Convergence Analysis and Error Estimates for a Second Order Accurate Finite Element Method for the Cahn-Hilliard-Navier-Stokes System

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1 Convergence Analysis and Error Estimates for a Second Order Accurate Finite Element Metod for te Can-Hilliard-Navier-Stokes System Amanda E. Diegel Ceng Wang Xiaoming Wang Steven M. Wise June 7, 06 Abstract In tis paper, we present a novel second order in time mixed finite element sceme for te Can-Hilliard-Navier-Stokes equations wit matced densities. Te sceme combines a standard second order Crank-Nicolson metod for te Navier-Stokes equations and a modification to te Crank-Nicolson metod for te Can-Hilliard equation. In particular, a second order Adams-Basfort extrapolation and a trapezoidal rule are included to elp preserve te energy stability natural to te Can-Hilliard equation. We sow tat our sceme is unconditionally energy stable wit respect to a modification of te continuous free energy of te PDE system. Specifically, te discrete pase variable is sown to be bounded in l 0, T ; L and te discrete cemical potential bounded in l 0, T ; L, for any time and space step sizes, in two and tree dimensions, and for any finite final time T. We subsequently prove tat tese variables along wit te fluid velocity converge wit optimal rates in te appropriate energy norms in bot two and tree dimensions. Keywords: Can-Hilliard equation, Navier-Stokes, mixed finite element metods, convex splitting, energy stability, error estimates, second order AMS subject classifications. 5K5 5K55 65M 65M60 Introduction In tis paper, we prove error estimates for a fully discrete, second order in time, finite element metod for te Can-Hilliard-Navier-Stokes CHNS model for two-pase flow. Let Ω R d, d =,, be an open polygonal or polyedral domain. For all φ H Ω, u L Ω, consider te energy { Eφ, u = φ ε 4ε φ } γ u dx,. Ω Department of Matematics, Louisiana State University, Baton Rouge, LA 7080 adiegel@lsu.edu Department of Matematics, Te University of Massacusetts, Nort Dartmout, MA 0747 cwang@umassd.edu Department of Matematics, Florida State University, Tallaassee, FL 06 wxm@mat.fsu.edu Department of Matematics, Te University of Tennessee, Knoxville, TN 7996 swise@mat.utk.edu

2 were φ represents a concentration field, u represents fluid velocity, and ε is a positive constant. Te CHNS system is a gradient flow of tis energy [7, 8,, 4]: t φ φ u = ε Mφ µ, in Ω T,.a µ = ε φ φ ε φ, in Ω T,.b t u η u u u p = γµ φ, in Ω T,.c u = 0, in Ω T,.d n φ = n µ = 0, u = 0 on Ω 0, T,.e were Mφ > 0 is a mobility, η = Re were Re is te Reynolds number, γ = W e were W e is te modified Weber number tat measures relative strengts of kenetic and surface energies, and µ is te cemical potential: µ := δ φ E = ε φ φ ε φ.. Here δ φ E denotes te variational derivative of. wit respect to φ. Te equilibria are te pure pases φ = ±. Te boundary conditions are of local termodynamic equilibrium and no-flux/noflow/no-slip type. A weak formulation of.a.e may be written as follows: find φ, µ, u, p suc tat and tere old for almost all t 0, T were φ L 0, T ; H Ω L 4 0, T ; L Ω, t φ L 0, T ; H Ω,.4a.4b µ L 0, T ; H Ω,.4c u L 0, T ; H 0Ω L 0, T ; L Ω, t u L 0, T ; H Ω, p L 0, T ; L 0Ω,.4d.4e.4f t φ, ν ε a µ, ν b φ, u, ν = 0, ν H Ω,.5a µ, ψ ε a φ, ψ ε φ φ, ψ = 0, ψ H Ω,.5b t u, v η a u, v B u, u, v c v, p γ b φ, v, µ = 0, v H 0Ω,.5c c u, q = 0, q L 0Ω,.5d a u, v := u, v, b ψ, v, ν := ψ v, ν,.6 c v, q := v, q, B u, v, w := [u v, w u w, v],.7 wit te compatible initial data φ0 = φ 0 H NΩ := { v H Ω n v = 0 on Ω }, u0 = u 0 V := { v H 0Ω v, q = 0, q L 0Ω },.8 and we ave taken Mφ for simplicity. Observe tat te omogeneous Neumann boundary conditions associated wit te pase variables φ and µ are natural in tis mixed weak formulation

3 of te problem. We define te space L 0 as te subspace of functions of L tat ave mean zero. Furtermore, we state te following definitions of wic te first is non-standard: H Ω := H Ω, H 0 Ω := [ H0 Ω] d, H Ω := H 0 Ω, and, as te duality paring between H and H in te first instance and te duality paring between H Ω and H 0 Ω in te second. Te notation Φt := Φ, t X views a spatiotemporal function as a map from te time interval [0, T ] into an appropriate Banac space, X. We use te standard notation for function space norms and inner products. In particular, we let u := u L and u, v := u, v L, for all u, v L Ω. Te existence of weak solutions to.5a.5d is well known. See, for example, [8]. It is likewise straigtforward to sow tat weak solutions of.5a.5d dissipate te energy.. In oter words,.a.e is a mass-conservative gradient flow wit respect to te energy.. Precisely, for any t [0, T ], we ave te energy law Eφt, ut t 0 ε µs η γ us ds = Eφ 0, u 0,.9 and were mass conservation for almost every t [0, T ], φt φ 0, = 0 of te system.5a.5d is sown by observing tat b φ, u, = 0, for all φ L Ω and all u V. Numerical metods for modeling two-pase flow via pase field approximation as been extensively investigated. See, for example, [4, 5, 8, 9, 0,,, 4, 4, 5, 6, 8, 9,,,, 4, 5, 6], and te references terein. Of te most recent, Sen and Yang [] proposed two new numerical scemes for te Can-Hilliard-Navier-Stokes equations, one based on stabilization and te oter based on convex splitting. Teir new scemes ave te advantage of being totally decoupled, linear, and unconditionally energy stable. Additionally, teir scemes are adaptive in time and tey provide numerical experiments wic suggest tat teir scemes are at least first order accurate in time. However, no rigorous error analysis was presented. Abels et. al. [] introduce a termodynamically consistent generalization to te Can-Hilliard- Navier-Stokes model for te case of non-matced densities based on a solenoidal velocity field. Te autors demonstrate tat teir model satisfies a free energy inequality and conserves mass. Te work of Abels et. al. builds on te pioneering paper of Lowengrub and Truskinovsky [0] wo use a mass-concentration formulation of te problem. Peraps te fundamental difference between te approaces is tat te model of Lowengrub and Truskinovsy end up wit a velocity field tat is not divergence free, in contrast wit tat of Abels et. al.. For tis reason, and oters, developing suitable numerical scemes for te model in [0] is a difficult task, but see te recent work of [6]. Garcke et. al. [] present a new time discretization sceme for te numerical simulation for te model in []. Tey sow tat teir sceme satisfies a discrete in time energy law and go on to develop a fully discrete model wic preserves tat energy law. Tey are furtermore able to sow existence of solutions to bot te time discrete and fully discrete scemes. Again, owever, no rigorous error analysis is undertaken for eiter of tese scemes. Grün et al. [, 4] provide anoter numerical sceme for te non-matced density model and tey carry out an abstract convergence analysis for teir sceme. Rigorous error analysis wit, say, optimal order error bounds for models wit non-matced densities seems to be a difficult prospect, but a very interesting line of inquiry for te future. Most of te papers referenced above present first order accurate in time numerical scemes. Second order in time numerical scemes provide an obvious advantage over first order scemes by decreasing te amount of numerical error. On te oter and, second-order in time metods are almost universally more difficult to analyze tan first-order metods. A few suc metods ave been developed in recent years [5, 8,, ]. Most notably, Han and Wang [8] present a second

