,  i,  ) + n(ij, 15, )  7 (W IVPII3WVeH —1— <  lW iqU3Ull  +  iiH ll’pii3U”li)  + eVe . 2  Also, n(,  ln(P, n(,  ,  ,  e)  5 + ejlVeU H , 2  CWil{  H + eWVe ipVi < 4 , 2  ij)  WVP3UV7)H <  15, ij)  + eWVeU , 2  and therefore, ejI2+IIVe2 Integrating proves the result.  <lleil2  . 4 +H  0  Lemma 2.7 There exists a constant C such that,  vu Veil 2+ L  , 4 ieuII d 2 s <H  (2.90)  for all t 0 <t < T. Proof: Letting ‘b  =  et in (2.83) implies  vd  -iI Veil 2 +iietli 2 = +n(p, , )  —  —  [n(p, j5,  n(p, q, )]  e) —  —  n(p, p, et) +  n(, ,  e)  —  n(,  n(15,  ,  15, )  i)  —  —  n(q, p, )  n(,  15, ‘j).  We would like to point out a little “trick” (using the inverse inequality). Since I Veil CH,  3 i 2 jVeIi ietii  <  CHUVe iietll  <  iCH 4 iVelllIetlI  i Ve CHil Veil 2+  2 + eiletil2, (if H iiVeii 70  1).  2 eiietll  We use this in what follows. n(p,  p, et)  n(p, p, et)  —  =  —n(, , e) + n(,  <  Cj V 2  p, et) +  jet li + Cj \7  n(,  ,  e)  3 + j ll) jet I (U Vjj  2 + clletjl2, CjjVe n(5,  ,  )  —  =  n(p, q,  —n(,  i,  ) + n(, j, ) + n(p, j, )  lletll + GIl Vll ljetll(jjVll 2 GjlVell 3 3 + 11Pl1 ) 00 < n(i,  p, )  —  n(q, p,  .) = (  CjjVejj + —n(q,  ,  llet j2,  ) + n(r, ii, )  + n(, , )  lletll —1— 7llVellljetil(ii\7Pll 2 jj\7ejl 3 3  +  llllc,o)  2 + lletij2, llVeil ri(q, q, et)  - n(p, q,  n(q, p, ‘ii)  =  d n(q, q, e) — n(qt, q, e) — ri(q, qj, e)  <  n(,e)+H5+llVell2,  =  n(p, q,  <  n(p,  — n(pt, q,  ij)  -  — n(p, q,  +jjVejj 4 )+H , 2  -  -  p, 77) — n(qj, p, 11) — n(q, pj,  =  )  i)  p, ) + H 4 + jj VeIl . 2 Combining, choosing  ullVeIl2+ jjetlj 2  appropriately,  ll VeIl 2  +H 4  —  {n(,  ,  e) + n(p, ,  i)  + n(, p,  Integrating,  2 I VeIl +  lletll2cls  4+ H  ln(,e)l + j(,77)l  +  )j. 77 n(,p,  Now,  ln(,e)l + I(P,,77)l  +  l(,77)I 71  + jjVejj,  i)}.  which proves the Lemma.  U  Lemma 2.8 There exists a constant C such that,  2 uHetB  d8 2 L 1It1I  (2.91)  , 4 <H  for all to <t < T. Proof: Letting  et in (2.87):  =  etI2+vjIVe2 —  {n(Pt, p, et)  +n(qt,  p, )  —fl(t,  ,  e)  =  n(p, p, et) + n(p,  —  —  —  n(qt, p, ) + n(Pt,  n(Pt,  ,  ij)  —  Pt, et) ,  )  n(OA, p, i)  n(p, Th, et) + n(, Pt, )  —  —  —  n(pt, q, ) + n(p,  n(,  ,  et)  —  )  ,  ri(p,  )  ,  n(q, Pt, )  —  —  —  n(p, qt,  n(, Pt,  Estimate the terms on the right hand side as follows: fl(t,  ,  et)  —  n(pt, p, et)  =  , et) + n(Pt, , e) — n(, et, P) VIHlVetU 3 cIeJi  —I—  —f  MII’etII t IIPIW  2 + (WetI 2 + 2 < eVet WVej ) ,  n(p, Pt, et)  —  n(p, pj, et)  =  —n(, , et) + n(p, , e) +  <  — n(, et, Pt)  PWJIVH etH  + PtjjWVe  ), 2 eVe + (UetH 2 + Ve  n(, Pt,  — n(q, Pt, t)  =  —n(77, t, ) + n(,  ) — n(i,  —I—  WD  < eVe + (Wet 2 + fl(qt,  P,  — n(qt, p,  =  S  —n(j, ,  etH —1—  1 72  —i-  WPtHc,DWHWVtW)  UVeI ) 2 ,  e) + n(qt, , ) — n(rj, W  ) t p  W \7e ( et  t,  P)  —f Ve)  IVeW n(Pt, i, )  —  n(pt, q,  =  <  n(, 1t,  —  n(p, qt,  2+ (Wetjl  +  WVeW ) 2 ,  + n(Pt, 11,  —n(, ii, W3W  <  eVejj +  =  —n(e, r,  fl(qt, j,  e)  n(15t, i,  + n(,  ,  e)  \7e  +  jIVejj ) 2 ,  2 ( etj + n(P,  —  + n(p, it, j)  eVej + etjj2 +  2 eVetH =  (jIet 1  —I—  i)  —I—  n(, , i)  Vetjj(jjetjl  —F <  t’  —  H—  IlveW)  WVeII ) 2 ,  , 5 + CH  —fl(Pt, nt, i)  —  n(p, nit,  h)  , 4 eVetjj + H < 2 fl(qt,  p, nie) + n(i, Pt, ‘ii)  < eVetjj . 4 2 + GH  Combining and choosing the appropriate e implies IIetW2 + vVetW 2 +H . 4 2 <jjetjj Therefore, aUet2  2 + + vajjVetW 2 <ujjetjj  and integrating, observing that the 1iminfajjet 2  =  2 jjet  + aH 4  0, proves the result.  0  Lemma 2.9 There exists a constant C such that,  2+ H  f  ds <H 2 jj , 4  (2.92)  for all t 0 <t < T. Proof: By assumption, our approximating spaces Xh and Lh satisfy the inf-sup condition and C sup ‘4’ eX  73  Consider the error equation (2.83): —b(C, )  =  ) + a(e, ) + n(, p, ) n(p, p, ) + n(, P) n(q, p, ) + n(p, Pb) n(p, q, P) + n(, ) + n(p, öb) + n(, öb).  (et,  ,  —  ,  —  —  ,  ,  ,  Estimating: CWetW JVb  (et, ‘ii’)  a(e,i&) n(p, p, b)  —  n(p, p,  < =  jVeIV’bI <H IVI, 2 —n(,  ,  b) + n(p, , b) + n(, p, )  +  n(, i5,P)  n(o, P)  n(q, p,  —  -  P)  n(q, p,P)  +  jjv  <  Hj\7j,  =  —n(,  <  Cjjjj V 3 elIIV’II + CVi,jV’IJ  =  -n(i,  ij,  P) + n(p,  ,  P)  P) + n(ö, P)  +  ,  ,  + n(, ,  P)  P)  VjVbI 7 G 3 +  +WVPU I 3 \7L’jj <H IV’bIj, 2 3 V V  HV  n(p, i, Q/’)  )IVL’II 2 jIPIicoIIiiHI?’U <H  n(i, p, ‘)  V  H I 2 I\7blj.  jf  Combining implies  HCU WV’’W  •<  7’’(ej  and hence, . 4 IIetI + OH  74  (2.93)  Integrating (2.93) from to to t and using estimate (2.90) implies the last half of the estimate. Multiplying (2.93) by a and using estimate (2.91) implies the other half. 0 Lemma 2.10 There exists a constant  2 hell  such that,  v  lVelI d 2 s  <  ds 2 2 + j êjj vhjVêW  <  (2.95)  L ahl2  <  (2.96)  +  j d 2 s  <  , 4 H  (2.94)  , 4 H  (2.97)  for all t 0 <t < T.  The proof is very similar to those of Lemmas 2.6, 2.7 and 2.9. Lemma 2.11 There exists a constant C such that,  ft  2 jejj  , 6 H  (2.98)  for all t 0 <t < T.  Proof: Consider the solution of the backwards Stokes’ equations zt + v  z  —  Vj3  =  ê, Vz  =  0, z(t)  =  312 0, zj  =  0.  (2.99)  Multiplying by ê and integrating over the domain,  2 hell Choosing b  = (Zt,  =  ê)  —  a(z, e) + b(, e)  =  (z,  ) + b(, è)  —  [(, z) + a(ê, z)j.  PHZ E XH C X,, as test function in (2.85),  (, FHZ) + a(e, PHz) —n(q, p, Phz)  + b(C, PHz) + n(, 5, PHz)  + n(p, j, PHz)  —  Th  n(p, q, PHz) + n(, j, PHz)  =  0,  since (PHz)  2 Well  =  0. Hence,  (z, ê) + b(,  e)  —  —n(q, p, PHz) + n(p,  [(at,  ,  z  PHz)  —  —  Pffz)  + a(ê, z  + n(, , PHz)  PFJz)  —  n(p, q, PHz) + n(,  PHz)  ,  —  b(, PHz)]  Estimating: b(/3,ê)  =  Ellzll (at,  z  -  +  llell jjz  PHz)  CHIIV/3IIHVêII  b(/3—jH/3,ê)  IlVll 2 CeH ,  -  llzIl 2 CIIêtJIH  PHzll  €zj + , 2 l 4 CH lêtIl a(ê,z  —  PHz)  n(, j, PHZ)—n(p, q, PHZ)  n(, p, PHz)—n(q, p, Pffz)  <  llVelillVz  =  n(, j, PHZ) + n(p, q, Pjzjz)  =  —n(, PHZ,  <  C  <  €jVPjjz + CH llVeU 2  =  n(ij,  —  llH  VPHzII  )  —  n(p, PHZ,  +  ljVll 2 CeH ,  i)  VPHzj j 3 jjj + jVp l\7PHzII II?)jl) 3  p, PHz) + n(q,  ll1Hzlloo(ll7lll I[ll  ,  +  llVPi-izIl + 0E116,  PHZ)  lll llVll)  , 6 llPHzH+H  I1HZllcolV1llllll  n(j,Pnz) b(,PHz)  =  b(C,Fjiz  -  z)  , 6 0 +Cj €IlPHzll  llllllVPhz  l{zll + 2 jj(jl H . Now, lFHzW,  3 IIVPHzII  76  Cjz  -  Vzll  and combining the previous estimates  2 leI  <  (z, ê) + 4 (H + H 2 H Hêt 2  Integrating from to to t, observing that  e(., to)  H° +  +  z(., t)  =  2.10 we obtain  j2  ef:  2)  =  IzII.  +  0, and applying Lemmas 2.7,  Wzds.  Now applying the backwards Stokes’ estimate (2.82) and choosing e appropriately,  ef IIzIIds This proves the desired result.  WlIds.  j  D  Lemma 2.12 There exists a constant  such that, , 3 H  jeW  (2.100)  for all t 0 <t < T. Proof: Recall (see (2.86)), (e, i) + a(ë, ) + b(, b) Letting  ‘/‘  =  n(p, p, Pb)  -  n(p, p,  Pm).  é 2 + vjjVéjj =  =  n(p,p,Pe)  n(Pé, p, Pè) +  —  n(p,,Pë)  n(P, p, P) + n(p, Fe, Fe)  n(Pé, p, Fe) + n(Fê,  p, e)  —  —  n(p, Fe, Pe)  n(p, Fe, Fe).  Notice, <  jVp  —  3 + jV Vj 3  Ch’jjVp  —  Vpjj + C 77  Cjj Ah (p C.  —  )jj +  and this implies,  n(Pe,p,iê)  2 +IIPê2, e{WVeIj <eVë  <  n(Pe, p, Fe) <  VeI  é,i 3 n(,i ê )  j  LeU ejVe , 2 eVë  êHVéW  . 2 +IIPeW  Therefore,  II2 + vWVe 2 Integrating, observing that  e(., t ) 0  =  2 + êU e . 2  0, and applying Lemma 2.11, proves the estimate. D  Corollary 2.4 Estimate (2.2) holds: there exists a constant C such that ej2  for all t <t  <  T.  Proof: By Lemma 2.7,  . 4 ds <H 2 H 3 e Also, by Lemmas 2.11 and 2.12,  ds 2 WeW  2j 2 WêH d s + 2]  Therefore, eI2 =  :  Ies2ds  <2  (f  2  ds 2 é  (L  . 6 H  esW2ds)  . 5 <H  fl  Lemma 2.13 There exists a constant C such that, UVetlj + v 2  L  2 a  for all t 0 <t < T. 78  , 4 ds <H 2 Ij  (2.101)  Proof: Let &  =  ett in (2.87):  2 2 + ettI —IVetI  —  2dt  +n(i,pt,t)  +n(Pt, , )  —  —  [n(Pt, p, ett)  —  n(pt, p, eu) + n(p, p,  n(q,pt,) +  ett)  n(p, Pt, ejt)  —  —  n(pt, q, ) + n(p, lt,  —  —n(t, , ett) —fl(t, , ‘q) —n(t, ji,  n(p, q,  n(,  ,  e)  —  n(p, t, ij)  —  n(, jit,  7itt)  Without going through the details,  [n(jit, ji, ett)  .  —  —  n(p, q,  2 + CVe  For the other terms: fl(qt,  ,  ett)  =  < —  n(, 1t, ett)  < —  fl(pt,  1,  =  < —  n(p, qt, 1tt)  < —  Ti(qt,  i5, i)  =  < —  d dt  —  n(qtt, j, et)  —  fl(t, t,  e)  n(t,,et) +H 5 + VetH , 2 dt d 5 + VetW —n(,t,et) + GH , 2 dt d —n(pt,,ij) n(Ptt, , i) fl(pt, dt —  —  + OH 4 + OjVetU , 2 dt d —n(p,t,q) + OH 4 + OWVetj , 2 dt d fl(qtt, ji, ‘j) fl(t, jit, dt d +4 OH + VejU , 2 dt d —n(,jit,q) + OH 4 + WVetH . 2 dt —  —  n(, jit, m)  <  vlIVetII2  4 22 IlettW <IVetjI +H  7))  j)  Thus, +  +n(Pt, , ‘i) + n(p, t, i) + 79  fl(qt,  {n(t,,et)  —  ji,  j)  + n(, jit,  +  n(t,et)  Since j  all Vell ds 2  <  cc (see Lemma 2.8),  vda2IlveIl2 + 2 BettB u  0 lirnsup  WetW a 2  =  , 2 0. Multiplying by a  4 H 2 2+2 2ullVetll IlVefl + a a  et)+n(,  ,  et)+n(Pj, , ij)+n(, t, )+n(t, p, t)+n(, t, i)}  +2aa’{n(t, , et)+n(,  t,  et)+n(Pt,  _a2{n(,  ,  Integrating from to to t, and  estimating  ,  proves the result.  Lemma 2.14 There exists a constant (C 2 a 3 for all to <1  <  such that, 2  + 3 , 4 )ds <H 2 W  (2.102)  T.  Proof: Recall that =  )+n(p, , m)+n(t, p, iit)+n(, p, 71)}.  satisfies (see 2.88):  (ett,) + a(et,)+n(,p,b)  —  n(pt,p,b) + n(j5,pb)  +n(, , Pi)—n(q, p, P)H-n(4, p, ib)—n(q, p, P’b)+n(, q,  b)+n(p,  P)—n(p, qt,  )+(t,  ,  )+fl(t,  ,  ,  —  b)  b)+n(t,  ,  ),  for all ‘,b E Xh. By the inf-sup condition, aCtHVW <a rhs. Estimating the terms on the right hand side:  a(ejt,,b)  CajetjllllVjJ,  aa(e)  <  allVetlllVl <H llVl, 2  rest of terms  <  llV’b. 2 CH  Therefore, g2lj 12 <ia2lle  80  12  . 4 H 2 +u  (2.103)  Integrating, using estimate (2.101), proves that j  II o I 8 d s l f2  jsI 2 o d s  < CH’.  The estimate for  is similar.  Lemma 2.15 There exists a constant  f  such that,  u ê 2 ds  for all t 0 <t <T.  Proof: Consider the solution to the backwards Stokes’ equations z+vAz—V/3=aê, Vz=0, z(t)=0, zan  0.  Multiplying by ê and integrating over the domain,  2 aIIêtlI  (zj,  =  Notice, choosing (tt,  t)  ?J’  PHZ) +n(i,  —  a(z, e) + b(,  PHZ  =  (z,  t)  + b(, ê)  [(e, z) + a(êj, z)].  —  E XH C Xh as test function in (2.88),  + a(êt, PHz) + ,  =  j)  P-z)  + n(i, it, PHz)  —  PHz)  =  —  [fl(j,  n(q, p,, Pj-z) + ri(Pt, n(p, qt, PHz)]  —  —  ,  n(O,  5, PHz)  Pffz)  —  j, PHz)  —  n(qt, p, Pffz)  n(p,, q, PHz) —  n(,  PHz).  Thus, 2 aUêt  =  (z,  + b(, t)  —n(qt, p, PHz) + n(, +n(,  ,  PHz)  —  —  ,  {(tt,  PHz)  z  —  —  PHz) + a(êt, z  —  Pffz) + [n(, , PHz)  n(q, Pt, PHZ) + n(Pt, i, PHZ)  n(p, qt, PHz)] +  fl(j, ,  PHz) + n(,  ,  —  n(pt, q, PHz)  PHZ) + b(C, Pffz)}.  