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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-STOKESEQUATIONS USING FINITE ELEMENTSByOwen WalshB.Sc.(Mathernatics) University of Victoria, 1986M.Sc.(IVlathernatics) University of British Columbia, 1988A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESi)EPARTMENT OF MATHEMATICSWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBiAApril 1994© Owen Walsh, 1994In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)______________________Department of mcthc.’The University of British ColumbiaVancouver, CanadaDate_________DE.6 (2/88)abstractThe nonlinearity in the Navier-Stokes equations couples the large and small scales ofmotion in turbulent flow. The nonlinear Galerkin method (NGM) consists of insertinginto 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 theoretical justification has been recently thrown into question. However, its actual performanceas a computational method has remained largely untested. Temam and collaboratorshave reported a 50% speed p in their spectral code for spatially periodic flow but theirexperiments have been recently criticized. In any case, spatially periodic computationsare of little practical use. The aim of this thesis has been to test the NGM in the morepractical context of the finite element method.Using finite elements, there is ambiguity and difficulty because the coarse grid hasno natural supplementary space. We analyze a family of supplementary spaces andit is found that the quality of the asymptotic error estimates depends on the choice.Choosing the space by theL2-projection, we prove that the resulting approximation is“asymptotically good”. These results extend and improve upon recent error estimates ofMarion and collaborators. For any other choice, the estimates are weaker and if -- as wesuspect— they are optimal it seems possible that the NGM may actually decrease theaccuracy of calculations. We also analyzed a variant of the NGM that we call “microscalelinearization” (MSL). We prove that the MSL is “asymptotically good” for any member ofthis family of supplementary spaces. Turning to calculations, choosing the supplementaryspace by the Ritz projection, we implemented the NGM by modifying a 2-D NavierStokes code of Turek; it performed very poorly. We implemented a variant of the MSL.iiIt performed better, but still not as well as the original code. We sought a furtherunderstanding 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 finiteelement method. In fact, unless the coarse mesh is itself very fine, all versions performedpoorly.illabstractTable of ContentsTable of Contents11ivList of FiguresList of Tablesacknowledgementglossary1 IntroductionviVI”ixx18111522222729343448582 The Navier-Stokes equations2.1 The NGM and MSL using finite elements2.2 Results of the error analysis2.3 the Numerical schemes2.3.1 The finite elements spaces2.3.2 Turek’s solver2.3.3 The implementation of the NCM and2.4 Numerical results2.4.1 Flow around a cylinder2.4.2 Diffuser calculations2.5 Proof of Theorem 2.1MSLiv2.5.1 Preliminaries.582.5.2 Organization 662.5.3 Proof of the Theorem 693 Burgers’ equation 863.1 The NGM 873.2 The MSL 893.3 Results of the error analysis of the NGM and MSL 903.4 Proof of Theorem 3.1 953.4.1 Prelirniiaries 953.4.2 Proof of the Theorem 993.5 Numerical results 1103.6 Implementation 1274 Conclusion 130A Proof of Theorem 2.2 132B Proof of weighted error estimates for the Navier-Stokes equations 141C Error estimates for Burgers’ equation 151Bibliography 158VList of Figures2.12.22.32.52.62.72.82.92.102.112.122.132.142.152.162.172.182.192.202.212.22Cylinder,Cylinder,Cylinder,Cylinder,Cylinder,Cylinder,Cylinder,Cylinder,Cylinder,Cylinder,Cylinder,Refining strategiesBasis functions of JhParticle tracing of a flow around a2.4 Flow past a cylinder, grids= 100SM solution, grid level 3SM solution, grid level 4SM solution, grid level 5NOM solution grid levels 3-4NGM solution grid levels 3-4,MSL solution grid levels 3-4NGM grid levels 4-5NGM solution grid levels 4-5,NGM solution grid levels 4-5,MSL solution grid levels 4-5 .“high modes”1126cylinder, Re 3435373839424243444545464748505154545555“high modes”“low modes”MSL solution grid levels 4-5, “high modes”Diffuser, gridsDiffuser, SM at grid level 5Diffuser, SM at grid level 6Diffuser, Test 2, NGMDiffuser, Test 2, NGM, “high modes” . . .Diffuser, Test 2, MSLDiffuser, Test 2, MSL, “high modes” . . .vi• 56• 56• 57572.23 Diffuser, Test 3, NGM2.24 Diffuser, Test 3, NGM, “high modes”2.25 Diffuser, Test 3, MSL2.26 Diffuser, Test 3, MSL, “high modes”3.27 “Moving shock”; A series of snapshots of3.28 “Moving shock”: SM, h =3.29 “Moving shock”: SM, h3.30 “Moving shock”: NGM, H = , h =3.31 “Moving shock”: MSL, H = , h =3.32 “Moving shock”: MSL, H = , ii =the exact solution 1141201201211211241241263.33 “Moving shock”: MSL, H = , Ii =3.34 Test 3: n = sin t sin rx, comparison of L2 errorsviiList of Tables3.1 Testi, u = sin(7rx): errors from SM 1123.2 Testl, u = sin(irx): errors from NGM, p + q 1123.3 Testi, u = sin(7rx): errors from NGM, p . 1123.4 Test 1, u = sin(lO7rx): errors from SM 1133.5 Test 1, u = sin(lO7rx): errors from NGM, p + q 1133.6 Test 1, u = sin(lO7rx): errors from NGM, p 1133.7 “Moving shock”; standard method ju — 1163.8 “Moving shock”: NGM Lu — — q!I, H = 2k 1163.9 “Moving shock”: MSL u—p — q, H = 2k 1163.10 “Moving shock”: SM—ajj 1173.11 “Moving shock”: NGM !u—p— qW, H = 2k . . . 1173.12 “Moving shock”: MSL lLu—p— qW, H = 2k 1173.13 “Moving shock”: SM u — a 1183.14 “Moving shock”: NGM Lu — — qI! 1183.15 “Moving shock”: IVISL Lu — p — q 1183.16 “Moving shock”; L2-errors at t = .3 1193.17 “Moving shock”: 7h values at t .3 1193.18 “Moving shock”:-rn values at t = .3 1193.19 “Moving shock”: L2-errors at t = .3, H = 2h 1223.20 Higher order convergence of Lu — p — q 1253.21 Higher order convergence of !Lu — Px — qj 1253.22 Higher order convergence of a—— qjI 125viiiacknowledgementI would like to thank my thesis supervisor Dr. John Heywood. I thank him for introducingto this fascinating area of study, and I am grateful that he so freely shared his ideas andafforded me the possibility of undertaking this work. I am indebted to Dr. R. Rannacherand Dr. S. Turek from the university of Heidelberg. Without their help and the time Ispent in Heidelberg this thesis would not have been possible. A special thanks to Stefanand Monica for their generous hospitality. Finally, I would like to thank my parents andKaren. They were always there to help and encourage me.ixglossaryChapter 2u the fluid velocity, 80 the fluid pressure, 8f an external force, 8ii the kinematic viscosity, 8a domain in R2 or R3, 8u0 the initial velocity field, 8C°°() the space of smooth vector fields in 2C°(2) the space of smooth vector fields with compact support inL2(f) the space of vector fields with finite L2-norm in 12H(12) the completion C°(12) in the Dirichiet-normW2’(12) the completion Cc0(12) in (z<2 IID 2)J(12) the divergence free space, 8A the Stokes’ operator, 8a the it_eigenfunction of the Stokes’ operator, 8(., .) theL2-inner product(V., V.) the Dirichiet inner productV(12) the space spanned by the first n-eigenfunctions, 8W’(12) the complement space of V, 8u the spectral Calerkin approximation, 8pfl, qfl the spectral NCM approximation, 9or the spectral MSL approximation, 9xh, H constants, 1 > H> h> 0Gh a grid, h is some measure on the size of the elementsXh a “velocity approximating” finite element spaceLh a “pressure approximating” finite element spaceJh the space of discretely divergence free finite elements, 10a, the standard finite element approximation, 9n(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 > 0W’ the “high mode” finite element space, 11the “low” mode projection, 12the “high” mode projection, 12PH, qh, 7t the finite element NGM approximation, 13or the finite element MSL approximation, 14r(t) the weight function, r(t) = min(t, 1)to a time (fixed), to > 0u(t) the weight function, a = min(t — to, 1)k the discrete time step0 a constant depending on the time discretization, 28interpolation operator onto Xh, 58jh interpolation operator, onto Lh, 58Ph L2-projection onto Xh, 58Rh H1-projection onto Xh, 58rh projection operator onto J1, 59xiAh the discrete Laplacian, 63II Ih theL3-norm116 theL6-norrnthe112 theV2’-norrne,C 16,66=Fe,i=Qes,fr 66ê, 67,t 67Chapter 3u solution to Burgers’ equation, 86the initial condition for Burgers’ equation, 86the interval [0, 1]f an external force, 86v the “viscosity” in Burgers’ equation, 86Gh a gridding of 1Xh a finite element spacethe finite element approximation to u, 86c3Wh the high mode space, 87F’ the “low” mode projection, 87the “high” mode projection, 87147h the space spanned by the induced basis, 88Ph, qh the finite element NCI\’I approximation, 88or the finite element MSL approximation, 89XIIto a time (fixed), to > 0u(t) the weight function, a = rnin(t—to, 1)Ph L2-projection onto Xh, 95Rh H’-projection onto Xh, 95e e=u—ü,99s 101ê=t—s,102=ljeè=p+q—s,104k the discrete time step size0 constant describing the one-step theta schemexliiChapter 1IntroductionA turbulent fluid flow contains a large range of scales of motion. The largest scalesdescribe the main flow itself and the large vortices in it. Smaller scales describe increasingly tiny “turbulent” vortices. An inertial manifold, as conceptualized by Foias, Selland Temam [11], determines the small scale of motion as a function of the large scale ofmotion. In a sense, an inertial manifold is a physical “law” which connects the motionof the small scales to that of the large scales. The velocity and pressure in a viscous incompressible fluid are believed to satisfy a system of partial differential equations calledthe (viscous incompressible) Navier-Stokes equations. In the special case of the two-dimensional, spatially periodic equations, Kwak [18] has recently proved the existence ofan inertial manifold. Otherwise, it is not known if an inertial manifold exists. Despitethis, there has been a great interest in the development of computational methods looselybased on the concept of an inertial manifold.Inertial manifolds have limitations as computational tools. As we have remarked, itis not known if one exists; even if one does there is no guarantee that it would have aclosed form or be of small dimension. To work around these limitations, approximate inertial manifolds along with a related computational method called the nonlinear Galerkinmethod (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 papersaim to give a theoretical justification to approximate inertial manifolds and the nonlinear Galerkin method. In this vein, a recent result in a paper of Dedulvier, Marion and1Titi [8] appeared to demonstrate a strong theoretical basis for the NGM. In the contextof spectral eigenfunctions and Dirichiet boundary conditions, for approximations calculated using the NGM (see equation (2.3)), they proved an order of convergence whichwas higher than that possible for approximations calculated using the standard spectralGalerkin method (see equation (2.2)). This higher order convergence was cited as a theoretical justification for the NGM and ultimately for approximate inertial manifolds. Thishas been thrown into question in a recent paper of Heywood and Rannacher [16]. Theydo not dispute the higher order convergence; it is true. They dispute any connection tothe 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 usingthe NGM. However, their experiments have been recently criticized in a new paper ofGarcia-Archilla and Frutos [7]. Thus, the NGM has generated disagreement over bothits theoretical justification and its practicality as a computational tool. We feel that asa computational tool the NGM has not been truly tested since spatially periodic calculations are of little practical importance. The major goal of this thesis is to study thenonlinear Galerkin method for the Navier-Stokes equations using finite elements.We looked at it from both the theoretical and computational sides, and we find thereis trouble from each side. The trouble, in each, case stems from the fact that the methodforces us to ignore significant terms involving the time derivative. We felt there was nocompelling reason to omit these terms, so we developed another method called MicroscaleLinearization, or MSL. The MSL, like the NGM, separates the “high” and “low” modesof the solution we refer to “modes” rather than “scales of motion” when discussing thesolution — but with MSL, there is no longer any connection to an approximate inertialmanifold.The NGM and MSL require function spaces for the “low” and “high” modes of asolution. In the spectral case, the low mode space is spanned by the first n eigenfunctions2of the Stokes operator while the “high” mode space is spanned by the rest. With finiteelements, the “low” mode space, XH, is spanned by finite element functions associatedwith a coarse grid while the “high” mode space (or supplementary space), isspanned by finite element functions associated with a more refined grid. A difficulty, notpresent when using spectral eigenfunctions, is defining this supplementary space; it is notclear that any one is better than another. As a consequence, we study a class of possiblechoices.Our main theoretical result is Theorem 2.1. Here, we considered the difference between 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 calculated on the fine grid. We prove this difference satisfies higher order estimates than thedifference 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 calculatedon a coarse grid and a correction calculated on a fine grid, are as accurate as thestandard approximation, where all is calculated on the fine grid. This result doesnot depend on the choice ofThe theoretical results for the NGM were mixed. We considered the difference between the NGM approximation, with the main part calculated on a coarse grid anda “correction” calculated on a fine grid, and the standard approximation, where all iscalculated on the fine grid. On the positive side,• for one special case of the NGM, a case very recently studied by Ammi and Marionin [1], we were able to prove higher order estimates, similar to those of Theorem2.1. These are proved as Corollary 2.2. In this case, XH and W’ are orthogonalwith respect to theL2-inner product.3The estimates of Corollary 2.2 improve the results of [1] since we prove higher orderconvergence for the L2-norm of the difference. WithL2-orthogonality, several termsinvolving the time derivative are identically zero. This is the mechanism that leads tothe higher order in these estimates. On the negative side,• If W’ is chosen in another way, using a different orthogonality, we are unable toprove these higher order estimates. The best general results for the NGM is theweaker Theorem 2.2.In the general case, the time derivative terms mentioned above are not zero and they arethe cause of the problems. Theorem 2.2 is not strong enough to imply a result similar toCorollary 2.1. In fact, as we remark in Section 3.3, these estimates are not strong enoughto guarantee that the “correction” calculated in the NGM will not, in fact, damage thesolution.Computationally, we considered the NCM and MSL for the two-dimensional NavierStokes equations. The implementation was difficult. We require a supplemental spacethat results in a “high” mode equation which is solvable and moreover, solvable in anefficient way. We suspect that implementational problems are endemic to the NCM (andMSL) for the Navier-Stokes equations using finite elements. They were present in thecase we considered.Using discretely divergence-free finite elements we implemented both a NGM and aMSL. In our numerical tests (described in Section 2.4), we found that:• Our version of the NGM performed very poorly, much worse than the standardmethod.• Our version of MSL performed better than the NCM. However, compared to thestandard method the results were disappointing and we saw no advantage in usingour version of MSL.4An illustrative test was our calculations of flow around a cylinder at Reynolds’ number100. (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 standardapproximations calculated on the equivalent coarse grid, despite the fact that when usingMSL 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 resultsand 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 classof possible choices for the “high” mode space, 1’V’.Our study of this mirrors that of the Navier-Stokes equations. We considered theNGM; of particular interest was one proposed by Marion and Temam in [20]. In parallelwe considered the MSL. Our theoretical results were similar to those for Navier-Stokesequations. We considered the difference between MSL approximations and the standardapproximation calculated on the associated fine grid. We prove, in Theorem 3.1, thisdifference satisfies higher order estimates than the difference between the exact solutionand this standard approximation. Its consequence, Corollary 3.1 states (paraphrased):• MSL approximations of the solution to Burgers’ equation, with a main part calculated on a coarse grid and a correction calculated on a fine grid, are asymptoticallyas accurate as the standard finite element method in which everything is calculatedc3on the fine grid. This result is true independent of 147hThe same was not true for the NCM.• For a special case of the NOM, we can prove versions of these higher order estimates.In this particular case, XH and T’V,’ are orthogonal with respect to the L2-inner5product.