4 order in time, uniquely solvable, unconditionally stable numerical sceme for te CHNS equations wit matc density. Teir sceme is based on a second order convex splitting metodology for te Can-Hilliard equation and pressure-projection for te Navier-Stokes equation. Te autors sow tat te sceme satisfies a modified discrete energy law wic mimics te continuous energy law and prove tat teir sceme is uniquely solvable. However, no rigorous error analysis is presented and stability estimates are restricted to tose gleaned from te energy law. Te overall sceme is based on te Crank Nicolson time discretization and a second order Adams Basfort extrapolation. Cen and Sen [5] ave very recently refined te sceme of Han and Wang [8]. In tis paper, we study a second-order in time mixed finite element sceme for te CHNS system of equations wit matced densities. Te metod essentially combines te recently analyzed secondorder metod for te Can-Hilliard equation from [7, 5] and te pioneering second-order in time linear, Crank-Nicolson metodology for te Navier-Stokes equations found in []. Te Can- Hilliard sceme from [7, 5] is based on convex splitting and some key modifications of te Crank- Nicolson framework. Te mixed finite element version of te sceme was analyzed rigorously in [7]. Te sceme erein is coupled, meaning te Can-Hilliard and Navier-Stokes must be solved simultaneously. But, te metod is almost linear, wit only a single weak nonlinearity present from te cemical potential equation. In particular, second order Adams-Basfort extrapolations are used to linearize some terms and maintain te accuracy of te metod, witout compromising te unconditional energy stability and unconditional solvability of te sceme. Te convergence analysis of a fully decoupled sceme, suc as tose in [5, 8] is far more callenging. Te present work may be viewed as a first step towards analyzing suc metods. Teoretical justification for te convergence analysis and error estimates of numerical scemes applied to pase field models for fluid flow equations as attracted a great deal of attention in recent years. In particular, te recent work [6] provides an analysis for an optimal error estimate in te energy norms for a first-order-accurate convex splitting finite element sceme applied to te Can-Hilliard-Nonsteady-Stokes system. Te key point of tat convergence analysis is te derivation of te maximum norm bound of te pase variable, wic becomes available due to te discrete l 0, T ; H stability bound of te velocity field, at te numerical level. However, a careful examination sows tat te same tecniques from [6] cannot be directly applied to te second-order-accurate numerical sceme studied in tis paper. Te primary difficulty is associated wit te /4 and /4 coefficient distribution in te surface diffusion for te pase variable, at time steps t n, t n, respectively. In turn, an l 0, T ; H estimate for te pase variable could not be derived via te discrete Gronwall inequality in te standard form. We terefore present an alternate approac to recover tis l 0, T ; H estimate for te pase field variable. A backward in time induction estimate for te H norm of te pase field variable is applied. In addition, its combination wit te l 0, T ; L estimate for te cemical potential leads to an inequality involving a double sum term, wit te second sum in te form of m j= m j. Subsequently, we apply a very non-standard discrete Gronwall inequality, namely Lemma A. in Appendix A, so tat an l 0, T ; H bound for te numerical solution of te pase variable is obtained. Moreover, te growt of tis bound is at most linear in time, wic is a remarkable result. It turns out tat tis stability bound greatly facilitates te second order convergence analysis in te energy norms for te numerical sceme presented in tis paper. We point out tat because of te l 0, T ; H bound for te discrete pase variable, we are able to carry out te analysis on te Navier-Stokes part of te system tat is muc in te spirit of tat wic appears in Baker s groundbreaking paper []. Due to te increased complexity of numerical calculations and te appearance of te nonlinear convection terms, a few more tecnical lemmas are required for te analysis included in tis paper compared to te work presented in [6] for te Can-Hilliard-Nonsteady-Stokes 4

5 system. Te use of tese lemmas results in a numerical sceme wic attains optimal convergence estimates in te appropriate energy norms: l 0, T ; H for te pase variable and l 0, T ; H for te cemical potential. Moreover, suc convergence estimates are unconditional: no scaling law is required between te time step size τ and te spatial grid size. Te remainder of te paper is organized as follows. In Section, we define our second order in time mixed finite element sceme and prove tat te sceme is unconditionally stable and solvable wit respect to bot te time and space step sizes. In Section, we provide a rigorous error analysis for te sceme under suitable regularity assumptions for te PDE solution. Finally, a few discrete Gronwall inequalities are reviewed and analyzed in Appendix A. A Second-Order-in-Time, Mixed Finite Element Sceme. Definition of te Sceme Let M be a positive integer and 0 = t 0 < t < < t M = T be a uniform partition of [0, T ], wit τ = t i t i and i = 0,..., M. Suppose T = {K} is a conforming, sape-regular, quasi-uniform family of triangulations of Ω. For r Z, define M r := { v C 0 Ω v K P r K, K T } H Ω and M r,0 := M r H 0 Ω. For a given positive integer q, we define te following: S := M q, S := S L 0Ω, { X := v [ C 0 Ω ] } d vi M q,0, i =,, d, } V := {v X v, w = 0, w S. Wit te finite element spaces defined above, our mixed second-order convex splitting sceme is defined as follows: for any m M, given φ m, φm S, u m, um X, p m S find φ m, µ m S, u m X, and p m S suc tat ε were χ φ m, φ m δ τ u m, v δ τ φ m, ψ ε η a, ν ū m, v ε a µ m, ν b φ m, ū m, ν = 0, ν S,.a φ m, ψ ε a ˇφ m, ψ µ m, ψ = 0, ψ S,.b B ũ m, ū m, v c v, p m γ b φ m, v, µ m = 0, v X,.c c ū m, q = 0, q S,.d δ τ φ m ˇφ m := φm := 4 φm φ m τ, φm 4 φm, χ φ m, φ m := φm φm, φm := φm φm φ := m, φ m φm.. 5