As we have many times before we must carefully estimate the right hand side. b(/3,è)  =  bCB—jn/3,êt) < CHV/WVetW 81  ‘2 a —ill2 zj 1 + 2 alVêtH CH ,  z  (tt,  —  IIêtIIIIz  PHz)  <  z [fl(t,  —  ,  2 + 2 1zw ajttU 4 CH ,  VefVz  PHz)  IIzII 2 CIIetjIE[  PH2II  ‘‘2  za  p, PHZ)—  n(O,  —  —  a’WzIj  VPjjz  a’zII  —n(p, q, z)] 11 P  + 2 aUVêtW CH ,  2 (uVe2 + H4), +H  , 6 WzW +CH  PHz)  IIPHzIIIiIHIVtW  n(i, ã., PHz)  b((, PHZ)  b(t,PHz  =  —1’  —  “2  1z112 +  IzII  z)  , 6 +H  tHVPhz  —  VzU  2 °lictli“2 a ii  Therefore,  2 aIIêI  < (z,  j)  +  J1Hz2  +6 H +2 (aIICtII + H  2 all Ve  +  allêtiIl 2 H ) .  Notice d a(z, et)  =  d o(z, et)  =  a(z, et)  —  a  (z, ej)  0.  (z, e)  d  —  + a (z, e)  Thus, d a(z, et)  —  a  (z, e) + a (z, e)  ±z + H 6  +2 (tll + a H  (2.104)  2 jIVetII  +  llêjtll 2 H ) .  We would like to proceed by integrating. However we are forced to be a little careful and first consider the case when t I  jto  a2 llell 2 ds  a(z.et)jt 0  +€j:  lzI  to + 1. For times t < to + 1, a’ —  =  1. Integrating,  (z,e)It +jIto (zt,e)ds 0  +H 6 +H 2  82  fa (llc 2  12 +  2 1 8 11Ve 1  +  llêttll 2 H ) ds.  Now, 0 a(z,êt)l since z(t)  =  a(to)êt(to)  =  x:  ê(to)  =  0 (z,ê)  =  0  0. Also, applying Lemmas 2.14 and 2.13,  =  a2(t2  +  2 ! 3 lIve !+  Hence we have the partial result: if t  to + 1 then <H°.  Before considering the case t  to + 1, we need the following consequence of what we  have just proved: flê(to +  2 ‘)W  <  CH°.  (2.105)  This follows since  IIê(to +  1)112 =  f°  aI!eIl2ds  =  tto+l  2Jto Suppose t  f°’(a’lleIl2 +  pto+l  ds + Jto 2 !êIl  Il)ds 3 2a1!êIIlIê —  11è 2 a 1 3 d s ! <CH°.  to + 1. Integrate (2.104) from to to t and notice that a’  -  Lj232ds  +j  =  0 if t > to + 1.  (z,e)l’ +j°zt,eds  jz +H°  +H2j  (llC 2 l 3 l+  2 1 5 11Ve ) ds. 1  Consider the right-hand side terms:  (z, ê)l’  =  0,  =  (z(to + 1),  ê(to + 1))  a2llêsII2ds+H6.  83  !!zv(to + 1)112  +  llêt(to + 1)112  Here, we have used the backwards Stokes estimate (2.81) to estimate z(to + 1)11 and (2.105) to estimate  Bêt(to + 1)W . 2  f  Thus,  f IIzIds  aê d 2 s  +H . 6  Using the Backwards Stokes estimate (2.81) and choosing  €  appropriately we are done.  U  Corollary 2.5 The estimate (2.,.9) is valid: there exists a constant C such that,  oiIeII2 for all to <t  <  , 6 CH  T.  Proof: Now,  2 WeW  2+ <2ajêj  2 2uIeII  <0116+  2 uIIêII  and =  ft to  <  2j  f(u1We2  S  to  Iê d 2 s+j  +3 2WeIIIIê ) ds  . 6 IIêI 2 u d s <H  Proof of Corollary 2.2 A consequence of Theorem 2.1, we are able to prove estimates for the special nonlinear Galerkin method proposed by Ammi and Marion in [1]. Corollary 2.2 (Error estimates for a special NGM,,) Consider the nonlinear Galerkin method proposed by Ammi and Marion in [1] where Wi, and XH are orthogonal with respect to the L -inner product. Approximations calculated using this scheme satisfy: 2  VeIj + u  IleW for all t 0 <t  <  + HIleIj  012  <  T. 84  , 2 H  (2.31)  , 3 H  (2.32)  Proof: We will not prove estimate (2.31) since it is proved in [1]. We will prove (2.32).  The key point is that when the spaces W and XH are orthogonal with respect to the L 2 inner product, (pL,QHP)  Let  (pH, qh,  (q,Pff)  =  =  0.  ‘7r) E XH xW, xLh be the solution to the MSL method and let  (jq, qh, h)  E  1 be the solution to the NC method. It is clear that the difference of the XH x Wh x L solutions w  =  p+q  —  satisfies  —  IIwII2+IIVwII2  =  [n(,,Pw) +n(,  ,  n(p,p,Pw) + n(,,Pw)  —  Pw)  —  n(p, q,  Pw)]  —  —  n(q, p,Pw)  (qt, Qw).  Now,  [n(,  ,  Pw)  -...  -  WW  (qt, Qw)  n(p,  q, Pw)]  lIwII2 + VwI , 2  qt + 2 ]1  QwH  , 2 Vw  and therefore  wI2  . q 2 2+H IIwW  + jVwI 2  As a consequence of estimate in Lemma 2.7  f  q d 2 s 4 <OH Thus integrating from to .  to t we prove that  jw 112 <OH . 6 Now  lieu  ilu  -  p  -  q + liwli  Ilu p qil -  -  + OH . 6  Estimate (2.32) is proved since u—p—qj can be estimated using either (2.22) or (2.23).  85  Chapter 3  Burgers’ equation  We consider Burgers’ equation in one-dimension. Much of the motivation for considering this problem was to supplement our Navier-Stokes results, particularly the numerical ones. For Burgers’ equation we can implement the NGM and MSL exactly as stated. A particular NOM of interest is one proposed by Marion and Temam in [20]. The initial-boundary-value problem for Burgers’ equation in t external force solutions u  =  f  =  f(x, t), initial condition u , and constant 0  ii  =  [0, 1] with a given  leads one to consider  u(x, t) to Ztt  —  vu + ‘uu  u(0, t)  n(l, t)  = =  ,  3 1  0, u(x, 0)  =  . 0 u  Consider a grid, Gh, which is a regular refinement of [0, 1] into subintervals of length  h.  Let Xh the finite element space spanned by continuous functions linear on each  subinterval, zero on the boundary. It is clear that X  =  Xh C  . The finite element 2 WJ’  (Galerkin) approximation to (3.1), over the space Xh, is the solution ü (Ut, h)  +  v(zt, bh,) +  (UUx, /‘h) =  (I, ‘) V  bh E X, U(x, 0)  =  Üh(t) E Xh  =  U.  to  (3.2)  We often refer to (3.2) as the standard method (SM). Let OH be a length H  =  coarse grid which is a regular refinement of [0, 1] into subintervals of  and let Gh be a fine  tervals of length H  i =  grid which is a regular refinement of [0, 1] into subin  1. We say Gh is the  th_  refinement of GH. Let XH be  ,  the piecewise linear finite element space associated with Gj.j and let Xh be the piecewise 86  linear finite element space associated with Gh. Clearly XH C Xh. Let solution of (3.2) when X We often refer to  ftH  =  XH and let  üjq  E X- be the  E Xh be the solution of (3.2) when X  th  =  Xh.  as the coarse standard approximation and to Üh E Xh as the fine  standard approximation.  The NGM  3.1  In the NGM we require a supplemental space for the “high modes” and, as it was for the Navier-Stokes equations, no one choice of supplemental space seems natural. We consider general supplementary spaces  W’ where a,  =  {  € X  a(,  )+  x)  0 for all  0 (not both zero). It is easily seen that XH +  The a, p3-projection P’ :  —*  XH}, =  X maps each n E H uniquely to  Xh for all a,  > 0.  E XH through  the relation  a(’u, H) + Let  3 Qr  =  I  =  a(u, r) + (u,  H,x),  VH E XH.  —  Remark 3.1 (The spaces l4/’) In the finite element case, different choices of a,/3 lead  to different spaces T’V,” . In the spectral caes, the eigenfunctions of the Laplacian are 3 a  =  sin(n7rx), n  =  l,2,•••, and these are orthogonal in both the Dirichlet and L -inner 2  products and as a consequence, different choices of a,  define the same space.  In [20], Marion and Temam consider a NGM where the supplemental space Wh is spanned by the hierarchical (or induced) basis. This basis is the set of “hat functions” E Xh, i  =  1,3,  , 2n  —  1 with v(ih)  =  1 and v(jh)  =  0 for all  j  i. Less formally,  this basis consists of functions in Xh which are piecewise linear, 1 at a fine grid point 87  vi  U coarse grid point •  fine grid point  not belonging to the coarse grid and zero at all other grid points. It is easily seen that XH+Wh=Xh.  It turns out that Wh  {  I  E Xh  e  Ofor all  XH},  and hence Wh and XH are orthogonal with respect to the Dirichiet inner product. The proof is straightforward. Consider a coarse subinterval, I, of length 2h divided into two fine subintervals of length Ii. Any function qjq E XH is linear on the subinterval I and hence its derivative is a constant c. Any function and linear on each fine subinterval  -  /‘h  E 147 h  is zero at the endpoints of I  a “hat function”. Thus, its derivative is k on one  subinterval and —k on the other. Therefore,  j H,xbh,xdX = h(ck  —  ck)  =  0.  Clearly, this property is also true when Gh is the ithrefinement of GH (defining the induced basis in the obvious way). Given a coarse space, , 11 and a supplemental space, T4/,’, the NOM approximation X is the solution (p,q)  =  (pH,qh)  e  to  XH X  (pt,)+v(p+qx,q ) 5 +(ppx +pqx+qpx,q) = (f,q), v(qx+px,x)+(ppx,) = (f,), 88  Vq E XH,  (3.3)  VeW’,  (3.4)  satisfying p(x,O)  = Po.  We call (3.3) the low-mode equation and (3.4) the high-mode  equation. In our numerical tests we consider the NGM of Marion and Temarn (a  =  0,  =  1).  This particular case is easy to implement computationally. In particular, the solution to the complement equation (3.4) can be easily calculated since the induced basis and the matrices related to it can be easily constructed. Remark 3.2 It is interesting to note that the variant of the NGM in our computations  of the Navier-Stokes equations also had a supplementary space defined with respect to the Dirichiet inner product.  3.2  The MSL  The MSL approximation (p, q) E Xjq x  (p  + q,  satisfying p(x, 0)  =  h)  +  v(px  + q,  0 and q(x, 0) Pu  is the solution to  h,x)  =  + (pp,  1i)  =  (f,cH), V  =  (f, ),  for  E XH,  all  h  e  (3.5) W’ (3.6)  . A more compact form of this system is 0 Q’u V  89  h  E Xh. (3.7)  Results of the error analysis of the NGM and MSL  3.3  Our main result is Theorem 3.1 which is proved in Section 3.4. It is enlightning to notice the parallels between our theoretical results for Burgers’ equation and our theoretical results for the Navier-Stokes equations (in Section 2.2). Let f =  h  E Xh be the standard finite element approximation calculated on the fine  grid, the solution of bh)  satisfying ü(x, 0) =  (u, /h) = (f,  + v(, bh,) + .  Let  /‘i  (3.2)  E Xh,  be the previously defined projections. For  and  notational convenience, we omit the superscript a, let  Ir all  in what follows. Let  = PH and  = QHã. For this approximation t the following standard estimate holds: there exist  Csm such that  lu  —  Il  +  hIIu  ll  , 2 Csm h  (3.8)  0. The constant sm is independent of h but dependent on u. We say this is a  for all t  standard estimate though we know of no source for its proof. For completeness we have proved this estimate in Appendix C. The subscript sm highlights the connection to the “standard method”. Error estimates for the MSL Fort  0 > 0, consider the MSL approximation (p,q) = (PH,qh) E XH x t  defined  as the solution of: Vh E Xh,  satisfying p(x,to) = PHU(x,to) and q(x,to) e=  eh  =  —  =  (p + q). Let the weight factor o(t  90  —  QHt(x,to). ) = min(t 0 t  We analyze the difference —  to, 1). The main result is:  Theorem 3.1 (Error estimate for the MSL) there exists a constant  ms1  independent of  h, depending only on u and v such that  IeW  jj2  i  uIeIj + HeU for all I  (3.9)  ms1  msi  , 3 H  (3.10)  . 0 t  The proof of Theorem 3.1 is given through a series of Lemmas and Corollaries in Subsec  tion 3.4.2. Estimate (3.9) is proved in Lemma 3.8. The estimate  <  CH is proved as  Corollary 3.2. Subsequently, by bootstrapping we are able to prove estimates for quan tities like  IIetI, and  hence, we are able to prove the full estimate (3.10) as Corollary 3.3.  