• The best general result for the NGM was the weaker Theorem 3.2. Our numericalresults suggest the weaker estimates in this theorem are optimal.Though we characterize Theorem 3.2 as weak, it does imply a positive order of convergence for NGM approximations. For the particular Marion/Temam NGM, this resultis stronger than the original one proved in [20] in which approximations were shownto converge to the correct solution in a very weak sense; no order of convergence wasproven. However, the estimates of Theorem 3.2 are not strong enough to guarantee thatthe “correction” obtained in the NOM will improve the calculation. In fact our numericalresults 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 versionof the NGM tested was the Marion/Temam one. We tested them in a wide variety ofsituations. (For instance, we tested cases where the fine grid was obtained from severalrefinements 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 ofSection 3.5, theL2-norm of the error associated with the NGM approximation was,for all grid sizes tested, more than 120 times larger than the error associated withthe standard method of approximation calculated on the equivalent fine grid. Thisis more than 30 times larger than the error associated with the standard methodof approximation when calculated on the equivalent coarse grid! In other words,the NGM “correction” damaged the calculations. The results are summarized inTables 3.16, 3.17 and 3.18. The effect of these large errors on the approximationsis visible; NGM approximations have unnatural “wiggles”. Figure 3.30 is a seriesof snapshots, at fixed times, of one such approximation.6• The results for the NGM did not improve if we refined the fine grid while keepingthe same coarse grid (see Table 3.19).• As our theoretical results predicted, the MSL worked well. If the grid size wassmall enough, the MSL approximation, which is calculated on a coarse and a finegrid, is as good as the standard approximation in which everything is calculatedon the fine grid.• However, we found no advantage to using the MSL. The requirement that the gridsize be “small enough” limits its use. We found that if the coarse grid was fineenough so that a standard approximation calculated on this grid was pretty wellable to fully resolve the solution, then the MSL approximation was good. If thecoarse grid was out of this range the MSL approximation was not so good.7Chapter 2The Navier-Stokes equationsThe initial boundary value problem for the Navier-Stokes in a bounded domain C R,n = 2, 3, with a given external force f = f(x, t) and kinematic viscosity v is to find thevelocity, u = u(x,t), and the pressure, 0 = 0(x,t), satisfyingu+u•VuvAu—V0+f, V•u=O (2.1)uj=O, u(x,O)=u0. JLet,J()=e L2() = 0, n = 0 in the weak sense}and let P be theL2-projection onto J. The Stokes operator, A : W2’(!) fl H(f) —J(Q), uniquely maps u to Au = P A u. Let & be the ithleigenfunction of the Stokesoperator. it is well know that there is a countable number of these eigenfunctions densein J and orthogonal in both theL2-inner product, (., .), and the Dirichlet inner product,(V.,V.). LetV = V() = span {a1,.. . ,a’}, and W = W(Q) = span.}Let Pv be theL2-projection onto V. The standard spectral Galerkin approximation ofthe velocity u, is the solution u’ V of the ordinary differential equations(u,) + v(Vu’,V) + (u.Vu,) = (f,y) ,V € V, (2.2)8satisfying u(O,.) = Pvu(O,.). The spectral “nonlinear” Calerkin approximation of thevelocity is the solution (pTh, q”) E V x W of(p,) + v(Vp,V) + (pTh.Vp + pVq +qTh•Vp,) = (f,) ,V E V (23)v(Vq,V) +(pTh.Vp,) = (f,) ,V E W, Jsatisfying p(O,.) Pvu(O,.).Remark 2.1 (Orthogonality) Because the eigenfunctions are orthogonal in both the L2-and Dirichiet inner products: (VpV) = (p’, ) = 0, for all W’ and (Vq’,Vç) =(q, ) = 0, for all e V’. Hence, in (2.3,) the only linear term omitted is (q, ).The spectral MSL approximation of the velocity is the solution (p”, qfl) (Vi’, W) of(p,) + v(Vp,V) + (pTh.Vp +pTh•Vq + = (f,) ,V E VTh, (24)(q, ) + v(Vq, V) + (p.Vp, ) = (f, ) ,V E W, Jsatisfying p(O,.) Pvu(0,) and q(0,.) = Pwu(0,.).Remark 2.2 (Spectral MSL) We have presented a spectral version of the MSL to highlight that no linear terms are omitted; in particular, no terms involving time derivativeshave been dropped.Consider finite element approximations of the Navier-Stokes equations. Supposeis subdivided into a grid Gh with associated finite element spaces: Xi,, a velocity approximating space, and Li,, a pressure approximating space. We assume Xh and Li, are“good” spaces possessing some necessary approximability and stability properties. Letn(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 approximationof (2.1) is the solution a = Üh e Xh, = ‘in, E Lh of(Ut, h) + v a(U, mi,) + n(u, U, ) — b(, mi,) = (f, h), Vi, E Xh, (2 5)b(qh,U) = 0, Vqh ELi, J9satisfying u(O,.) = ao. Let Jh C Xh be the space of discretely divergence-free finiteelements:Jh = {h e Xh b(qh, h) = 0, Vqh E Lh}.An alternative formulation of (2.5) is to seek solutions a = Üh E J,, = lrh e Lh of(at, h) + v a(ã, h) + n(u, a, ) = (f, h) Vh E Jh, (2.6)b(, bh) = (ut, h) + v a(ü, I’h) + n(a, a, ‘) — (f, h) Vh e Xh/Jh. J102.1 The NGM and MSL using finite elementsSuppose we have a domain Q with a given coarse grid, GH. A standard way of producinga finer grid, GH/2, is to systematically refine GH. One refining strategy, used by thesoftware package FEAT (see [3]), is illustrated in Figure 2.1. This process can be repeatedFigure 2.1: refining strategiesto obtain afine grid, Gh. A consequence is the cascade (or hierarchy) of grids GH, GH/2This 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 usingfinite elements. Basically, the low modes are modes which can be approximated byfinite element functions associated with a coarse grid while the high modes are modeswhich can be approximated in some supplementary finite element space associated with amore refined grid. More precisely, let XH be a “standard” velocity approximating spaceassociated with the coarse grid, GH, and let Xh be a “standard” velocity approximatingspace associated with the fine grid Gh. Assume H < h and C Xh C H. Wedefine the low modes as finite element functions in XH. A major difficulty is faced indefining the high modes. Unlike the spectral case, there does not seem to be a naturalsupplementary space to choose. There are other possibilities, but our idea is to considera class of spaces, defined for c, > 0 (not both zero), whereW= { Xh (, ) + a(, ) =0, V E XH}. (2.7)11It is clear that Xh = XH + W’ for any a, /9 0. The a, /9-projection P’ : H —* XHis the operator which maps u H to the unique element P’u E XH which satisfiesa (P’u, H) + /9 a(’u, H) = a (u, H) + /9 a(u, H) VH E XH.Also, let : Xh —4 WZ’ where = I — For notational convenience wesometimes suppress the superscripts a, /9 and refer to Wh, PH, and QH.Remark 2.3 (Special cases of P’) If a = 0, /9 = 1 then PH is the Ritz projection andXH arid V7h are orthogonal with respect to the Dirichlet inner product. If a = 1, /9 = 0then PH is theL2-projection and XH and W1 are orthogonal with respect to theL2-innerproduct.Remark 2.4 (The spaces V’) In the finite element case, different choices of a,/3 leadto different spaces This is a major difference from the spectral case where, aswe know, the spectral eigenfunctions are mutually orthogonal in both the Dirichiet andL2-inner products.12The NGMGiven spaces XH and W’ for the low and high modes of the velocity and Lh for thepressure, we obtain a NGM by mimicking the spectral version (2.3). Let t > to > 0. ANGM 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, +b(r,b),Vb EXH, (2.8)va(p+q, )+n(p, p, b) = (f, b)+b(7r,b), V W’ (2.9)(qh,V(p+q)) = O,VqhE Lh, (2.10)satisfying p(.,to) = Fu(.,t0) Equations (2.8) and (2.9) can be rewritten more compactly as(Pt, b)+va(p+q, &)+n(p, p, &)+n(p, q,b)+n(q, p,b)=(f, b)+b(7r,b), (2.11)for all b e Xh. Alternatively, (2.8), (2.9), (2.10), can be rewritten in terms of thediscretely divergence-free functions:(pt, P)+va(p+q, )+n(p, p, )+n(p, q,)+n(q, p,) = (f, ), VEJh, (2.12)b(7r, ) (pt, &)+va(p+q, )+n(p, p, )+n(p, q,b)+n(q, p,P) — (f, ), (2.13)for all ‘çb e Xh/Jh.13The MSL methodAs mentioned in the introduction, MSL is like the NOM except that all terms involvingtime 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) isthe 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) (f, /)+b(ir,’z/’), V’zb E XFJ, (2.14)(pt+qt, )+va(p+q, )+n(p, p, = (f, )+b(,) V eW’, (2.15)(qh,V(p+q)) = O,Vqhe Lh, (2.16)satisfying po = FHU(,to), q0 = Hu(,to). The velocity equations (2.14), (2.15), maybe 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, &) = (pt+qt, &)+va(p+q, ‘)+n(p, p, b)+n(p, q,b)+n(q, p,Pb)—(f, ), (2.19)for all ‘ E Xh/Jh.142.2 Results of the error analysisIn this section we state the results of our error analysis and consider some of the consequences. 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 theapproximation calculated using the standard finite element method:(Üt,) + va(ü,) + n(ã,ã,) + b(,) = (f,), V E Xh, (25)b(qh,a) = 0, Vqh E Lh, Jwith u(.,0) = uo. Let p = Pfa, and let = Qj3a. We suppose that the spaces Xh, Lhare “nice” approximating spaces satisfying some necessary approxirnability and stabilityproperties. These properties are discussed in more detail in Subsection 2.5.1. We supposethat u, 0 satisfy some standard regularity properties for 0 < t < T < cc. Let sm —sm to highlight that this constant is linked to the “standard method” — be a positiveconstant independent on ii but depending on u, T, t1 and some properties related to thepair Xh, Lh. Let r(t) = min(t, 1). We supposeIlu — all + hllvu — VaR + rll0 — <srnh2. (2.20)For a proof of this estimate we refer the reader to [15]. We also suppose, some “weightederror” estimates hold for the u — a and Utt — a (see Subsection 2.5.1). In fact we provethe 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 witho < h < H < 1 and further, we suppose the associated “velocity approximating spaces”satisfy XH C Xh C H.15Error estimates for the MSL methodConsider 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) = 0, for all q E Lh, satisfying p(, t0) P”°ã(, to) andq(.,to) = /ã(.,to). Let us now introduce e = a—(p + q), the difference between thefinite element approximation obtained on the fine grid Gh and the MSL approximationobtained using the coarse mesh GH and the fine mesh Gh, and the difference — 7r.We also define the weight function a(t) = rnin(t — to, 1).Theorem 2.1 (Error estimates for the MSL method) There exists a constant Cmsi, independent of h but dependent on u, T, and some properties related to the pair Xh, Lh,such thatIVeW rnsi H2, (2.21)jell msiH, (2.22)uejj < msi H3, (2.23)lCH jmsiH2, (2.24)for all t0 <t < T.We have used the notation Cmsi to highlight its connection to MSL. The proof of Theorem2.1 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 velocity16error, u— (pH + q), and the full pressure error, 0 — 7rh, associated with MSL approximations and consider the corresponding errors u— Uh and 0— h associated with the“standard” finite element approximation calculated on the fine grid. Corollary 2.1 provesthat the absolute size of the errors u— PH — q and u — Üh, measured in either the L2-or the Dirichiet norm, are, for all intents and purposes, the same if H is small enough.Similarly for 0— 7rh and 0— h measured in theL2-norm.Corollary 2.1 Let and Jmsi be as previously defined. Then,IlVu — VPH — Vqh 1sm h + Jmsl H2, (2.25)a2u— pH— qhW sm h2 + ms1 H3, (2.26)—h + H2, (2.27)for all to t < 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 isH = 2h. In this case, the approximation obtained using the MSL method (calculatedpartly on the coarse grid, partly on the fine grid) is as accurate as the standard finiteelement approximation (where all is calculated on the fine grid) in the following sense:given any e > 0 there exist h such that if h < h, thenVu— VPH — Vqhj (sm + e)h, (2.28)— PH — q, < (Jsm + e)h2, (2.29)J2W0— lrhW < sm + e)h. (2.30)Proof: The proofs of (2.25), (2.26), (2.27) are straightforward consequences of the estimates 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 removed at the cost of one half a power of H on the right hand side.17When H = 2”h, (2.25) is equivalent toVu— VPH — Vqh sm h + msI 2”h,and if h < h = h =—then Vu— VPH — Vqh (“ + e)h. The proofs ofmsl(2.29) and (2.30) are similar. For the estimate (2.29) use Remark 2.5. DError estimates for the NGMLet a, , and e1 be as before. Consider the NGM approximation (p. q, r) = (pH, qh, 7rh)EXH x Xh x W, the solution to(Pt, P)+va(p + q, )+n(p, p, )+n(p, q,) + n(q, p, P)=(f, ) + b(, ),for all b e Xh, b(qh, p + q) = 0, for all q E Lh, satisfying p(., to) = PHu(., to). Lete = a—p — q and ( = 0 — r where p, q and r are now the NOM solution above. In theNGM, 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 Wis chosen with respect to the L2-inner product (a = 1, 41 = 0), the resulting nonlinearGalerkin 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 Galerkinmethod proposed by Ammi and Marion in [1] where Wh and XH are orthogonal withrespect to theL2-inner product. Approximations calculated using this scheme satisfy:VeW + C12 H2, (2.31)ue+He < uH3, (2.32)‘Sfor all t0 < t < T. Here l2 — the subscript 12 highlights the connection to the specialNGM with “L2-orthogonality” — 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 Corollary2.1. In [1], they proved (2.31). We have extended their results and proved the higher orderestimates for the L2-norm in (2.32). This Corollary is proved at the end of Subsection2.5.3. The key point is that, in this special case,(Pt, QH) = (q, PH) 0, V X.This is not true without theL2-orthogonality and in any other case, because these termsare non-zero, we can only prove the weaker Theorem 2.2.Theorem 2.2 (Error estimate for the NGM,) There exists a constant1ngm, independentof h but dependent on u, T, S2 and some properties related to the pair Xh, Lh, such thataIeW + HeW + HVejj + uHIICII ngm H2, (2.33)for all t0 <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 = 2h(as we did in Corollary 2.1), and consider U— PH — q, thenIPH —qhI JU_ ÜhW + Üh PH —qhU sm h2 +ngm H2 = CsrnH2 +ngmH.This suggests that:• the size of lu— PH— qhll depends on the approxirnability properties of the coarsegrid. Fixing the coarse grid but taking more refined fine grids (larger n), we expect Ilu— PH— qhll Cngm H2. In other words we see no reason to expect theapproximation to improve.19• Consider the standard approximation calculated on the coarse grid, ÜH. We know,— aH!I sm H2. It is not clear which is smaller, sm H2 or H2 + ngmH2.It is possible that u— PH + q,W > u — UH. In other words, the “correction” qmay damage the approximation. Our numerical experiments seem to indicate thatthis happens.A further remarkFrom 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 asthe main approximation to the solution u?Supposing the higher order estimates in Theorem 2.1 hold, the following is a heuristicargument which shows why p + q is the main approximation. For simplicity considerMSL approximations. Consider the relative sizes of u— p and u — p — q.since PHq 0. Now, using Lemma 2.1 and estimate (2.23),l1H(u- P-Cu-p- q <CH3.Using Lemma 2.1,Mu — PHUII OH2and hence, for small enough H,Mu- pM OH2.On the other hand,20andIIFh(u—p—q) < CWu—p — q <CH3,U—PhUjj Oh.Thus, for small enough H,-p-II Oh2.Clearly, since h < H, for H small enough, ju—l1 > lu — p — qll. In the particular casewhere H = 2h, we expectlu—ll CH2 while lu — p — ll OH2.For instance if n = 1, we expect lu—p 4llu—— qll•212.3 the Numerical schemesMuch of the numerical work was a collaborative work with S. Turek. For the two-dimensional Navier-Stokes equations, Turek has designed and implemented an excellentfinite element solver (see [30], [28], [29], [27], [23]). We have taken his basic code andmodified it to allow the possibility of solving using either a nonlinear Galerkin or aMicroscale Linearization method. To start we give a brief overview of the finite elementspaces involved and some of the basic ideas of Turek’s solver. After this, we describe theimplementation of the NGM and MSL that we used.2.3.1 The finite elements spacesThe velocity approximating space is the nonconforming finite element space Xh spannedby functions which are, in each coordinate, “rotated bilinear” on each quadrilateral ofthe grid with degrees of freedom at the midpoints. If the grid is composed of trianglesthen Xh is spanned by functions which are, in each coordinate, linear on each triangle ofthe 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 (ortriangle). The triangular case has been studied in [13] while the quadrilateral case hasbeen studied in [23]. The beauty of these nonconforming spaces is they allow a naturalconstruction 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 calculateusing 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. Wewrite Gh = UT where T is an individual triangle or quadrilateral. The parameter h > 0is a measure on the size of the elements of Gh. The family of grids, {Gh}, is assumed22to satisfy a uniform shape and size condition and for the case of quadrilaterals each Tmust be convex. The common edge between adjacent T, T is denoted by and themidpoint of this edge is We let uGh = UFj he the set of all edges and denote theboundary edges by F = (0G fl 811). Since the spaces are nonconforming, we have towork with bilinear forms that are defined elementwise:ah(uh,vh) f VUh : Vvhdx,TEGhq.f Vuhdx. (2.34)T;EGhRemark 2.6 It is convenient to work with a slightly different bh for one of the casesof quadrilateral elements. T’Ve discuss this particular case at the end of this section (see(2.35)).Let:Lh = {qh E L2(11) I q = const, VT e Gh, j qhdx =Xh = Sh x Sh,Jh = {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 P1(T) = span {1,x,y}.On T, the degrees of freedom are determined by the nodal functionals {F- P E 8T}where Fp(v) = v(mp). Any v E P1(T) is uniquely determined by the values of the Fi-’(v).Let,vIT E P1(T), VT E Gh, s.t. v is continuous at all edges, Pjj,Sh = V E L2(11) in the sense that Fr(v)T = Fr,(v)I, VP c 8Gh,and Fp(v) = 0, VP e P.23Remark 2.7 If we choose asfunctionals Fr(v) = FL’ j v()d7, in this triangular case,we obtain the same space S,.Suppose we have a grid composed of convex quadrilaterals. In this situation weconsider the parametric version of Xh since this is what we use in practice. Parametricmeans the functions in the finite element space are defined with respect to mappingsto the reference element. In the case of triangles the parametric and non-parametricforms are equivalent (see Remark 2.8). Let T be the reference square, T = [—1, 112.Let Qi = span {1, x, y, xy}. Since T is convex, there exists a unique invertible mapping?