6 Te notation involving te pressure and velocity approximations are similar. For initial conditions we take φ 0 := R φ 0, φ := R φτ, u 0 := P u 0, u := P uτ, p 0 := P p 0, p := P pτ,. were R : H Ω S is te Ritz projection, a R φ φ, ξ = 0, ξ S, R φ φ, = 0,.4 and P, P : V L 0 V S is te Stokes projection, η a P u u, v c v, P p p = 0, v X, c P u u, q = 0, q S..5 It will be useful for our stability analyses to define te cemical potential at te time step via µ, ψ := χ φ ε, φ 0, ψ φ ε, ψ ε a φ, ψ, ψ S..6 We also define te residual function ρ S tat solves ρ, ν := δ τ φ, ν ε a µ, ν b φ, ū, ν, ν S..7 Wile we do not expect te residual ρ to be identically zero for finite, τ > 0, it will be stable in te relevant norms wit te assumption of sufficiently regular PDE solutions. Remark.. We ave assumed exact expressions for φ and u. Tis is done to manage te lengt of te manuscript. We can employ a separate initialization sceme, but te analysis becomes far more tedious. See, for example, [7]. We point out tat, because of te properties of te Ritz projection, φ 0, = φ,,.8 under te natural assumption tat φ0, = φτ,. δ τ φ, = 0. Furtermore, note tat tis implies, Proposition.. Suppose tat ψ S and v X are arbitrary. Ten Proof. Using integration-by-parts, we get b ψ, v, = 0..9 b ψ, v, = ψ, v = ψ, v = ψ, v ψ ψ, v = ψc, v c ψ ψ, v = 0..0 Observe tat c, v = 0 by te divergence teorem, using v n = 0 on Ω, and c ψ ψ, v = 0 since ψ ψ S and v X. 6

7 Remark.. Te last result relies on te fact tat ψ ψ S. In oter words, te pase field space sould be a subspace of te pressure space, wic is restrictive. If tis does not old, mass conservation is lost. It may, owever, be possible to prove wat we wis using a variation of te trilinear form b. For example, we may take te alternate form bψ, v, q := ψ v, ν v, ψν. Tis allows us to decouple te pressure space from te pase space. Te analysis of tis case will be considered in a future work. Remark.4. For te Stokes projection, if te family of meses satisfy certain reasonable properties, we ave P u u P u u P p p C s u H s p H s,. provided tat u, p H 0 Ω Hs Ω H s Ω, for all 0 s q. In fact, for our analysis, we do not need te optimal case s = q. We only require tat te sub-optimal case s = q olds, in oter words, we will only assume u, p H 0 Ω Hq Ω H q Ω. See Assumption.. Following similar arguments to wat are given in [6], we get te following teorem, wic we state witout proof: Teorem.5. For any m M, te fully discrete sceme.a.d is uniquely solvable and mass conservative, i.e., φ m φ0, = 0.. Unconditional Energy Stability We now sow tat te solutions to our sceme enjoy stability properties tat are similar to tose of te PDE solutions, and moreover, tese properties old regardless of te sizes of and τ. Te first property, te unconditional energy stability, is a direct result of te convex decomposition. Consider te modified energy were Eφ, u is defined as above. F φ, ψ, u := Eφ, u 4ε φ ψ ε 8 φ ψ, Lemma.6. Let φ m, µ m, u m, p m S S X S be te unique solution of.a.d, for m M. Ten te following energy law olds for any, τ > 0: F φ l, φ l, ul τ [ l m= 4ε φ m for all l M. l ε m= µm φ m φm ε 8 η γ ūm φ m ] φ m φm = F φ, φ0, u,. 7

8 Proof. Setting ν = µ m in.a, ψ = δ τ φ m in.b, v = γ ūm in.c, and q = γ pm in.d gives δ τ φ m, µ m ε χ φ m δ τ u m γ, ū m η γ ε, φ m ε a µm, δτ φ m b φ m, ū m, µ m = 0,. ε ˇφ m, δ τ φ m γ B ūm ū γ c m, p m b Combining..6, using te following identities χ φ m, φ m, δτ φ m φ m, δ τ φ m = 4τ a ˇφ m, δ τ φ m = τ 8τ φ m, δ τ φ m µ m, δ τ φ m ũ m, ū m, ū m φ m, ū m, µ m γ c φ m 4τ φ m φ m 8τ φ m 4τ φ m φ m φ m ū m, p m = 0,.4 = 0,.5 = 0..6 φ m φ m φ m φm φ m φm,.7 φ m φm φ m φ m φm,.8 and applying te operator τ lm= to te combined equations, we get.. Assumption.7. From tis point, we assume te following reasonable stabilities independent of and τ: E φ 0, u0 C, F φ, φ0, u = Eφ, u φ 4ε φ0 ε φ 8 φ0 C, τ µ τ ū C, were C > 0 is independent of. Remark.8. In te sequel, we will not track te dependences of te estimates on te interface parameter ε > 0 or te viscosity η > 0, toug tese may be important. Te next result follows from energy stability and Assumption.7. We omit te proof. 8

9 Lemma.9. Let φ m, µ m, u m, p m S S X S be te unique solution of.a.d, for m M. Ten te following estimates old for any, τ > 0: [ ] max φ m 0 m M φ m u m C,.9 ] max [ φ m 0 m M 4 L 4 φ m φ m H C,.0 [ φ m φm φ m φm ] C,. M m= [ φ m max m M τ M m=0 [ µm ūm ] C,. φ m φm φ m φ m φm ] C,. for some constant C > 0 tat is independent of, τ, and T. We are able to prove te next set of a priori stability estimates witout any restrictions on and τ. We define te discrete Laplacian, : S S, as follows: for any v S, v S denotes te unique solution to te problem In particular, setting ξ = v in.4, we obtain v, ξ = a v, ξ, ξ S..4 v = a v, v. We also need te following discrete Gagliardo-Nirenberg inequalities. See, for example, [9, 9]. Proposition.0. If Ω is a convex polygonal or polyedral domain, and T is a globally quasiuniform triangulation of Ω, we ave ψ L C ψ d 6 d 4 d 6 d ψ C ψ L 6 L 6, ψ S, d =,,.5 ψ L 4 C ψ ψ d 4 ψ 4 d 4, ψ S, d =,,.6 for some constant C > 0 tat is independent of. Assumption.. From tis point, we will assume tat Ω is a convex polygonal or polyedral domain, and T is a globally quasi-uniform triangulation of Ω. Furtermore, we assume te following initial stabilities old: τ δ τ φ τ δ τ φ τ µ C,.7 H, were C > 0 is independent of. See [, 6, 9] for a definition of te norm,. 9