A consequence of Theorem 3.1 is: Corollary 3.1 Let  sm  be the best possible constant of estimate (3.8), and let  msl  be  the constant of Theorem 3.1. Then,  lIux uWu for all I  —  —  0. In particular, if H  p p  —  —  =  qjf  q  smh  Osm  +  2+ h  ms1  ms1  , 2 H  (3.11)  , 3 H  (3.12)  Th (i.e. the fine grid is obtained via n-refinements of 2 h  the coarse one) then the MSL approximation is, asymptotically, as good as fine grid SM approximation in the following sense: given any e > 0, there exist h such that if h < h then —  —  p p  —  —  qI  q  (sm  (sm  for all I > 0.  91  + e)h,  + €)h , 2  (3.13)  (3.14)  Proof: (3.11) and (3.12) are a simple result of the triangle inequality and Theorem 3.1.  IIu  —  —  q  + Wu  —  —  —  h+  qj  . 2 smh+msiH un—p—q{ °sm  If H  =  2+ h  . 3 H  msl  ’h then 2  Iux and choosing h  —  Px  —  2 = Csm h + Cmsi H  qW  e  srn  h 4 h + ms1 2  proves (3.14). Similarly,  = msl —  —  qW  h+ Csm 2  2 e  and choosing h( =  proves  Cmsi  (3.13).  H  =  h + Csm 2  rnsi  U  msl  Remark 3.3 If one varies the MSL and includes in equation (3.6,) the (nonlinear) term  (pqx,)  one can prove:  Ia  —  p  —  qj  + ha  —  p  —  . 4 CH  q  One can check this by carefully considering the proofs of the Lemmas in Subsection 34.2. We will not provide the details. Error estimates for the NGM  Fort  to >0, consider the NOM  approximation  (p,q)  =  (pH,qh)  E XH  defined  x  as the solution of: (ps, q) + v (p + q,  H,x)  + (PPx + pqx + qpx,  v(qx + Ps, satisfying p(x, 0)  =  , 0 Fiju(t  .).  h,s)  + (pps,  H)  (f,  V  )  (f, ),  V  cbH E  XH,  E W’  The best general result we obtain for the NGM is: 92  Theorem 3.2 (Error estimate for the J\TGM) There exists a constant ngm such that iiuh  —  PH  —  qhil + HIIth,  —  PH,x  —  qh,  , 2 CH  (3.15)  for all t > t . 0 This is a weak estimate. However, it is an improvement over the error estimate originally proved in [20] where no order of convergence was proved. We suspect these estimates are of optimal order and our numerical results seem to back this up. If NOM approximations did satisfy higher order estimates (like Theorem 3.1) one should see for H small enough,  un —p— il  ii  —  ühii. In our “moving shock” numerical experiment this was not true.  In fact, see Tables 3.16, 3.17, and 3.18,  un  —  p  l2Ozt  —  —  Suppose, as we suspect, these estimates are of optimal order. This suggests that:  lu  • the size of  —  PH  —  qhii  depends on the approximability properties of the coarse  grid. Fixing the coarse grid but taking more refined fine grids (larger n), we expect  un  —  PH  —  qhll  Cngm  . 2 H  In other words we have no reason to expect the  approximation to improve. Our numerical tests suggest this is true (see Table 3.19 and the remarks after it). • Consider the standard approximation calculated on the coarse grid, that  un  —  UHIi  Gm  seems possible that  H.  We know  . It is not clear which is smaller, sm H 2 H 2 or ngmH . It 2  un  —  PH +  qhil >  Ii  —  LHll.  In other words, the approximate  inertial manifold “correction” q may damage the approdmation. Our numerical experiments seem to indicate this happens. In the “moving shock” test qhii  Oiiu 3  —  in  —  PH  +  UHII for all grid sizes we tested (see Tables 3.16, 3.18 and Figure  3.30). The cause of the problems in the analysis (and in the numerical calculations) stems from the omission of terms involving the time derivative: (ps, 93  h), (qj, kH).  These terms  are too large (of too small an order of H) and pollute the estimates (and the numerics). We will not provide the details of the proof of Theorem 3.2. The result is easily obtained by copying the proof of Theorem  3.1 while keeping track of these larger terms on the  right-hand side. r Remark 3.4 (A special NGM) If XH and Whc3 are orthogonal in the L 2 -inner product,  then higher order estimates (in the sense of Theorem 3.1,) can be proven. In this case (pt,h) = (qt,H) =  0.  At the end of Section 2.2, we had a heuristic argument which showed that was a better approximation than  p  p  + q  to the solution u. We can repeat this argument  word for word to show that exactly the same thing is true for Burgers’ equation. As a consequence if Gh is the i-refinement of GH, we expect  IIu. —p  2u  —p  —  Izt  —  p  u11u 2  —  p  —  q  and  qW. Our numerical results of Test 1 of Section 3.5 support this  conclusion.  94  3.4  Proof of Theorem 3.1  3.4.1  Preliminaries  This subsection contains some necessary background estimates needed in the course of proving Theorem 3.1. Some of these are special cases of estimates proved previously in Section 2.5.1. Let  =  (0, 1). Let {Gh} be a family of grids satisfying a uniform size condition. For  a given Gh, we associate Xh the piecewise linear finite element space. Let Ph be the L 2 projection onto Xh and let R, 1 be the Ritz projection onto H . In the following le C, i 1  =  1,... , 4, be constants independent of h but perhaps depending on the uniform properties of the family {Gh}. The following results are well known. There exists constants C , C 1 , 2 3 such that G  Mu  —  Phu + Mu  —  RhUM  lu Phull -  +  + Mu  h(Ilu -  llu  RhuM)  <  lluW (3.16) Cih , 2  Rhull + h(llPhuM + HRhuSM)  <  hllulf, 2 C  (3.17)  <  11u11. 3 C  (3.18)  —  PhxW +  lRhull  —  + lPhuI  If ‘I’h E Hh, the inverse inequality holds: Ch’ llhIL  lIh,xll Let  be the c, 3-projection where for each  it  (3.19)  E H, P,’u E Xh is the unique  element such that +  /3(’itr,h,x)  =  (u,h) +  Lemma 3.1 Let a,,6 > 0, not both zero and let  it  Vh E Xh. E H fl  14/2,2  then there exists a  constant C independent of h, ci, /3 such that  lu  —  P’ull + hllu  —  95  11u11 <Ch . 2  (3.20)  In fact, C = C 1 is the best possible constant of estimates (3.16). , where C 1 This Lemma is a special case of Lemma 2.1. Lemma 3.2 There exists a constant C independent of h such that for all u 6  In  -  P’u <ChIIuLI.  (3.21)  In particular, let P’ be the a,/3 projection onto XH and let XH C Xh  , for all finite element function 1 C H IQuhIj =  IUh  -  nh  3 = I Q/  Since  —  6 Xh,  P’uh <CHIIuhI.  (3.22)  This Lemma is a special case of Lemma 2.2. Lemma 3.3 Let  nh  11 X E ,  pj-j  = P’u and qj = Qj’ u then there exists a constant C 3  independent of h such that IpHjj  IIpH,II  +  +  IIqhII Iqh,  CIIuhU,  (3.23)  Cjjui,j.  (3.24)  This Lemma is a special case of Lemma 2.3. Let  uh  be the standard finite element approximation, tile solution to (3.2). There  exists a constant °sm, independent of h but depending on  —  for all t  uH  + hIIu  —  txIj  n  and  ii  such that  . 2 smh  0. There exists a constant C independent of h hut depending on u,  (3.25) ii  and t 0  such that  —  —  tII + hIInt  ihtll  + hjjntt  96  —  —  uj <  , 2 Oh  (3.26)  , 2 Oh  (3.27)  for all t  to > 0. We know of no reference for these particular estimates. We prove  estimate (3.25) in Appendix C. We do not prove (3.26) or (3.27) since we proved a similar estimate to (3.26) for the Navier-Stokes equations in Appendix B. Lemma 3.4 Let XH C Xh, let ü , be the solution to (3.2) and let 1 qh  =  Q/ ã 3 i  =  Üh  —  to > 0, there exists a constant  PH. For any t  PH  ,  =  FH”uh and  independent of h  but depending on u and t 0 such that  + for alit  It W  +  jIII  + IJ(  WqH  +  +  WttII)  (3.28)  0 > 0. t  We require estimates for solutions to the backwards heat equation. Lemma 3.5 Suppose  zt + Az  z(t)  0,  =  2 -I-]to IIz(to)I  jjzW d 2 s  C]  L(z  zIf ) 2 ds  cj  =  f,  =  0.  (3.29)  Then  2+ Iz(to)U  +  to  WfII d 2 s  (3.30)  fM d 2 s.  (3.31)  Proof: Multiplying (3.29) by —z one can show, _zR2  + HzxW . 2 2 <Cf  Integrating this from to to t implies (3.30). Multiplying (3.29) by Az one can show _zx2  +11  A zW 2  =  2 CUfW  and hence,  2+ IIzx(to)11  fI  A  zjJ d 2 s <cj 2 IIfI d s. 97  The full estimate (3.31) is obtained since  2 1z11  Cf A zj (a standard elliptic estimate)  and since  z=WAz-fII<IjAzI+WfU. Since the domain is one-dimensional, if zz  ItLI  H(fl) then <Cu.  (3.32)  We use the following a priori (energy) estimate for solutions of the MSL (3.5), (3.6). Lemma 3.6 Let p + q be the solution to (3.5), (3.6), then  jp + qjj + Proof: Choosing as test function /‘ +  Now (ppx,p)  =  +  vp  ds 2 i + qj  =  (3.33)  <.  p + q in (3.5), (3.6) we obtain  + qxt (qpx,p) 2 + (ppx,p) + (pq,p) + 2  0 and (pqx,p) d  =  vf  =  (f,p + q).  —2(qp,p). Hence, 2 + qj +  vp  + qH 2 <CJf 2  and integrating  IIp+ q11 2 + vj  +2 q d s <  cj  98  ds + ju(to)U 2 f 2  =  D  Proof of the Theorem  3.4.2  In the following, t 0  >  u and t 0 while depending on u,  0 is fixed. C represents a generic positive constant independent of  represents a generic positive constant independent of h but possibly and to. GH is a coarse grid and Gh is a fine grid with associated finite  ii  element spaces XH and Xh respectively. We assume XH C X,L and 0 P = P ’ and 1  Q  <  are the previously defined projections. c, 3  =  H  <  h  <  1.  0 are arbitrary  (not both zero).  a =  Let (p, q) = e=  Üh  (pH,  q)  be the solution to the standard finite element method (3.2) and let be the solution to the Microscale Linearization method (3.7).  eh =a—(pH+qh)  and 1et  H =p—p  and  77  =7]h =—q.  Let  Subtracting (3.7),  from (3.2) we obtain the error equation: (et,  &)  +  v(e,  br) + (jiji  —  ppx,  &)+  3 34  (pqx—pqx+qpx—qpx, P) + (qqx, ) +  (pqx+qpx,  Qb) = 0, V b E  Xh,  satisfying e(to,.) = 0. Lemma 3.7’ There exist a constant C such that lie 2+vjexW2ds  for all t  , 4 <H  (3.35)  to.  Proof: Letting  =  e  in (3.34) we obtain  IIeW2+viiexll2=_[(ppx  —  PPx + qqx,e)+(pqx  —  pqx+qpx—qpx,) + (pqx+qpx,7])].  Estimate the right hand side as follows: (ppm —ppx,e)  =  (p +p,e)  liIlllPllllell +  + j5  )WeU + eiiexil2 W 2  jpx  99  IIpIIclll!lleIt <iiell2  +  eiiexll2,  —pq,)  =  (q +Plx,)  <  2+ C€(jq  =  (7Px + qx,)  <  Cpjj +  e)  <  lIe  (P + P,’i)  =  (ii  —  qp,)  5)II77XIHI’i  +  )U eM 2 [M 2 + jeM 2 177 coWPxWWlj  2 + IeM cMeH , 2  IIcIlxlIMM  +  )Well + eMeH 2  I  IiI Ili  <  2 CMeM  + II2,  eexll2 + eej + H . 4  Combining these, while choosing  €  appropriately, implies  MeW2 + vIIe2 <lleII2 + H , 4 which integrated from to to t, noting that cIt 0  0, proves the result.  0  We are now ready to prove the estimate (3.9) of Theorem 3.1. Actually we prove a little more: Lemma 3.8 there exists a constant C such that  2 vMesll for all t  , 4 ds <H 2 et  +  (3.36)  o• t  Proof: Letting i?L’  =  ej in (3.34) implies  Estimate the right hand side as follows:  (ppx —ppx,et)  =  (px  2 (IIll  (pqx —pq,t)  =  IIILIIilllletII  +p,et) +  +  )IleII + eIIetjj2 2 IpII  IIlIIqIllltII  (q +77x,t)  100  +  lIpIIIIxIIIIetII  <llexII2  +  ejletII2,  lpIIll77IllltII  l[xII2)IIexII2 + c{letll 2  2 + C(q (j5  — qp,t) = —(,et)  U + etll2 2  C(p +  =  —(,e)+ (@,e)+(,t,e)  <  ——(, e) + C(lltll lll llell  <  6 + Wexll , 2 —-(, e) + Gil  =  (1x,t,) = (i7x,i5)  I  ) We 2  Combining, choosing the appropriate  ,  etlj2,  IlB ll,II lIexlI)  +  <  —  llll  2.  