Error Analysis of a Mixed Finite Element Method for a Cahn-Hilliard-Hele-Shaw System

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1 Error Analysis of a Mixed Finite Element Metod for a Can-Hilliard-Hele-Saw System Yuan Liu Wenbin Cen Ceng Wang Steven M. Wise June 1, 016 Abstract We present and analyze a mixed finite element numerical sceme for te Can-Hilliard- Hele-Saw equation, a modified Can-Hilliard equation coupled wit te Darcy flow law. Tis numerical sceme was first reported in [19], wit te weak convergence to a weak solution proven. In tis article, we provide an optimal rate error analysis. A convex splitting approac is taken in te temporal discretization, wic in turn leads to te unique solvability and unconditional energy stability. Instead of te more standard l 0, T ; L l 0, T ; H error estimate, we perform a discrete l 0, T ; H 1 l 0, T ; H error estimate for te pase variable, troug an L inner product wit te numerical error function associated wit te cemical potential. As a result, an unconditional convergence for te time step τ in terms of te spatial resolution is derived. Te nonlinear analysis is accomplised wit te elp of a discrete Gagliardo-Nirenberg type inequality in te finite element space, gotten by introducing a discrete Laplacian of te numerical solution, suc tat φ S, for every φ S, were S is te finite element space. Keywords. Can-Hilliard equation, Hele-Saw Flow, Converegence analysis. AMS subject classifications. 5K5 5K55 65K10 65M1 65M60 1 Introduction Let Ω R d, d =,, be an open, bounded and convex polygonal or polyedral domain. We consider te following Can-Hilliard-Hele-Saw problem wit natural and no-flux/no-flow boundary conditions: t φ = ε φu, in Ω T := Ω 0, T, 1.1a = ε 1 φ φ ε φ, in Ω T, 1.1b u + p = γφ, in Ω T, 1.1c u = 0, in Ω T, 1.1d n φ = n = 0, u n = 0, on Ω 0, T, 1.1e wit initial data φ 0 = φ0, H 1 Ω. We assume tat te model parameters satisfy ε, γ > 0. Scool of Matematics; Fudan University, Sangai, Cina @fudan.edu.cn Scool of Matematics; Fudan University, Sangai, Cina 00 wbcen@fudan.edu.cn Matematics Department; University of Massacusetts; Nort Dartmout, MA 077, USA cwang1@umassd.edu Matematics Department; University of Tennessee; Knoxville, TN 7996, USA corresponding autor: swise1@utk.edu 1

2 We can reformulate te model by eliminating te velocity: t φ = ε + φ p + γφ, in Ω T, 1.a = ε 1 φ φ ε φ, in Ω T, 1.b p = γ φ, in Ω T, 1.c n φ = n = n p = 0, on Ω 0, T. 1.d If needed, te velocity may be back-calculated as u = p + γφ. A weak formulation of te problem may be expressed as t φ, ν + ε, ν + p + γφ, φ ν = 0, ν H 1 Ω, 1.a ε 1 φ φ, ψ + ε φ, ψ, ψ = 0, ψ H 1 Ω, 1.b p + γφ, q = 0, q H 1 Ω, 1.c for almost every t 0, T. We will also consider a weak formulation tat keeps te velocity as separate variable: were We consider t φ, ν + εa, ν b φ, u, = 0, ν H 1 Ω, 1.a ε 1 φ φ, ψ + εa φ, ψ, ψ = 0, ψ H 1 Ω, 1.b u, v + c v, p γb φ, v, = 0, v L Ω, 1.c c u, q = 0, q H 1 Ω, 1.d a u, v := u, v, b ψ, v, ν := ψv, ν, c v, q := v, q. 1.5 Eφ = 1 ε φ 1 + ε φ = 1 ε φ L 1 ε φ + Ω ε + ε φ, 1.6 wic is defined for all φ H := { φ H 1 Ω φ φ 0, 1 = 0 }, were φ 0 = 1 Ω Ω φ 0xdx. From now on, we denote by te standard L norm, provided tere is no ambiguity. Clearly, Eφ 0 for all φ H. It is straigtforward to sow tat weak solutions dissipate te energy 1.6. In oter words, 1.1a 1.1e is a conserved gradient flow wit respect to te energy 1.6. Precisely, for any t [0, T ], we ave te energy law t 1 Eφt + γ us + ε s ds = Eφ 0, and, in addition, te following mass conservation law: φt,, 1 = φ 0, 1 = φ 0 Ω. Formally, one can also easily demonstrate tat in 1.1b is te variational derivative of E wit respect to φ. In symbols, = δ φ E. Definition 1.1. Define Te projection P : L Ω W is defined via W := { u L Ω u, q = 0, q H 1 Ω }. 1.8 Pw = p + w, 1.9 were p H 1 Ω := { φ H 1 Ω φ, 1 = 0 } is te unique solution to p + w, q = 0, q H 1 Ω. 1.10