10 Lemma.. Let φ m, µ m, u m, p m S S X S be te unique solution of.a.d, m M. Ten te following estimates old for any, τ > 0: [ M τ δ τ φ m ] δ τ φ m C,.8 m=0 τ M m= [ ˇφ m H τ, M µm m=0 CT,.9 ε ˇφ m µm C, m M,.0 ε φ µ C,. 46 d ] ˇφ m d CT,. for some constant C > 0 tat is independent of, τ, and T. L Te proof of Lemma. is very similar to proofs of [7, Lemma.7] and [6, Lemma.]. We omit te details for te sake of brevity.. Unconditional l 0, T ; L Stability of te Discrete Pase Variable Lemma.. Let φ m, µ m, u m, p m S S X S be te unique solution of.a.d, for m M. Ten te following estimates old for any, τ > 0: φ m 8 m k ˇφk m φ 0,. k= φ m 8 m k ˇφk m φ..4 k= Proof. Using te definition of ˇφ m, for m M, we ave te following inequality: ˇφ m = 4 φm 4 φm = = 8 φ m φ m φ m Its repeated use gives te result. φ m 8, φ m φ m 6 φ m 6 φ m 6 φ m 6 φ m 8..5 Assumption.4. From tis point on, we assume te following initial stabilities τ δ τ φ µ ρ φ 0 φ C,.6 were C > 0 is independent of. 0

11 We are now ready to sow te main result for tis section. Lemma.5. Let φ m, µ m, u m, p m S S X S be te unique solution of.a.d, for m M. Ten te following estimates old for any, τ > 0: max 0 m M µm max m M [ ˇφm max 0 m M τ M m=0 for some constant C > 0 tat is independent of, τ, and T. δ τ φ m Proof. Te proof will be completed in two parts. Part : m = Subtracting.6 from.b wit m =, we obtain µ µ, ψ = ε a ˇφ φ, ψ φ ε φ, ψ ε = ε a 4 τδ τ φ 4 τδ τ φ, ψ τδ τ φ ε, ψ χ φ ε, φ CT,.7 46 d ] ˇφ m d CT,.8 L [ ] φ m φ m 46 d d L CT,.9 χ φ, φ χ φ, φ 0, ψ χ φ, φ 0, ψ..40 Additionally, we take a weigted average of.a wit m = and.7 wit te weigts 4 and 4, respectively, to obtain, 4 δ τ φ 4 δ τ φ, ν = ε a 4 µ 4 µ, ν 4 b φ, ū, ν 4 b φ, ū, ν 4 ρ, ν..4

12 Taking ψ = 4 µ 4 µ in.40, ν = τ 4 δ τ φ τ 4 δ τ φ in.4, and adding te results yields µ µ, 4 µ 4 µ τ 4 δ τ φ 4 δ τ φ = φ ε φ0, 4 µ 4 µ χ φ 4ε, φ χ φ, φ 0, µ µ 4 b φ, ū, τ 4 δ τ φ τ 4 δ τ φ 4 b φ, ū, τ 4 δ τ φ τ 4 δ τ φ ρ, τ4 δ τ φ τ4 δ τ φ 4 4 µ C µ C φ C φ 0 C χ φ, φ C χ φ, φ0 τ 4 φ L L ū δ τ φ 4 δ τ φ τ 4 φ L L ū δ τ φ 4 δ τ φ τ 4 ρ 4 δ τ φ 4 δ τ φ C 4 µ Cτ 4 δ τ φ 4 δ τ φ ū φ φ Cτ 4 δ τ φ 4 δ τ φ ū φ φ τ 6 4 δ τ φ 4 δ τ φ C 4 µ τ 6 4 δ τ φ 4 δ τ φ Cτ ū Cτ ū C 4 µ τ 4 δ τ φ 4 δ τ φ Cτ ū, were we ave used Young s inequality, te embedding H L 6, estimates.0 and.6 and Assumption.4. Considering Assumption.4, estimate., and te following estimates 4 δ τ φ 4 δ τ φ 8 δ τ φ 8 δ τ φ similar to.5, µ µ, 4 µ 4 µ = 4 µ µ, µ 4 µ µ µ, we ave, 4 µ τ 6 δ τ φ C µ τ 6 δ τ φ Cτ ū C C..4 Now, using.0,.5, te embedding H Ω L 6 Ω, and.0, we ave ˇφ 46 d ˇφ d C. L Using Lemma.,.5, te embedding H Ω L 6 Ω, and.0, we obtain φ φ 46 d d C. L

13 Part : m M For m M, we subtract.b from itself at consecutive time steps to obtain µ m µ m, ψ = ε a ˇφ m ˇφ m, ψ φ m ε φ m, ψ χ φ m ε, φ m χ φ m, φ m, ψ = ε a 4 τδ τ φ m 4 τδ τ φ m, ψ ε τδ τ φ m τδ τ φ m, ψ ω m 4ε φ m φ m, ψ,.4 for all ψ S, were ω m := ω φ m, φ m, φm := φ m φ m φ m φ m φ m φm φ m φ m φm. We note tat using te H Ω L 6 Ω embedding, we acieve te following bound, ω m L = φ m φ m φ m φ m φ m φm φ m φ m φm C φ m C φ m L 6 L 6 C φ m L 6 C φ m C φ m H H C φ m C. H Now, for all m M, we take a weigted average of te m and m time steps wit te weigts 4 and 4 of.a, respectively, to obtain, 4 δ τ φ m 4 δ τ φ m, ν = ε a 4 µm 4 µm, ν 4 b φ m, ū m, ν 4 b φ m, ū m, ν, ν S..44 Taking ψ = 4 µm 4 µm te results yields τ 4ε in.4, ν = τ µ m µ m, 4 µm 4 µm τ 4 δ τ φ m 4 δ τ φ m = τ ε δ τ φ m δ τ φ m, 4 µm 4 µm τ 4ε ω m δ τ φ m, 4 µm 4 µm τ 4 b φ m, ū m, 4 δ τ φ m 4 δ τ φ m τ 8ε δ τ φ m τ 6ε ωm L µm δ τ φ m L 4 δ τ φ m 4 δ τ φ m in.44, and adding ω m δ τ φ m, 4 µm 4 µm τ4 b φ m, ū m, 4 δ τ φ m 4 δ τ φ m µ m τ 8ε δ τ φ m µm µ m µm µ m δ τ φ m µm µ m L 6 L 6 τ4 b φ m, ū m, 4 δ τ φ m 4 δ τ φ m τ 4 b φ m, ū m, 4 δ τ φ m 4 δ τ φ m.