e)} +  +  to t,  let Il ds 2  (t77x,P)  —  implies that  -  Integrating this inequality from to  j  <llexll2 +  + C(jl Ili5U lL’iU +  4+ —(ii) + H  vlIex 112 +  clIejII2,  —  5)  2 vlexlJ2 + jetII  +  llcoxlltW  PlHlW +  <  d  —(p +  lillco  (llpx + x,t)  2 llCrU  . 4 2+H IieU  noticing that e(•, to)  0,  we obtain  . 4 ll(P)U + H( e)II + OH  (3.37)  Now,  WexH2+H4,  (lll which, combined with (3.37), proves the It  is helpful to consider s (Si,  = j3 +  Lemma.  6 +CH U  E Xh satisfying  ) 5 ) + v(s, ) + (j5 + pq + qp, q  =  (f, ),  for all q  ) + v(s, ) + (ppx, )  =  (f, ),  for all  (Si,  with (x, t ) = Pã(x, t 0 ) and (x, 0) = 0  ü(x,  t ) 0 .  101  XH,  E W,  (3.38) (3.39)  Remark 3.5 One should not get confused in the definition of s. ji  from the solution to the standard method (3.2) while  p  =  =  are  and q are from the solution to the  MSL method (3.7). Let ê  =  ü  —  .s  = —  3 j  +  —  (at, b) + v(e, x) + (j5 —  and ê(to,.)  =  1 pq+P  —  +  =  —  .  ppx,  Subtracting (3.38), (3.39) from (3.2), one obtains:  )+  (3.40)  )+(, )+( +  qpx,  =  0, V  E X,  0.  Lemma 3.9 There exists a constant C such that  (3.41) 2  2 ê  +  for all t  (3.42)  t.  We will not supply the details of the proof of Lemma 3.9 since the proof of (3.41) is similar to the proof of Lemma 3.7 and the proof of (3.42) similar to that of Lemma 3.8. Lemma 3.10 There exists a constant C such that  2 iIeI for all t  <CH°  (3.43)  0 t  Proof: Consider the solution to the backwards heat equation Zt  Multiplying by ê  +z Az  =  ê, z(t)  =  zIaO = 0.  0,  (3.44)  and integrating over the domain results in  IeU = (zt, e)  —  v(z, e)  =  d —(z, e) 102  —  [(et, z) + ii(er,  zr)]  Notice that choosing PHz)  (at,  +  = 0.  since c(PHz)  2 = (z, e) hell  —  ‘  = Pjz  E XH  as test function in (3.40) then  v(e, PHz) + (iNs  —  pq +  —  qp, PHz) +  PHz)  =  0,  Hence,  [(at,  z—PHz)+(ê,  z)+(, 1 P 1 PHz).  Estimating:  (êt,z  —  PHZ)  -  <  (&, z (p:  —  iiêhlWz  PHz)  -  qpx, PHz)  —  €z  PHzXW  pqx, PHz) = ( +p,Pz)  —  <  (q  WzU <CIIêjWH Itll liz PHzlh 2 €z + 2 Wethi 4 CcH ,  =  ChlPHzgW(hlii ihl  +2 WêU CH ,  (C,PHz)  —  (p(PHz)  —  (PHz)p,)  ileli CH iip hlii) <cHzhh + 2  +  = (p + ,PHz) <ChIPfjzli(lilllill + llpUiihl) <  PHz)  jjz  +2 Weii GH  6 2+H iiFHzXhl  , 6 Ilzii + H  and combining we obtain  (lleii + 2 2 <(z, e) + H ell Integrating from to to t, noticing that  hiehl2 + H ) +H 4 4 11Ct11 2 + iizll.  e(., to) =  z(., t)  =  0, and using estimates from  Lemmas 3.8 and 3.9 we obtain  f hlêhids  6 H  +  hlzhlds.  Using the backwards heat equation estimate (3.31) and choosing the Lemma.  0  103  appropriately proves  Lemma 3.11 Let  =  (p  + q)  —  s =  +  There exists a constant  .  such that (3.45)  for all t  to.  Proof: Subtracting (3.38), (3.39) from (3.7) and choosing as test function )L’ =  we  obtain: ld  2 + vII2 —-Hl  = (iii  Now (ppP,)  =  (Px+&x+x+x,)  =  (px+px+px,)  <  ejjj  +  2 jj{  +  <2  +  —  (px+px,e)  and hence + vl2  . 2 eW  Integrating from to to t, using Lemma 3.10 proves the result.  (3.46) D  Lemma 3.12 There exists a constant C such that 2  <H°  (3.47)  for all t > t . 0 Proof: This is a direct consequence of Lemmas 3.10 and 3.11 since  2 I  2f  weH + 2  )ds <H°. 2  D  At this point estimate (3.10) of Theorem 3.1 is a simple corollary. 104  Corollary 3.2 Estimate (3.10) is valid: there exists constant for all t >  such that ejf CH,  t0.  Proof: Since, Ile!12 =  (3.10) follows because  :  L  e2ds  <2  (L  6 and ds <?iH 2 eW  IesI2ds)2  j  t (f  eds)  ,  0  ds <H 2 3 e . 4  The rest of the section is devoted to extending this partial result for the L —norm to 2 the full estimate (3.10). In order to do this we consider estimates for time derivatives of the error. A major step is to prove  f  êtU u d 2 s, Lemma 3.15, because having obtained  this, estimate (3.10) is a simple consequence. Differentiating the error equation (3.34) with respect to time we obtain:  (ett,  ) + v(e, ) + (jkt5  —  ptp +  (ptqx —ptqx+pqtx —pqtx+qpx +(qtqx  —  j5t  —  pp, b)  qtp +qptx  —  qpt,  + qqt, ) + (ptqx+pqtx+qtpx+qx, Qk)  Lemma 3.13 There exists a constant  IetII2 +  V  P’/) =  0, V /) E Xh.  such that , 4 ujet d 2 s <H  (3.49)  for all t > t . 0 Proof: Letting b  = e  in (3.48) we obtain  ld  IetW 2 -f-vWeW 2 =  —  (3.48)  [(ii  —  PtPx + PPxt  +(p —ptqx +j5 —pqtx +tj5  —  qtp  +  —  PPix, et) —  (qtq + qq, et) + (ptqx+pqtx+qtpx+qpjx, 7t)j.  105  Estimate each term on the right hand side as follows: (PP —ptpx,et)  =  (tPx + Pt, et)  (Ppxt —pptx,et)  =  (&tx  (ptqx —ptq,’t)  =  (pq—pqt, t)  =  (qtpx —qp,  =  (liipx +  qp,  =  (  (qtqr+ qqt,et)  <  +GH 2 eWetU , 6  (ptq + qptx,  =  (pqt + qtpx, lit)  =  (qptx  —  +  +  ) + eIIetII 4 2+H (IIetII , 2 tx,  ) + 4 2 + H (UetW  et)  , 2 IetW  H + eejf ) , 2 2+4 (UeW  tx+liPtx+tx,i)  + (et ) 4 , 2 +eet H  CH + cjjet , 2  (7t,j5)  4 + <GH  . 2 eIet  Combining and choosing the appropriate e implies + vetW 2  <eI2  +H . 4  (3.50)  Notice ci 2 -aejW  ci d ‘‘2 +2 aetW < je + uetIl,  =  and hence, aIIetW2 + vuWetW 2  2+ <CjIe  etII2 + CaH . 4  (3.51)  Integrating this, noting that 2 = 0 we obtain the result. 0 liminfoetW t—o As before, consider e = ê + ë. Differentiating (3.40) with respect to time: (tt,  ) + u’(eS,  (pq  (qtq  —  +  ptqx+pqtz  (tx  —  —  PtPx  pqtx+qtpx  + qqt, )+(ptqx +  pqtx  + —  PPtx  PPtx,  qtpx+qptx  —  Q)+ qptx,  + qp + qpt, Qb)  Copying the proof of Lemma 3.13, one can show: 106  =  (3.52)  P/))+  0, V  E Xh.  Lemma 3.14 There exists a constant C such that  +  for all I  vf e ds 2  (3.53)  . 0 t  Lemma 3.15 There exists a constant C such that  , 6 H  WêjW 2 d s  (3.54)  for all I > t o. Proof: Consider the solution to the backwards heat equation zL + Az Multiplying by  z(L)  = Jt,  =  0,  Z8O =  0.  and integrating over the domain results in  t  aljetl  =  (zi,  Using as test function L’ (tt,  Ct)  —  (zr, e)  =  d —(z, et)  Pz) + v(êtT, Pz) + —  zr)]  [(eu, z) +  Pz E V in (3.52) implies, noting that (Pz)  = PHZ  +(ji  —  qtpx +  j5ix  —  —  qpt  ptqx +  tx 5 1  + Jt +  0,  pqtx, Pz)  —  Pz)  =  0.  Hence, 2 aUêtW  =  (Pt  (zt,  t)  —  —  (zr, êj)  =  (z,  —  t)  —  —  [(tt,  z  qp+j5t  —  —  Pz) +  (tr,  qptx, Pz) +  z  Pz)] +  —  (tx  +  ts,  Pz).  Estimate the right hand side: (tt,  z  —  Pz)  =  z  —  Pz)  —  (at,  zj  107  —  Pz)  =  z  —  Pz)  —  (t  —  Pat,  Zt)  (ê,z  z  <  (et,z—Pz) (tq  (p  (i5  ptq, Pz)  —  pqt, Pz)  —  —  qtp,Pz)  Pz) + Ztjjt  —  WeIHIz—PzlI Pz)  (tqx, Pz)  =  (tqx +  <  ClIPzgj(ftW Wq +  =  (tx  <  CIFzW(W WixII + PIjj txI) 71  =  (‘iji  =  —  77j  xjII71tW  +  Zj  +C 2  (WII +  (WzIl +  ). 2 WztW  j) 2 + h  u(êt,Pz)  +  6+ H  CPzt  We need to multiply this by  Consider first t  etxI2 +  6 + zU CH  jjj  Pz) + H 2  =  (  2U +  (U71tXU2  Combining, choosing the appropriate  u(et,Pz)  2 IIzW + CH  W HW)  + qtr/,Pz)  +H 2  C Pz  <  (t,Pz)  <  (5(Pz) + (Pz)pt, 1)  =  UzW  2 u)IêtU  —  + P?ltx, Pz)  +H 2 (W71  (i?j,Pz)  FêtW  Pz) +  —  CWPzW(II71 (q5—qp,Pz)  —  ,  we obtain +  [UeW2  UetxU2]  +H 6 +  (355)  and integrate. With this in mind notice —  ‘(êt,Pz)  to + 1. In this case a’  L  Pz) = (U(t,  Pz)  =  =  (e, —  —  u’(ê,Pz) + ‘(ê,Fz).  1 and  (e, —  u(et,Pz)  Pz) +  0 + Pz)) 108  (e,  Pz)]  j(e, Pzt)ds  =  f  L(e,Pzt)ds  Weds  + ef  ds 2 IIztII  6 + ef 2 <H WêW d s. Thus if t < to + 1, multiplying (3.55) by  and integrating from t 0 to t, we see that  <H + eJ2etI2ds  La2et2d5  and choosing e appropriately proves the result if t  Be(to  to + 1. Using this one can show  3 (copy the proof of Corollary 3.3 below) and by the backward heat H  + 1)W  estimate (3.30)  Iz(to +  2 l)j  ‘.to+i  <J  io  IIet u d 2 s.  Now suppose t > to -I- 1. Fz) — a’(ê, Pz) + u’(ê, Pzt)]ds = U(t,  <—  Pz)I  —  f  +l 0  (ê,Pz) + cJ°’ êI ds + 2  <(e(to + 1), Pz(to + <  d (e, Pz)ds +  l))I  6 + eWz(to + 1)2 + H  €j  6 + +H  +1 0 t  f ef ef 2 IIêiI d s  (e, Pzt)ds ds 2 jfztH  <H + ds 6 ê 2 u  Now, an appropriate choice of e proves the Lemma.  IiêtW d 2 s.  U  Corollary 3.3 Estimate (3.10) holds: there exists a constant for all t  ef  , 3 such that ueW <H  . 0 t  Proof: Now,  2 uIIeW  =  uIlêIIds 2f jêW ds + 2  =  f  Lu2eS2d5  This combined with the fact that sult.  (a’e + 2ulWe I) ds 3 . 6 H  proved in Lemma 3.11, implies the re  U  109  3.5  Numerical results  We test the schemes in several ways. We compare the 2 L , H’- and L°°-norms of the difference between the solution u and the MSL approximation, the MSL  error,  the dif  ference between the solution u and the NOM approximation, the NGM error and the difference between  zt  and the standard approximation, the SM error. We also consider  the qualitative properties of the solutions. We considered several solutions to calculate: Test 1, two steady exact solutions, Test 2, an interesting “moving shock” solution, Test 3, a time periodic solution. Test 1 The first test is actually two different steady calculations. We considered: • the exact steady solution and  f  =  zt =  f  =  =  2 ir  sin(irx) + ir sin(irx) cos(7rx).  • The exact steady solution u and  sin(7rx). This is an exact solution to (3.1) if v’  =  sin(lO7rx). This is an exact solution if z’’  =  2 1007r  sin(lOirx) + lOir sin(lO7rx) cos(lO7rx).  We considered these steady calculations for several reasons. In a steady situation, the NG and MSL methods are the same (since any terms involving a time derivative is zero). In particular, we expect both methods to result in good approximations, if H is small enough, since Theorem 3.1 holds. We considered only the case where H  =  2h, i.e the  grid Gh is one refinement of GH. From the results we conclude: • if H was small enough, the 2 L , H’-, and L-norms of u same size as the the 2 L , H’-, and L°°-norms of  zi  —  tth.  —  p  —  q was basically the  As we expect the order of  convergence is 0(h ) in the L 2 2 norm and 0(h) in the H’-norm. In the L-norm it is 2 0(h ) . 