/)T E Qi x Qi mapping t to T. Let (T) be the rotated bilinear elements(T) = {qo’ I q e span{1,x,y,x2— y2}}Remark 2.8 For triangles define (T) in the obvious way. The invertible mapping,‘/T,from the reference triangle to any triangle is an affine mapping and hence po’ E P1(T)for each p E P,(T). Hence, P,(T) = P1(T) which implies that the parametric and non-parametric cases are the same.On a given quadrilateral T, we consider two types of functionals:F(v) = IL1 j v() d or F(v) = v(mp).Either choice is unisolvent in Q,(T) but will lead to different spaces, S, rn = 1 or 2:E Q,(T), VT Gh, s.t. v is continuous at all edges, F,= v E L2() in the sense that F(v)IT = F,(v)I, VF3 C äGh,and F1(v) = 0, VP E F.Let us consider the space Jh C Xh. In particular, we are interested in its basisfunctions. A function v, e Jh satisfies bh(h,vh) = 0, for all \h E Lh. Consider24the quadrilateral case where Xh = x S (defined using the midpoint functionals).Consider a general quadrilateral T E Gh with vertices a1, a2, a3, a4, edges e1, e2, e3, e4,midpoints m1, rn2, rn3, m4, tangential vectors t1, t2, t3, t4, and unit normal vectors n1,n2, n3, n4. In this particular case, it is convenient to work with a bilinear form bh(.,.)e1slightly 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) fl. (2.35)The following helps to understand the connection between this definition and the firstone.f V.vdx=TfTLet çj Sh, j = 1, . . . , 4, be the nodal basis functions on the quadrilateral T, that is,j(rn)= 6j, i,j = 1,••. ,4. The functions i = ,4, are linearly independentand satisfy:= AIT 7j (?n) t . nj = 0.y3Thus, a first group of basis functions are the tangential basis functions, v = qjt,corresponding to the edges (or midpoints) of T. A second group of basis functions,a125corresponding to the vertices of T, are given by=— i = 1,... ,4, j = (i+ 2) 4 + 1.V/i HYlThese are the streamfunction (or rotational,) basis functions. It can be seen that thecoefficient of the basis functions associated with the vertex a, is an approximation to thevalue of the streamfunction at this vertex. Physically, the basis function v correspondsto a flow parallel to the side e while v corresponds to a flow around the vertex a.tangential-type streamfunction-typeFigure 2.2: basis functions of JhThe construction of the basis for Jh in the other two cases is very similar.1e262.3.2 Turek’s solverTo introduce the solver, it is best to first consider the stationary Navier-Stokes equations:uVu=vAu—VO+f, V•u=O, ulan=O.The finite element approximation is the solution u11 E J, ir E Lh ofva(ui, h) + n(uh, Uh, çh) (f, h) Vh E Jh, (2.36)b(lrh, I’h) = va(u1,) + 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 canbe calculated using a marching process from element to element without solving a linearsystem of equations (see [13]). Consider the velocity equation more carefully.In fact, there are several possible variants of the velocity equation (2.36), corresponding to one of several possible discrete versions, n(u,v,’), of (u.Vv,w):• is centrally discretizedf OV/n(ui, Vh, Wh)ITUh,i Wh,j dx.• n(.,.,.) is discretized using a first order upwinding proposed in [21].• n(.,., ) is discretized using a higher order Sarnarskij upwinding proposed in [25].Let be the ith_basis functions in the basis of Jh. Let U,. be the vector with componentsU = (u, ) and let F be the vector with components F = (f, j. Let Ah be thematrix with components A = a(, ,) and let N,.(U,.) be the matrix, associated withthe nonlinear term, with components N,.(U,.)3 = n(u,, ). Then, (2.36) is equivalentto finding Uh such that:vAU + N/1(Uh)Uh = Fh.27This nonlinear algebraic equation is solved using a fixed-point defect correction methodwhere the (n + 1)t_iterate, U’, is obtained from the flth by solvingu = u+w[vAh+h(u)r1 (Fh — vAhU — N(U)U).Here, w is a damping parameter which can be set beforehand, generally close to 1, orcalculated using some sort of optimization scheme. The matrix ]cTh(UJ) is built using oneof the upwinded discretizations, not the central one. This is done to take advantage ofthe nice algebraic properties of such matrices (see [21], [26]) which allow the use of a fastmultigrid 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 temporaldiscretization, Turek’s solver allows the possibility of using either the fractional-steptheta scheme (see [12], [4]) or one-step theta schemes. The properties of such schemes arediscussed in [22]. Consider one-step theta schemes and let k he the discrete time step. Inthis discussion, let Um(.) = U(., mk). We note that 0 = , 0 0 and 6 = 1 correspond tothe Crank-Nicholson, Forward Euler and Backward Euler schemes respectively. Applyinga one-step theta scheme to the discretized nonstationary equations (2.6), suppressing thesubscript h, we obtain the algebraic system(I-1- kOvA + kON(Um+l))Um+l = Gm,where Gm = (I + (6 — 1)k(vA + N(Um)))Urn+OkFrn+l + (1 — 6)kFm. Suppressing thetime step superscripts, this is simply(I + kOvA + kON(U))U = G.This nonlinear algebraic system is solved using a fixed-point defect correction methodwhere the (n + 1)st_iterate is obtained from the by solvingU’ = Utm + w {i + kvA + kêN(U)]’ (G — (I + kOvA)U — k0N(U)Uj.282.3.3 The implementation of the NGM and MSLAs we have seen, things that were natural for spectral eigenfunctions are not true for finiteelement functions. For instance it is not possible to choose the finite element spaces XHand Wh in such a way that they are mutually orthogonal with respect to both the L2-and Dirichiet inner products. This is why in our analysis we considered a whole class ofpossibilities. When implementing a NGM (or a MSL) further obstacles are faced. Wenow have the added problem of choosing XH and Wh in such a way that the resultingNOM 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 areorthogonal in theL2-norm). In general, we suspect this difficulty is endemic to the NOM(and MSL) when using finite elements and this may be reason enough not to considerusing the method. Nevertheless, in our chosen setting of discretely divergence-free finiteelements, we did implement and test some schemes.We chose discretely divergence-free finite elements for two major reasons. The firstwas our initial feeling that a variational formulation that separates the velocity fromthe pressure was somehow “closer” to the spectral eigenfunction equations (in which thevelocity and pressure are separated). The second reason, perhaps the most important,was that we saw a possibility of implementing a NGM (and MSL) making somesimplifications— as a modification of Turek’s basic code. This was especially appealingbecause the implementation from ground zero of a Navier-Stokes solver is a very largetask.29Implementing the NGMConsider 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 isone refinement of GH. The refining process is done by joining opposing midpoints (seeFigure 2.1). When working with the discretely divergence-free basis functions, we workfrom a formulation where the velocity is separated from the pressure. Thus, consider thevelocity formulations (2.12) which can be rewritten as(Pt, Pço)+va(p+q,p)+n(p, p, Ph)+n(P,q,o11)+n(q, P,P’h)=(f, Sh), (2.38)iía(p+q, Qh)+n(p, p, Qco)=(f, Qh), (2.39)for all cp E Jh. (2.38) is the “low mode” equation and (2.39) the “high mode” orsupplementary equation. We make the following simplification. We suppose PJhOne consequence of this is that (2.38) is approximated by seeking p e JH such that(ps, H) + va(p, H) + n(p, p, H) + n(p, q, H) + n(q, p, H) = (f, H)Another consequence is that for h e Jh, Qh— I’H where ?I)H is the solution toa(H, H) + a(, H) VH E JH.We consider only the case where c = 0 and = 1 (the case where the spaces areorthogonal with respect to the Dirichlet inner product) because then (2.38) , (2.39) isapproximated by finding the solutions p, q = — , where (p,, ) JH x Jh Xsatisfy(Pt, H) + va(p+q, H) + n(p, p, H) + n(p, q, H) + n(q, P, H) = (f, H) (2.40)for all‘Hva(p+h, h)+n(p, p, Sh) = (f, °h) (2.41)= va(h,p), VsZ’HEJH. (2.42)30Remark 2.9 We consider only the case where the “high” and “low” mode spaces areorthogonal with respect to the Dirichlet inner product because in this case we have thismethod of solving the high mode equation as the difference of two Stokes problems. Thisdoes not work for any other cases.For the temporal discretization we use one step-theta schemes with a slight modification. The two nonlinear terms involving q are discretized explicitly. This is done inorder to be able to use a robust fixed-point solver. We let k be the discrete time stepand again use the notation where pfl=p(., nk) (and similarly qfl, f2,•. .). Applying thisdiscretization, we obtain our fully discrete NCM solver. At a fixed time I = (n + l)k, weseek the solution p1, t’ to(pP’, ci)+kÔv a(p’, coH)+k n(p’, p’, H) = gfl(i), VH E JH, (2.43)va(p1 ,j+n(p’, = (fn+l, ç) VSOh J, (2.44)= va(’,ff), VçoJJ-j. (2.45)wheregfl(i)= (p,H)+k(f,H)+k(O —1) (va(p,yH)+n(p,p,yH))—k ((pfl, qfl, y)+n(q, pTh, SH))Remark 2.10 As initial conditions we often use po = ü(.,to), where ü(.,to) E JH is thesolution, at time to, calculated using the standard method (2.5), and q0 0.The MSL methodTo 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, ),31for all p e Jh. Discretize this system using a one-step theta scheme with an explicitdiscretization of the two nonlinear terms involving q. The result is(p’ + q1,P)+ kÔa(p7’ +q’, Pço)+ kOn(p1,pfl+l P(p) g’Qp), (2.46)(p’+q1,)+ka(q1+p,co)+kên(p1 , pr’,) = ii(ç), (2.47)wheregfl()= (p + q,) + k(f, )+- i){va(pj+q, P) + n(p, p, Pp)] - k [n(q, p, P)+n(p, q, co)}and= (p + + k(f,c) + k(ê — 1) [va(p+q)+n(pH,pH,)].We now fix a = 1, /3 = kOz-’ and we assume P4’Jh Jj-. The fully discrete MSLapproximation, at a fixed time t = (n + 1)k, is the solution PH E JH, q = q— q, whereq E Jh, clEf E JEf to(P’,H) + kOa(p1,H) + k0fl(p’,p1,H) gfl(), VH E JR, (2.48)= 1(h), V/h E Jh, (2.49)(1,H) + k8va(’,H) =(1,H) + kua(1,ii), V JH, (2.50)with Po = u0, qo = u0.Remark 2.11 We often use as initial conditions Po = ã(.,to) e JH, qo = 0 where a isobtained 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 avoidconfusion omit the superscript associated with the time step. Let PH be the vector32of coefficients of p with respect to the standard basis of Jjj. Let GH the vector withcomponents 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 ]3+1 is obtained from the rnth iterate P# by solvingprn+l= + [ + Nh(P)] (GH - vA11P- N(P)P).Having obtained the solution P (when the fixed-point iteration converges) the linearsupplementary problems (2.44), (2.45) (or (2.49), (2.50)) can be easily solved.2.4 Numerical resultsWe test the NG and MSL methods by comparing the qualitative properties of solutionscomputed using these methods with the qualitative properties of those calculated usingTurek’s code, which we often refer to as the “standard method”.2.4.1 Flow around a cylinderWe consider flow around a cylinder with a Reynolds’ number of 100. A Reynolds’ numberof 100 corresponds (approximately) to towing a 1 cm. diameter cylinder through waterat a rate of 1 cm/sec. In this situation, the long time solution to the Navier-Stokesequations is known to be the stable time periodic von Kármán vortex street. In hisbook 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 = 100The computational domain is a rectangular box of length 32, height 16 with a circle ofdiameter 1 centered at the point (7,8). In the test calculations we prescribe:•‘: • ••—..• •.,, . •• :34at 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 isnaturally defined by the variational problem. Let n be the normal vector at the outflowboundary. Let u = u n and let uT = u — u. It is shown in [14], that at the outflow8u7boundary, p — ii— = c, = 0, where c is a constant. This boundary condition hasanbeen analyzed and carefully tested in [14]. In all of our calculations we used the CrankNicholson scheme (the one-step theta scheme with U= ). Figure 2.4 pictures the coarsegrid and we used several refinements of it. The grids we calculated with are: Grid level 3= 1984 elements, 6160 unknowns, Grid level 4 = 7936 elements, 24224 unknowns, Gridlevel 5 = 31744 elements, 96064 unknowns.Figure 2.4: cylinder grids, levels 1, •.., 4Results for the standard methodWe 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 oscillationsgetting smaller with time. There is too much artificial viscosity from a combinationof 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 improvement from the Level 3 calculation. The streamlines of the results are depictedin 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 streamlinesare shown in Figure 2.7 and Figure 2.3 is a particle tracing visualization of thecalculation.36==I..Figure 2.5: SM solution, grid level 3, t =0, 20, , 180370--5(wFigure 2.6: SM solution, grid level 4, t = 0, 20, , 18038a_aaoFigure 2.7: SM solution, grid level 5, t 0, 20, , 18039a a1WResults for the NGM and MSLWe propose two tests:Test 1 -. Calculate using grid level 3 as coarse grid and grid level 4 as fine grid. Theinterest in this test is whether either method somehow “bridges the gap” betweenthe no vortex street solution we obtained with the standard method at grid level 3and the fairly well developed vortex street solution we obtained with the standardmethod at grid level 4 . As initial condition use the final solution of the standardmethod level 3 calculation (a symmetric solution).Test 2 - Calculate using grid level 4 as coarse grid and grid level 5 as fine grid. Here, themain interest is whether either method produces a fuller von IKármán vortex streetlike the solution to the standard method at grid level 5. The initial condition isthe final solution of the standard method grid level 4 calculation (vortex street isdeveloped).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 longerconverged.Test 1, MSL- The results are shown in Figure 2.10. The so]ution was stable and wecalculated for 20 seconds (2000 time steps). At the end of this time there was novortex street. There were very small oscillations but they were decreasing withtime.Test 2, NOM - The results are in Figure 2.11. The solution developped unnatural“wiggles” and by time t = 1.65 (165 time steps) the solver no longer converged.40In Figure 2.12, we have included some streamlines of the “high modes” q and inFigure 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 timesteps). The vortex street continued unimpeded. However, it retained the characteristics of the original vortex street. It did not become “fuller” or oscillate witha shorter period. In Figure 2.15, we have included some streamlines of the highmodes q.41aq CD I.Ij CDFigure 2.10: Test 1; MSL grid levels 3-4, t = 0, 5, 10, 15, 2043a-w0Figure 2.11: Test 2; NGM grid levels 4-5, t = 0, 1, 1.5, 1.62, 1.63, 1.6444Figure 2.12: Test 2; NOM grid levels 4-5, “high modes” q, L = 1, 1.5, 1.64Figure 2.13: Test 2; NOM grid levels 4-5, “low modes” p, times near blow up45owFigure 2.14: Test 2; MSL grid levels 4-5, t = 0, 5, 10, 15, 20, 25, 30\%rear46CD I2.4.2 Diffuser calculationsWe consider flow into a diffuser, that is jet flow into a widening pipe. The intake is ofwidth 1 and the outflow is of width 5 while the pipe is of length 32. In figure 2.16, thedomain, 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.