14 Now we bound te trilinear form b,,. Using.6, Hölder s inequality, and.9, te following estimates are available: b φ m, ū m, 4 δ τ φ m 4 δ τ φ m φ m ūm L 4 L 4 4 δ τ φ m 4 δ τ φ m C 4 δ τ φ m 4 δ τ φ m ūm φ m φm 4 δ τ φ m 4 δ τ φ m C ūm C ūm φm,.45 and, similarly, b φ m, u m, 4 δ τ φ m 4 δ τ φ m 4 δ τ φ m 4 δ τ φ m C ūm for any m M. Terefore, we arrive at C ūm φm,.46 µ m µ m, 4 µm 4 µm τ 4 δ τ φ m 4 δ τ φ m τ 8 δ τ φ m τ δ τ φ m τ δ τ φ m Cτ µm Cτ µm H H Cτ ūm Cτ ūm φm Cτ ūm Cτ φm, m M..47 ūm Furtermore, we use Lemma. and.0 to derive te following inequalities: φ k φ k C φ k C φ k C C j= k k j ˇφj k k j µ j j= C φ k C φ k C C j= k k j ˇφj k k j µ j j= ˇφ j C µj C,.48 ˇφ j C µj C..49 4

15 Applying lm= to.47 and using te following properties µ m µ m, 4 µm 4 µm = we conclude tat µl 4 δ τ φ m 4 δ τ φ m m=0 µ m µ m, µ m µ m µ m 4 µ m, µ m µ m µ m = µm µm 8 µm µ m 8 µm µ m 8 µm µ m µ m 8 δ τ φ m 8 δ τ φ m similar to.5, τ l 6 δ τ φ m m= 8 µ µ τ δ τ φ 5τ δ τ φ l Cτ l µm Cτ m ūm m j µ j m=0 H m=0 j= l Cτ m ūm m j µ j l Cτ ūm CT Cτ l j= ūm m=0 j= m=0 m m j µ j µj, for any l M, were we ave used Part,.9 and.. Moreover, wit an application of te discrete Gronwall inequality from Lemma A. wit α = <, we arrive at µl τ l 6 δ τ φ m l CT exp CA α τ ūm m= m=0 CT,.50 were. as been repeatedly applied. Now, using.0,.5, te embedding H Ω L 6 Ω, and.0, we get ˇφ l 46 d ˇφ l d CT, l M. L By Lemma., te following bound is available: φ l CT, l M. Using.5 again, te embedding H Ω L 6 Ω, and.0, we arrive at 46 d φ l d CT, l M. L Te proof is completed by combining Parts and., 5

16 Error Estimates for te Fully Discrete Sceme Assumption.. For te error estimates tat we pursue in tis section, we sall assume tat weak solutions ave te additional regularities φ L 0, T ; W,6 Ω H 0, T ; H q Ω H 0, T ; H Ω H 0, T ; L Ω,. φ H 0, T ; H Ω, µ L 0, T ; H q Ω,.. u H 0, T ; L Ω L 0, T ; L 4 Ω L 0, T ; H q Ω H 0, T ; H q Ω,.4 p L 0, T ; H q Ω L 0Ω L 0, T ; H q Ω,.5 were q corresponds to te finite element spaces defined at te beginning of Section. Te norm bounds associated wit te assumed regularities above are not necessarily global-in-time and terefore can involve constants tat depend upon te final time T. We also assume tat te initial data are sufficiently regular so tat te stability from Assumptions.7,., and.4 old. Weak solutions φ, µ, u, p to.5a.5d wit te iger regularities..5 solve te following variational problem: for all t [0, T ], t φ, ν ε a µ, ν b φ, u, ν = 0, ν H Ω,.6 µ, ψ ε a φ, ψ ε φ φ, ψ = 0, ψ H Ω,.7 t u, v η a u, v B u, u, v c v, p γ b φ, v, µ = 0, v H 0Ω,.8 c u, q = 0, q L 0Ω..9 We define te following: for any real number m [0, M], t m := m τ, and ψ m := ψt m. Tis definition applies to vector valued functions of time as well. Note tat, in general, ψ m := ψtm ψ m ψ m =: ψ m. An over-bar will always indicate a simple central average in time. Denote E φ,m a := φ m R φ m, E µ,m a := µ m R µ m, E u,m a := u m P u m, E p,m a := p m P p m..0 Te following definitions are given for any integer 0 m M : δ τ φ m := φ m φ m τ, δ τ u m u m u m :=, τ σ φ,m := δ τ R φ m δτ φ m, σ u,m := δ τ P u m δτ u m, σ φ,m := δ τ φ m t φ m, σ u,m := δ τ u m t u m, σ φ,m := φ m φ m u,m, σ := ū m u m, σ φ,m 4 := χ φ m, φ m φ m p,m, σ := p m p m. 6

17 Ten te PDE solution, evaluated at te alf-integer time steps t m, satisfies δ τ R φ m, ν ε a R µ m, ν = ε a R φm, ψ R µ m, ψ = ε a, ν σ φ,m σ φ,m ε ε δ τ P u m, v η a P ū m, v c v, P p m = η a c P ū m, q = c σ φ,m, ψ b φ m, u m, ν E µ,m a, ψ σ φ,m 4, ψ,.a χ φ m, φ m, ψ ε φ m, ψ,.b, v σ u,m, v σ u,m σ u,m c v, σ p,m γ b φ m, v, µ m B u m, u m, v,.c σ u,m, q,.d for all ν, ψ S, v X, and q S, for any 0 m M. Restating te fully discrete splitting sceme.a.d, we ave, for m M, and for all ν, ψ S, v X, and q S, ε a δ τ φ m, ν ε a φ m, ψ ε 4 a τ δτ φ m, ψ δ τ u m, v η a ū m, v µ m, ν µ m, ψ = b φ m = ε, ū m, ν χ φ m, φ m, ψ ε c v, p m = γ b φ m, v, µ m B ũ m c ū m, q were δ τ ψ m := τ ψ m ψ m ψ m. Now let us define te following additional error terms,.a φ m, ψ,.b, ū m, v,.c = 0,.d E φ,m := R φ m φ m, Eφ,m := φ m φ m, Eµ,m := R µ m µ m, E u,m := P u m u m, Eu,m := u m u m, Ep,m := P p m p m, Ep,m := p m p m.. We also define, for m M, 5 := χ φ m, φ m χ φ m, φ m,.4 σ φ,m σ φ,m 6 := φ m φm,.5 σ u,m 6 := u m ũ m..6 7