110  • We also see that p + q is a better approximation to u than p. This confirms what we stated at the end of Section 3.3. In fact, as we argued, Lt and  —  iM  2u  —  p  —  qM  111  —  pM  4Mu  —  p  —  q  Bu  h  Table 3.1: Test 1,  H  h  5  10  10  --  -i-  40  I  80  80  160 320  -_  Table 3.2: Test 1,  H I 5  -i-10  1 20  1 -I80 1 40  160  h 1  uj .4058923 .2018046 .1007732 .0503708 .0251835 .0125915 .0062957  rate  —  1.01 1.00 1.00 1.00 1.00 1.00  sin(irx); errors from standard method:  rate  —  -—  L  =  —  1  o1  u  2.01 2.00 2.00 2.00 2.00 2.02  u p q .0043430 .0011040 .0002776 .0000695 .0000174 .0000043  20  1  rate  t .0179237 .0044560 .0011129 .0002782 .0000695 .0000174 .0000043 —  u  =  1.98 1.99 2.00 2.00 2.02  fu  p q .2021616 .1007853 .0503712 .0251835 .0125915 .0062957 —  —  rate 1.00 1.00 1.00 1.00 1.00  sin7rx); errors for NGM, p + q  u—p  rate  UxP  rate  .4006181 .0225651 .0057323 1.98 .2011727 .99 20 J2.00 .0014390 .1006950 1.00 40 1 .0003601 2.00 .0503611 1.00 80 2.00 .0251822 .0000901 1.00 160 1 2.00 .0000225 .0125913 1.00 2_______ 10  --  Table 3.3: Test 1,  ‘u  =  sinQ7rx); errors from NGM,  11  p  h  {{u—uW  rate  .0191726 .0042924 .0010627 .0002653 .0000663  2.16 2.01 2.00 2.00  Table 3.4: Test 1, u  H  h  20  40  --  --  40  80  80  —i-160  .320  6D  L  Mu  —  p  Table 3.5: Test 1, ‘a  h 1  40  -i-  1  I  L 160 —i---  40 80  I..  160  1  -  q  .0427327 .0078798 .0011190 .0002657 .0000663  1 1 i 1  ö  —  =  80  320 640  Table 3.6: Test 1,  =  5.1582378 2.5314807 1.2607378 0.6297691 0.3148099  1.03 1.01 1.00 1.00  sin(107rx); errors from SM  rate 2.44 2.82 2.07 2.00  Hzt p qII 5.6072437 2.6793891 1.2763767 0.6303594 0.3148289 —  —  rate 1.07 1.07 1.02 1.00  sin(l0irx); errors from NGM, p + q  Mu—pM .0990926 .0266822 .0078606 .0020063 .0005035  ‘a =  u—uj rate  rate  M’u—pM  rate  1.89 1.76 1.97 2.00  9.9974301 5.0106141 2.5132876 1.2584812 0.6294877  1.00 1.00 1.00 1.00  sin(10rx); errors from NGM, p + q  113  Test 2  The next test is a more complicated time dependent one. We did not want to resort to a time dependent force to produce such a solution. We chose to use constant non homogeneous boundary values. It is hard to find a time dependent problem, let alone an interesting one, that has an exact explicit solution without resorting to a time dependent force. We overcome this by calculating a very good approximation (basically an exact solution) using the standard method and use this for comparison.  -0.6 v  0.2  0.4  0.6  0.8  Figure 3.27: “moving shock”; A series of snapshots of the “exact” solution We consider (3.1) with and  u(t,  1)  =  —  ii =  .01,  f  0, non-homogeneous boundary values u(t, 0)  and initial condition uo(x)  =  —  =  2x. Initially, the solution is smooth  but quickly develops into something resembling a moving “shock”. The “shock” moves along for awhile at a steady speed then slows down fairly abruptly at around t by t  =  1.2 it is nearly steady-state. To calculate the “exact” solution,  Crank-Nicholson scheme with an extremely small time step k 114  =  zt,  =  1.1;  we used the  .0001 and we used a  very refined spatial grid with h  We stored the data every .1 unit of time (every 5120• 1000 time steps) until time t=1.2. This series of snapshots is depicted in Figure 3.27. =  Comparison calculations were made using the standard method, the NGM and the MSL on coarser grids. In these calculations, to limit possible outside sources of error, we again used the Crank-Nicholson scheme with k iteration to be very small  (llDW/JD I 0 j  =  .0001 and forced the defect in the nonlinear  < 1014)  (see Section 3.6). It is worth noting that  since the initial condition is linear, it is in the finite element space Xh and there is no initial source of error in any of the calculations.  115  + —  0.3 0.6 0.9 1.2  160  80  0.000149 0.003112 0.003793 0.006598  0.000037(2.0) 0.000783(2.0) 0.000957(2.0) 0.001895(1.8)  h—L 320 0.000009(2.0) 0.000196(2.0) 0.000240(2.0) 0.000483(2.0)  0.000002(2.0) 0.000049(2.0) 0.000059(2.0) 0.000120(2.0)  Table 3.7: “moving shock”; standard method u  ime 0.3 0.6 0.9 1.2  —--  —  80  0.017253 0.038689 0.030483 0.010111  —  160  0.004547(1.9) 0.010298(1.9) 0.008092(1.9) 0.001978(2.4)  320  —  0.001155(2.0) 0.002619(2.0) 0.002068(2.0) 0.000495(2.0)  Table 3.8: “moving shock”; NOM u  +  1_i 80  0.3 0.6 0.9 1.2  0.000178 0.003678 0.004679 0.009972  i_L —  160  0.000040(2.2) 0.000789(2.2) 0.000969(2.3) 0.001964(2.3)  i_L —  320  0.000010(2.1) 0.000195(2.0) 0.000239(2.0) 0.000481(2.0)  Table 3.9: “moving shock”; MSL u  116  h——i— 1280 0.000001(1.9) 0.000012(2.0) 0.000014(2.0) 0.000029(2.0)  640  —  —  1 h—640 0.000290(2.0) 0.000657(2.0) 0.000520(2.0) 0.000121(2.0) —  —  p  —  q, H  =  h—L 640 0.000002(2.1) 0.000049(2.0) 0.000059(2.0) 0.000120(2.0)  h—--— 1280 0.000073(2.0) 0.000164(2.0) 0.000130(2.0) 0.000029(2.0) —  2h  —  —  p  —  q, H  —  1280  0.000001(1.9) 0.000012(2.1) 0.000014(2.0) 0.000029(2.0) =  2h  ‘ —  0.3 0.6 0.9 1.2  80  0.0561611 1.0739022 1.3043985 2.3722568  —  160  0.0280714(1.0) 0.5397615(1.0) 0.6555578(1.0) 1.3147671(0.9)  —  320  640  —  0.0140152(1.0) 0.2699064(1.0) 0.3278527(1.0) 0.6650024(1.0)  Table 3.10: “moving shock”; standard method  —I-—— .  0.3 0.6 0.9 1.2  i.  “f  I flf  —  1.1827644 2.9016019 2.9392503 2.6105484  1  1.  —  0.6049941(1.0) 1.3544537(1.1) 1.4550711(1.0) 1.3159412(1.0)  ‘If  0.3045043(1.0) 0.6559526(1.0) 0.7186771(1.0) 0.6652290(1.0)  0.3 0.6 0.9 1.2  i.  —  i._1  T5 0.0762865(1.0) 0.1612361(1.0) 0.1776831(1.0) 0.1618978(1.0) f t ‘  —  0.1525138(1.0) 0.3245924(1.0) 0.3575114(1.0) 0.3315514(1.0) —  I.  —  qg, H  =  —  2h  1  ,  1280  0.0033992(1.0) 0.0654942(1.0) 0.0795584(1.0) 0.1618976(1.0)  i._i  f t ‘  Table 3.11: “moving shock”; NGM  —I-—— I  —  0.0069663(1.0) 0.1342100(1.0) 0.1630288(1.0) 0.3315437(1.0)  1  IIf—  ‘if—j  f— t ‘  fj t I  fjjj t ‘  0.0563524 1.1122328 1.3633529 2.6830306  0.0280773(1.0) 0.5414911(1.0) 0.6580272(1.1) 1.3252708(1.0)  0.0140154(1.0) 0.2699582(1.0) 0.3279296(1.0) 0.6652570(1.0)  0.0069663(1.0) 0.1342116(1.0) 0.1630311(1.0) 0.3315517(1.0)  0.0033992(1.0) 0.0654943(1.0) 0.0795585(1.0) 0.1618980(1.0)  Table 3.12: “moving shock”; MSL  117  IIz  —  —  H  =  2k  T ‘  0.3 0.6 0.9  1.2  —  80  0.0005126 0.0204400 0.0245810 0.0709505  —  160  0.0001287(2.0) 0.0059386(1.8) 0.0080409(1.6) 0.0216504(1.7)  —  320  0.0000322(2.0) 0.0015617(1.9) 0.0020618(2.0) 0.0062738(1.8)  —  640  1280  0.0000081(2.0) 0.0003935(2.0) 0.0005168(2.0) 0.0016109(2.0)  Table 3.13: ‘moving shock”; standard method u  0.3 0.6 0.9 1.2  1 1. 1If—  i._1 itf— 1  0.0611453 0.2657431 0.2271326 0.0926590  0.0160487(1.9) 0.0804664(1.8) 0.0719312(1.7) 0.0225577(2.0)  1  1  0.0040565(2.0) 0.0214517(1.9) 0.0196385(1.9) 0.0064216(1.8)  —--—  0.3 0.6 0.9 1.2  z.  ....1  0.0006368 0.0226275 0.0284915 0.0867396  0.0001392(2.2) 0.0065323(1.8) 0.0080536(1.8) 0.0230547(1.9)  In  —  —  1  0.0002546(2.0) 0.0013621(2.0) 0.0012550(2.0) 0.0004080(2.0)  qilco  1  1.  IIf—  L  0.0010177(2.0) 0.0054355(2.0) 0.0050131(2.0) 0.0016213(2.0)  Table 3.14: “moving shock”; NGM  —I-——  —  1  1.  (tf  0.0000020(2.0) 0.0000990(2.0) 0.0001301(2.0) 0.0004072(2.0)  j.  1  f— 1t  f—j t i  “fTö  0.0000330(2.1) 0.0015584(2.1) 0.0020347(2.0) 0.0061648(1.9)  0.0000081(2.0) 0.0003923(2.0) 0.0005150(2.0) 0.0015969(2.0)  0.0000020(2.0) 0.0000989(2.0) 0.0001300(2.0) 0.0004066(2.0)  Table 3.15: “moving shock”; MSL  118  In  —  p  —  qIIco  Clearly the NO method has large errors when the solution is nonsteady. Focus on the L 2 -data at t  =  u—pH—qh  .3 and consider  7h  =  -  h—’ 160 0.0000372 0.0045474 0.0000396  j  and  ‘u—pH—qh  7H  =  -  from both the  NO and MSL data.  —  SM NGM MSL  80  —  0.0001489 0.0172531 0.0001776  —  1 320  1 — 640  0.0000093 0.0011553 0.0000095  Table 3.16: L -errors at t 2  j_ “ —  NGM MSL  1  80  1 j_ — 160  115.8 1.2  1 160  30.5 0.27 Table 3.18:  As h  —*  0, for the MSL ‘y  other hand for the NGM times larger than that of  —*  1  — 320  122.9 1.1  Table 3.17:  NGM MSL  1—  =  =  1 j_ — 320  1 j_ — 640  31.1 0.26  31.2 0.25  7H  ,_  —  values at t  1 1280  121.0 1.0  126.1 1.0  values at t  1280  0.0000006 0.0000726 0.0000006  .3  1 j_ — 640  124.2 1.0 Yh  —  0.0000023 0.0002901 0.0000023  .3  j_ —  1 1280  31.6 0.25 =  .3  1. This is what we expect in virtue of Theorem 3.1. On the 121. The error from the NC approximation is over 120  u,!! In fact,  31. This implies the NCM approximation  7jq  is 31 times worse than the coarse approximation  tH!  As we feared, the “correction”  q is badly damaging the calculations! The damage this causes is clearly evident in the wiggles of the NGM approximation in Figure 3.30.  119  1.5  0.5  0  -0.5 0.2  0.4  0.6  0.8  Figure 3.28: standard method, h  =  Figure 3.29: standard method, ii  =  1.5  0.5  0  -0.5  120  1.5  0.5  0  -0.5 0  0.2  0.4  0.6  Figure 3.30: NOM, H  0.2  0.4  =  --  40’  0.8  h  —  --  80  0.6  Figure 3.31: MSL, H  121  40’  0.8  h  —  80  The following are some results for more general cases where Gh is obtained by refining GH two or more times. In particular, we have results for the and the  3-refinement case H  =  s-refinement case  8k. Table 3.19 tabulates the L 2 errors, at time t  H  =  =  .3, in  4k  all the relevant cases. j_1 80  SM NGM  MSL  H H H H H H  = = = = = =  j,_ —  0.0001489 0.0172531 0.0727605  2k 4k 8k 2k 4k 8h  1 160  j, —  0.0000372 0.0045474 0.0148293 0.0749600 0.0000396 0.0001517 0.0014745  0.0001776 0.0013286  1 320  —  0.0000093 0.0011553 0.0055969 0.0218624 0.0000095 0.0000198 0.0001618  Table 3.19: L -errors at t 2  =  .3, H  640  —  0.0000023 0.0002901 0.0014272 0.0058501 0.0000023 0.0000031 0.0000188 =  1280  0.0000006 0.0000726 0.0003587 0.0014946 0.0000006 0.0000006 0.0000023  2’k  • The NGM does not improve if we take more levels. We do a lot more work to obtain worse results. For example, consider the three cases when H  3-refinement  cases (h  0.0058501. When H  =  ,  h  =  h  =  =  ãhM  For the 1, 2,  ),the errors are: 0.0045474, 0.0055969,  the errors are: 0.0011553, 0.0014272, 0.0014946.  =  • For the MSL approximation—if H is “small enough” (see below)  Mu  .  Iu—pH—qhf  in all cases.  • However, the MSL approximation is only good if H is small enough. As we see in Figures 3.29, if k  =  ,  the resulting standard method approximation resolves  the true solution quite well. On the other hand, if k  =  the standard method  approximation does not resolve the solution very well at around time t  =  1.2 (see  Figure 3.28). Consider the MSL approximations pictured in Figures 3.31, where H  =  and h  =  (1-refinement), and 3.32, where H  =  and k  =  (2-  refinements). The approximation in Figure 3.31 does not resolve the steep gradient 122  of the solution when t  =  1.2. The approximation in Figure 3.32 is just not good at  all. In conclusion, though both cases use the MSL with a finest grid with h neither is close to as good as the standard method approximation with h we see in Figure 3.33, where H  =  and h  =  =  =  As  (3-refinements), the situation  does not improve by increasing the refinement of the fine grid. • Summarizing the MSL results: though the MSL approximation is good if H is “small enough”, it is only good if H is so small that the standard approximation calculated on this grid basically fully resolves the problem. With this limitation, it is hard to see a great use for MSL.  123  1.5  0.5  -0.5 0.2  0.4  0.6  Figure 3.32: MSL, H  =  ---  20’  0.8  h  —  80  2  1.5  0.5  0  -0.5  Figure 3.33: MSL, H  124  =  I  20’  h  1 —  160  For interest, we checked to see if the numerical results would reflect the higher order convergence we proved in Theorem 3.1. Tables 3.20, 3.21 and 3.22 confirm is higher order convergence for the difference ü  —  p  —  q. Interestingly, in the one-refinement case our  estimates do not seem to be optimal, though they seem to be in the other cases. We do not have an explanation for this.  