____________IIIIIiFigure 2.16: diffuser grids, levels 1, 2, 4, 5llh1tllhilI ‘‘‘ I I Ill lU’I11I 411t IIiLWIiI IllSIn 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 calculations we used the Crank-Nicholson scheme.48The results for the standard methodStarting 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 varioustimes are depicted in Figure 2.17.Level 6 - We also calculated until t 550. Streamlines at various times are depicted inFigure 2.18.Compare the two results. For instance, compare the streamlines at t = 550 in Figures2.17 and 2.18. The shape of the large scale structures is fairly similar though the largervortices 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 thestreamlines 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 calculationhas 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 capacityof our workstations.49CDIoLID0CD CD —01CDc<—CD CI1cD CD Th C000,II0______0Results for the NGM and MSLWe considered several tests:Test 1 - At a Reynolds’ number of 5000, we calculated using as initial conditions thestandard method level 5 solution from various times.Test 2 - As a result of the tests above, we lowered the Reynolds’ to 1500 by increasingthe viscosity. At this Reynolds’ number, we calculated using as initial data thestandard method level 5 solution at t = 100 (with q = 0 initially). At this timethe solution is still fairly symmetric and our interest was to see what happenedthrough the “bifurcation” to the non-symmetric solution.Test 3 - Again at Reynolds’ number 1500, we calculated used as starting data the level5 standard method solution at t = 400 (with q = 0 initially). At this time thesolution is not symmetric but seems to have a fairly stable large scale structurewith, 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 longerconverged.Test 1 - At this Reynolds’ number of 5000 both the NG and MSL methods performedvery 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 timesteps). The results are in Figures 2.21, 2.22.Test 3, NGM- The solution was unstable and blew up by time t = .11 (11 time steps).The results are in Figures 2.23, 2.24.52Test 3, MSL - The solution was also unstable and blew up by time t = .56 (56 timesteps). The results are in Figures 2.25, 2.26.53ccccIIII-0Ejict.IjIC,)HIIL*11IL\ /11,40n Li/,/({H-1II SDb bI\JIIIIIII SiIs._IiitrnIiIIIIIIIIIiIIIIIlIIII—‘)1IiII!1IJ1IIII/J)IIriItflJ\c/LAl(((NlIWT/)1i\V/(/ftJf-IICSll1-CIDI\’SIIHihi)IIIISI2.5 Proof of Theorem 2.12.5.1 PreliminariesWe let Gh = T be a finite decomposition of mesh size h, 0 < h < 1, of the domainC R, n 2,3, into closed subsets T. Let C, i = 1,... , 8, be positive constantsindependent 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 L2 withmean zero, a pressure approximating space. We make some standard assumptions. Thespace Xh is assumed to satisfy the inverse inequality:VbhW <C1hhH (2.51)for all Jh in Xh. For each u E H(1l) fl W2’(l) and q H1 there exist ihU Xh,jq E Lh such that— VihuW G2hluW, jq — jhqIL2/R < G3hIIqH1/R. (2.52)The pair Xh, Lh satisfy the classical inf-sup condition for mixed finite element methods:inf supb(qh, h) > 4 > 0. (2.53)qhLh qhU WVi’hH —Let Ph be theL2-projection onto X11 and let Rh be the Ritz projection onto Xh. We alsoassume (though some of the following can be shown to be consequences of or previousassumptions):— PhuU+ju — Rhuj + h(W\7u— VPhu+IlVu —\7RhuU) < C5h2WuW,(2.54)- PhUj + lu - RhujI + h(ll\7Phull + lVRhull) < C6h11 Vu!!, (2.55)!Rhu!I + !Phu!l 07!Iull. (2.56)58Let J= { E H : V• = o}, we assume for every v E J1 Fl W2’ there exists anapproximation rhv E Jh such that:Mv — rv + hWy— rhvM <Csh2vW. (2.57)In the “two-grid” situation which arises in the NOM and MSL, we assume that GH is acoarse 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.Let F, be the aI3-projection where for each u e Pu E X is the uniqueelement such thata(Pu,1)+ /3(Vu,Vh) = o(u,h) + /3(Vu,Vh), V X.Lemma 2.1 Let ,/3 > 0, not both zero and let u H nW2’. There exists a constantC independent of h, c, /3 such thatIlu— PuM + hWVu — VPuII Ch2UuII. (2.58)In fact C = C5 where C5 is the best possible constant of estimate (2.54’).Proof: To simplify the notation we drop the superscripts , /3. By definition,— Phu, h) + /3(Vu — VPhu, V) 0, Vb,1 X. (2.59)Letting /‘h = u — Fhu — (u — Phu) in (2.59), we obtainallu - PhM2 + /3Vu- VPhuU2 == a(u— Phu, U— Phu) + /3 (Vu — VFhu, Vu — VPhu)- FhUW2 + u - Phj2 + /3MVu - VPVu- VPhUM.Hence,[u- FhuU2- Mu - phuM2] + /3HVu - VPujj2 /3Vu -Vi1uMIjVu- VPhuII59and since lu — Phull lu Phull (because theL2-projection minimizes the L2-norm)we obtain/WVu - VPhull2 Vu- VPhuIIWVu - VFhUll.Cancelling terms and using (2.54) impliesIlVu - Vhull <IlVu - VPhull <GhIjujI2. (2.60)TheL2-estimate is proved similarly. Choosing /‘h = U — Fhu — (u — Rhu) as test functionin (2.59), implies- FhUII2 + lvu - VPhuj2(u— PhU, u— Rhu) + 9 (Vu — VPhu, Vu — VRiu)Ilu - PhuIilIu- RhuIl + llVu - Vhull2 + çIIVu - VRhu2.Now lVu—VRhull < lVu—VPhuIl (because the Ritz projection minimizes the Dirichietnorm) and hence, using (2.54),lu - Phull <Ilu - Rhujj ChUull2. (2.61)Combining estimates (2.60), (2.61) proves the Lemma. ERemark 2.13 It is clear that Lemma 2.1 holds in more general circumstances thanconsidered above. For instance Xh could be nonconforming. In this case we must modifythe 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 satisfiesllU-PU < llU-RhUj,llVu—VPull < lVu—VPiull.60Lemma 2.2 Let u E Hj thenlu- PhUII <GhljVull. (2.62)In particular, suppose XH C X,. Let F’ be the cq3-projection onto XH and Qj =I — P’ then, for all finite element functions Uh Xh,IIQFfUhM = llUh — PHUhll <CHllVuhI. (2.63)Proof: Using Corollary 2.3 and (2.55), (2.62) follows sincelu PuW Mu- RhulI ChWVuU.Clearly, (2.63) is just a special case.Lemma 2.3 Let Uh E X11, PH = Pu and q = Qfu thenMPHW + MqhM GMuhW, (2.64)WVPHW + Vqhl CWVuhW. (2.65)Proof: The proof relies on the inverse inequality (2.51) and estimate (2.62). Firstly,MHhM < MPiu - uhM + MuhM <ChllVuhll + luhil <ClluhlandllQffUhj = huh - PHuhI <ChhlVuh <Chluhiimply (2.64). Secondly, sinceII VP = lVPJJUhl < VFuh - \7RHuhj + IVRHuh< ChllPuh — RHuhll + IIVUhICli1 (hlPu5 — uhhl + huh — RFJuhhl) + bVuhhl< Ch’hlbVuhhl + bVuhhl <Cl Vuihl6jand= llVuh - VPuhlI Ch1u- FHUhli <CliVuhil,(2.65) is true. 0We require estimates for standard finite element approximations to the Navier-Stokesequations. Let Üh, t9, be the solution to (2.5). Let T(t) = rnin(t, 1) then for 0 < t < T,in — ü + hi! Vu — Vu + riO — <2 (2.66)For the proof of this estimate we refer the reader to [15]. The constant C is independent ofh but dependent on T, u, Q and some properties of the grid. Under these circumstancesone may further prove a sequences of weighted error estimates, much like the sequenceof weighted estimates proved in [15] for the continuous problem. For our purposes, it isenough to know that, for all T > t t > 0, there exists constants C (depending on to,t as well as other quantities) such thatHut — ãtii + hi! Vu— Vat!! (2.67)ilutt — u!l + hiiVujt— Vütt!i <h2. (2.68)Many of the ingredients for proving such an estimates can be found in [15]. However, itis still a fairly involved technical argument to prove these weighted estimates. We provethe first of these in Appendix B. In [1], they use such estimates and in this paper thereference for the proof of these estimates is [2] (a reference we do not have).Lemma 2.4 Let XH C Xh, let Üh, 7Th be the solution to (2.5) and let PH = PÜh andqh = = Üh— PH For all T> t to > 0, there exist constants , independent ofH, but depending on u, i’ and T, such thati!qhIl + H!!Vhii C1H, (2.69)iqh,t!i + HiiVh,tii < 2H, (2.70)Hqh,ttii + H!iVh,!i (2.71)62Proof: (2.69) follows from (2.66), and Lemma 2.1 sinceiqhii = Wuh - PHuhW - uli + Mu - + iiP(ah - u)ll H2.In a similar way, (2.70) follows from (2.67) and while (2.71) follows from (2.68). Westress that estimates (2.70) and (2.71) require no weighting factor since the time t0 isspecifically taken away from t = 0. üWe require some further technical results proved in [15]. Let A,, : Xh —* X,, be theinvertible operator which maps each Vh e Xh to Ahvh Xh wherea(v,,,,b,,)= —(Ahvh,bh), Jh Xh.For any ‘h/h E Xh,ii A b,,ii <Gh’ijVhii. (2.72)This is easily proved since,M Ah hi2 = -a(Ah,,,,,) <Ch’ii Ah hIIIhiiWe also note that, for all t > t0,A ahM + U A,, uh,tii + 1 A a,ttii . (2.73)This follows sinceU Ah a,,i12 = —a(Ahüh, ü,,) = —a(A,,ü,,, Üh — u) — a(Aü,,, u — Rhã)< G Ah ã,,lih (liVu - Vail + UVu — VRhÜII) CU A uii <.Similarly for other two terms. To handle the (discrete) nonlinear term n(.,, .), we willneed discrete analogues of several Sobolev inequalities (see pg. 298 [15]). For all v’,, E X,,,i’/-’hii6 ClV’4’,,ii, (2.74)63Ih3 GhIVh, (2.75)IV’’hW6 < Cjj A,. (2.76)WhI{ + IhM3 CIVhI4Ah hW2. (2.77)Note, as a consequence,IR’Boo + IjVuI3 + IpW00 + lPW3 , (2.78)+ WVW3 < CVh Ah (2.79)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 Supposezt + Az + Vp = f, z(t) = 0, zIan = 0. (2.80)ThenWz(to)W2+ j Vzjds cj fW2ds (2.81)BVz(to)W2+ L(IzI + BziU2 + jVp2)ds C] HfW2ds. (2.82)Proof: Multiplying (2.80) by —z and integrating over the domain, we obtain_IIzII2 + IIVzH2 CjlfH2.Integrating this from to to I implies (2.81). Multiplying (2.80) by Az, where A is theStokes projection, and integrating over the domain one can show_IIVzW2 + WAzH2 = CfH264and, integrating from to to t,Vz(to)I2+ j AzW2ds < Cf fW2ds.The full estimate (2.82) is obtained since IzI2 CIAzf by the Cattabriga-Solonnikovestimate ([5], [24]) and sinceIIztIi + Vp =11 A z - zI2 + jjW652.5.2 OrganizationThis subsection is included to familiarize the reader with some of the different quantitieswe study in the course of proving Theorem 2.1. In Theorem 2.1 we estimate eh =Uh— (PH + q), the difference between the standard approximation on the fine grid andthe MSL approximation on a coarse and this fine grid, and Ch = 7rh — 7Th, the differenceof the two discrete pressures. For simplicity we drop the subscripts h, H and also letP = Pf and QH = Q. Subtracting (2.17) from the velocity equation of (2.5) weobtain the error equation for e, C: For all t to,(et, ) + a(e, ) + n(p, p, ) — n(p, p, ) + n(, p, P) — n(q, p,+ n(p,,P)- n(p, q, + b(C, (2.83)= —n(,, ) — n(p, , Q) — n(, p, V e Xh,b(qh,e) = 0, Vqj. e Lh,with e(.,to) = 0. We divide e into its “low mode part”, eH = Pi-je = PH — PH and its“high mode part”, 11h = QHeh = qi — q,.Let (s, *) E Jh x Lh be the solution to(sj, ) + a(s, ) + b(, ) = -n(p, Vp, P)-n(p, q, P) - n(q, p, P) - n(p, p,) + (f, ) (2.84)for all & Xh, satisfying (x, to) = Pã(x, to) and (x, 0) = Qü(x, t0). Loosely speaking,(s, *) is a solution “half-way-between” the approximations (a, ) and (p + q, ir). Letê=a—s, C=—*, =Fê and i7=Qê.Similarly, letè=p+q—s, =7r—, =Pê and i=Qé.66Notice e ê — é and= —. We obtain an equation for ê, ( by subtracting (2.84)from (2.17), the result is:(et, ) + a(ê, ) + b(, ) = - p,) - n(p, p, b)+n(,,) — n(q, p, + n(p, , P) — n(p, q, (2.85)—n(,, )—n(,,b)— n(, 5, b), V E Xh,with e(to,.) = 0. An equation for è, is obtained by subtracting (2.84) from (2.17):(t, ) + a(é, ) + b(, b) = n(p, p, - n(p, fi,) (2.86)with é(to,.) = 0.Splitting the error e, C into the parts ê, and é, helps us in some instances; inothers we work directly with e, . We mention the following rule of thumb: any estimatethat can be proven for e, can be proven for ê, and é, and therefore, at times, havingestimates for e, (, we will conclude the same estimates are true for ê, ( and ê, withoutsubjecting 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, Pt, b) —ri(p, p, ‘II’)+n(, p, P) — n(q, Pt,) + n(, p, b) — n(qt, p,+n(pt, ,) — n(pt, q,) + n(p, ,) — n(p, q,) (2.87)=—fl(qt, , ) — n(Pt, ,) —, p,)—n(, , ) — n(p, ,) — n(, Pt, Q), V E X1.67Similarly, differentiating (2.85) with respect to t, we obtain, for all t > t0,(tt, &) +a(ê, b) H- b(C1,) = — [fl(t, 1,) — n(pt, p, b)+n(, Pt, b)—n(p, Pt,) + fl(t, p, Pb) — n(qt, p,+n(, Pt,) — n(q, p, P) + n(Pt, , P) — n(pt, q, (2.88)+ n(, t, &)—n(p, qt, Pb)]—n(t, , &) — n(, t, b) —n(Pt, ,)—n(p,,b) — n( p,) — n(, p, b), V e Xh.682.5.3 Proof of the TheoremIn the following, 0 < to < T < cc are fixed. C represents a generic positive constantindependent of h, u and to while represents a generic positive constant independentof h but possibly depending on u, to and T. (Both C and C may depend on andsome parameters, uniform in h, associated with the family of grids). GH is a coarse gridand Gh is a fine grid with associated finite element spaces XH and Xh respectively. Weassume XH C C H and 0 < H < h < 1. P — P’ and Q = Q/3 are the previouslydefined projections. a, /3 0 are arbitrary (not both zero).Lemma 2.6 There exists a constant such that,heW2 + vj hlVehI2ds <H4, (2.89)for all t0 <t < T.Proof: Letting b = e in (2.83) we obtainld 2 2IheW +WVejh = — [n(ji, , e) — n(p, p, e) + n(q, P ) — n(q, p, )+n(p,, ) — n(p, q, )J — n(, j, e) — n(, , j) — n(, p, i).Now,n(p, , e) — n(p, p, e) =—n(e, , e) + n(, p, e) + n(p, , e),=—n(, e ,) + n(, j3, e) + n(p, e i)n(p,, ) — n(p, q, ) = —n(, ii, ) + ri(, j, ) + n(p, ii,= n(,,ij) + n(,, ) — n(p, , q)n(, i, ) — n(q, p, ) = —n(ij, , ) + n(’q, p, ) + n(, , )= n(’q,p,).69Hence,[n(p, 15, e) — .} = n(, 15, e) + n(, i, ) + n(ij, 15, )7 (W IVPII3WVeH —1— lW iqU3Ull + iiH ll’pii3U”li)< + eVe2.Also,n(, , e) CWil{ H5 + ejlVeU2,ln(P, , ij) ipVi < H4 + eWVe2,n(, 15, ij) WVP3UV7)H < + eWVeU2,and therefore,ejI2+IIVe2 <lleil2 +H4.Integrating proves the result. 0Lemma 2.7 There exists a constant C such that,vu Veil2 + L ieuII2ds <H4, (2.90)for all t0 <t < T.Proof: Letting ‘b = et in (2.83) impliesvd 2 2-iIVeil +iietli = — [n(p, j5, e) — n(p, p, et) + n(, 15, ) — n(q, p, )+n(p,, ) — n(p, q, )] — n(, , e) — n(15, , i) — n(, 15, ‘j).We would like to point out a little “trick” (using the inverse inequality). Since II VeilCH,jVeIi2iieti3 < CHUVe iietll ii Ve< CH4iiVelllIetlI CHil Veil2 + eiietll2iiVeii2+ eiletil2, (if H 1).70We use this in what follows.n(p, p, et) — n(p, p, et) = —n(, , e) + n(, p, et) + n(, , e)< CjV2 jet li + Cj \7 jet II (U Vjj3 + j ll)CjjVe2+ clletjl2,n(5,, ) — n(p, q, = —n(, i, ) + n(, j, ) + n(p, j, )GjlVell2lletl3+ GIl Vll ljetll(jjVll3+ 11Pl100)< CjjVejj + llet j2,n(i, p, ) — n(q, p, .) = —n(q, , ) + n(r, ii, ) + n(, , )( jj\7ejl2lletl3—1— 7llVellljetil(ii\7Pll3+ llllc,o)llVeil2+ lletij2,dri(q, q, et) = n(q, q, e) — n(qt, q, e) — ri(q, qj, e)- -< n(,e)+H5+llVell2,-n(p, q, = n(p, q,— n(pt, q, ij) — n(p, q, )- -< n(p, )+H4+jjVejj2-n(q, p, ‘ii) = p, 77) — n(qj, p, 11) — n(q, pj, i)p, ) + H4 + jjVeIl2.Combining, choosing appropriately,ullVeIl2+ jjetlj2 llVeIl2 + H4—{n(, , e) + n(p,,i) + n(, p, i)}.Integrating,II VeIl2+lletll2cls H4 + ln(,e)l + j(,77)l + n(,p,77)j.Now,ln(,e)l + I(P,,77)l + l(,77)I + jjVejj,71which proves the Lemma. ULemma 2.8 There exists a constant C such that,uHetB2 L 1It1I2d8 <H4, (2.91)for all to <t < T.Proof: Letting = et in (2.87):etI2+vjIVe2 =— {n(Pt, p, et) — n(p, p, et) + n(p, Pt, et) — n(p, Th, et) + n(, Pt, ) — n(q, Pt, )+n(qt, p, ) — n(qt, p, ) + n(Pt, , ) — n(pt, q, ) + n(p, , ) — n(p, qt,—fl(t, , e) — n(Pt, , ij) — n(OA, p, i) — n(, , et) — ri(p,, ) — 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)cIeJi3VIHlVetU —I— —f IIPIWtMII’etII< eVet2 + (WetI2 + WVej2),n(p, Pt, et) — n(p, pj, et) = —n(, , et) + n(p, , e) — n(, et, Pt)< + PWJIVH etH + PtjjWVeeVe + (UetH2+ Ve2),n(, Pt, — n(q, Pt, t) = —n(77, t, ) + n(, ) — n(i, pt)—I— WD etH —1— WPtHc,DWHWVtW)< eVe + (Wet2+ UVeI2),fl(qt, P, — n(qt, p, =—n(j, , e) + n(qt, , ) — n(rj, t, P)S W 1 —i- W \7e ( et —f Ve)72IVeW + (Wetjl2+ WVeW2),n(Pt, i, ) — n(pt, q, = —n(, ii, + n(Pt, 11,— t’ i)< W3W —I— \7e (jIet 1 —I—< eVejj + ( etj2 + jIVejj2),n(, 1t, — n(p, qt, = —n(e, r, + n(P, — n(, , i)—F Vetjj(jjetjl H— IlveW)< eVej + etjj2 + WVeII2),fl(qt, j, e) + n(, , e) eVetH2 + CH5,n(15t, i, + n(p, it, j) = —fl(Pt, nt, i) — n(p, nit, h)< eVetjj2 + H4,fl(qt, p, nie) + n(i, Pt, ‘ii) < eVetjj2 + GH4.Combining and choosing the appropriate e impliesIIetW2 + vVetW2 <jjetjj2+ H4.Therefore,aUet2 + vajjVetW2<ujjetjj2+ jjet2 + aH4and integrating, observing that the 1iminfajjet2= 0, proves the result. 0Lemma 2.9 There exists a constant C such that,H2 + f jj2ds <H4, (2.92)for all t0 <t < T.Proof: By assumption, our approximating spaces Xh and Lh satisfy the inf-sup conditionandC sup‘4’ eX73Consider the error equation (2.83):—b(C, ) = (et, ) + 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).Estimating:(et, ‘ii’) CWetW JVba(e,i&) < jVeIV’bI <H2IVI,n(p, p, b) — n(p, p, = —n(, , b) + n(p, , b) + n(, p, )+ + jjv< Hj\7j,n(, i5,P) — n(q, p, P) = —n(, ij, P) + n(p, , P) + , , P)< Cjjjj3VelIIV’II + CVi,jV’IJn(o, P) - n(q, p,P) = -n(i, P) + n(ö, P) + n(, , P)G73VjVbI ++WVPU3I\7L’jj <H2IV’bIj,3VV HVn(p, i, Q/’) jIPIicoIIiiHI?’U <H2)IVL’IIn(i, p, ‘) V jf H2II\7blj.Combining impliesHCU WV’’W •< 7’’(ejand hence,IIetI + OH4. (2.93)74Integrating (2.93) from to to t and using estimate (2.90) implies the last half of theestimate. Multiplying (2.93) by a and using estimate (2.91) implies the other half. 0Lemma 2.10 There exists a constant such that,hell2 + v lVelI2ds < H4, (2.94)vhjVêW2+ j êjj2ds < (2.95)L j2ds < (2.96)ahl2 < H4, (2.97)for all t0 <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 jejj2 H6, (2.98)for all t0 <t < T.Proof: Consider the solution of the backwards Stokes’ equationszt + v z — Vj3 = ê, Vz = 0, z(t) = 0, zj312 = 0. (2.99)Multiplying by ê and integrating over the domain,hell2 = (Zt, ê) — a(z, e) + b(, e)=(z, ) + b(, è) — [(, z) + a(ê, z)j.Choosing b = PHZ E XH C X,, as test function in (2.85),(, FHZ) + a(e, PHz) + b(C, PHz) + n(, 5, PHz)—n(q, p, Phz) + n(p, j, PHz)—n(p, q, PHz) + n(, j, PHz) = 0,Thsince (PHz) = 0. Hence,Well2 (z, ê) + b(, e) — [(at, z — Pffz) + a(ê, z — PFJz) + n(, , PHz)—n(q, p, PHz) + n(p, , PHz) — n(p, q, PHz) + n(, , PHz) — b(, PHz)]Estimating:b(/3,ê) = b(/3—jH/3,ê) CHIIV/3IIHVêIIEllzll + CeH2IlVll,(at, z- PHz) llell jjz- PHzll CIIêtJIH2llzIl€zj + CH4llêtIl2,a(ê,z— PHz) < llVelillVz— VPHzII llH + CeH2ljVll,n(, j, PHZ)—n(p, q, PHZ) = n(, j, PHZ) + n(p, q, Pjzjz)= —n(, PHZ, ) — n(p, PHZ, i)< C VPHzj3jjjj + jVp l\7PHzII3II?)jl)< €jVPjjz + CH2llVeU llVPi-izIl + 0E116,n(, p, PHz)—n(q, p, Pffz) = n(ij, p, PHz) + n(q, , PHZ)ll1Hzlloo(ll7lll I[ll + lll llVll)llPHzH+H6,n(j,Pnz) I1HZllcolV1llllll€IlPHzll0+Cj6,b(,PHz) = b(C,Fjiz - z) llllllVPhz- Vzlll{zll +H2jj(jl.Now,lFHzW, IIVPHzII3 Cjz76and combining the previous estimatesleI2 < (z, ê) +H2(H4+ H2êt j2 + 2) + IzII.Integrating from to to t, observing that e(., to) = z(., t) = 0, and applying Lemmas 2.7,2.10 we obtainH° +ef: Wzds.Now applying the backwards Stokes’ estimate (2.82) and choosing e appropriately,ef IIzIIds j WlIds.This proves the desired result. DLemma 2.12 There exists a constant such that,jeW H3, (2.100)for all t0 <t < T.Proof: Recall (see (2.86)),(e, i) + a(ë, ) + b(, b) = n(p, p, Pb) - n(p, p, Pm).Letting‘/‘ é+ vjjVéjj2 = n(p,p,Pe) — n(p,,Pë)= n(Pé, p, Pè) + n(P, p, P) + n(p, Fe, Fe)—n(p, Fe, Pe)n(Pé, p, Fe) + n(Fê, p, e) — n(p, Fe, Fe).Notice,< jVp — Vj3 + jV3 Cjj Ah (p—)jj +Ch’jjVp — Vpjj + C C.77and this implies,n(Pe,p,iê) < e{WVeIj <eVë2+IIPê2,n(Pe, p, Fe) < VeI ejVeLeU,n(,i3é,ê) j êHVéW eVë +IIPeW2.Therefore,II2 + vWVe2 e2 + êU2.Integrating, observing that e(., t0) = 0, and applying Lemma 2.11, proves the estimate. DCorollary 2.4 Estimate (2.2) holds: there exists a constant C such thatej2for all t <t < T.Proof: By Lemma 2.7,e3H2ds <H4.Also, by Lemmas 2.11 and 2.12,WeW2ds 2j WêH2ds + 2] é2ds H6.Therefore,eI2= : Ies2ds <2 (f 2 (L esW2ds) <H5. flLemma 2.13 There exists a constant C such that,v2UVetlj + L a2 Ij2ds <H4, (2.101)for all t0 <t < T.78Proof: Let & = ett in (2.87):—IVetI2+ ettI2 — [n(Pt, p, ett) — n(pt, p, eu) + n(p, p, ett) — n(p, Pt, ejt)2dt+n(i,pt,t) — n(q,pt,) + —+n(Pt,,) — n(pt, q, ) + n(p, lt, — n(p, q,—n(t, , ett)—fl(t, , ‘q) —n(t, ji, n(, , e) — n(p, t, ij) — n(, jit, 7itt)Without going through the details,For the other terms:[n(jit, ji, ett) — . — n(p, q, + CVe2Thus,fl(qt, , ett)n(, 1t, ett)fl(pt, 1,n(p, qt, 1tt)Ti(qt, i5, i)n(, jit, m)d= dt< n(t,,et)— dtd<—n(,t,et)— dtd=—n(pt,,ij)dt<— dtd<—n(p,t,q)— dtd= dtd<— dtd<—n(,jit,q)dt— n(qtt, j, et) — fl(t, t, e)+H5+ VetH2,+ GH5 + VetW2,—n(Ptt, , i) — fl(pt, 7))+ OH4 + OjVetU2,+ OH4 + OWVetj2,—fl(qtt, ji, ‘j)—fl(t, jit, j)+ OH4 + VejU2,+ OH4 + WVetH2.vlIVetII2 + IlettW2 <IVetjI2+H4 — {n(t,,et) + n(t,et)+n(Pt,, ‘i) + n(p, t, i) + fl(qt, ji, j) + n(, jit,79Since j allVell2ds < cc (see Lemma 2.8), lirnsup0a2WetW = 0. Multiplying by a2,vda2IlveIl2 +u2BettB 2ullVetll +a2IlVefl +a2H4_a2{n(,, et)+n(, , et)+n(Pj, , ij)+n(, t, )+n(t, p, t)+n(, t, i)}+2aa’{n(t, , et)+n(, t, et)+n(Pt, , )+n(p, , m)+n(t, p, iit)+n(, p, 71)}.Integrating from to to t, and estimating proves the result.Lemma 2.14 There exists a constant such that,a2(C3 2 + 3W2)ds <H4, (2.102)for all to <1 < T.Proof: Recall that 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(, , b)q, b)+n(p, P)—n(p, qt, )+(t, , )+fl(t, , 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 <H2llVl,rest of terms < CH2llV’b.Therefore,g2lj 12 <ia2lle 12 +u2H4. (2.103)80Integrating, using estimate (2.101), proves that jo2jsIds < CH’. The estimate forfo2II8lds is similar.Lemma 2.15 There exists a constant such that,fu2êdsfor all t0 <t <T.Proof: Consider the solution to the backwards Stokes’ equationsz+vAz—V/3=aê, Vz=0, z(t)=0, zan 0.Multiplying by ê and integrating over the domain,aIIêtlI2 = (zj, t) — a(z, e) + b(, j) = (z, t) + b(, ê) — [(e, z) + a(êj, z)].Notice, choosing ?J’ = PHZ E XH C Xh as test function in (2.88),(tt, PHZ) + a(êt, PHz) + PHz) = — [fl(j, 5, PHz) — n(qt, p, Pffz)+n(i, , P-z) — n(q, p,, Pj-z) + ri(Pt, , Pffz)—n(p,, q, PHz)+ n(i, it, PHz) — n(p, qt, PHz)] — n(O, j, PHz) — n(, PHz).Thus,aUêt2 = (z, + b(, t)—{(tt, z— PHz) + a(êt, z — Pffz) + [n(, , PHz)—n(qt, p, PHz) + n(, , PHz) — n(q, Pt, PHZ) + n(Pt, i, PHZ) — n(pt, q, PHz)+n(, , PHz) — n(p, qt, PHz)] + fl(j, , PHz) + n(, , 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/WVetW81(tt, z — PHz)z— PHz)[fl(t, p, PHZ)—n(O, , PHz)n(i, ã., PHz)b((, PHZ)Therefore,—ill ‘2a 1zj2 + CHalVêtH,IIêtIIIIz— PH2II CIIetjIE[2I z I‘‘2< za 1zw2 + CH4ajttU,VefVz — VPjjz a’WzIj + CH2aUVêtW,—n(p, q,P11z)] a’zII + H2 (uVe2 + H4),WzW +CH6,IIPHzIIIiIHIVtW IzII +H6,= b(t,PHz — z) tHVPhz — VzU—1’ “2+a2 ii “21z112 °lictliaIIêI2 < (z, j) + J1Hz2 + H6 +H2(aIICtII + all Ve2 +H2allêtiIl).NoticeThus,d da(z, et) = o(z, et) — a (z, ej)d= a(z, et) — 0. (z, e) + a (z, e)da(z, et) — a (z, e) + a (z, e)±z + H6 +H2a(tll + jIVetII2 +H2llêjtll).(2.104)We would like to proceed by integrating. However we are forced to be a little careful andfirst consider the case when t to + 1. For times t < to + 1, a’ = 1. Integrating,I 2 2 t t Ij a llell ds a(z.et)j0 — (z,e)I0 +j (zt,e)dsto to+€j:lzI + H6 + H2fa2(llc 12 + 11Ve812+H2llêttll)ds.82Now,a(z,êt)l0 = (z,ê)0 = 0since z(t) = a(to)êt(to) = ê(to) = 0. Also, applying Lemmas 2.14 and 2.13,x: a2(t2 + lIve3!!2+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 wehave just proved:flê(to + ‘)W2 < CH°. (2.105)This follows sinceIIê(to + 1)112= f° aI!eIl2ds = f°’(a’lleIl2 + 2a1!êIIlIê3Il)dstto+l pto+l—2J !êIl2ds + J a211è3!ds <CH°.to toSuppose t to + 1. Integrate (2.104) from to to t and notice that a’ = 0 if t > to + 1.Lj232ds - (z,e)l’ +j°zt,eds+j jz +H° +H2j 2(llC3l + 11Ve512)ds.Consider the right-hand side terms:= 0,(z, ê)l’ = (z(to + 1), ê(to + 1)) !!zv(to + 1)112 + llêt(to + 1)112a2llêsII2ds+H6.83Here, we have used the backwards Stokes estimate (2.81) to estimate z(to + 1)11 and(2.105) to estimate Bêt(to + 1)W2. Thus,f aê2ds f IIzIds + H6.Using the Backwards Stokes estimate (2.81) and choosing€appropriately we are done. UCorollary 2.5 The estimate (2.,.9) is valid: there exists a constant C such that,oiIeII2 CH6,for all to <t < T.Proof: Now,WeW2 <2ajêj + 2uIeII <0116+ uIIêII2and= ft f(u1We2 + 2WeIIIIê3)dsto S to< 2j Iê2ds+ ju2IIêIds <H6.Proof of Corollary 2.2A consequence of Theorem 2.1, we are able to prove estimates for the special nonlinearGalerkin method proposed by Ammi and Marion in [1].Corollary 2.2 (Error estimates for a special NGM,,) Consider the nonlinear Galerkinmethod proposed by Ammi and Marion in [1] where Wi, and XH are orthogonal withrespect to theL2-inner product. Approximations calculated using this scheme satisfy:VeIj + u 012 H2, (2.31)IleW + HIleIj < H3, (2.32)for all t0 <t < T.84Proof: 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 theL2-inner product,(pL,QHP) = (q,Pff) = 0.Let (pH, qh, ‘7r) E XH xW, xLh be the solution to the MSL method and let (jq, qh, h) EXH x Wh x L1 be the solution to the NC method. It is clear that the difference of thesolutions w=p + q — — satisfiesIIwII2+IIVwII2 = [n(,,Pw) — n(p,p,Pw) + n(,,Pw) — n(q, p,Pw)+n(, , Pw) — n(p, q, Pw)]— (qt, Qw).Now,[n(, , Pw) -... - n(p, q, Pw)] lIwII2 + VwI2,(qt, Qw) WW QwH ]12qt+ Vw2,and thereforewI2 + jVwI2 IIwW2 +H2q.As a consequence of estimate in Lemma 2.7 f q2ds <OH4. Thus integrating from toto t we prove thatjw 112 <OH6.Nowlieu ilu-p- q + liwli Ilu-p- qil + OH6.Estimate (2.32) is proved since u—p—qj can be estimated using either (2.22) or (2.23).85Chapter 3Burgers’ equationWe consider Burgers’ equation in one-dimension. Much of the motivation for consideringthis problem was to supplement our Navier-Stokes results, particularly the numericalones. For Burgers’ equation we can implement the NGM and MSL exactly as stated. Aparticular NOM of interest is one proposed by Marion and Temam in [20].The initial-boundary-value problem for Burgers’ equation in t = [0, 1] with a givenexternal force f = f(x, t), initial condition u0, and constant ii leads one to considersolutions u = u(x, t) toZtt — vu + ‘uu = ,3 1u(0, t) n(l, t) = 0, u(x, 0) = u0.Consider a grid, Gh, which is a regular refinement of [0, 1] into subintervals of lengthh. Let Xh the finite element space spanned by continuous functions linear on eachsubinterval, zero on the boundary. It is clear that X = Xh C WJ’2. The finite element(Galerkin) approximation to (3.1), over the space Xh, is the solution ü = Üh(t) E Xh to(Ut, h) + v(zt, bh,) + (UUx, /‘h) = (I, ‘) V bh E X, U(x, 0) = U. (3.2)We often refer to (3.2) as the standard method (SM).Let OH be a coarse grid which is a regular refinement of [0, 1] into subintervals oflength H = and let Gh be a fine grid which is a regular refinement of [0, 1] into subintervals of length H= ,i 1. We say Gh is the th_ refinement of GH. Let XH bethe piecewise linear finite element space associated with Gj.j and let Xh be the piecewise86linear finite element space associated with Gh. Clearly XH C Xh. Let üjq E X- be thesolution of (3.2) when X = XH and let th E Xh be the solution of (3.2) when X = Xh.We often refer to ftH as the coarse standard approximation and to Üh E Xh as the finestandard approximation.3.1 The NGMIn the NGM we require a supplemental space for the “high modes” and, as it was for theNavier-Stokes equations, no one choice of supplemental space seems natural. We considergeneral supplementary spacesW’= { € X a(, ) + x) 0 for all XH},where a, 0 (not both zero). It is easily seen that XH + = Xh for all a, > 0.The a, p3-projection P’ : —* X maps each n E H uniquely to E XH throughthe relationa(’u, H) + = a(u, r) + (u, H,x), VH E XH.Let Qr3 = I —Remark 3.1 (The spaces l4/’) In the finite element case, different choices of a,/3 leadto different spaces T’V,”3. In the spectral caes, the eigenfunctions of the Laplacian area = sin(n7rx), n = l,2,•••, and these are orthogonal in both the Dirichlet andL2-innerproducts 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 isspanned 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 point87viU coarse grid point• fine grid pointnot belonging to the coarse grid and zero at all other grid points. It is easily seen thatXH+Wh=Xh.It turns out thatWh { E Xh I Ofor all e XH},and hence Wh and XH are orthogonal with respect to the Dirichiet inner product. Theproof is straightforward. Consider a coarse subinterval, I, of length 2h divided into twofine subintervals of length Ii. Any function qjq E XH is linear on the subinterval I andhence its derivative is a constant c. Any function /‘h E 147h is zero at the endpoints of Iand linear on each fine subinterval - a “hat function”. Thus, its derivative is k on onesubinterval 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 theinduced basis in the obvious way).Given a coarse space, X11, and a supplemental space, T4/,’, the NOM approximationis the solution (p,q) = (pH,qh) e XH X to(pt,)+v(p+qx,q5)+(pp +pqx+qpx,q) = (f,q), Vq E XH, (3.3)v(qx+px,x)+(ppx,) = (f,), VeW’, (3.4)88satisfying p(x,O) = Po. We call (3.3) the low-mode equation and (3.4) the high-modeequation.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 tothe complement equation (3.4) can be easily calculated since the induced basis and thematrices related to it can be easily constructed.Remark 3.2 It is interesting to note that the variant of the NGM in our computationsof the Navier-Stokes equations also had a supplementary space defined with respect to theDirichiet inner product.3.2 The MSLThe MSL approximation (p, q) E Xjq x is the solution to= (f,cH), V E XH, (3.5)(p + q, h) + v(px + q, h,x) + (pp, 1i) = (f, ), for all h e W’ (3.6)satisfying p(x, 0) = Pu0 and q(x, 0) = Q’u0. A more compact form of this system isV h E Xh. (3.7)893.3 Results of the error analysis of the NGM and MSLOur main result is Theorem 3.1 which is proved in Section 3.4. It is enlightning to noticethe parallels between our theoretical results for Burgers’ equation and our theoreticalresults for the Navier-Stokes equations (in Section 2.2).Let f = h E Xh be the standard finite element approximation calculated on the finegrid, the solution ofbh) + v(, bh,) + (u, /h) = (f, Ir all /‘i E Xh, (3.2)satisfying ü(x, 0) = . Let and be the previously defined projections. Fornotational convenience, we omit the superscript a, in what follows. Let = PH andlet = QHã. For this approximation t the following standard estimate holds: there existCsm such thatlu— Il + hIIu ll Csm h2, (3.8)for all t 0. The constant sm is independent of h but dependent on u. We say this is astandard estimate though we know of no source for its proof. For completeness we haveproved this estimate in Appendix C. The subscript sm highlights the connection to the“standard method”.Error estimates for the MSLFort t0 > 0, consider the MSL approximation (p,q) = (PH,qh) E XH x definedas the solution of:Vh E Xh,satisfying p(x,to) = PHU(x,to) and q(x,to) = QHt(x,to). We analyze the differencee = eh =—(p + q). Let the weight factor o(t—t0) = min(t — to, 1). The main result is:90Theorem 3.1 (Error estimate for the MSL) there exists a constant ms1 independent ofh, depending only on u and v such thatIeW i ms1 jj2 (3.9)uIeIj + HeU msi H3, (3.10)for all I t0.The proof of Theorem 3.1 is given through a series of Lemmas and Corollaries in Subsection 3.4.2. Estimate (3.9) is proved in Lemma 3.8. The estimate < CH is proved asCorollary 3.2. Subsequently, by bootstrapping we are able to prove estimates for quantities 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 bethe constant of Theorem 3.1. Then,lIux—p—qjf smh + ms1 H2, (3.11)uWu—p— q Osm h2 + ms1 H3, (3.12)for all I 0. In particular, if H = 2Thh (i.e. the fine grid is obtained via n-refinements ofthe coarse one) then the MSL approximation is, asymptotically, as good as fine grid SMapproximation in the following sense: given any e > 0, there exist h such that if h < hthen—p—qI (sm + e)h, (3.13)—p — q (sm + €)h2, (3.14)for all I > 0.91Proof: (3.11) and (3.12) are a simple result of the triangle inequality and Theorem 3.1.IIu——q—+ Wu——qj h +smh+msiH2.un—p—q{°sm h2 + msl H3.If H = 2’h thenIux— Px—qW Csm h + Cmsi H2 = srn h + ms1 24hand choosing h = e proves (3.14). Similarly,msl—— qW Csm h2 + Cmsi H = Csm h2 + rnsiand choosing h( = e2 proves (3.13). UmslRemark 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 — q CH4.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 NGMFort to >0, consider the NOM approximation (p,q)= (pH,qh) E XH x definedas the solution of:(ps, q) + v(p + q, H,x) + (PPx + pqx + qpx, H) (f, V cbH E XH,v(qx + Ps, h,s) + (pps, ) (f, ), V E W’satisfying p(x, 0) = Fiju(t0,.). The best general result we obtain for the NGM is:92Theorem 3.2 (Error estimate for the J\TGM) There exists a constant ngm such thatiiuh— PH— qhil + HIIth, — PH,x — qh, C H2, (3.15)for all t > t0.This is a weak estimate. However, it is an improvement over the error estimate originallyproved in [20] where no order of convergence was proved. We suspect these estimates areof optimal order and our numerical results seem to back this up. If NOM approximationsdid 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:• the size of lu— PH— qhii depends on the approximability properties of the coarsegrid. Fixing the coarse grid but taking more refined fine grids (larger n), we expectun— PH— qhll Cngm H2. In other words we have no reason to expect theapproximation to improve. Our numerical tests suggest this is true (see Table 3.19and the remarks after it).• Consider the standard approximation calculated on the coarse grid, H. We knowthat un — UHIi Gm H2. It is not clear which is smaller, sm H2 or ngmH2. Itseems possible that un— PH + qhil > Ii — LHll. In other words, the approximateinertial manifold “correction” q may damage the approdmation. Our numericalexperiments seem to indicate this happens. In the “moving shock” test in— PH +qhii 3Oiiu—UHII for all grid sizes we tested (see Tables 3.16, 3.18 and Figure3.30).The cause of the problems in the analysis (and in the numerical calculations) stemsfrom the omission of terms involving the time derivative: (ps, h), (qj, kH). These terms93are 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 obtainedby copying the proof of Theorem 3.1 while keeping track of these larger terms on theright-hand side.r c3 2Remark 3.4 (A special NGM) If XH and Wh are orthogonal in the L -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 p + qwas a better approximation than p to the solution u. We can repeat this argumentword for word to show that exactly the same thing is true for Burgers’ equation. As aconsequence if Gh is the i-refinement of GH, we expect Izt—p 2u11u — p — q andIIu. —p 2u —p — qW. Our numerical results of Test 1 of Section 3.5 support thisconclusion.943.4 Proof of Theorem 3.13.4.1 PreliminariesThis subsection contains some necessary background estimates needed in the course ofproving Theorem 3.1. Some of these are special cases of estimates proved previously inSection 2.5.1.Let = (0, 1). Let {Gh} be a family of grids satisfying a uniform size condition. Fora given Gh, we associate Xh the piecewise linear finite element space. Let Ph be the L2-projection onto Xh and let R,1 be the Ritz projection onto H1. In the following le C, i =1,... , 4, be constants independent of h but perhaps depending on the uniform propertiesof the family {Gh}. The following results are well known. There exists constants C1, C2,G3 such thatMu — Phu + Mu — RhUM + h(Ilu — PhxW + llu — RhuM) < Cih2lluW, (3.16)lu - Phull + Mu - Rhull + h(llPhuM + HRhuSM) < C2hllulf, (3.17)lRhull + lPhuI < C311u11. (3.18)If ‘I’h E Hh, the inverse inequality holds:lIh,xll Ch’ llhIL (3.19)Let be the c, 3-projection where for each it E H, P,’u E Xh is the uniqueelement such that+ /3(’itr,h,x) = (u,h) + Vh E Xh.Lemma 3.1 Let a,,6 > 0, not both zero and let it E H fl 14/2,2 then there exists aconstant C independent of h, ci, /3 such thatlu — P’ull + hllu — <Ch211u11. (3.20)95In fact, C = C1, where C1 is the best possible constant of estimates (3.16).