3 Clearly Pw W for any w L Ω. Furtermore, we ave Lemma 1.. P is linear, and, given w L Ω, it follows tat Pw w, v = 0, v W In particular, since Pw W, and, consequently, for all w L. Pw w, Pw = 0, 1.1 Pw w, 1.1 Wit te projection, we ave te following alternate weak formulation: t φ, ν + ε, ν + Pγφ, φ ν = 0, ν H 1 Ω, 1.1a ε 1 φ φ, ψ + ε φ, ψ, ψ = 0, ψ H 1 Ω. 1.1b Equivalently, wit u = Pγφ, we ave t φ, ν + ε, ν u, φ ν = 0, ν H 1 Ω, 1.15a ε 1 φ φ, ψ + ε φ, ψ, ψ = 0, ψ H 1 Ω. 1.15b Te well-posedness of tis weak form, as well as te basic regularity of te weak solution, can be found in [19]. In more detail, a convex splitting numerical sceme, wic treats te terms of te variational derivative implicitly or explicitly according to weter te terms corresponding to te convex or concave parts of te energy, was formulated in [19], wit a mixed finite element approximation in space. Suc a numerical approac assures two matematical properties: unique solvability and unconditional energy stability; also see te related works for various PDE systems, including te pase field crystal PFC equation [, 5, 7,, 5, 9], epitaxial tin film growt model [8, 10, 1, ], and oters [1, ]. Moreover, for a gradient system coupled wit fluid motion, te idea of convex splitting can still be applied and tese distinguised matematical properties are retained, as given by a few recent works [9, 1, 1, 19, 8]. In particular, a weak convergence of te finite element numerical approximation to a global-in-time weak solution was establised in [19], using certain compactness arguments. In addition to tis weak convergence result, a convergence analysis wit an associated convergence order, for tese gradient flows coupled wit fluid motion, as attracted a great deal of attentions in recent years. For instance, a convex splitting finite element sceme applied to Can- Hilliard-Stokes equation was analyzed in [1] and an optimal rate convergence analysis was provided in detail. Suc a convergence result was derived by an H 1 error estimate, combined wit unconditional energy stability and oter iger order stability properties for certain numerical variables. Meanwile, a careful examination sows tat, tis convergence analysis relies eavily on te l 0, T ; H 1 stability bound of te velocity field, at te numerical level. Wit tis stability available, te maximum norm bound of te pase variable φ could be derived, wic leads to a great simplification in te convergence analysis. However, for te CHHS system 1.1a 1.1e, only an l 0, T ; L bound for te velocity field is valid. As a result, a global-in-time L bound is not available to te pase variable; see more detailed PDE analyses in [6, 7], etc. Witout tis estimate, an error estimate for te CHHS equation 1.1a 1.1e becomes very callenging, due to te appearance of a igly nonlinear convection term; te velocity error term turns out to be a Helmoltz projection of te nonlinear error associated wit γφ. In turn, even te igest