H=2h H=4h H=8h  i,_I 80  i__L  .0000474 .0012654  .0000042 (3.5) .0001347 (3.2) .0014601  —  160  i—L —  i—._L  320  h—-i80 .0045459 .1217086  h—---160 .0005632 (3.0) .0300208 (2.0) .1375035 —  —  320  .0000702 (3.0) .0074605 (2.0) .0342822 (2.0)  —  80  H=2h H=4h H=8h  .0002144 .0074887  —  160  .0000171 (3.6) .0008203 (3.2) .0078402  h—_L 320 .0000012 (3.8) .0000923 (3.2) .0008597 (3.2) —  Table 3.22: higher order convergence of  125  p  —  q at t  —  at t  h—1 640 .0000001 (3.6) .0000104 (3.2) .0000955 (3.2) —  IFu  —  p  —  at t  .3  =  —  —  1280 -  h—1 640 .0000088 (3.0) .0018622 (2.0) .0085406 (2.0)  Table 3.21: higher order convergence of  h—1  —  .0000000 ( ) .0000002 (3.2) .0000021 (3.1)  -  Table 3.20: higher order convergence of  H=2h H=4h H=8h  640  —  .0000000 ( ) .0000018 (3.1) .0000180 (3.1)  .0000003 (3.8) .0000152 (3.1) .0001584 (3.2)  h—— 1 — 1280 .0000011 (3.0) .0004653 (2.0) .0021330 (2.0) —  =  .3  h—-1 -1280 .0000000 ( ) .0000012 (3.1) .0000106 (3.2) —  -  =  .3  Test 3  u(x,t)  We considered the exact solution -  and  sintsinirx which is a solution to (3.1) with  =  f (time-dependent) suitably chosen. Notice: 0.0005  0.00045  ++ ++ + ++  +÷ + + +  ++  ++* + +  + *+  0.0004 ++  0.00035  /  0.0003  +  ÷  + +  + + +  + +  +  + +  +  +  0.00025  + + +  +  + S  +  ÷  +7  +  0.0002  ÷“ + +  “. 5\  +  ±  /  ‘  + 5’  -f +  +  ‘S  + +  N  ,+ 5,  *  ‘S  “S  0.00015  + +  / ‘  5” 5,  +  + +  S’S  0.0001  + +  S’  + +  + +  “S 5”  ‘,-“  ÷  +  +  “S  +  5oo:  TIME  +++  u  Figure 3.34:  • The size of • The size  lu  lu  —  p  —  —  —  =  SM SM NGM  llu—uh llu—uHll u—p—qll  MSL ju—p—q  sintsinirx, comparison of L 2 errors  qj from the MSL is the same as  ll  lu  —  thll.  from is correlated to the time derivative of the solution. At  times where the solution has a large time derivative, around the times t and t  =  2ir,  around t  =  lu  —  p  —  and t  ll  =  =  0, t  =  is extremely large. Where the time derivative is very small,  ,  this same term is about equal to  126  lu  —  ZLhII.  3.6  Implementation  Consider first the MSL method (3.5), (3.6). As before, consider one-step theta schemes for temporal discretization. In our calculations, we mainly used the Crank-Nicholson Applying a one-step-theta scheme reduces (3.5), (3.6), to finding  ).  scheme (0  (pfl+l,  qfl+l) (p’, q’)e XH x W, such that (pfl+l  qfl+l,  +  +q q)+ 1 q) + kOv(p  e  kO(p’p’ + p + q”’p’, q)=g’(q!), Vq q 1  (p”’+ q’,)+kOv(q’+p’, ) + kO(p’ p’,)= with p°  =  0 and q° Pu  gfl+l((pfl  =  , 0 u  (),V E W, (3.57) 1  where  )+kO(f)+k(1  q  (3.56)  XH  -  0)[(f  )-v(p+q,  and fl+1()  (pfl  + qfl,  ) + kO(f’, ) + k(1  -0)  ((fn,  )  -  v(p + q,  )  -  (pflp,  i)).  At this point we choose W, (i.e. fix a, 3) for implementational ease. Letting  W  =  {  Xh  (h, H)  + k0v(h,,  H,x) =  0 for all  H  E XH},  equations (3.56), (3.57) are reduced to (pfl+l, q) k0ii(p’, + )+ k0(p’p’ + p’q’ + q’p’,q) (qfl+l, )+k8v(q’, with p°  = iu , 0  q°  = u 0  =  (q5) V e XH 1 g  ,) 1 ) + k8(pp  =  h’() V  (3.58) W,  (3.59)  and g’ and h” 1 as before. This reduced form (3.58), (3.59)  has several advantages. One is the possibility of using a fixed point defect correction 127  method to solve (3.58) and most importantly, we can solve for (3.59) since its solution can be obtained as the difference of a solution to a generalized Stokes problem and a solution to a certain projection problem. The solution is obtained by letting q where  E Xh, t7 E XH.  (h)  —  ’,bh), Vbh E Xh, k8p p 1  (3.60)  is obtained by solving  (, H) + kv0(, H) Lemma 3.16 Letv then q  —  is the solution to  (bh) + k6v(q,h,) while  =  =  =  V  (3.61)  E XH.  H  is the solution to (3.60) and the solution to (3.61)  where  =  H) (, H) + kv8(, 6  v is the unique solution to (3.59). W, satisfies  Proof: The solution to (3.59) is unique for if w (w, ç) + k0v(wr,  =  0 Vq  e  j473  then jwj2  0. From (3.61) it is clear v  and w  (v,h) + k0(v,h) for all  c3  h  E Wh  .  Hence, q  =  =  + kv0jwxW 2 =0  e 147,. Using this and (3.60) implies  (,h)  + k0v(,h,)  =  h)  —  k0(p’p’,h),  v.  Consider the NGM (3.3), (3.4). Recall that in the Marion/Temarn scheme one uses the induced basis and Xjj and Wh are orthogonal in the Dirichiet inner product 0, 3  =  1). Hence  (H,x, h,x) =  0 for all qjq E XH and all  theta scheme, reduces (3.3), (3.4), to finding  (pTl+l, qTh+l)  128  H  e  Wh.  (p’  (  =  Applying a one step  q’)e  11 x Wh, such X  that (pfl+l,g)  + kOv(p’,q)+  1 + p’’q’+ q kO(p’’p p’, ç)=g’(ç’) Vq 4 v(qfl+1,)  with  p°  0 Pu  and  , 0 q° = Qu  +  (pfl+lpfl+l,)=(ffl+l,)  Vq  e  Wh,  e XH  (3.62) (3.63)  where  g’()=(p)+kO(f)+k(l  -  We can use a fixed point defect correction to solve for p in (3.62) and (3.63) is very simple to solve since we have a basis for Wh.  129  Chapter 4  Conclusion  Philosophically, we do not see a turbulence model at the basis of the two-grid finite element methods we studied. Rather we see the NGM and the MSL as methods obtained from the standard finite element by dropping certain small terms. One hopes that in doing so, computational time can be saved. Right from the start, for any of these twogrid methods, it is not clear to us how much computational time  —  if any  —  there is to  save. Nevertheless, we can make some definite conclusions about which of the two-grid methods should be and which should not be considered. Our theoretical results suggest one should only consider • the MSL (with any reasonable high mode space Wh), • the NOM when the high mode space Wh and the low mode space XH are orthogonal in the L -inner product. 2 The NGM with any other combination of spaces XH and Wh should not be considered. We have both theoretical evidence and strong numerical results indicating that approxi mations calculated using these schemes will not be very accurate and hence the schemes themselves will not be efficient. What about the methods which look good theoretically? Are any of them practical? This question has not been definitively answered in this thesis but we do have some ideas. Firstly, consider the special NGM. This scheme may not be practical because we 130  suspect one cannot solve the high mode equation efficiently. One reason is the lack of a “computational good” basis for the high mode space. Of course, this suspicion can only be proved (or disproved) by actually testing the method. This leaves one to consider the MSL. The requirement of an “efficiently solvable” high mode equation leads one, perhaps, to consider a hierarchical high mode space. By a hierarchical space we mean a space spanned by some type of hierarchical basis (for examples see some described in [20]). Such a space has the advantage of a “computationally good” set of basis functions. This combination, we feel, has the best chance of being efficient and it may be of interest to test it. However, even for this case we have some doubts. We have some numerical evidence indicating that the asymptotic range where the approximations are “good” is not practical.  Our numerical results in Section 2.4 and our results for the MSL for  Burgers’ equation seemed to indicate the solution is good only if the coarse grid is itself very refined.  131  Appendix A  Proof of Theorem 2.2  In the following, 0 < t 0 < T < oo are fixed. C represents a generic positive constant independent of h, u and to while  represents a generic positive constant independent  of h but possibly depending on u, v, to and T. (Both C and  may depend on Q and  some parameters, uniform in h, associated with the family of grids). GH is a coarse grid and Gh is a fine grid with associated finite element spaces XH and Xh respectively. We assume XH C X, C H and 0 < H < h < 1. P defined projections. =  1, j3  =  Let eh  ,  3  P’ and  =  Q  are the previously  =  0 are arbitrary (not both zero), though for the special case  0 we have proved Corollary 2.2. ãh  =  —  (PH  +  q)  and  Ch  =  lrh  —  7rh  where  Üh, 7 Th  is the solution to the  standard finite element method (2.5) calculated on the fine grid and PH, solution to the NGM (2.8), (2.9), (2.10). Let defined projection operators and let j5 lh T  =  Qeh  =  qh  —  =  P  Pa,  =  P =  and QH  QU,  eN  =  Qf  =  Fe  ri is the  be the previously =  PH  —  PH  and  qh.  Theorem 2.2 (Error estimate for the NGIY1 There exists a constant ueW  for  qh,  +  Hejj  such that  + HjVeW + uHHCj  all t > to.  The construction of the proof is very similar to the construction of the proof of Theorem 2.1 (see Subsection 2.5.3). What is new is the presence of some terms involving the time derivative. These terms are of the form (ps, 132  (j, ‘) where ‘ç& E Xh. In all estimates  (Pt, ‘)  (or some version of it), is the leading order term. By leading order, we mean  that for some fixed test function & (chosen at the time) the absolute value of (j5i, Qb) is majorized with the lowest order H. e, (et, ) + a(e, ) + n(p, p, b) + n(, —  ,  )  n(,ob)  —  -  —  n(p, p, ) + n(, ,  n(p, q, ) + b((, )  n(p,,  )  b(qh,e)  with  e(.,to) =  satisfy:  Pt,  n(p,b),  —  =  =  0,  Vqh  b)  —  n(q, p,  P)  +  (A.i)  V E Xh,  e  0.  Lemma A.1 There exists a constant C such that 2 ieii  for all 0  <to  V  ds 2 jiVell  (A.2)  , 2 <H  <t < T.  Proof: Letting ‘zb id  +  = e in  llell 2 + II Veil 2  (A.i) we obtain  (Pt,  —n(q, p, ) + n(,  Qe)  ,  )  + —  (, Pc)  —  n(p, q, )]  [n(, —  p,  e)  (n(, i,  —  e)  n(p, p,  e)  + n(p, ,  + n(, p, ) ij)  + n(, p, ij)).  Estimating the right hand side as follows: 2 + eli Veil H , 2  [n(p, p, e) (n(, , e) +  -  .  n(p, q, )]  + n(, p,  ))  <  4 + eli Veil H , 2  <  iieil2  <  4 + GH  +  , 2 il Veil  eli Veil . 2  Choosing e appropriately and integrating from to to t proves the desired result.  133  D  Lemma A.2 There exists a constant  such that  vI Veil 2+ for all 0  <to  f  ejds <H , 2  (A.3)  <t < T.  Proof: Letting ç’  =  et in (A.1) implies  vd  lI Veil 2 + lletU 2  =  (Pt, Qet) +  —n(q, p, ) + n(p,  ,)  —  (, Pet)  [n(p, P, e)  —  n(p, q,  —  (n(i,  —  , e)  n(p, p, et) + n(, p, ) + n(,  , m)  + n(, p, i)).  Estimating: (Pt, et)  =  (Pt, e)  Pee)  <  4+ CH  [n(p, p, et)  (pa, e)  2 (pj, e) + H  + VelI , 2  , 2 eWe  2 + eWe C.