18 Subtracting.a.d from.a.d, yields, for m M, δ τ E φ,m, ν ε a E µ,m, ν = σ φ,m σ φ,m, ν b φ m, u m, ν b φ m, ū m, ν,.7a ε a Ē φ,m, ψ ετ 4 a δτ E φ,m, ψ E µ,m, ψ = ε a σ φ,m, ψ E µ,m a, ψ σ φ,m 5, ψ σ φ,m 4, ψ ε ε σ φ,m 6, ψ ετ ε 4 a δτ φ m, ψ,.7b δ τ E u,m, v η a Ē u,m, v c v, Ēp,m c Ē u,m, q =, v σ u,m σ u,m c v, σ p,m γ b B u m, u m, v B ũ m, ū m, v = c σ u,m, q η a σ u,m, v φ m, v, µ m γ b φ m, v, µ m,.7c = 0,.7d Setting ν = E µ,m in.7a, ψ = δ τ E φ,m in.7b, v = γ Ēu,m in.7c, q = γ Ēp,m 8

19 in.7d and adding te resulting equations, we ave ε E φ,m τ E φ,m E u,m γτ E u,m ε Eµ,m ετ 4 a δτ E φ,m, δ τ E φ,m η γ Ēu,m = σ φ,m σ φ,m, E µ,m ε a σ φ,m, δ τ E φ,m ε γ b b σ φ,m 4 σ φ,m 5 σ φ,m 6, δ τ E φ,m ηγ a σ u,m σ u,m, Ēu,m φ m, u m µ,m, E φ m, Ē u,m, µ m γ B u m, u m, Ē u,m ετ 4 a σ u,m, Ēu,m b φ m, ū m, E µ,m b φ m, Ēu,m, µ m E µ,m a, δ τ E φ,m δ τ φ m, δ τ E φ,m γ B ũ m, ū m, Ēu,m γ c Ē u,m, σ p,m,.8 for all m M. Expression.8 is te key error equation from wic we will define our error estimates. Observe tat te error equation is not defined for m = 0. Te following estimates are standard and te proofs are omitted. Lemma.. Suppose tat φ, µ, u, p is a weak solution to.a.d, wit te additional regularities in Assumption.. Ten for all t m [0, T ] and for any, τ > 0, tere exists a constant C > 0, independent of and τ and T, suc tat for all 0 m M, σφ,m σφ,m σφ,m σφ,m φ m φm φ m τ H 96 In addition, for all m M, τ δ τ φ m τ τ δ τ φ m τ σφ,m 6 τ tm C q s φs Hq ds,.9 τ t m τ tm sss φs ds, t m τ tm ss φs ds,. 96 t m τ tm ss φs ds,. 96 t m tm tm t m ss φ s H ds.. t m ss φs ds,.4 tm t m ss φs ds,.5 tm t m ss φs ds..6 9

20 Lemma.. Suppose tat φ, µ, u, p is a weak solution to.a.d, wit te additional regularities in Assumption.. Ten for all t m [0, T ] and for any, τ > 0, tere exists a constant C > 0, independent of and τ and T, suc tat for all 0 m M, σu,m σu,m σu,m σu,m σp,m In addition, for all m M, tm C q s us Hq ds,.7 τ t m τ tm sss us ds, t m τ tm ss us ds,.9 96 t m τ tm ss us ds,.0 96 t m τ tm ss ps ds.. 96 t m τ δ τ u m τ τ δ τ u m τ σu,m 6 τ τ δ τ p m τ Te following estimates are proved in [7]. tm t m ss us ds,. tm t m ss us ds,. tm t m ss us ds,.4 tm t m ss ps ds..5 Lemma.4. Suppose tat φ, µ, u, p is a weak solution to.a.d, wit te additional regularities in Assumption.. Ten, tere exists a constant C > 0 independent of and τ but possibly dependent upon T troug te regularity estimates suc tat, for any, τ > 0, tm tm σφ,m 4 Cτ σφ,m 5 were E φ,m := φ m φ m. ss φs ds Cτ ss φ s ds,.6 H t m t m C E φ,m E C φ,m,.7 Lemma.5. Suppose tat φ, µ, u, p is a weak solution to.a.d, wit te additional regularities in Assumption.. Ten, tere exists a constant C > 0 independent of and τ suc tat, for any, τ > 0, tm σφ,m 6 Cτ ss φs ds C E φ,m E C φ,m,.8 t m tm σu,m 6 Cτ ss us ds C E u,m C E u,m,.9 t m were E φ,m := φ m φ m and Eu,m := u m u m. 0

21 Proof. For m M, using te truncation error estimate.6, we obtain σφ,m 6 τ tm Estimate.9 similarly follows. ss φs ds 7 t m 4 Te following tecnical lemma is proved in [6]. Lemma.6. Suppose g H Ω, and v S. Ten for some C > 0 tat is independent of. E φ,m 4 E φ,m..40 g, v C g v,,.4 We use only some very basic estimates for te trilinear form B: Lemma.7. Suppose u, v, w H 0 Ω. Ten If u L Ω and v, w H 0 Ω, ten If u L Ω, v, w H Ω L Ω, ten B u, v, w C u v w..4 B u, v, w C u L v w..4 B u, v, w u v w L w v L..44 We also recall some basic inverse inequalities. ϕ W m q C d /q d/p l m ϕ W l p, ϕ M r, p q, 0 l m,.45 From tis and te Gagliardo-Nirenburg and Poincaré inequalities it follows tat [] for all ϕ M r,0. ϕ L C d ϕ ϕ, d =,,.46 Lemma.8. Let P, P : V L 0 V S be defined as in.5 and suppose tat φ, µ, u, p is a weak solution to.a.d, wit te additional regularities in Assumption.. Ten, for any, τ > 0 tere exists a constant C > 0, independent of and τ, suc tat, for 0 m M, P u L 0,T ;L Ω C,.47 and, as a simple consequence, Ea u L 0,T ;L Ω C..48