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 6In-P’u <ChIIuLI. (3.21)In particular, let P’ be the a,/3 projection onto XH and let Q/3 = I — SinceXH C Xh C H1, for all finite element function nh 6 Xh,IQuhIj = IUh - P’uh <CHIIuhI. (3.22)This Lemma is a special case of Lemma 2.2.Lemma 3.3 Let nh E X11, pj-j = P’u and qj = Qj’3u then there exists a constant Cindependent of h such thatIpHjj + IIqhII CIIuhU, (3.23)IIpH,II + Iqh, 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). Thereexists a constant °sm, independent of h but depending on n and ii such that— uH + hIIu—txIj smh2. (3.25)for all t 0. There exists a constant C independent of h hut depending on u, ii and t0such that— tII + hIInt — uj Oh2, (3.26)—ihtll + hjjntt — < Oh2, (3.27)96for all t to > 0. We know of no reference for these particular estimates. We proveestimate (3.25) in Appendix C. We do not prove (3.26) or (3.27) since we proved a similarestimate to (3.26) for the Navier-Stokes equations in Appendix B.Lemma 3.4 Let XH C Xh, let ü1, be the solution to (3.2) and let PH = FH”uh andqh = Q/3ãi = Üh— PH. For any t to > 0, there exists a constant , independent of hbut depending on u and t0 such that+ It W + jIII + IJ( + WqH + WttII) (3.28)for alit t0 > 0.We require estimates for solutions to the backwards heat equation.Lemma 3.5 Supposezt + Az = f, z(t) = 0, = 0. (3.29)ThenIIz(to)I2 -I-] jjzW2ds C] WfII2ds (3.30)to toIz(to)U2+ L(z + zIf2)ds cj fM2ds. (3.31)Proof: Multiplying (3.29) by —z one can show,_zR2 + HzxW2 <Cf2.Integrating this from to to t implies (3.30). Multiplying (3.29) by Az one can show_zx2 +11 A zW2 = CUfW2and hence,IIzx(to)112 + f II A zjJ2ds <cj IIfI2ds.97The full estimate (3.31) is obtained since 1z112 Cf A zj (a standard elliptic estimate)and sincez=WAz-fII<IjAzI+WfU.Since the domain is one-dimensional, if zz H(fl) thenItLI <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), thenjp + qjj + vf i + qj2ds <. (3.33)Proof: Choosing as test function /‘ = p + q in (3.5), (3.6) we obtain+ + vp + qxt2 + (ppx,p) + (pq,p) +2(qpx,p) = (f,p + q).Now (ppx,p) = 0 and (pqx,p) = —2(qp,p). Hence,d 2 2 2+ qj + vp + qH <CJfand integratingIIp+ q112 + vj +q2ds < cj f2ds + ju(to)U2 = D983.4.2 Proof of the TheoremIn the following, t0 > 0 is fixed. C represents a generic positive constant independent ofu and t0 while represents a generic positive constant independent of h but possiblydepending on u, ii and to. GH is a coarse grid and Gh is a fine grid with associated finiteelement spaces XH and Xh respectively. We assume XH C X,L and 0 < H < h < 1.P = P1’ and Q = are the previously defined projections. c, 3 0 are arbitrary(not both zero).Let a = Üh be the solution to the standard finite element method (3.2) and let(p, q) = (pH, q) be the solution to the Microscale Linearization method (3.7). Lete= eh =a—(pH+qh) and 1et H =p—p and 77 =7]h =—q. 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 thatlie 2+vjexW2ds <H4, (3.35)for all t to.Proof: Letting = e in (3.34) we obtainIIeW2+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 + IIpIIclll!lleItjpx + j5 W2)WeU + eiiexil2 <iiell2 + eiiexll2,99—pq,) = (q +Plx,) + 5)II77XIHI’i< C€(jq2+ [M2)UeM2 + jeM2 cMeH2 + IeM2,(ii — qp,) = (7Px + qx,) 177 coWPxWWlj + IIcIlxlIMM< Cpjj + II )Well + eMeH2 CMeM2 + II2,e) < lIe IiI Ili < eexll2 +(P + P,’i) = eej + H4.Combining these, while choosing€ appropriately, impliesMeW2 + vIIe2 <lleII2 + H4,which integrated from to to t, noting that cIt0 0, proves the result. 0We are now ready to prove the estimate (3.9) of Theorem 3.1. Actually we prove a littlemore:Lemma 3.8 there exists a constant C such thatvMesll2 + et2ds <H4, (3.36)for all t to•Proof: Letting i?L’ = ej in (3.34) impliesEstimate the right hand side as follows:(ppx —ppx,et) = (px +p,et) IIILIIilllletII + lIpIIIIxIIIIetII(IIll2+ IpII2)IleII + eIIetjj2 <llexII2 + ejletII2,(pqx —pq,t) = (q +77x,t) IIlIIqIllltII + lpIIll77IllltII100C(q2+ l[xII2)IIexII2 + c{letll2 llCrU2 + clIejII2,(j5— qp,t) = (llpx + x,t) lillco PlHlW + llcoxlltW< C(p + II 2) We U2 + etll2 <llexll2 + etlj2,—(,et) = —(,e)+ (@,e)+(,t,e)< ——(, e) + C(lltll lll llell + IlB ll,II lIexlI)d—< —-(, e) + Gil6 + Wexll2,—(p + = (1x,t,) = (i7x,i5)—(t77x,P)—5) + C(jl Ili5U lL’iU + llll—(ii) + H4 + 2.Combining, choosing the appropriate , implies thatvlexlJ2 + jetII2- + e)} + IieU2 + H4.Integrating this inequality from to to t, noticing that e(•, to) 0, we obtainvlIex 112+ j let Il2ds < ll(P)U + H( e)II + OH4. (3.37)Now,WexH2+H4,(lll +CH6which, combined with (3.37), proves the Lemma. UIt is helpful to consider s = j3 + E Xh satisfying(Si, ) + v(s, ) + (j5 + pq + qp, q5) = (f, ), for all q XH, (3.38)(Si, ) + v(s, ) + (ppx, ) = (f, ), for all E W, (3.39)with (x, t0) = Pã(x, t0) and (x, 0) = ü(x, t0).101Remark 3.5 One should not get confused in the definition of s. ji = = arefrom the solution to the standard method (3.2) while p and q are from the solution to theMSL method (3.7).Let ê = ü — .s =—j3 + — = + . Subtracting (3.38), (3.39) from (3.2), one obtains:(at, b) + v(e, x) + (j5 — ppx, )+ (3.40)—pq+P1— qpx, )+(, )+( + = 0, V E X,and ê(to,.) = 0.Lemma 3.9 There exists a constant C such that(3.41)2+ê2 (3.42)for all t t.We will not supply the details of the proof of Lemma 3.9 since the proof of (3.41) issimilar 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 thatiIeI2 <CH° (3.43)for all t t0Proof: Consider the solution to the backwards heat equationZt + z A z = ê, z(t) = 0, zIaO = 0. (3.44)Multiplying by ê and integrating over the domain results indIeU = (zt, e) — v(z, e) = —(z, e) — [(et, z) + ii(er, zr)]102Notice that choosing ‘ = Pjz E XH as test function in (3.40) then(at, PHz) + v(e, PHz) + (iNs — pq + — qp, PHz) + PHz) = 0,since c(PHz) = 0. Hence,hell2 = (z, e) — [(at, z—PHz)+(ê, P11z)+(, PHz).Estimating:Itll liz - PHzlh <CIIêjWH2WzU< €z + CcH4Wethi2,iiêhlWz - PHzXW €z + CH2WêU,= (+p,Pz) = (C,PHz) — (p(PHz) — (PHz)p,)< ChlPHzgW(hlii ihl + iip hlii) <cHzhh + CH2ileli= (p + ,PHz) <ChIPfjzli(lilllill + llpUiihl)< jjz + GH2WeiiPHz) iiFHzXhl2+ H6 Ilzii + H6,and combining we obtainell2 <(z, e) +H2(lleii + hiehl2 + H4) + H411Ct112 + iizll.Integrating from to to t, noticing that e(., to) = z(., t) = 0, and using estimates fromLemmas 3.8 and 3.9 we obtainf hlêhids H6 + hlzhlds.Using the backwards heat equation estimate (3.31) and choosing appropriately provesthe Lemma. 0(êt,z — PHZ)(&, z — PHz)(p:— pqx, PHz)(q— qpx, PHz)103Lemma 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’ = weobtain:ld—-Hl2+ vII2 = (iiiNow(ppP,) = (Px+&x+x+x,)= (px+px+px,) — (px+px,e)< ejjj + jj{2 +and hence+ vl2 <2 + eW2. (3.46)Integrating from to to t, using Lemma 3.10 proves the result. DLemma 3.12 There exists a constant C such that2 <H° (3.47)for all t > t0.Proof: This is a direct consequence of Lemmas 3.10 and 3.11 sinceI2 2f weH2 + 2)ds <H°. DAt this point estimate (3.10) of Theorem 3.1 is a simple corollary.104Corollary 3.2 Estimate (3.10) is valid: there exists constant such that ejf CH,for all t > t0.Proof: Since,Ile!12= : e2ds <2 (L IesI2ds)2 (ft eds) ,(3.10) follows because L eW2ds <?iH6 and j e32ds <H4. 0The rest of the section is devoted to extending this partial result for theL2—norm tothe full estimate (3.10). In order to do this we consider estimates for time derivatives ofthe error. A major step is to prove fu2êtUds, Lemma 3.15, because having obtainedthis, estimate (3.10) is a simple consequence. Differentiating the error equation (3.34)with respect to time we obtain:(ett, ) + v(e, ) + (jkt5 — ptp + j5t — pp, b)(ptqx —ptqx+pqtx —pqtx+qpx — qtp +qptx — qpt, P’/) (3.48)+(qtqx + qqt, ) + (ptqx+pqtx+qtpx+qx, Qk) = 0, V /) E Xh.Lemma 3.13 There exists a constant such thatIetII2 + V ujet2ds <H4, (3.49)for all t > t0.Proof: Letting b = e in (3.48) we obtainld 2 2IetW -f-vWeW =—[(ii — PtPx + PPxt — PPix, et)+(p—ptqx +j5 —pqtx +tj5—qtp +—(qtq + qq, et) + (ptqx+pqtx+qtpx+qpjx, 7t)j.105ci 2 ci d ‘‘2-aejW = + aetW2 < je + uetIl,Copying the proof of Lemma 3.13, one can show:106(3.50)Estimate each term on(PP —ptpx,et)(Ppxt —pptx,et)(ptqx—ptq,’t)(pq—pqt, t)(qtpx —qp,(qptx — qp,(qtqr+ qqt,et)(ptq + qptx,(pqt + qtpx, lit)the right hand side as follows:= (tPx + Pt, et) (IIetII2 + H4) + eIIetII2,= (&tx + + tx, et) (UetW2+ H4) + IetW2,=== (liipx + (UeW2+ H4) + eejf2,= ( tx+liPtx+tx,i)(et+H)+eet< eWetU2+GH6,= (7t,j5) CH + cjjet2,= <GH4+ eIet2.Combining and choosing the appropriate e implies+ vetW2<eI2 + H4.Noticeand hence,aIIetW2 + vuWetW2<CjIe2+ etII2 + CaH4.Integrating this, noting that liminfoetW2= 0 we obtain the result. 0t—oAs before, consider e = ê + ë. Differentiating (3.40) with respect to time:(tt, ) + u’(eS, + (tx — PtPx + PPtx PPtx, Q)+(pq— ptqx+pqtz — pqtx+qtpx— qtpx+qptx— qptx, P/))+(qtq + qqt, )+(ptqx + pqtx + qp + qpt, Qb) = 0, V E Xh.(3.51)(3.52)Lemma 3.14 There exists a constant C such that+ vfe2ds (3.53)for all I t0.Lemma 3.15 There exists a constant C such that2WêjWds H6, (3.54)for all I > to.Proof: Consider the solution to the backwards heat equationzL + Az = Jt, z(L) = 0, Z8O = 0.Multiplying by t and integrating over the domain results indaljetl = (zi, Ct) — (zr, e) = —(z, et) — [(eu, z) + zr)]Using as test function L’ = PHZ Pz E V in (3.52) implies, noting that (Pz) 0,(tt, Pz) + v(êtT, Pz) +— ptqx + 15tx — pqtx, Pz)+(ji— qtpx + j5ix—qpt + Jt + Pz) = 0.Hence,aUêtW2 = (zt, t) — (zr, êj) = (z, t) — [(tt, z — Pz) + (tr, z — Pz)] +(Pt — — — qp+j5t — qptx, Pz) + (tx + ts, Pz).Estimate the right hand side:(tt, z — Pz)=z — Pz)—(at, zj — Pz)=z — Pz)—(t — Pat, Zt)107(ê,z — Pz) + Ztjjt— FêtW< z — Pz) +(et,z—Pz) WeIHIz—PzlI(tq — ptq, Pz) = (tqx + Pz) = (tqx, Pz) — (5(Pz) + (Pz)pt, 1)< ClIPzgj(ftW Wq + W HW) IIzW + CH2 ( etxI2 +(p— pqt, Pz) = (tx + P?ltx, Pz)< CIFzW(W WixII + PIjj71txI) Zj + C2 (WII +(i5 — qtp,Pz) = (‘iji — 77j + qtr/,Pz)CWPzW(II71 + xjII71tW +h2j)+ H2 (W71 U2 +(q5—qp,Pz) =UzW + H2 (U71tXU2 +(i?j,Pz) < C Pz jjj CH6 + zU(t,Pz) CPzt H6 +Combining, choosing the appropriate , we obtainu)IêtU2 < Pz) + H2 [UeW2 + UetxU2] + H6 + (WzIl + WztW2). (355)We need to multiply this by and integrate. With this in mind noticeu(et,Pz) = u(êt,Pz) — ‘(êt,Pz) = u(et,Pz) — u’(ê,Pz) + ‘(ê,Fz).Consider first t to + 1. In this case a’ = 1 andL Pz) — (e, Pz) + (e, Pz)]=(U(t, Pz)—(e, Pz))0 + j(e, Pzt)ds108= L(e,Pzt)ds f Weds + ef IIztII2ds<H6+ ef2WêWds.Thus if t < to + 1, multiplying (3.55) by and integrating from t0 to t, we see thatLa2et2d5 <H + eJ2etI2dsand choosing e appropriately proves the result if t to + 1. Using this one can showBe(to + 1)W H3 (copy the proof of Corollary 3.3 below) and by the backward heatestimate (3.30)‘.to+iIz(to + l)j2 <J u2IIetds.ioNow suppose t > to -I- 1.Fz) — a’(ê, Pz) + u’(ê, Pzt)]ds0+l d t0+1= U(t, Pz)I—f (e, Pz)ds + f (e, Pzt)ds<— (ê,Pz) + cJ°’ êI2ds + ef jfztH2ds<(e(to + 1), Pz(to + l))I + H6 + ef2IIêiIds< H6 + eWz(to + 1)2 +€j u2êds <H6+ ef2IiêtWds.Now, an appropriate choice of e proves the Lemma. UCorollary 3.3 Estimate (3.10) holds: there exists a constant such that ueW <H3,for all t t0.Proof: Now,uIIeW2 = uIlêIIds= f (a’e + 2ulWe3I)ds2f jêW2ds + Lu2eS2d5 H6.This combined with the fact that proved in Lemma 3.11, implies the result. U1093.5 Numerical resultsWe test the schemes in several ways. We compare the L2-, H’- and L°°-norms of thedifference between the solution u and the MSL approximation, the MSL error, the difference between the solution u and the NOM approximation, the NGM error and thedifference between zt and the standard approximation, the SM error. We also considerthe qualitative properties of the solutions. We considered several solutions to calculate:Test 1, two steady exact solutions, Test 2, an interesting “moving shock” solution, Test3, a time periodic solution.Test 1The first test is actually two different steady calculations. We considered:• the exact steady solution zt = sin(7rx). This is an exact solution to (3.1) if v’ = ir2and f = sin(irx) + ir sin(irx) cos(7rx).• The exact steady solution u = sin(lO7rx). This is an exact solution if z’’ = 1007r2and f = sin(lOirx) + lOir sin(lO7rx) cos(lO7rx).We considered these steady calculations for several reasons. In a steady situation, theNG 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 smallenough, since Theorem 3.1 holds. We considered only the case where H = 2h, i.e thegrid Gh is one refinement of GH. From the results we conclude:• if H was small enough, the L2-, H’-, and L-norms of u — p — q was basically thesame size as the the L2-, H’-, and L°°-norms of zi—tth. As we expect the order ofconvergence is 0(h2) in the L2 norm and 0(h) in the H’-norm. In the L-norm itis 0(h2).110• We also see that p + q is a better approximation to u than p. This confirms whatwe stated at the end of Section 3.3. In fact, as we argued, Lt— pM 4Mu — p — qand— iM 2u — p — qM111h Bu—t rate— uj rate.0179237 .4058923.0044560 2.01 .2018046 1.01.0011129 2.00 .1007732 1.00.0002782 2.00 .0503708 1.00.0000695 2.00 .0251835 1.00.0000174 2.00 .0125915 1.00.0000043 2.02 .0062957 1.00Table 3.1: Test 1, u = sin(irx); errors from standard method:H h u—p — q rate fu — p — q rate.0043430 .20216165 10-i--- .0011040 1.98 .1007853 1.0010 201 1.0002776 1.99 .0503712 1.00o1.0000695 2.00 .0251835 1.0040 80I-—.0000174 2.00 .0125915 1.0080 160L.0000043 2.02 .0062957 1.00-_320Table 3.2: Test 1, u = sin7rx); errors for NGM, p + qH h u—p rate UxP rateI 1.0225651 .40061815 10-i-- --.0057323 1.98 .2011727 .9910 201 J-.0014390 2.00 .1006950 1.0020 401 1.0003601 2.00 .0503611 1.0040 80-I-.0000901 2.00 .0251822 1.0080 1601 1.0000225 2.00 .0125913 1.00160 2_______Table 3.3: Test 1, ‘u = sinQ7rx); errors from NGM, p1 1h {{u—uW rate u—uj rate.0191726 5.1582378.0042924 2.16 2.5314807 1.03.0010627 2.01 1.2607378 1.01.0002653 2.00 0.6297691 1.00.0000663 2.00 0.3148099 1.00Table 3.4: Test 1, u = sin(107rx); errors from SMH h Mu—p— q rate Hzt — p — qII rate.0427327 5.607243720 40-- --.0078798 2.44 2.6793891 1.0740 80L—i-- .0011190 2.82 1.2763767 1.0780 1601 1.0002657 2.07 0.6303594 1.02i1.0000663 2.00 0.3148289 1.00.320 6DTable 3.5: Test 1, ‘a = sin(l0irx); errors from NGM, p + qh Mu—pM rate M’u—pM rate1.0990926 9.9974301ö 40-i- 1.0266822 1.89 5.0106141 1.0040 80I L.0078606 1.76 2.5132876 1.0080 160I.. —i---.0020063 1.97 1.2584812 1.00160 3201.0005035 2.00 0.6294877 1.00-640Table 3.6: Test 1, ‘a = sin(10rx); errors from NGM, p + q113Test 2The next test is a more complicated time dependent one. We did not want to resortto a time dependent force to produce such a solution. We chose to use constant nonhomogeneous boundary values. It is hard to find a time dependent problem, let alone aninteresting one, that has an exact explicit solution without resorting to a time dependentforce. We overcome this by calculating a very good approximation (basically an exactsolution) using the standard method and use this for comparison.We consider (3.1) with ii = .01, f 0, non-homogeneous boundary values u(t, 0) =and u(t, 1) = — and initial condition uo(x) = — 2x. Initially, the solution is smoothbut quickly develops into something resembling a moving “shock”. The “shock” movesalong for awhile at a steady speed then slows down fairly abruptly at around t = 1.1;by t = 1.2 it is nearly steady-state. To calculate the “exact” solution, zt, we used theCrank-Nicholson scheme with an extremely small time step k = .0001 and we used a-0.6v 0.2 0.4 0.6 0.8Figure 3.27: “moving shock”; A series of snapshots of the “exact” solution114very refined spatial grid with h= 5120• We stored the data every .1 unit of time (every1000 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 MSLon coarser grids. In these calculations, to limit possible outside sources of error, we againused the Crank-Nicholson scheme with k = .0001 and forced the defect in the nonlineariteration to be very small (llDW/JD0Ij < 1014) (see Section 3.6). It is worth noting thatsince the initial condition is linear, it is in the finite element space Xh and there is noinitial source of error in any of the calculations.115+ h—L h——i—— 80 160 — 320 640 12800.3 0.000149 0.000037(2.0) 0.000009(2.0) 0.000002(2.0) 0.000001(1.9)0.6 0.003112 0.000783(2.0) 0.000196(2.0) 0.000049(2.0) 0.000012(2.0)0.9 0.003793 0.000957(2.0) 0.000240(2.0) 0.000059(2.0) 0.000014(2.0)1.2 0.006598 0.001895(1.8) 0.000483(2.0) 0.000120(2.0) 0.000029(2.0)Table 3.7: “moving shock”; standard method u——-- h—-1- h—--—ime— 80 — 160 — 320 — 640 — 12800.3 0.017253 0.004547(1.9) 0.001155(2.0) 0.000290(2.0) 0.000073(2.0)0.6 0.038689 0.010298(1.9) 0.002619(2.0) 0.000657(2.0) 0.000164(2.0)0.9 0.030483 0.008092(1.9) 0.002068(2.0) 0.000520(2.0) 0.000130(2.0)1.2 0.010111 0.001978(2.4) 0.000495(2.0) 0.000121(2.0) 0.000029(2.0)Table 3.8: “moving shock”; NOM u—p—q, H = 2h+ 1_i i_L i_L h—L80 — 160 — 320 — 640 — 12800.3 0.000178 0.000040(2.2) 0.000010(2.1) 0.000002(2.1) 0.000001(1.9)0.6 0.003678 0.000789(2.2) 0.000195(2.0) 0.000049(2.0) 0.000012(2.1)0.9 0.004679 0.000969(2.3) 0.000239(2.0) 0.000059(2.0) 0.000014(2.0)1.2 0.009972 0.001964(2.3) 0.000481(2.0) 0.000120(2.0) 0.000029(2.0)Table 3.9: “moving shock”; MSL u—p — q, H = 2h116‘— 80 — 160 — 320 — 640 — 12800.3 0.0561611 0.0280714(1.0) 0.0140152(1.0) 0.0069663(1.0) 0.0033992(1.0)0.6 1.0739022 0.5397615(1.0) 0.2699064(1.0) 0.1342100(1.0) 0.0654942(1.0)0.9 1.3043985 0.6555578(1.0) 0.3278527(1.0) 0.1630288(1.0) 0.0795584(1.0)1.2 2.3722568 1.3147671(0.9) 0.6650024(1.0) 0.3315437(1.0) 0.1618976(1.0)Table 3.10: “moving shock”; standard method ——I-—— i. I 1. 1 i._i i._1. “f — flf—‘tf ‘If—‘tf—T50.3 1.1827644 0.6049941(1.0) 0.3045043(1.0) 0.1525138(1.0) 0.0762865(1.0)0.6 2.9016019 1.3544537(1.1) 0.6559526(1.0) 0.3245924(1.0) 0.1612361(1.0)0.9 2.9392503 1.4550711(1.0) 0.7186771(1.0) 0.3575114(1.0) 0.1776831(1.0)1.2 2.6105484 1.3159412(1.0) 0.6652290(1.0) 0.3315514(1.0) 0.1618978(1.0)Table 3.11: “moving shock”; NGM— — qg, H = 2h—I-—— i. I. , 1 1I IIf— ‘if—j ‘tf— Itfj ‘tfjjj0.3 0.0563524 0.0280773(1.0) 0.0140154(1.0) 0.0069663(1.0) 0.0033992(1.0)0.6 1.1122328 0.5414911(1.0) 0.2699582(1.0) 0.1342116(1.0) 0.0654943(1.0)0.9 1.3633529 0.6580272(1.1) 0.3279296(1.0) 0.1630311(1.0) 0.0795585(1.0)1.2 2.6830306 1.3252708(1.0) 0.6652570(1.0) 0.3315517(1.0) 0.1618980(1.0)Table 3.12: “moving shock”; MSL IIz——H = 2k117T‘ — 80 — 160 — 320 — 640 12800.3 0.0005126 0.0001287(2.0) 0.0000322(2.0) 0.0000081(2.0) 0.0000020(2.0)0.6 0.0204400 0.0059386(1.8) 0.0015617(1.9) 0.0003935(2.0) 0.0000990(2.0)0.9 0.0245810 0.0080409(1.6) 0.0020618(2.0) 0.0005168(2.0) 0.0001301(2.0)1.2 0.0709505 0.0216504(1.7) 0.0062738(1.8) 0.0016109(2.0) 0.0004072(2.0)Table 3.13: ‘moving shock”; standard method u —1. 