€llVell , t  —  (n(, , et) +  —  d  )  {n(, ,  e) + n(p, , ij)  —  —  + n(, p, ij)} + CH 4 + CWVeW . 2  Combining, choosing e appropriately, vBVeW2+iletW2  2 Vejl +H  —  {(Pt, e) + n(, , e) + n(p, , ij) + n(, p,  Integrating,  2+j IIVeW t lletlIds  2+ H  (Pt, e)j +  ln(, , e) + n(p, ,  )I +  n(, P, .  Now,  I(Pt,e)l which proves the lemma.  +  +  ln(,p,)l  2+ <H  lI Veil,  D  Lemma A.3 There exists a constant C such that  alIetII2 + v  ds 2 f allVetll  for all 0 < to < t < T.  134  (A.4)  Proof: Taking the time derivative of (A.1) and choosing as test function b  =  et in the  resulting equation, we obtain: WetII2+yIIVetII2 —  {n(Pt, P, et)  +fl(qt,  —  j5, )  (fl(t,  (Pt, et) +  —  —  pet)  n(pt, p, et) + n(p, Pt, et)  n(qt, p,  + ri(Pt, i, )  j, et) + n(Pt, , j) +  fl(t,  —  —  n(p, Pt, et) + n(j, Pt,  n(pt, q, ) + n(p,  t,  )  —  —  n(q, Pt,  n(p, q,  p, ij) + n(i, t, et) + n(p, t, ij) + n(, 15t, 1))  Estimating:  Cpj + (Ptt, et) < 2 (jj,Pet) [n(Pt, ji, e)  -  <  2 eIIvetII  GH + eWVetl{ 2  2 ejVe  (n(t,,et) +)  +  + Iet2 +  ) 2 jVej  . 2 efjVe  Combining and choosing the appropriate e implies IIetU2 + vJ!VetB 2 <etI2 + 2, Therefore, 2 + vaWVetM 2 <aetI2 + WetII 2 + aH  and integrating, observing that the 1irninfuletW 2  =  0, proves the result.  0  Lemma A.4 There exists a constant C such that  for all 0 <t 0  < t <  ds 2 IICII  <  (A.5)  2 WlI  <  (A.6)  T. 135  Proof: By assumption, Xh and Lh satisfy the inf-sup condition and  WCHIIVII  C sup i1, EXh  Consider the error equation (A.1): —b(C, )  =  (et, ) + {a(e, ) + n(, , ) n(p, , b)  +(, b)  +  —  —  n(p, p,  ) + n(,  n(p, q, P) + n(, , ) + n(,  ,  ,  )  —  n(q, p,  b) + n(, ,  (, ).  Estimating:  (et, &)  CIIeII  [a(e,b)+...n(,p,b)]  <  (p,b)  <  (i,P)  <  WV. 2 H  Thus,  1CM  (MetM + H)  and estimates (A.5), (A.6) are easily proved using our previous results. D Let (s, *) E  Jh  x Lh be the solution to  (Psi, P) + a(s, ) + b(, ) -n(p, q, Pb) for all ib  Xh,  ê=a—s, Similarly, let  with (x,to)  C=—*,  =  -  =  n(q, p, )  -n(, Vp, -  n(p, p,  Pã(x,to) and (x,O) =Pê  and  =  P)  ) + (f, ),  ã(x,to). Let  ñ=Qê.  (A.7)  ë=p+q—s, Notice e  ê  =  —  è and  C=ir—*, =  . —  and  e=Fe ê,  ij=Qé.  satisfy:  ) + a(ê, ) + b(, ) -(Pt, ) (t, P) [n(p, P, ) —n(p, p, b) + n(, p, P) n(q, p, P) + n(p, )  (Pe,  =  -  -  —  —n(p, q, with ê(to,.)  =  —  =  )  n(p, ,  —  )  —  n(, ,  Q)+,  (A.8)  V E Xh,  0. An equation for é, ç is:  ) with e(to,.)  n(, ,  ,  + a(ê, ) + b(, )  =  n(p, p,  -  n(, p, P)  (A.9)  0.  Lemma A.5 There exists a constant ?J such that  2 feW  + vf  2 + vWVeW  ds 2 fvêfl  f L  ds 2 jfê  (A.10) (A.11)  <  ds 2 jf  (A.12) <  4 JJ  (A.13)  for all 0 < to < t < T. The proof is very similar to those of Lemmas A.1, A.2 and A.4. Lemma A.6 There exists a constant C such that  f for all 0 <t 0 < t  , 4 H  T.  137  (A.14)  Proof: Consider the solution of the backwards Stokes’ equations zt +  ‘  Az  —  V/3  =  ê, V•z  0, z(t)  =  =  0,  zç  (A.15)  0.  =  Multiplying by ê and integrating over the domain,  2 IêI Choosing b  = (Zt,  =  e)  —  a(z, ê) + b(,  e)  (z,  =  e) + b(, ê)  —  [(t,  z) + a(ê, z)].  C Xh as test function in (A.8),  PHZ E XH  (Pe, PHz) + a(ê, PHz) + b(, PHz) + (, z) + n(, j5, PHz) —n(q, p, PHz)  since (PHz)  III2  =  =  + n(p, , PHz)  —  n(p, q, PHz)  + n(,  , PHz)  =  0,  0. Hence,  (z, ê) + b(, ê) —n(q, p, PHz)  —  [(t,  z  PHz)  —  + n(, j, PHz)  —  —  (, Pz) + a(ê, z  —  PHz)  n(p, q, PHz) + n(, j, PHZ)  —  Estimating:  Pz)  CH + zj  b(3,ê)  (êt,z  —  +2 WVêW CH ,  êtH 4 CH , eUzI + 2  PHz)  z+2 IjVêjI CH ,  a(ê,z—PHz)  iIFFIzW + 2 UVeW H  n(13, j, PHZ)—n(p, q, PHz) n(,  , Pffz)—n(q, p, PHz) n(,  , PHz)  b(, PHz)  , 6 €PHzj+CH <  €Uz +  I(I H . 2  Combining we obtain  2 WeI  (z,e)  (H + êjj +H 2 2+ 138  j2)  +  €jzh12  II2  + n(, p, PHz) b(, PHz)]  Integrating from t 0 to t, observing that ê(.,to)  =  z(.,t)  =  0, and applying Lemmas A.2,  A.5 we obtain  j  ds 2 ê  zds.  +  Now applying the backwards Stokes’ estimate (2.82) and choosing e appropriately,  <j This proves the desired result.  ds. 2 ê  D  Lemma A.7 There exists a constant  such that , 2 H  (A.16)  for all 0 <Io < t < T. Proof: Letting b  =  è in (A.9),  ItI2 + vVeI 2 =  =  n(p,p,)  n(p,)  -  n(, p, ) + n(E, p, )  -  n(p, ,  E).  Now, p, ) + n(, p, )  -  n(p  )  <  72  +  and as a consequence, < + vWVéW 2  Integrating, observing that e(., t ) 0 the fact that  IIQéW  HjVèW  =  0 proves that  +  ILe  . This, combined with 2 H  2 proves the desired result. H  At this point, it is clear we have proved  2 <H[ Bell 139  To prove the full estimate for  jeW  with one more half power of H, we copy the proofs of  Lemmas 2.13, 2.14, 2.15 while keeping track of the two new terms. The term (pt, Q’b) is, as it was in all the estimates above, the leading order term. As a consequence we prove estimates similar to those in Lemmas 2.13, 2.14, 2.15, however with a lower order of H (lower order by 2) on the right hand side. These estimates finish the proof of the Theorem.  D  140  Appendix B  Proof of weighted error estimates for the Navier-Stokes equations  In the following, 0 <T <cc is fixed. C is a generic positive constant independent of h, u while on u,  represents a generic positive constant independent of h but possibly depending ii  and T. (Both C and  may depend on  and some parameters, uniform in h,  associated with the family of grids). Let u, 0 be the solution to the navier-Stokes equations (2.1) and let solution to (2.5), the standard finite element method. Let u(t)  =  Üh, 7rh  be the  min(t, 1).  Theorem B.1 (Weighted error estimate for the Navier-Stokes equations) Suppose u and O satisfy some (standard) regularity properties for all 0  < t  $ T < cc. Suppose the  external force f, and its time derivatives, are as smooth as needed and suppose the finite element spaces  Xh  and Lh satisfy some standard stability and approximability properties.  Under these conditions, there exists a constant,  —  2 a  +h BVut 2  ,  —  such that  ) 2 VütW  for all 0 <t < T. Remark B.1 We are not sure if the power on the weight factor a is optimal. Remark B.2 If we assume global existence of the solution u, 0, some version of the es  timate holds (with some exponential weight factors) for T this situation see [15].  141  =  cc. For an ideas concerning  Remark B.3 (The infinite sequence of estimates) It is clear that the ideas presented generalize and we can prove an infinite sequence of estimates. For n  =  l,2,•.., there  exists a constant C such that  o(IIDu  —  2 + hWVDu DuW  —  ) 2 VD’uII  forallOtT,wherec(n)—*ccasn—-oc.  The main building blocks of the proof can be found in Heywood and Rannacher [15]. In their paper, they allow the possibility of nonconforming Xh. We will prove Theorem B.1 only for the conforming case. The equivalent result is also true for the nonconforming case but the proof is technically more difficult. To start, we require a priori estimates for solutions to the Navier-Stokes equations. Again for details, we refer the reader to [15] (pages 285-7). Lemma B.1 (A priori estimates for the Navier-Stokes equations) Under standard as sumptions, u and 0 satisfy, for n  1,  IIDuW + Db0HH1/n}  sup a 22 {D’u +  ,  (B.1)  f u 2_2Dfluf2ds <oc, sup t  (11.2)  O<t<T  <  0  sup o ’Du < oc, 2  (B.3)  0<t<T  sup 0<t<T  f 0  D’ul + jD0MH1,R} ds <oc.  {u2hIDuJj +  (BA)  Let v, 0 be the solution to the Stokes equations: Av—V0=g, and let  Vh  E  Jh  V•u=O,  =O, 3 v  (B.5)  be the solution to a(vh,  4’h)  =  (g,  &h),  14  V/’h E  Jh.  (B.6)  Lemma B.2 (Steady Stokes error estimate) Let v, 0 be the solution to (B.5) and let vh  be the solution to (B.6). Then,  Iv  -  + hIlVv  VhW  -  gj. 2 Ch  VvhU  (B.7)  This is proved (for a more general case) on page 295 in [15]. Let ShU E  Jh  be the solution to a(Shu,’bh)  =  (f  —  uj  Corollary B.1 There exists a constant  f  2  (IIShUt  —  (IISiut (IISutt 4 a  —  (IlShutt  utjj —  —  V’cbh E  u.Vu,,bh),  Jh.  (B.8)  such that,  ,  + hW VShu  Ut!!  uttll  —  + hW ‘hUt  + hI! VShutt  —  ut + hi! VShut  ds  , 2 h  VutW)  , 2 i; h  VUtU)  —  —  VuttIl) ds —  VujtII)  <  for all 0 <<T. Proofr One can show that:  IISh’’t llShutt  —  —  Ut!!  + hi! VShut  UttII +  hI! VShutt  —  —  VutlI  <  Vuttli  lift 2 Ch  IIftt 2 Ch  —  —  Du Du  —  —  Dt(ut•Vut)II D(u.Vu)II,  for all 0 < T. Each estimate is easily proved by considering the equations satisfied by DShu  and mu and applying Lemma B.1. From this it is clear that to prove the first  two estimates we need to bound need to bound  Ilftt  —  Du  —  lift  —  Du  —  Dtu.Vull and for the last two estimates we  Du.VulI. Consider the first two estimates. By assumption,  f is smooth, and its integral is bounded. By Lemma B.1,  a2(s)IID2uII2ds t j 143  <00.  11Th {u(s).Vu(s)}  Similarly, by first estimating  and then using the appropriate a priori  estimates from Lemma B.1,  The second estimate follows because the extra power of a implies terms whose integrals were previously bounded are now themselves bounded for 0 estimates are proved similarly. Let v  Let e  =  Vh  U  estimating e  D  be the solution to  —  (f  (Vht, h)  + a(vh,  h) =  = U  vh and  ij = Vh  Üh,  —  T. The last two  t  —  u•Vu, h)  —  Üh.  —  V?,bh E  Clearly, e  +  =  Our approach to  j.  and then estimate  is to first estimate  actually its time derivative  (B.9)  Jh.  In [15] is is proved that:  Bell  lllI  +  +  llll  +  h(IlVejl + llVli  Lemma B.3 There exists a constant  s.t.  (,b) + (ve,v,b) Choosing ‘ci’ Ut  lVll)  ds 2 j a(s)lltl{  . 2 h  , for all 0 2 h  t  T.  satisfies:  Proof:  of  +  onto  Jh,  —  u -4-  PhUt  =  (O,V.), Vb e  (B.10)  Jh.  as test function in (B.10), where  PhUt  is the L -projection 2  we obtain:  IItIl2 + llVll2  =  1 (e,u  —  Phut) + (V,V(uj  —  Phut))  + (U,V•Ph).  Estimating the right-hand side: (O,V•Ph)  (, ut (,V(u  —  —  P/Lug)  Put))  =  (8,V•Ph)  <  (O,V.Ph) +  =  ljVulj + Ch 2  llutll 2 Ch 144  —  (8t,V•Ph)  llVOtll + llell Ch 2 2  , 2 llll  . 2 + Ch  Thus,  f2  ieu + II 2  (1 + jUt + 2 (O, V•P) + Ch  ), 2 IIV&tW  and multiplying by u,  uIIII2 +  uIlVW2 < V (1 + lut + IVOtU 2 2 + u(O, V.Ph) + Cuh ). 2  We have used that  . 2 a(&,VPh) + (&,VPh < u(6,VPhe) + Ch  u(O,V•Ph)  Integrating, from 0 to t, and estimating, implies Ch + u(O, V•Phe)  2 W I t l + uVl  . 2 Ch  Differentiating (B.10) with respect to time, we obtain + a(.,L’) Lemma B.4 There exists a constant l+ 2 J  =  (O,V•’), Vb E J.  such that  j  a2  ds 2 I Vjl  for all 0 <t <T. Proof: Choosing as test function, ,b  IIell2 + IvJl 2  =  —  =  —  Ut  +  PhUt E  Jh  Phut) + (V,V(ut  —  in (B.11), Phut)) + (Ot,V.Fht).  One can show: Phuj)  =  ld -—llUt  Phut))  <  llutlj + ellVll 2 Ch , 2  t,VPht) 8 (  =  (O —ihOt,VPht)  (,ut V(u  —  —  —  Phutlj  145  2  , 2 UV8 + eVj Ch 2  (B.11)  and, as a consequence,  + VI 2 <Ch (Iut + I{VOjW 2 )+ 2  -  . 