22 Proof. Let w = I u X, te standard Lagrange nodal interpolant of u. Following Baker s unpublised paper [] and using standard finite element approximations, including., inverse inequalities, and Sobolev s embedding teorem, we ave P u L = P u w w u u L P u w L w u L u L C d P u w w u L u L C d P u u u w w u L u L C u w L d u w C d P u u u L u L C q d u H q p H q. Taking te L norm over 0, T and noting tat q, te proof is concluded. We now proceed to estimate te terms on te rigt-and-side of.8. Lemma.9. Suppose tat φ, µ, u, p is a weak solution to.a.d, wit te additional regularities in Assumption.. Ten, for any, τ > 0 and any α > 0 tere exist a constant C = Cα, T > 0, independent of and τ, suc tat, for m M, ε E φ,m τ E φ,m E u,m τγ E u,m ετ 4 a δτ E φ,m, δ τ E φ,m ε 4 Eµ,m η γ Ēu,m C E φ,m C E φ,m C E φ,m C E u,m C E u,m α δ τ E φ,m CR m,.49 were R m := q τ tm t m τ tm s φs q Hq ds τ t m sss us ds τ tm, tm s us Hq ds τ t m t m ss φ s H ds τ tm tm tm t m t m tm sss φs ds ss φs ds tm τ ss us ds τ ss φs ds τ ss ps ds t m t m t m q µ m H q φ m q φ m q H q H q q φ m q H u m q H q q u m H q q u m H q q p m H q q p m H q q p m H q..50 Proof. Define, for m M, time-dependent spatial mass average E µ,m := Ω E µ,m,..5 Using te Caucy-Scwarz inequality, te Poincaré inequality, wit te fact tat σ φ,m σ φ,m, = 0,

23 and te local truncation error estimates.9 and.0, we get te following estimate: σ φ,m σ φ,m, E µ,m = σ φ,m σ φ,m, E µ,m E µ,m σφ,m σ φ,m Eµ,m E µ,m C σφ,m σ φ,m Eµ,m C σφ,m C σφ,m ε 8 Eµ,m C q tm s φs τ tm Hq ds C sss φs ds ε τ t m 640 t m 8 Eµ,m..5 Meanwile, standard finite element approximation teory sows tat Eµ,m a = R µ m µ m C q µ m. H q Applying Lemma.6 and te last estimate, we ave E µ,m a, δ τ E φ,m C Eµ,m a δ τ E φ,m, C q µ m H q δ τ E φ,m, C q µ m α H q 6 δ τ E φ,m Using Lemma.6 and estimate., we find ε a σ φ,m, δ τ E φ,m = ε σ φ,m, δ τ E φ,m C σφ,m δ τ E φ,m, C τ tm ss φs ds α 96 t m 6 δ τ E φ,m Now, using Lemmas.4 and.6, we obtain ε σ φ,m 4, δ τ E φ,m C σφ,m 4 δ τ E φ,m, C σφ,m 4 α 6 δ τ E φ,m tm Cτ ss φs ds t m,, tm Cτ ss φ s ds α H t m 6 δ τ E φ,m..5,,

24 Similarly, using Lemmas.4 and.6, te relation E φ,m = Ea φ,m E φ,m, and a standard finite element error estimate, we arrive at ε σ φ,m 5, δ τ E φ,m C σφ,m 5 α 6 δ τ E φ,m, C E φ,m E C φ,m α 6 δ τ E φ,m, C Ea φ,m C E φ,m C Ea φ,m C E φ,m α 6 δ τ E φ,m, C q φ m C H E φ,m q C q φ m H q C E φ,m α 6 δ τ E φ,m..56 Applying Lemmas.5 and.6, te relation E φ,m = Ea φ,m E φ,m, and a standard finite element error estimate, we find tat ε σ φ,m 6, δ τ E φ,m C σφ,m 6 α 6 δ τ E φ,m, tm Cτ ss φs ds C t m, E φ,m C C q φ m H q Cq φ m α H q 6 δ τ E φ,m Te following inequality is a direct consequence of.4: ετ 4 a δτ φ m, δ τ E φ,m C τ tm ss φs ds α t m 6 δ τ E φ,m Using Lemma., we also obtain σ u,m σ u,m, Ēu,m C Next, using., Ē γ c u,m, σ p,m σu,m η γ Ēu,m tm C τ 640 η γ η γ C σu,m C q τ η γ tm t m E φ,m, Ēu,m, s us Hq ds t m sss us ds..59 Ēu,m Ēu,m C σp,m τ tm ss ps ds, t m 4

25 Now let s consider te convection trilinear terms. Adding and subtracting te appropriate terms, for all m M, b φ m, u m µ,m, E b φ m, ū m, E µ,m b φ m, Ē u,m, µ m b φ m, Ēu,m, µ m b σ φ,m 6, u m µ,m, E E µ,m b σ φ,m 6, Ēu,m, µ m b φ m, Ēu,m a σ u,m, E µ,m E µ,m b φ m, Ēu,m, E µ,m a σφ,m 6 u m L 4 Eµ,m E µ,m σφ,m 6 Ēu,m µ m L 4 L 4 L 4 φ m Ēu,m a σ u,m Eµ,m E µ,m L 4 L 4 φ m Ēu,m Eµ,m a L 4 L 4 ε 8 Eµ,m η γ Ēu,m C E φ,m C E φ,m q φ m H q q φ m q u m q u m H q H q H q q u m H q q µ m tm tm Cτ ss φs ds Cτ ss us ds,.6 H q t m t m were we ave used Lemmas.,.,.5. Additionally, after adding and subtracting te appropriate terms, for m M, we ave γ B u m, u m, Ē u,m B ũ m, ū m, Ēu,m = γ B σ u,m, u m, Ē u,m τ B δ τ u m, u m, Ē u,m B ũ m u,m, σ, Ēu,m B ũ m, Ē u,m a, Ēu,m B Ẽ u,m a, Ēu,m a, Ēu,m B Ẽ u,m a, ū m, Ē u,m B Ẽ u,m, Ēu,m a, Ēu,m B Ẽ u,m, ū m, Ē u,m B ũ m, Ēu,m, Ēu,m..6 Tis is te same basic decomposition considered in Baker s paper []. We immediately see tat te last term vanises by anti-symmetry in te last two terms of B: B ũ m, Ēu,m, Ēu,m = 0. 5

26 We examine te oter eigt terms individually. Using estimate.4 of Lemma.7, γ B σ u,m, u m, Ē u,m C σu,m u m Ēu,m η γ Ēu,m C τ tm ss us ds;.6 96 t m γ γ B τ δτ u m, u m, Ē u,m C τ δτ u m u m Ēu,m η γ Ēu,m τ tm ss φs ds;.64 t m γ B ũ m u,m, σ B ũ m, Ē u,m, Ēu,m a, Ēu,m C η γ η γ C ũ m Ēu,m Ēu,m Ēu,m C σu,m C τ tm ss us ds; t m σu,m ũ m Ēu,m a Ēu,m η γ Ēu,m C q u m H u m q H q C q p m p m H q H q ;.66 γ B Ẽ u,m a, ū m, Ē u,m Ẽu,m a ū m Ēu,m η γ Ēu,m C q u m H q u m H q C q p m H q p m..67 H q Using te stability estimate.48, γ B Ẽ u,m a, Ēu,m a, Ēu,m Ẽu,m a Ēu,m a Ēu,m L C Ēu,m a Ēu,m η γ Ēu,m C q u m H u m q H q C q p m p m H q H q ;.68 6