1 i._1 1 1 1. 1 L 11If— itf—1 (tf0.3 0.0611453 0.0160487(1.9) 0.0040565(2.0) 0.0010177(2.0) 0.0002546(2.0)0.6 0.2657431 0.0804664(1.8) 0.0214517(1.9) 0.0054355(2.0) 0.0013621(2.0)0.9 0.2271326 0.0719312(1.7) 0.0196385(1.9) 0.0050131(2.0) 0.0012550(2.0)1.2 0.0926590 0.0225577(2.0) 0.0064216(1.8) 0.0016213(2.0) 0.0004080(2.0)Table 3.14: “moving shock”; NGM In—— qilco—I-—— z. ....1 1. 1 j. 1—--— IIf— tf— itf—j “fTö0.3 0.0006368 0.0001392(2.2) 0.0000330(2.1) 0.0000081(2.0) 0.0000020(2.0)0.6 0.0226275 0.0065323(1.8) 0.0015584(2.1) 0.0003923(2.0) 0.0000989(2.0)0.9 0.0284915 0.0080536(1.8) 0.0020347(2.0) 0.0005150(2.0) 0.0001300(2.0)1.2 0.0867396 0.0230547(1.9) 0.0061648(1.9) 0.0015969(2.0) 0.0004066(2.0)Table 3.15: “moving shock”; MSL In—p — qIIco118Clearly the NO method has large errors when the solution is nonsteady. Focus on the2 u—pH—qh ‘u—pH—qhL -data at t = .3 and consider 7h = - and 7H = - from both theNO and MSL data.h—’ j 1 1— 80 — 160— 320 — 640 — 1280SM 0.0001489 0.0000372 0.0000093 0.0000023 0.0000006NGM 0.0172531 0.0045474 0.0011553 0.0002901 0.0000726MSL 0.0001776 0.0000396 0.0000095 0.0000023 0.0000006Table 3.16: L2-errors at t = .3j_ 1 j_ 1 1— 1 j_ 1 ,_ 1“— 80 — 160 — 320 — 640 — 1280NGM 115.8 122.9 124.2 126.1 121.0MSL 1.2 1.1 1.0 1.0 1.0Table 3.17: Yh values at t = .31 j_ 1 j_ 1 j_ 1160 — 320 — 640 — 1280NGM 30.5 31.1 31.2 31.6MSL 0.27 0.26 0.25 0.25Table 3.18: 7H values at t = .3As h —* 0, for the MSL ‘y —* 1. This is what we expect in virtue of Theorem 3.1. On theother hand for the NGM 121. The error from the NC approximation is over 120times larger than that of u,!! In fact, 7jq 31. This implies the NCM approximationis 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 thewiggles of the NGM approximation in Figure 3.30.1191.50.50-0.51.50.50-0.50.2 0.4 0.6 0.8Figure 3.28: standard method, h =Figure 3.29: standard method, ii =1201.50.50-0.501210.2 0.4 0.6 0.8Figure 3.30: NOM, H =-- h — --40’ 800.2 0.4 0.6 0.8Figure 3.31: MSL, H h —40’ 80The following are some results for more general cases where Gh is obtained by refiningGH two or more times. In particular, we have results for the s-refinement case H = 4kand the 3-refinement case H = 8k. Table 3.19 tabulates the L2 errors, at time t = .3, inall the relevant cases.j_1 j,_ 1 j, 180 — 160 — 320 — 640 — 1280SM 0.0001489 0.0000372 0.0000093 0.0000023 0.0000006NGM H = 2k 0.0172531 0.0045474 0.0011553 0.0002901 0.0000726H = 4k 0.0727605 0.0148293 0.0055969 0.0014272 0.0003587H = 8k 0.0749600 0.0218624 0.0058501 0.0014946MSL H = 2k 0.0001776 0.0000396 0.0000095 0.0000023 0.0000006H = 4k 0.0013286 0.0001517 0.0000198 0.0000031 0.0000006H = 8h 0.0014745 0.0001618 0.0000188 0.0000023Table 3.19: L2-errors at t = .3, H = 2’k• The NGM does not improve if we take more levels. We do a lot more work to obtainworse results. For example, consider the three cases when H = . For the 1, 2,3-refinement cases (h = , h = h = ),the errors are: 0.0045474, 0.0055969,0.0058501. When H=the errors are: 0.0011553, 0.0014272, 0.0014946.• For the MSL approximation—if H is “small enough” (see below) Iu—pH—qhfMu ãhM in all cases.• However, the MSL approximation is only good if H is small enough. As we seein Figures 3.29, if k = , the resulting standard method approximation resolvesthe true solution quite well. On the other hand, if k = the standard methodapproximation does not resolve the solution very well at around time t = 1.2 (seeFigure 3.28). Consider the MSL approximations pictured in Figures 3.31, whereH = and h = (1-refinement), and 3.32, where H = and k = (2-refinements). The approximation in Figure 3.31 does not resolve the steep gradient122of the solution when t = 1.2. The approximation in Figure 3.32 is just not good atall. 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 = Aswe see in Figure 3.33, where H = and h = (3-refinements), the situationdoes 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 approximationcalculated on this grid basically fully resolves the problem. With this limitation, itis hard to see a great use for MSL.1231.5-0.521.50.50-0.50.50.2 0.4 0.6 0.8Figure 3.32: MSL, H = --- h —20’ 80Figure 3.33: MSL, H = I h — 120’ 160124For interest, we checked to see if the numerical results would reflect the higher orderconvergence we proved in Theorem 3.1. Tables 3.20, 3.21 and 3.22 confirm is higher orderconvergence for the difference ü—p — q. Interestingly, in the one-refinement case ourestimates do not seem to be optimal, though they seem to be in the other cases. We donot have an explanation for this.i,_I i__L i—L i—._L80 — 160 — 320 — 640 — 1280H=2h .0000474 .0000042 (3.5) .0000003 (3.8) .0000000 ( - ) .0000000 ( - )H=4h .0012654 .0001347 (3.2) .0000152 (3.1) .0000018 (3.1) .0000002 (3.2)H=8h .0014601 .0001584 (3.2) .0000180 (3.1) .0000021 (3.1)Table 3.20: higher order convergence of—p — q at t = .3h—-i- h—---- h—-1-- h——1-80 — 160 — 320 — 640 — 1280H=2h .0045459 .0005632 (3.0) .0000702 (3.0) .0000088 (3.0) .0000011 (3.0)H=4h .1217086 .0300208 (2.0) .0074605 (2.0) .0018622 (2.0) .0004653 (2.0)H=8h .1375035 .0342822 (2.0) .0085406 (2.0) .0021330 (2.0)Table 3.21: higher order convergence of——at t = .3h—1 h—_L h—-1-- h—--1--80 — 160 — 320 — 640 — 1280H=2h .0002144 .0000171 (3.6) .0000012 (3.8) .0000001 (3.6) .0000000 ( - )H=4h .0074887 .0008203 (3.2) .0000923 (3.2) .0000104 (3.2) .0000012 (3.1)H=8h .0078402 .0008597 (3.2) .0000955 (3.2) .0000106 (3.2)Table 3.22: higher order convergence of IFu—p — at t = .3125Test 3We considered the exact solution u(x,t) = sintsinirx which is a solution to (3.1) with- and f (time-dependent) suitably chosen. Notice:0.00050.00045 ++ +÷ ++ ++*++ + ++ + + +++ + *+0.0004++0.00035 / ÷ + ++ + + ++ + + + ++0.0003 + +++ + -f+ +0.00025 + ‘S ++ S÷+ / ÷“+7 +‘+0.0002 + “. + N5\,+5’ ± + 5,‘S *0.00015 + + “S 5” + +/ + + 5, S’ + +S’S ++ “S5” ÷ + “S0.0001 ‘ +‘,-“ + +5oo:TIMESM llu—uhSM llu—uHll+++ NGM u—p—qllMSL ju—p—qFigure 3.34: u = sintsinirx, comparison of L2 errors• The size of lu—— qj from the MSL is the same as lu — thll.• The size lu—p— ll from is correlated to the time derivative of the solution. Attimes where the solution has a large time derivative, around the times t = 0, t =and t = 2ir, lu—p— ll is extremely large. Where the time derivative is very small,around t = and t = , this same term is about equal to lu — ZLhII.1263.6 ImplementationConsider first the MSL method (3.5), (3.6). As before, consider one-step theta schemesfor temporal discretization. In our calculations, we mainly used the Crank-Nicholsonscheme (0 ). Applying a one-step-theta scheme reduces (3.5), (3.6), to finding (pfl+l,qfl+l) (p’, q’)e XH x W, such that(pfl+l+qfl+l, q) + kOv(p1qq)+kO(p’p’ +p1qq”’p’, q)=g’(q!), Vq e XH (3.56)(p”’+ q’,)+kOv(q’+p’, ) + kO(p’p’,)= 1(),V E W, (3.57)with p° = Pu0 and q° = u0, wheregfl+l((pfl q )+kO(f)+k(1-0)[(f )-v(p+q,andfl+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. LettingW= { 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) =g1(q5) V e XH (3.58)(qfl+l, )+k8v(q’, ) + k8(pp1,) = h’() V W, (3.59)with p° = iu0, q° = u0 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 correction127method to solve (3.58) and most importantly, we can solve for (3.59) since its solutioncan be obtained as the difference of a solution to a generalized Stokes problem and asolution to a certain projection problem. The solution is obtained by letting q = —where E Xh, t7 E XH. is the solution to(bh) + k6v(q,h,) (h) — k8p1p’,bh), Vbh E Xh, (3.60)while is obtained by solving(, H) + kv0(, H) = (, H) + kv8(, 6H) V H E XH. (3.61)Lemma 3.16 Letv = where is the solution to (3.60) and the solution to (3.61)then q = v is the unique solution to (3.59).Proof: The solution to (3.59) is unique for if w W, satisfies(w, ç) + k0v(wr, = 0 Vq e j473thenjwj2 + kv0jwxW2=0and w 0. From (3.61) it is clear v e 147,. Using this and (3.60) implies(v,h) + k0(v,h) = (,h) + k0v(,h,) = h) — k0(p’p’,h),c3for all h E Wh . Hence, q = v.Consider the NGM (3.3), (3.4). Recall that in the Marion/Temarn scheme one usesthe 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 H e Wh. Applying a one steptheta scheme, reduces (3.3), (3.4), to finding (pTl+l, qTh+l) (p’ q’)e X11 x Wh, such128that(pfl+l,g)+ kOv(p’,q)+kO(p’’p1+ p’’q’+q4p’, ç)=g’(ç’) Vq e XH (3.62)v(qfl+1,) + (pfl+lpfl+l,)=(ffl+l,) Vq e Wh, (3.63)with p° Pu0 and q° = Qu0, whereg’()=(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 simpleto solve since we have a basis for Wh.129Chapter 4ConclusionPhilosophically, we do not see a turbulence model at the basis of the two-grid finiteelement methods we studied. Rather we see the NGM and the MSL as methods obtainedfrom the standard finite element by dropping certain small terms. One hopes that indoing so, computational time can be saved. Right from the start, for any of these two-grid methods, it is not clear to us how much computational time — if any — there is tosave.Nevertheless, we can make some definite conclusions about which of the two-gridmethods should be and which should not be considered. Our theoretical results suggestone 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 orthogonalin theL2-inner product.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 approximations calculated using these schemes will not be very accurate and hence the schemesthemselves 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 someideas. Firstly, consider the special NGM. This scheme may not be practical because we130suspect 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 onlybe proved (or disproved) by actually testing the method. This leaves one to considerthe 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 meana 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 interestto test it. However, even for this case we have some doubts. We have some numericalevidence indicating that the asymptotic range where the approximations are “good” isnot practical. Our numerical results in Section 2.4 and our results for the MSL forBurgers’ equation seemed to indicate the solution is good only if the coarse grid is itselfvery refined.131Appendix AProof of Theorem 2.2In the following, 0 < t0 < T < oo are fixed. C represents a generic positive constantindependent of h, u and to while represents a generic positive constant independentof h but possibly depending on u, v, to and T. (Both C and may depend on Q andsome parameters, uniform in h, associated with the family of grids). GH is a coarse gridand Gh is a fine grid with associated finite element spaces XH and Xh respectively. Weassume XH C X, C H and 0 < H < h < 1. P = P’ and Q = are the previouslydefined projections. , 3 0 are arbitrary (not both zero), though for the special case= 1, j3 = 0 we have proved Corollary 2.2.Let eh = ãh— (PH + q) and Ch = lrh — 7rh where Üh, 7Th is the solution to thestandard finite element method (2.5) calculated on the fine grid and PH, qh, ri is thesolution to the NGM (2.8), (2.9), (2.10). Let P = P and QH = Qf be the previouslydefined projection operators and let j5 = Pa, = QU, eN = Fe = PH — PH andTlh = Qeh = qh — qh.Theorem 2.2 (Error estimate for the NGIY1 There exists a constant such thatueW + Hejj + HjVeW + uHHCjfor all t > to.The construction of the proof is very similar to the construction of the proof of Theorem2.1 (see Subsection 2.5.3). What is new is the presence of some terms involving the timederivative. These terms are of the form (ps, (j, ‘) where ‘ç& E Xh. In all estimates132(Pt, ‘) (or some version of it), is the leading order term. By leading order, we meanthat for some fixed test function & (chosen at the time) the absolute value of (j5i, Qb)is majorized with the lowest order H. e, satisfy:(et, ) + a(e, ) + n(p, p, b) — n(p, p, ) + n(, , b) — n(q, p,+ n(,,) - n(p, q,) + b((, ) = Pt, + P) (A.i)— n(,ob) — n(p,,) — n(p,b), V E Xh,b(qh,e) = 0, Vqh ewith e(.,to) = 0.Lemma A.1 There exists a constant C such thatieii2 + V jiVell2ds <H2, (A.2)for all 0 <to <t < T.Proof: Letting ‘zb = e in (A.i) we obtainid 2 2llell + II Veil (Pt, Qe) + (, Pc) — [n(, p, e) — n(p, p, e) + n(, p, )—n(q, p, ) + n(, , ) — n(p, q, )] — (n(, i, e) + n(p, , ij) + n(, p, ij)).Estimating the right hand side as follows:H2 + eli Veil2,< H4 + eli Veil2,[n(p, p, e) - . n(p, q, )] < iieil2 + il Veil2,(n(, , e) + + n(, p, )) < GH4 + eli Veil2.Choosing e appropriately and integrating from to to t proves the desired result. D133Lemma A.2 There exists a constant such thatvI Veil2+ f ejds <H2, (A.3)for all 0 <to <t < T.Proof: Letting ç’ = et in (A.1) impliesvd 2 2lIVeil + lletU = (Pt, Qet) + (, Pet) — [n(p, P, e) — n(p, p, et) + n(, p, )—n(q, p, ) + n(p, , ) — n(p, q, — (n(i, , e) + n(, , m) + n(, p, i)).Estimating:(Pt, et) = (Pt, e)— (pa, e) (pj, e) + H2 + VelI2,Pee) < CH4 + eWe2,[n(p, p, et) — C.€llVell2+ eWet,d — —(n(, , et) + ) {n(, , e) + n(p, , ij) + n(, p, ij)} + CH4 + CWVeW2.Combining, choosing e appropriately,vBVeW2+iletW2 Vejl +H2— {(Pt, e) + n(, , e) + n(p, , ij) + n(, p,Integrating,IIVeW2 + jt lletlIds H2 + (Pt, e)j + ln(, , e) + n(p, , )I + n(, P, .Now,I(Pt,e)l + + ln(,p,)l <H2+ lI Veil,which proves the lemma. DLemma A.3 There exists a constant C such thatalIetII2 + v f allVetll2ds (A.4)for all 0 < to < t < T.134Proof: Taking the time derivative of (A.1) and choosing as test function b = et in theresulting equation, we obtain:WetII2+yIIVetII2 (Pt, et) + pet)— {n(Pt, P, et) — n(pt, p, et) + n(p, Pt, et) — n(p, Pt, et) + n(j, Pt, — n(q, Pt,+fl(qt, j5, ) — n(qt, p, + ri(Pt, i, ) — n(pt, q, ) + n(p, t, ) — n(p, q,— (fl(t, j, et) + n(Pt, , j) + fl(t, p, ij) + n(i, t, et) + n(p, t, ij) + n(, 15t, 1))Estimating:(Ptt, et) < Cpj2 + eIIvetII2(jj,Pet) < GH + eWVetl{2[n(Pt, ji, e) - ejVe2 + Iet2 + jVej2)(n(t,,et) +) + efjVe2.Combining and choosing the appropriate e impliesIIetU2 + vJ!VetB2 <etI2 + 2,Therefore,+ vaWVetM2<aetI2 + WetII2 + aH2and integrating, observing that the 1irninfuletW2= 0, proves the result. 0Lemma A.4 There exists a constant C such thatIICII2ds < (A.5)WlI2 < (A.6)for all 0 <t0 < t < T.135Proof: By assumption, Xh and Lh satisfy the inf-sup condition andWCHIIVII C supi1, EXhConsider the error equation (A.1):—b(C, ) = (et, ) + {a(e, ) + n(, , ) — n(p, p, ) + n(, ,) — n(q, p,n(p, , b) — n(p, q, P) + n(,, ) + n(, , b) + n(, ,+(, b) + (, ).Estimating:(et, &) CIIeII[a(e,b)+...n(,p,b)] <(p,b) <(i,P) < H2WV.Thus,1CM (MetM + H)and estimates (A.5), (A.6) are easily proved using our previous results. DLet (s, *) E Jh x Lh be the solution to(Psi, P) + a(s, ) + b(, ) = -n(, Vp, P)-n(p, q, Pb)-n(q, p,) - n(p, p,) + (f, ), (A.7)for all ib Xh, with (x,to) = Pã(x,to) and (x,O) = ã(x,to). Letê=a—s, C=—*, =Pê and ñ=Qê.Similarly, letë=p+q—s, C=ir—*, e=Fe and ij=Qé.Notice e = ê — è and= —. ê, satisfy:(Pe,) + a(ê, ) + b(, ) = -(Pt,) - (t, P) - [n(p, P,)—n(p, p, b) + n(, p, P) — n(q, p, P) + n(p,,) (A.8)—n(p, q, — n(,, ) — n(p, ,) — n(, , Q)+, V E Xh,with ê(to,.) = 0. An equation for é, ç is:) + a(ê, ) + b(, ) = n(p, p, - n(, p, P) (A.9)with e(to,.) = 0.Lemma A.5 There exists a constant ?J such thatfeW2 + vf fvêfl2ds (A.10)vWVeW2 + f jfê2ds < (A.11)L jf2ds (A.12)< JJ4 (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 thatf H4, (A.14)for all 0 <t0 < t T.137Proof: Consider the solution of the backwards Stokes’ equationszt + ‘ A z — V/3 = ê, V•z = 0, z(t) = 0, zç = 0. (A.15)Multiplying by ê and integrating over the domain,IêI2 = (Zt, e) — a(z, ê) + b(, e) = (z, e) + b(, ê) — [(t, z) + a(ê, z)].Choosing b = PHZ E XH C Xh as test function in (A.8),(Pe, PHz) + a(ê, PHz) + b(, PHz) + (, z) + n(, j5, PHz)—n(q, p, PHz) + n(p, , PHz) — n(p, q, PHz) + n(, , PHz) = 0,since (PHz) = 0. Hence,III2 = (z, ê) + b(, ê) — [(t, z — PHz)— (, Pz) + a(ê, z — PHz) + n(, p, PHz)—n(q, p, PHz) + n(, j, PHz) — n(p, q, PHz) + n(, j, PHZ) — b(, PHz)]Estimating:Pz)b(3,ê)(êt,z — PHz)a(ê,z—PHz)n(13, j, PHZ)—n(p, q, PHz)n(, , Pffz)—n(q, p, PHz)n(, , PHz)b(, PHz)WeI2 (z,e) +H2( +138CH + zj+ CH2WVêW,eUzI + CH4êt2,z + CH2IjVêjI,iIFFIzW +H2UVeW€PHzj+CH6,<€Uz +H2I(I.Combining we obtainêjj2 + j2) + €jzh12II2Integrating from t0 to t, observing that ê(.,to) = z(.,t) = 0, and applying Lemmas A.2,A.5 we obtainj ê2ds + zds.Now applying the backwards Stokes’ estimate (2.82) and choosing e appropriately,<j ê2ds.This proves the desired result. DLemma A.7 There exists a constant such thatH2, (A.16)for all 0 <Io < t < T.Proof: Letting b = è in (A.9),ItI2 + vVeI2 = n(p,p,) - n(p,)= n(, p, ) + n(E, p, ) - n(p, , E).Now,p, ) + n(, p, ) - n(p ) < 72 +and as a consequence,+ vWVéW2 < +Integrating, observing that e(., t0) = 0 proves that ILe H2. This, combined withthe fact that IIQéW HjVèW H2 proves the desired result.At this point, it is clear we have provedBell2 <H[139To prove the full estimate for jeW with one more half power of H, we copy the proofs ofLemmas 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 weprove estimates similar to those in Lemmas 2.13, 2.14, 2.15, however with a lower orderof H (lower order by 2) on the right hand side. These estimates finish the proof of theTheorem. D140Appendix BProof of weighted error estimates for the Navier-Stokes equationsIn the following, 0 <T <cc is fixed. C is a generic positive constant independent of h, uwhile represents a generic positive constant independent of h but possibly dependingon u, 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 Üh, 7rh be thesolution to (2.5), the standard finite element method. Let u(t) = min(t, 1).Theorem B.1 (Weighted error estimate for the Navier-Stokes equations) Suppose u andO satisfy some (standard) regularity properties for all 0 < t $ T < cc. Suppose theexternal