4 order diffusion term in te standard Can-Hilliard part is not able to control te numerical error term associated wit te nonlinear convection. In tis paper, we provide an optimal rate convergence analysis for te mixed finite element sceme applied to te CHHS equation 1.1a 1.1e, as reported in [19]. Instead of te standard l 0, T ; L l 0, T ; H error estimate for te pure Can-Hilliard equation [1,, 1, 15, 16, 18, 0, ], we perform an l 0, T ; H 1 l 0, T ; H error estimate in an alternate way. Tis error estimate is necessary to make te error term associated wit te nonlinear convection ave a nonpositive inner product wit te corresponding error test function, wic is crucial to te convergence analysis. In particular, we note tat, altoug te l 0, T ; H 1 error estimates ave been available for te pure Can-Hilliard equation in te existing literature [, 17,, 9], an l 0, T ; H error estimate remains open for te finite element approximation applied to te related PDE systems, in te autors knowledge. To overcome te difficulty associated wit te lack of regularity for te velocity field in te Darcy law, a discrete Gagliardo-Nirenberg inequality is needed in te finite element analysis, in bot - D and -D cases. Meanwile, suc an inequality is involved wit an H norm of te numerical solution, wic is beyond its regularity in te standard finite element space. In tis paper, we establis te desired inequality in a modified version, wic plays a key role in te nonlinear error estimate. First, a discrete Laplacian operator,, is introduced for any H 1 function in te finite element space. Subsequently, by applying various Sobolev inequalities for continuous function, combined wit a few error bounds in te finite element space, te maximum norm bound of te numerical solution could be establised in terms of a discrete Gagliardo-Nirenberg inequality. Anoter key point of te analysis presented in tis paper is tat, te l 0, T ; H 1 error estimate is performed troug an L inner product wit te numerical error associated wit te cemical potential term. Suc an inner product yields an L 0, T ; H 1 stability of te cemical potential error term, wic contains certain nonlinear parts. Tese nonlinear errors are analyzed via appropriate Sobolev inequalities, so tat its growt is always controlled. Furtermore, by applying a subtle W 1 estimate for te temporal derivative of te numerical solution at a discrete level, we could convert all te nonlinear error terms at te current time step into te ones at te previous one. Wit tis approac, an l 8 0, T ; L estimate of te numerical solution for te pase variable φ could be applied so tat an unconditional convergence for te time step τ in terms of te spatial resolution is available, and a constraint for bot τ and turns out to be very mild. Te rest of te paper is organized as follows. Te fully discrete finite element sceme is reviewed in Section. Terein we recall an unconditional energy stability and a few oter refined stability estimates, and a discrete Gagliardo-Nirenberg inequality is establised in te finite element space. Subsequently, te detailed convergence analysis is given by Section, wic results in an optimal rate error estimate. Finally, a useful discrete Gronwall inequality is restated in Appendix A. Some Mixed Finite Element Convex Splitting Scemes.1 Definitions of te Scemes Let M be a positive integer and 0 = t 0 < t 1 < < t M = T be a uniform partition of [0, T ], wit τ = t i t i 1, i = 1,..., M. Suppose T = {K} is a conforming, sape-regular, globally 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 1 Ω. Define L 0 Ω := { φ L Ω φ, 1 = 0 }. We set S := M q and S := S L 0 Ω, were q is a positive integer. Te mixed convex-splitting sceme is defined as follows [19]: for any

5 1 m M, given φ m 1 S, find φ m, m S and p m S suc tat δ τ φ m, ν + ε a m, ν + φ m 1 p m + γφ m 1 m, ν = 0, ν S,.1a ε 1 φ m φ m 1, ψ + ε a φ m, ψ m, ψ = 0, ψ S,.1b were δ τ φ m Te operator R : H 1 Ω S is te Ritz projection: p m + γφm m, ζ = 0, ζ S,.1c := φm φm 1, φ 0 τ := R φ 0.. a R ϕ ϕ, χ = 0, χ S, R ϕ ϕ, 1 = 0.. Te velocity may be defined from te oter variables as u m Now we define a discrete projection. Definition.1. Define := pm γφm 1 m L.. W := { u L Ω u, q = 0, q S }..5 Observe tat W W. Te projection P : L Ω W is defined via were p S is te unique solution to Clearly P W. Furtermore, we ave P w = p + w,.6 p + w, q = 0, q S..7 Lemma.. P is linear, and given any w L Ω, it follows tat In particular, since P w W, P w w, v = 0, v W..8 P w w, P w = 0,.9 and, consequently, for all w L. P w w,.10 Tere is an estimate for te difference between te projections P and P. Lemma.. Suppose tat w H q Ω wit te compatible boundary conditions w n = 0 on Ω and p H q+1 Ω were p = Pw w..11 Ten P w Pw C q p H q