27 and, wit te estimate.44, te inverse-sobolev inequality.46, te Poincaré inequality, and estimate.48 again, γ B Ẽ u,m, Ēu,m a, Ēu,m C Ẽu,m Ēu,m a Ēu,m L C Ẽu,m Ēu,m Ēu,m a L C Ẽu,m q u H q p H q d Ēu,m C Ẽu,m Ēu,m η γ Ēu,m C E u,m C E u,m ;.69 γ B Ẽ u,m, ū m, Ē u,m ū m ū m L η γ Ēu,m L 4 Ẽ u,m C E u,m C E u,m Ēu,m..70 Combining te estimates.5.70 wit te error equation.8, te result follows. Lemma.0. Suppose tat φ, µ, u, p is a weak solution to.a.b, wit te additional regularities in Assumption.. Ten, for any, τ > 0, tere exists a constant C > 0, independent of and τ, suc tat, for m M, δ τ E φ,m, ε E µ,m 5 C Ēu,m C E φ,m C E φ,m CR m,.7 were C = C0 C, C 0 is te H Ω L 4 Ω Sobolev embedding constant, C is a bound for max 0 t T φm, and Rm is te consistency term given in.50. Proof. Define T : S S via te variational problem: given ζ S, find ξ S suc tat a T ζ, ξ = ζ, ξ for all ξ S. Ten, setting ν = T δ τ E φ,m in.7a and combining, we 7

28 ave δ τ E φ,m, = ε a E µ,m, T φ m, u m, T δ τ E φ,m σ m σ m, T δ τ E φ,m b δ τ E φ,m b φ m, ū m, T δ τ E φ,m = ε a E µ,m, T δ τ E φ,m σ m σ m, T δ τ E φ,m b σ φ,m 6, u m, T δ τ E φ,m b φ m, Ēu,m u,m σ, T δ τ E φ,m ε Eµ,m δ τ E φ,m σm σ m T δ τ E φ,m, σφ,m 6 u m L 4 T δ τ E φ,m L 4 φ m u,m Ēu,m σ T δ τ E φ,m L ε E µ,m 4 δ τ E φ,m C σm σ m, C σm 6 T δ τ E φ,m C Ēu,m T δ τ E φ,m C Ē u,m a σ u,m T δ τ E φ,m ε E µ,m δ τ E φ,m C E φ,m C E φ,m, 5C Ēu,m CR m,.7 for m M, were we ave used Lemmas.,., and.5. Te result now follows. Lemma.. Suppose tat φ, µ, u, p is a weak solution to.a.d, wit te additional regularities described in Assumption.. Ten, for any, τ > 0, tere exists a constant C > 0, independent of and τ, but possibly dependent upon T, suc tat, for any m M, ε E φ,m τ E φ,m E u,m τγ E u,m ετ 4 a δτ E φ,m, δ τ E φ,m ε Eµ,m η 4γ Ēu,m C E φ,m C E φ,m C E φ,m C E u,m C E u,m CR m..7 Proof. Tis follows upon combining te last two lemmas and coosing α in.49 appropriately. Using te last lemma, we are ready to sow te main convergence result for our second-order splitting sceme. 8

29 Teorem.. Suppose φ, µ, u, p is a weak solution to.a.d, wit te additional regularities described in Assumption.. Ten, provided tat 0 < τ < τ 0, wit some τ 0 sufficiently small, max m M E φ,m E u,m τ M m= for some CT > 0 tat is independent of τ and. Eµ,m Ēu,m CT τ 4 q,.74 Proof. Using Lemma., we ave E φ,m E φ,m E u,m τγ E u,m 4 η 4γ Ēu,m E φ,m 8τ E φ,m E φ,m E φ,m C E φ,m C E φ,m C E φ,m C E u,m C τ CR m. Eµ,m E u,m.75 Now, applying τ l m= to.75, and observing tat Eφ,m 0 and E u,m 0, for m = 0,, leads to E φ,l E u,l τ l Eµ,m Ēu,m C τ m= l R m C4 τ m= l E φ,m m= C 5 τ l E u,m m= If 0 < τ τ 0 := C 4 < C 4, since C 4 τ, it follows from te last estimate tat E φ,l E u,l τ l Eµ,m Ēu,m C τ C 4 τ m= m= l R m C 4 τ C 4 τ C C 6 τ 4 q C 7 τ l E φ,m m= C 5τ C 4 τ l..76 E u,m m= l E φ,m E u,m,.77 m= were we ave used te fact tat τ M m= Rm C 6 τ 4 q and were C 7 := max C 4, C 5. Appealing to te discrete Gronwall inequality A., it follows tat, for any E φ,l E u,l for any l M. τ l m= Eµ,m Ēu,m C C 6 τ 4 q expc 7 T,.78 9

30 Remark.. From ere it is straigtforward to establis an optimal error estimate of te form E max φ,m E u,m τ m M M m= E µ,m Ēu,m CT τ 4 q.79 using E φ = Ea φ E φ, et cetera, te triangle inequality, and te standard spatial approximations. We omit te details for te sake of brevity. Acknowledgment Tis work is supported in part by te grants NSF DMS C. Wang, NSFC 78 C. Wang, NSF DMS-4869 S. Wise, NSF DMS X. Wang, and NSF DMS-70 X. Wang. A Some Discrete Gronwall Inequalities We will need te following discrete Gronwall inequality cited in [0, 7]: Lemma A.. Fix T > 0. Let M be a positive integer, and define τ T M. Suppose {a m} M m=0, {b m } M m=0 and {c m} M m=0 are non-negative sequences suc tat τ M m=0 c m C, were C is independent of τ and M. Furter suppose tat, a l τ l b m C τ m=0 l m=0 a m c m, l M, A. were C > 0 is a constant independent of τ and M. Ten, for all τ > 0, l l a l τ b m C exp τ c m C expc, l M. A. m=0 m=0 Note tat te sum on te rigt-and-side of A. must be explicit. In addition, te following more general discrete Gronwall inequality is needed in te stability analysis. Lemma A.. Fix T > 0. Let M be a positive integer, and define τ T M. Suppose {a m} M m=0, {b m } M m=0 and {c m} M m=0 are non-negative sequences suc tat τ M m=0 c m C, were C is independent of τ and M. Suppose tat, for all τ > 0 and for some constant 0 < α <, a l τ l b m C τ m=0 l m c m m=0 j=0 were C > 0 is a constant independent of τ and M. Ten, for all τ > 0, a l τ l m=0 α m j a j, l M, A. C b m C a 0 C exp, l M. A.4 α 0

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