2 PujW  Multiplying by a , 2 2Ij  22 l IIve  +a2IlUt  —  (IIutU + IlVOtIl a 2 Ch ) 2  112  2 Put jj +  + aut  —  . 2 Pu  Upon integrating, from 0 to t, 22 + a  VI a j2  IIUt 2 <U  Lemma B.5 There exists a constant  —  2 <Ch . 2 2 + Ch PutW  0  such that  J2W2ds t j  for all 0 <t <T. Proof: Let z, 3 be the solution to the backwards Stokes equations: zt+vAz—V/3=ae,V•z=0,z(t)=0,zlao=0. Multiplying by  and integrating over the domain, implies ajIlI2  =  (z, ) + b(/9, )  z) +  -  z)].  Consider some of the right-hand side terms. =  Let  PhZ  ullVll Ch . 2+2 eu’IIV/911  b(/9,Phe)  be the L -projection of z onto 2  (, z)  + a(t, z)  =  (, z  —  Phz)  Jh.  z  —  —  Phz)  —  b(O, Phz) <  z  —  llVell + Cf h +Ch 2 a(llutIl + lVOtjl 4 ) + ea’(IIV/911 2 2 + lzIl). 14  Phz)  Combining,  2 W t II  < (z, ) + 2 IVW + C€h Ch a(Wutj + WV0t 4 ) + eu’(WV 2 2 + zj).  Multiplying by a, integrating from 0 to  and estimating proves the desired result.  t,  D  Lemma B.6 There exists a constant C such that  WVU h ) (M + 2 3 2 W t  <  for all 0 <t <T. Proof: Differentiate equations (B.8) and (B.9) with respect to time and take the differ ence. The resulting equation is (VShut  —  VVht,  b)  (Dvh  —  Du, sb).  As a consequence, (DShu Choosing  i/’ =  ShUt  ShUt  —  —  —  2 VhtII  Vht  Vht  Integrating, from 0 to IlShut 3 a  Dvh, b) + (VShut  —  112  Vv, ‘/‘)  —  =  (DShu  Du, sb).  —  as test function and multiplying by o , one can show 3 llDShu 3 a  —  DuIlllShut  —  VhtIl  +  llShUt 2 Cu  —  Vht  112.  t,  llDShu 4 ja WDShu 3 ta j  —  —  j (Shu 2 a ds + j 3 2 DuII ds + 2 DulI  —  Vhds  —  ull2  Thus by previous estimates and Corollary B.1, IIll <a 3 u 2 llShUt 3  —  2 Vhtll  +a llShUt 3  147  —  u < Ch . 4  +  1 5 11e ) 2 ds. 1  The estimate for  IIVjI 3 o 2  IVII 3 a 2  IIVShut 3 u  —  follows since,  VuI + J WVShUt 3  —VhtW  . 2 h  satisfies:  i  .  2 + Cu Ch 1 WShUt 3h  VvhtI  D  This finishes the proof of the lemma. We now require estimates for  —  ‘,l’) + (V7j, Vb)  =  (u.Vu  —  ã.Vã,’b),  Vb E  (B.12)  Jh.  Lemma B.7 There exists a constant C such that  j  , 2 IWd <h  for all 0 <t <T. Proof: Choosing ,b  as test function in (B.12) and estimating,  =  2 + lvlI2 WW  (u.Vu  =  ü.Vü,7j)  —  =  (—eVe + e•Vu + u.Ve,?j)  II’iW2+Ch2.  Integrating proves the desired result.  C  Lemma B.8 There exists a constant  such that  j  , for allO <t < 2 2 V,j a ds <h 2  Proof: differentiating (B.12) with respect to time and choosing b 2 + VijJ  =  (urVu  —  üt.Vü,7)) + (u.Vut  as test function,  =  —  Now,  (u•Vu (ut.Vu  —  —  ã•Vã, ‘q)  =  üVü, ‘ii)  =  +  + Vu  (‘iV’i+ ‘icV+ iVu  —  —  148  + uVi •V’i + u•V’i  —  —  + •Vu + uV, ‘i)  •V+  + u•V, ‘ii).  We must estimate each of these terms (we will not show all of the details). (u.Vut  —  ü.Vü,ij) + (ut.Vu  —  + lIvjI 2+  ut.Vu,)  , 2 CmI  where A  =  IIVII + Ch 2+2 G1  IIVutII Ch . 2  Hence, g2j2  V 2 +a  aA() + Gj , 2  From our previous results and the estimates of Lemma B.1, j a A(s)ds 2 integrating the differential inequality proves the desired estimate. Lemma B.9 There exists a constant Proof:  such that  j  IIlI a d 2 s  , and 2 h  D  , for all 0 <t 4 <h  T.  Let z, /3 be the solution to the backwards Stokes equations: z+vzz—V/3=a,, V•z=0, z(t)=0,  Multiplying by  i,  ç3 ZI  =0.  integrating over the domain and then multiplying by a, we obtain  IW 2  =  a(z, j) + a b(/3, ,)  —  a  [(c, z) +  z)j.  Now, d a(z, i)  <  d 2 a(z, ij) + 2 , 4 eHzf + Ch Vij a Ch . eWV/9 + 2  ab(/3,7j)  Let PhZ be the L -projection of z onto Jh, then 2 (q,z) + a(ij,z) +(uVu  —  =  (ii,z  —  ü.Vüt,Phz)  149  Phz)  + a(jt,z  + (u.Vu  —  —  Phz)  aj•Vu,Phz),  (B.13)  and z  Phz)  =  0 (since ‘q  VPhz)  <  ejzj + 2 HVi,lj u Ch ,  u•Vü, Phz)  <  B+2 2 IzII + a IIij ea ,  ãVü,Phz)  <  ez + u B+2 2 ’qjj ea ,  o(Vi1, Vz u(uVut a(ut•Vu  —  —  —  —  Jh),  where B  =  2+2 Cj fjV + 3 Ch  WutIj. 4 Ch  (B.14)  Thus,  WmII 2  B(t), 2 I VtIl a 2 2+u ) + ez + ch  and integrating, from 0 to t, proves the desired result. Lemma B.1O There exists a constant C such that  D  2 j 3 O  , for all 0 4 Oh  Proof: differentiating (B.12) with respect to time and choosing ‘çb  =  =  (u.Vu  —  ã•Vü,i) + (u.Vu  —  Multiplying by o and estimating: a3jW2  <Ca + a j B, 3 +2 jjVW 2 3 a  where B is defined in (B.14). Integrating proves the desired estimate. Finally, , 2 h  +  and the estimate for jVetW follows since  IIViiJI  Ch’WViijI.  150  0  T.  as test function,  we obtain: + Vm 2  t  D  Appendix C  Error estimates for Burgers’ equation  In the following C is a generic positive constant independent of h and u while  is a  generic positive constant independent of h but possibly depending on u and v. (Both may depend on some parameters, uniform in h, associated with the family of grids). Let u be the solution to Burgers’ equations (3.1) and let  Üh  be the solution to (3.2),  the standard finite element method. We assume: (al) u E C°°((O,oo) x  ) fl  ([O,) x ’ 2 xW  (a2) f, f, f,... E L°°((O, oo) x 1) fl L ((O, oo) x 2 (a3) the family of grids satisfies a uniform size condition. There exists constants k , k 1 2 independent of h such that k h 1  IThI  <  kh for all Th  e  Gh and all h.  Theorem C.1 (Error estimate for Burgers’ equation) Suppose (al), (a2) and (aS) are satisfied, then, there exists a constant  Ilu UhII —  for all t  such that +  hIIu  —  0.  We know of no source for such an estimate. One can also prove a sequence of weighted error estimates. Let a(t)  =  rnin(1,t) be a weight function.  Theorem C.2 (weighted error estimates for Burgers’ equation) Suppose (a]), (a2) and (aS) are satisfied, then there exists a constants o(IIDu  —  Dühfj + hWDu 151  n —  =  such that  , 2 Dl) <h  for all t  0. a(n) > 0, is increasing and a(n)  —*  cc  as n  —*  cc.  This sequence resembles the one for the Navier-Stokes equations in Theorem B.1. We will not prove any of these here. These estimates can be proven using an approach similar to the proof of Theorem B.1. Lemma C.1 (A priori estimates for Burgers’ equations) Suppose (a]), (a2) and (a3)  are satisfied, then there exists costants M 2+ huM  j  =  , f, l) such that 0 M(u  , 0 ds <M 2 huhI  )ds 2 huIh + (IIuhI2 + ButM 2  Mi,  for all t > 0. The general formula for the rest of the estimates, n (BDuxM2 +  IDmuM2)  IhD u u 1 2 xiI +  +  (  f  j  is:  =  , 22 jDugM d 2 s <M  uph + 1 D 2  hD2uH2)  ds < . 23 M  These estimates are similar to those proved in [15] for the Navier-Stokes equations. A proof of this sequence of estimates has been done in [32]. Let w be the solution to: —  Let  wh  Aw  =  g,  =  0,  (C.1)  € Xh be the solution to: (Whr, chx) =  (g, q),  (C.2)  Vqh e Xh.  The following is a standard result. Lemma C.2 (Poisson error estimate) Let w be the solution to (C.]) and let wh be the so  lution to (C.2,). There exists a constant C such that hw for all t > 0. 152  —  whhh  + hwx  —  g, 2 Ch  Let Sh’u E Xh be the solution to a(Shu,bh)  =  (f  —  —  Vi’h E Xh.  uu,bh),  (C.3)  Corollary C.1 Suppose the assumptions above are satisfied, then there exists a constant ,  such that, + lIShu—uW l 2 IShu—uIl h (lIShut 2 a  f  (Wh  2+h utfl 2 WShut +h ilShut 2  2 utW  —  (lIhtt j3 t for all t  —  —  —  + hljShutt  tt  —  ) 2 utW  —  )ds 2 ut uW)ds  0.  Proof: By Lemma C.2, —  uii  + hWShu  Because of the assumptions on  If  —  ut  —  f  ulI  —  f 2 <Ch  —  —  and the a priori estimates for u,  uuf <C + M 2+  Again using Lemma C.2,  IlShut  —  utII  + hIIShut  —  2 lift Ch  ut  —  —  uu  —  Now,  lift  —  Utt  —  UtUx  —  C+  uuII  iIUttII  + CIluii  and the leading order term, the term requiring the highest order of a in the a priori estimates, is  Iluttil.  Thus,  iift 2  —  Utt  —  —  Utt  utu,, —  —  uu  153  2 uutj —  ds 2 uuiI  Hence, we have proved the first two estimates. The others are proved similarly. Let v  e  = vh  Xh the solution to  (vt, bh) + (vs, bh)  Let e  =  u  —  ü,  =  u  —  v and  =  v  i  (f —  uu, bh),  —  E Xh,  Vh  ü. Clearly, e  +  =  Vo  =  PhUQ.  .  Lemma C.3 There exists a constant C such that fuWI2ds <h , 4  Proof:  0. satisfies  (, b) + Choosing Ih  =  —  7-  bh)  =  Vb € Xh.  0,  -projection of u onto Xh, 2 + Pjn, where Phu is the L  III2 +  =  (t,u  -  Wu  Phu) + -  -  Phu)  2 + Fhu  + Ch Iu. 2  Integrating,  2+ WW  ds 2 L IIl  . 2 <h  Let z be the solution to the backwards heat equation: Zt  Multiplying by  + Az  =  , z(t)  =  0,  =  0.  and integrating over the domain,  II2  =  (z,)  -  —  + (,z)j  [(t,Z)  [(,z  (u  —  Phz) +  =  (Phz,  )  <  (Phz,  (h + Ch )+2  +  —  Phu, zt)  154  —  —  (, z  2)  (C.4)  To prove Theorem C.1, i.e.  i.  and then  to estimate e, our plan of attack is to first estimate  for all t  0  Fhz)] —  Phz)  + e( WztW 2+  Integrating from 0 to t, noting that  ds 2 tf III  (Phz, ) ds  j  4 + <h  0, we obtain  =  ci (Wztj 2  +  zjI)  The backwards heat equation estimate (3.30) implies the result.  D  Lemma C.4 There exists a constant C such that + hds for all t  0.  Proof: Subtracting (C.4) from (C.3) we obtain (Shu  —  v,  (vt, /‘)  =  —  (Ut,  b),  V/’  e  Xh.  Thus, (Shut  Choosing b  =  Shzt  —  —  vt, /‘) + (Shu  —  Vr,  =  (Shut  —  Ut,  k) VL’  € Xh.  v as test function,  —  2+ vU  Shu  —  2 vxW  =  (Shut  —  Ut,  Shu  —  v).  Multiplying by o, gWShu_vU2+aUShux _vI 2  <  u(Shut  —Ut,ShU—V)  2Sut  —  2 + CUShU utU  Before integrating, we would like to point two things out: • a IIShut 2  —  2 utII  and its integral are majorized by h . Thus 4 a2IISu t j  —  155  + Shu —vU 2  . 4 ds <ah 2 utll  —  . 2 vU  • Similarily, IShu  —  2 v  and its integral are bounded and  IShu  —  ds 2 v  . 4 h 1 <at  Upon integrating,  JllShu and we have shown IShu  —  —  vU  ,  ds 2 vll  —  <  uh  . This implies, 2 Ch  lu  IlI The estimate for  2+ v  ShuI +  -  Shu  -  vi  . 2 h  is a simple consequence of the inverse inequality.  lluv  li  + IlShux  —  —  vx  jiShu—v <h. 1 h+Ch Lemma C.5 There exists a constant  I for all t Proof:  i  ‘  such that  , 2 + hliids <h  0. satisfies: (ifl,  b) +  Choosing as test function /‘  (, )  =  (uu  —  ü,  ),  Vb E Xh.  =  ld  2  2  +  =  (uu  —  One can show, (uu  —  üüx, 77)  =  (—i  —  77x  +  U?lx  156  +  ?lUx  —  +u+  ),  and (,ii)  =  +  (—ii  zi  ( ) (u  0, +  —  2+h <j , 6  )  +  2+ WII  ru  +  Thus,  IIII2 +  2  +h 4  and integrating from 0 to t  2+j wI  . 4 ds <h  By the inverse inequality,  III  0’IIII  Ch.  Finally, we have proved Theorem C.1 since  hell  llhI  +  llll  2 and <h  157  llehl  l1ll  +  llll  h.  Bibliography  [1] Ait Ou Ammi A. and Marion, M. nonlinear Galerkin methods and mixed finite elements: two-grid algorithms for the Navier-Stokes equations. preprint, 1992. [2] Ait Ou Ammi. 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