6 Proof. By definition, were p H 1 Ω is te unique solution to and were p S is te unique solution to Tus by a standard approximation estimate. Pw = p + w,.1 p + w, q = 0, q H 1 Ω,.1 P w = p + w,.15 p + w, q = 0, q S..16 P w Pw = p p C q p H q+1,.17 We may re-express te sceme as δ τ φ m, ν + ε a m, ν + b φ m 1, P γφ m 1 m, ν = 0, ν S,.18a ε 1 φ m φ m 1, ψ + ε a φ m, ψ m, ψ = 0, ψ S,.18b or equivalently, wit u m := P γφ m 1 m L, as δ τ φ m, ν + ε a m, ν b φ m 1, u m, ν = 0, ν S,.19a ε 1 φ m φ m 1, ψ + ε a φ m, ψ m, ψ = 0, ψ S..19b We observe tat, in general, u m is a discontinuous function, its components are not in te finite element spaces so far described. To remedy tis we could formulate a sceme wic keeps te velocity as a separate variable in some appropriate finite element space. To tis end, we will also consider a sceme tat uses a mixed metod for te velocity and pressure: for any 1 m M, given φ m 1 S, find φ m, m S and u m X, p m Q suc tat δ τ φ m, ν + ε a m, ν b φ m 1, u m, ν = 0, ν S,.0a ε 1 φ m φ m 1, ψ + ε a φ m, ψ m, ψ = 0, ψ S,.0b u m, v + c v, p + γb φ m 1, v, m = 0, v X,.0c c u m, q = 0, q Q,.0d were X L and Q H 1 Ω are compatible and inf-sup stable finite element spaces. Here we ave used te so called primal mixed formulation. A finite element metod based on te dual mixed formulation is also available. We will not pursue tis furter at tis time. 6

7 . Unconditional Solvability and Energy Stability In tis subsection, we demonstrate some results from [1] and [19] tat are important for te proof in te following section. Tese results sow tat our scemes are unconditionally uniquely solvable. We begin by defining some macinery for te solvability, as well as te stability and convergence analyses discussed later. First, consider te invertible linear operator T : S S defined via te following variational problem: given ζ S, find T ζ S suc tat a T ζ, χ = ζ, χ, χ S..1 Tis clearly as a unique solution because a, is an inner product on S. We now wis to define a mes-dependent 1 norm, i.e., a discrete analogue to te H 1 norm. Te following result can be found in [1] and [19]. Lemma.. Let ζ, ξ S and set ζ, ξ 1, := a T ζ, T ξ = ζ, T ξ = T ζ, ξ.. Terefore,, 1, defines an inner product on S, and te induced negative norm satisfies ζ 1, := Consequently, for all χ S and all ζ S, Te following Poincaré-type estimate olds: ζ, χ ζ, ζ 1, = sup 0 χ S χ.. ζ, χ ζ 1, χ.. ζ 1, C ζ, ζ S,.5 for some C > 0 tat is independent of. Finally, if T is globally quasi-uniform, ten te following inverse estimate olds: ζ C 1 ζ 1,, ζ S,.6 for some C > 0 tat is independent of. Te result for te uniquely solvability of te sceme can be found in [19]. Te solutions to our sceme enjoy stability properties tat are similar to tose of te PDE solutions. 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 represented in te sceme [19]. Lemma.5. Let φ m, m, pm S S S be te unique solution of.1a.1b. Ten te following energy law olds for any, τ > 0: E φ l + τε l m + τ 1 γ m=1 l u m + τ m=1 { l ε δ τ φ m + 1 δ τ φ m ε m=1 + 1 ε φm δ τ φ m + 1 ε δ τ φ m } = E φ 0,.7 for all 0 l M. 7

8 Te discrete energy law immediately implies te following uniform in and τ a priori estimates for φ m, m, and um. Note tat, from tis point, we will not track te dependence of te estimates on te interface parameter ε > 0, toug tis may be of importance, especially if ε is made smaller [19]. Lemma.6. Suppose tat Ω is convex polyedral. Let φ m, m, pm S S S be te unique solution of.1a.1c. Assume tat E φ 0 < C0, independent of. Ten for any 0 m M, φ m dx = φ 0 dx,.8 Ω and tere is a constant C > 0 independent of and τ suc tat te following estimates old for any, τ > 0: [ max φ m 0 m M + φ m 1 ] C,.9 ] [ φ m L + φ m + φ m H 1 C,.0 M [ φ m m=1 φm 1 max 0 m M τ Ω M [ ] m + u m C,.1 m=1 + φ m φm 1 τ M m=1 + φ m φm φm 1 + φ m φ m 1 ] C,. [ ] φ m + m + φ m 6 d d L CT + 1,. τ M m=1 for some constant C > 0 tat is independent of, τ, and T. δ τ φ m W 1 C,. We are able to prove te next set of a priori stability estimates witout any restrictions of and τ. Before we begin, we will need te discrete Laplacian, : S S, wic is defined as follows: for any v S, v S denotes te unique solution to te problem In particular, setting χ = v in.5, we obtain v, χ = a v, χ, χ S..5 v = a v, v. Lemma.7. Te discrete Laplacian as te following properties. For any v S, and, tere is some constant C > 0 suc tat v v 1/ v 1/,.6 v C v,.7 and v C v..8 8