force f, and its time derivatives, are as smooth as needed and suppose the finiteelement spaces Xh and Lh satisfy some standard stability and approximability properties.Under these conditions, there exists a constant, , such that— a2 +h2BVut— VütW2)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 estimate holds (with some exponential weight factors) for T = cc. For an ideas concerningthis situation see [15].141Remark B.3 (The infinite sequence of estimates) It is clear that the ideas presentedgeneralize and we can prove an infinite sequence of estimates. For n = l,2,•.., thereexists a constant C such thato(IIDu — DuW2+ hWVDu— VD’uII2)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 TheoremB.1 only for the conforming case. The equivalent result is also true for the nonconformingcase but the proof is technically more difficult. To start, we require a priori estimates forsolutions 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 assumptions, u and 0 satisfy, for n 1,sup a22 {D’u + IIDuW + Db0HH1/n} < , (B.1)supftu2_2Dfluf2ds <oc, (11.2)O<t<T 0supo2’Du < oc, (B.3)0<t<Tsup f {u2hIDuJj + D’ul + jD0MH1,R} ds <oc. (BA)0<t<T 0Let v, 0 be the solution to the Stokes equations:Av—V0=g, V•u=O, v3=O, (B.5)and let Vh E Jh be the solution toa(vh, 4’h) = (g, &h), V/’h E Jh. (B.6)14Lemma B.2 (Steady Stokes error estimate) Let v, 0 be the solution to (B.5) and let vhbe the solution to (B.6). Then,Iv - VhW + hIlVv - VvhU Ch2gj. (B.7)This is proved (for a more general case) on page 295 in [15].Let ShU E Jh be the solution toa(Shu,’bh) = (f — uj— u.Vu,,bh), V’cbh E Jh. (B.8)Corollary B.1 There exists a constant , such that,f 2 (IIShUt — utjj + hW VShu — VUtU) ds h2,(IISiut— Ut!! + hW ‘hUt— VutW) i; h2,(IISutt— uttll + hI! VShutt— VuttIl) dsa4 (IlShutt — ut + hi! VShut — VujtII) <for all 0 <<T.Proofr One can show that:IISh’’t — Ut!! + hi! VShut — VutlI < Ch2lift — Du — Dt(ut•Vut)IIllShutt— UttII + hI! VShutt— Vuttli Ch2IIftt — Du— D(u.Vu)II,for all 0 < T. Each estimate is easily proved by considering the equations satisfied byDShu and mu and applying Lemma B.1. From this it is clear that to prove the firsttwo estimates we need to bound lift — Du — Dtu.Vull and for the last two estimates weneed to bound Ilftt — Du — Du.VulI. Consider the first two estimates. By assumption,f is smooth, and its integral is bounded. By Lemma B.1,jta2(s)IID2uII2ds <00.143Similarly, by first estimating 11Th {u(s).Vu(s)} and then using the appropriate a prioriestimates from Lemma B.1,The second estimate follows because the extra power of a implies terms whose integralswere previously bounded are now themselves bounded for 0 t T. The last twoestimates are proved similarly. DLet v Vh be the solution to(Vht, h) + a(vh, h) = (f — u•Vu, h) V?,bh E Jh. (B.9)Let e = U— Üh, = U — vh and ij = Vh — Üh. Clearly, e = + j. Our approach toestimating e — actually its time derivative is to first estimate and then estimateIn [15] is is proved that:Bell + lllI + llll + h(IlVejl + llVli + lVll) h2.Lemma B.3 There exists a constant s.t. j a(s)lltl{2ds h2, for all 0 t T.Proof: satisfies:(,b) + (ve,v,b) = (O,V.), Vb e Jh. (B.10)Choosing ‘ci’ — u -4- PhUt as test function in (B.10), where PhUt is theL2-projectionof Ut onto Jh, we obtain:IItIl2 + llVll2 = (e,u1 — Phut) + (V,V(uj — Phut)) + (U,V•Ph).Estimating the right-hand side:(O,V•Ph) = (8,V•Ph) — (8t,V•Ph)< (O,V.Ph) + Ch2llVOtll + llell2(, ut — P/Lug) = Ch2ljVulj + llll2,(,V(u — Put)) Ch2llutll + Ch2.144Thus,ieu2 + II f2 (O, V•P) + Ch2(1 + jUt + IIV&tW2),and multiplying by u,uIIII2 + uIlVW2 < V2 + u(O, V.Ph) + Cuh2(1 + lut + IVOtU2).We have used thatu(O,V•Ph) a(&,VPh) + (&,VPh < u(6,VPhe) + Ch2.Integrating, from 0 to t, and estimating, impliesWtIl + uVl2 Ch + u(O, V•Phe) Ch2.Differentiating (B.10) with respect to time, we obtain+ a(.,L’) = (O,V•’), Vb E J. (B.11)Lemma B.4 There exists a constant such thatJ2l+ j a2 IIVjldsfor all 0 <t <T.Proof: Choosing as test function, ,b =— Ut + PhUt E Jh in (B.11),IIell2 + IvJl2 = — Phut) + (V,V(ut — Phut)) + (Ot,V.Fht).One can show:ld 2(,ut — Phuj) = -—llUt — PhutljV(u — Phut)) < Ch2llutlj + ellVll2,(8t,VPht) = (O —ihOt,VPht) Ch2UV8 + eVj2,145and, as a consequence,+ VI2<Ch2(Iut + I{VOjW2)+ - PujW2.Multiplying by a2,2Ijl22IIve 112 Cha(IIutU + IlVOtIl2)+a2IlUt— Put jj2 + + aut — Pu2.Upon integrating, from 0 to t,a22 + ja2VI <U2II t — PutW2 + Ch2 <Ch2. 0Lemma B.5 There exists a constant such thatjtJ2W2dsfor 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, impliesajIlI2 = (z,) + b(/9, ) - z) + z)].Consider some of the right-hand side terms.= b(/9,Phe) eu’IIV/9112+Ch2ullVll.Let PhZ be theL2-projection of z onto Jh.(, z) + a(t, z) = (, z — Phz) — z — Phz) — b(O, Phz) < z — Phz)+Ch2llVell + Cfh4a(llutIl + lVOtjl2)+ ea’(IIV/9112+ lzIl).14Combining,IItW2 < (z,) + Ch2IVW + C€h4a(Wutj + WV0t2)+ eu’(WV2+ zj).Multiplying by a, integrating from 0 to t, and estimating proves the desired result. DLemma B.6 There exists a constant C such that3(MtW2+h2WVU)<for all 0 <t <T.Proof: Differentiate equations (B.8) and (B.9) with respect to time and take the difference. The resulting equation is(VShut — VVht, b) (Dvh — Du, sb).As a consequence,(DShu — Dvh, b) + (VShut — Vv, ‘/‘) = (DShu — Du, sb).Choosing i/’ = ShUt — Vht as test function and multiplying by o3, one can showShUt — Vht 112 a3llDShu— DuIlllShut — VhtIl + Cu2llShUt — Vht 112.Integrating, from 0 to t,a3IlShut — VhtII2 ja4llDShu — DulI2ds + j — Vhdsjta3WDShu — DuII2ds + ja2(Shu3— ull2 + 11e52)ds.Thus by previous estimates and Corollary B.1,u3IIll2<a3llShUt — Vhtll2 +a3llShUt — u < Ch4.147The estimate fora3IVII2 follows since,o3IIVjI2 u3IIVShut — VuI +J3WVShUt — VvhtI Ch2 + Cu3h1 WShUt —VhtW h2.This finishes the proof of the lemma. DWe now require estimates for . i satisfies:‘,l’) + (V7j, Vb) = (u.Vu — ã.Vã,’b), Vb E Jh. (B.12)Lemma B.7 There exists a constant C such thatj IWd <h2,for all 0 <t <T.Proof: Choosing ,b = as test function in (B.12) and estimating,WW2 + lvlI2 = (u.Vu — ü.Vü,7j) = (—eVe + e•Vu + u.Ve,?j)II’iW2+Ch2.Integrating proves the desired result. CLemma B.8 There exists a constant such that j a2 V,j2ds <h2,for allO <t <Proof: differentiating (B.12) with respect to time and choosing b = as test function,+ VijJ2 = (urVu — üt.Vü,7)) + (u.Vut —Now,(u•Vu — ã•Vã, ‘q) = + + Vu — + uVi — + •Vu + uV, ‘i)(ut.Vu — üVü, ‘ii) = (‘iV’i+ ‘icV+ iVu — •V’i + u•V’i — •V+ + u•V, ‘ii).148We must estimate each of these terms (we will not show all of the details).(u.Vut — ü.Vü,ij) + (ut.Vu— ut.Vu,) +lIvjI2 + CmI2,whereA = G12+ Ch2IIVII + Ch2IIVutII.Hence,g2j2 +a2V aA() + Gj2,From our previous results and the estimates of Lemma B.1, ja2A(s)ds h2, andintegrating the differential inequality proves the desired estimate. DLemma B.9 There exists a constant such that ja2IIlIds<h4,for all 0 <t T.Proof: Let z, /3 be the solution to the backwards Stokes equations:z+vzz—V/3=a,, V•z=0, z(t)=0, ZI3ç- =0.Multiplying by i, integrating over the domain and then multiplying by a, we obtain2IW = a(z, j) + a b(/3, ,) — a [(c, z) + z)j. (B.13)Now,d d 2a(z, i) < a(z, ij) + eHzf2 + Ch4,ab(/3,7j) eWV/9 +Ch2aVij.Let PhZ be theL2-projection of z onto Jh, then(q,z) + a(ij,z) = (ii,z — Phz) + a(jt,z — Phz)+(uVu — ü.Vüt,Phz) + (u.Vu — aj•Vu,Phz),149andz— Phz) = 0 (since ‘q Jh),o(Vi1, Vz — VPhz) < ejzj +Ch2uHVi,lj,u(uVut — u•Vü, Phz) < IzII + a2B + ea2IIij,a(ut•Vu — ãVü,Phz) < ez + u2B + ea2’qjj,whereB = Cj2+ Ch3fjV2+ Ch4WutIj. (B.14)Thus,2WmII ) + ez + ch2aIVtIl2 +u2B(t),and integrating, from 0 to t, proves the desired result. DLemma B.1O There exists a constant C such that O3j2 Oh4, for all 0 t T.Proof: differentiating (B.12) with respect to time and choosing ‘çb = as test function,we obtain:+ Vm2 = (u.Vu — ã•Vü,i) + (u.Vu —Multiplying by o and estimating:a3jW2 +a3jjVW2<Ca2j+a3B,where B is defined in (B.14). Integrating proves the desired estimate. DFinally,+ h2,and the estimate for jVetW follows sinceIIViiJI Ch’WViijI. 0150Appendix CError estimates for Burgers’ equationIn the following C is a generic positive constant independent of h and u while is ageneric positive constant independent of h but possibly depending on u and v. (Bothmay 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 xW2’([O,) x(a2) f, f, f,... E L°°((O, oo) x 1) fl L2((O, oo) x(a3) the family of grids satisfies a uniform size condition. There exists constants k1, k2independent of h such that k1h IThI < kh for all Th e Gh and all h.Theorem C.1 (Error estimate for Burgers’ equation) Suppose (al), (a2) and (aS) aresatisfied, then, there exists a constant such thatIlu— UhII + hIIu —for all t 0.We know of no source for such an estimate. One can also prove a sequence of weightederror 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 n = such thato(IIDu—Dühfj + hWDu — Dl) <h2,151for 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. Wewill not prove any of these here. These estimates can be proven using an approach similarto 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 = M(u0,f, l) such thathuM2 + j huhI2ds <M0,huIh2 + (IIuhI2 + ButM2)ds Mi,for all t > 0. The general formula for the rest of the estimates, n = is:(BDuxM2 + IDmuM2) + j2jDugMds<M22,u2IhD1xiI + f ( D1uph2+ hD2uH2) ds < M23.These estimates are similar to those proved in [15] for the Navier-Stokes equations. Aproof of this sequence of estimates has been done in [32].Let w be the solution to:— Aw = g, = 0, (C.1)Let wh€ Xh be the solution to:(Whr, chx) = (g, q), Vqh e Xh. (C.2)The following is a standard result.Lemma C.2 (Poisson error estimate) Let w be the solution to (C.]) and let wh be the solution to (C.2,). There exists a constant C such that hw — whhh + hwx — Ch2g,for all t > 0.152Let Sh’u E Xh be the solution toa(Shu,bh)= (f — — uu,bh), Vi’h E Xh. (C.3)Corollary C.1 Suppose the assumptions above are satisfied, then there exists a constant, such that,lIShu—uW2+huIla2(lIShut— utfl2 + h2 WShut — utW2)f (Wh — utW2 +h2ilShut — ut2)dsjt3(lIhtt — tt + hljShutt — uW)dsfor all t 0.Proof: By Lemma C.2,— uii + hWShu— ulI <Ch2f— —Because of the assumptions on f and the a priori estimates for u,If — ut — uuf <C + M2 +Again using Lemma C.2,IlShut— utII + hIIShut — ut Ch2lift — — uu —Now,lift — Utt — UtUx — uuII C + iIUttII + CIluiiand the leading order term, the term requiring the highest order of a in the a prioriestimates, is Iluttil. Thus,2iift — Utt — utu,, — uutj2— Utt — uu— uuiI2ds153Hence, we have proved the first two estimates. The others are proved similarly. 0Let v = vh e Xh the solution to(vt, bh) + (vs, bh) = (f — uu, bh), Vh E Xh, Vo = PhUQ. (C.4)Let e = u — ü, = u — v and i v — ü. Clearly, e = + i. To prove Theorem C.1, i.e.to estimate e, our plan of attack is to first estimate and then .Lemma C.3 There exists a constant C such thatfuWI2ds <h4,for all t 0.Proof: satisfies(, b) + bh) = 0, Vb € Xh.Choosing Ih = — 7- + Pjn, where Phu is theL2-projection of u onto Xh,III2 += (t,u - Phu) + - Phu)Wu-Fhu2 + + Ch2Iu.Integrating,WW2 + L IIl2ds <h2.Let z be the solution to the backwards heat equation:Zt + Az = , z(t) = 0, = 0.Multiplying by and integrating over the domain,II2 = (z,)-[(t,Z) + (,z)j— [(,z— Phz) + — Fhz)]= (Phz, ) + (u — Phu, zt) — (, z — Phz)< (Phz, ) + Ch2( + 2) + e( WztW2 +154Integrating from 0 to t, noting that j (Phz, ) ds = 0, we obtainft III2ds <h4+ ci (Wztj2 + zjI)The backwards heat equation estimate (3.30) implies the result. DLemma C.4 There exists a constant C such that+ hdsfor all t 0.Proof: Subtracting (C.4) from (C.3) we obtain(Shu — v, = (vt, /‘) — (Ut, b), V/’ e Xh.Thus,(Shut — vt, /‘) + (Shu — Vr, = (Shut — Ut, k) VL’ € Xh.Choosing b = Shzt — v as test function,— vU2 + Shu — vxW2 = (Shut — Ut, Shu — v).Multiplying by o,gWShu_vU2+aUShux _vI2 < u(Shut —Ut,ShU—V) + Shu —vU22Sut— utU2 + CUShU — vU2.Before integrating, we would like to point two things out:• a2IIShut — utII2 and its integral are majorized by h4. Thusjta2IISu— utll2ds <ah4.155• Similarily, IShu — v2 and its integral are bounded andIShu— v2ds <at1h4.Upon integrating,JllShu — v2 +— vll2ds < uhand we have shown IShu — vU Ch2. This implies,IlI lu - ShuI + Shu - vi h2.The estimate for , is a simple consequence of the inverse inequality.li lluv — + IlShux — vxh+Ch1jiShu—v <h. ‘Lemma C.5 There exists a constant such thatI + hliids <h2,for all t 0.Proof: i satisfies:(ifl, b) + (,) = (uu — ü, ), Vb E Xh.Choosing as test function /‘ =ld 2 2+ = (uu—One can show,(uu— üüx, 77)= (—i — 77x + U?lx + ?lUx — + u + ),156and(,ii) = 0,(—ii + zi + ru— WII2 +( ) <j2+h6,(u + ) +Thus,IIII2 + 2 + h4and integrating from 0 to twI2 + j ds <h4.By the inverse inequality,III 0’IIII Ch.Finally, we have proved Theorem C.1 sincehell llhI + llll <h2 and llehl l1ll + llll h.157Bibliography[1] Ait Ou Ammi A. and Marion, M. nonlinear Galerkin methods and mixed finiteelements: two-grid algorithms for the Navier-Stokes equations. preprint, 1992.[2] Ait Ou Ammi. PhD thesis, Ecole Centrale de Lyon, 1993.[3] Blum, H., Hang, J., Muller, S., and Turek, S. Finite element analysis tools release1.3, 1992.[4] Bristeau,M. 0., Glowinski, R., and Periaux, J. Numerical methods for the NavierStokes equations. Applications to the simulation of compressible and incompressibleviscous flows, pages 73—187. North Holland, Amsterdam-New York, 1987.[5] Cattabriga, L. Su un problema al contorno relativo al sistema di equazioni di Stokes.Rend. Sem. Mat. Univ. Padova, 31:308—340, 1961.[6] Constantin, P., Foias, 0., Nicolaenko, B., and Temam R. Integral manifolds andinertial manifolds for dissipative equations. Appl. Math. Sci., 70, 1988.[7] de Frutos, J. and Garcia-Archilla, B. On the time integration of the nonlinearGalerkin method. preprint, 1993.[8] Dedulvier, C., Marion, M., and Titi, E. S. On the rate of convergence of the nonlinearGalerkin methods. submitted to Math. Comp.[9] Foias, C., Manley, 0., and Temam, R. Modelling of the interaction of small andlarge eddies in two dimensional turbulent flows. M2AN, 22:93—114, 1988.[10] Foias, C., Sell, G. R., and Titi, E.S. Exponential tracking and approximation ofinertial manifolds for dissipative nonlinear equations. J. Dyn. Duff. Eq., 1:199—244,1989.[11] Foias, C., Sell, R., and Temam, R. Inertial manifolds for nonlinear evolutionaryequations. J. Duff. Eq., 73:309—353, 1988.[12] Glowinski, R. and Periaux, J. Numerical methods for nonlinear problems in fluiddynamics. In Proceed. Intern. Seminar on Scientific Supercomputers, Paris, Feb2-6, 198’1 North Holland, 1987.[13] D.F. Griffith. Finite elements for incompressible flow. Math. Meth. Appi. Sci.,1:16—31, 1979.158[14] Heywood, J. G., Rannacher, R., and Turek, S. Artificial boundaries and flux andpressure conditions for the incompressible Navier-Stokes equations. (to appear) mt.J. Num. Meth. in Fluids, 1994.[15] Heywood, John G. and Rannacher, Roif. Finite element approximation of the non-stationary Navier-Stokes problem. i. regularity of solutions and second order errorestimates for spacial discretization. SIAM J. of Num. Anal., 19(2):275—311, 1982.[16] Heywood, John 0. and Rannacher, Roif. On the question of turbulence modelingby approximate inertial manifolds and the nonlinear Galerkin method. SIAM J. ofNum. Anal., 30(6):1603—1621, 1993.[17] Jauberteau, F., Rosier, C., and Temam, R. A nonlinear Galerkin method forthe Navier-Stokes equations. Comp. Meth. in Applied Mechanics and Engineering,80:245—260, 1990.[18] Kwak, M. Finite dimensional inertial manifolds for the 2D Navier-Stokes equations.Technical Report 91-30, AHPCRC, University of Minnesota, 1991.[19] Marion, M. and Temam, R. Nonlinear Galerkin methods. SIAM J. Numer. Anal.,2:1139—1157, 1985.[20] Marion, M. and Temam, R. Nonlinear Galerkin methods: the finite element case.Numer. Math., 57:205—226, 1990.[21] Ohmori, K. and Ushijima, T. A technique of upstream type applied to a linear nonconforming finite element approximation of convective diffusion equations.R.A.I.R.O Nummer. Anal., 18:309—332, 1984.[22] R. Rannacher. Numerical analysis of nonstationary fluid flow. Technical report,Universität Heidelberg, 1988.[23] Rannacher, R. and Turek, S. Simple nonconforming quadrilateral stokes’ element.Num. methods for F.D.E., 8:97—111, 1992.[24] Solonnikov, V.A. On differential properties of the solutions of the first boundaryvalue problem for nonstationary systems of Navier-Stokes equations. Trudy Mat.Inst. Steklov., 73:221—291, 1964.[25] L. Tobiska. Full and weighted upwind finite element methods. In J.W. Schmidt andH. Spath, editors, Spliries in Numerical Analysis. Internationales Seminar ISAM 89in Wessig, Spriger, Berlin, 1989.159[26] Tobiska, L. and Schieweck, F. A nonconforming finite element method of upstreamtype applied to the nonstationary Navier-Stokes equations. MMAN, 23:627—647,1989.[27] Stefan Turek. A multigrid solver using discretely divergence free finite elements.Technical report, Uriiversität Heidelberg, 1989.[28] Stefan Turek. Em robustes und effizientes Mehrgitterverfahren zur Lôsung der instationären inkompressiblen, 2-D Navier-Stokes-Gleichungen mit diskret divergenzfreien finite Elementen. PhD thesis, Universitãt Heidelberg, 1992.[29] Stefan Turek. Visualization tools for the nonstationary Navier-Stokes equations.Technical report, Universität Heidelberg, 1992.[30] Stefan Turek. Tools for simulating nonstationary incompressible flow via discretelydivergence free finite elements. (to appear) mt. J. Num. Meth. in Fluids, 1994.[31] Van Dyke, Milton. An Album of Fluid Motion. The Parabolic Press, 1982.[32] 0. Walsh. On the existence and uniqueness of solutions to Burgers’ equation. University of British Columbia, Masters paper 1988.160

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