9 Proof. Te first inequality follows from. and te Caucy-Scwarz inequality. For te second inequality, starting from te first and using a standard inverse inequality, we ave v v v C 1 v v..9 Applying te inverse inequality again, te tird inequality follows as well. Next we need a kind of discrete Gagliardo-Nirenberg inequality in te finite element space. Noting tat te functions in te conforming finite element space only ave te regularity up to H 1, it is impossible to directly apply standard Gagliardo-Nirenberg inequalities involving iger order norms, suc as H or H. Now tat we ave te definition of, we can prove te following discrete Gagliardo-Nirenberg inequality. Similar tecniques can be found in te existing works [5, 8] for related finite element estimates involved wit iger order derivatives. Teorem.8. Suppose tat Ω is convex and polyedral. Ten, for any ψ S d d ψ L C ψ 6 d 6 d ψ + C ψ L 6 L 6, d =,,.0 ψ L C ψ d 6 ψ 6 d 6 + C ψ, d =,,.1 and, consequently, ψ ψ L C ψ d 6 d ψ 5d 6 d + C ψ, d =,,. ψ L C ψ d 1 ψ 1 d 1 + C ψ, d =,,. using te Poincaré inequality and estimate.6. Proof. Define HN := { φ H Ω n φ = 0 }. By elliptic regularity, for any ψ S, tere is a unique function ψ HN suc tat ψ, w = ψ, w, w H 1, ψ ψ, 1 = 0.. According to te definitions of R in. and te discrete Laplacian in.5, ψ = R ψ. Moreover, ψ = ψ in L Ω. Terefore, tere is a constant C > 0 suc tat ψ H C ψ = C ψ..5 We summarize some standard inverse inequalities, wic can be found in [6, 11]: ϕ W m q C d /q d/p l m ϕ W l p, ϕ S, 1 p q, 0 l m 1,.6 for some constant C > 0. By I : H Ω S we denote te C 0 Ω, piecewise-polynomial nodal interpolation operator, and we recall te following approximation estimate from [6, 11]: for any φ H Ω, and any q, φ I φ W m q C d /q d/ m φ H,.7 for m = 0, 1, and some constant C > 0. Ten, by approximation properties, an inverse inequality, and elliptic regularity, we ave ψ ψ L 6 ψ I ψ L 6 + I ψ ψ L 6 C d/ ψ I ψ + C d/ ψ H C d/ ψ ψ + C d/ ψ I ψ + C d/ ψ H C d/ ψ H Ω C d/ ψ..8 9

10 Terefore, by te triangle inequality, On te oter and, using.7 and.6, we ave ψ L 6 ψ L 6 + C d/ ψ..9 ψ C + d/ ψ L 6,.50 and combining te last two inequalities, we ave te reciprocal stability bound ψ L 6 C ψ L 6,.51 for some constant C > 0. Using te Gagliardo-Nirenberg inequality, we ave d 6 d ψ L C ψ L 6 d d 6 d ψ + C ψ H L 6 6 d C ψ L 6 ψ d 6 d + C ψ L 6..5 Using inverse inequalities, te approximation properties above, and te last inequality, we find ψ L ψ I ψ L + I ψ ψ L + ψ L C d/ ψ I ψ + C d/ ψ H + ψ L C d/ ψ ψ + C d/ ψ I ψ + C d/ ψ + ψ L C d/ ψ + ψ L C d/ ψ d 6 d ψ d 6 d 6 d + C ψ L 6 ψ d 6 d + C ψ L 6 d 6 d C ψ L 6 ψ d 6 d + C ψ L 6,.5 were te inequality.50 is applied in te last step. Te result.0 is proven. Since ψ is te Ritz projection of ψ, te forward stability ψ ψ follows easily. To obtain te inequality in te oter direction, by te definition of ψ, te triangle inequality, a standard approximation estimate for te Ritz projection, and te inverse inequality.7, it follows tat ψ ψ φ + ψ ψ + ψ d C ψ + ψ = C ψ,.5 wic is anoter type of reciprocal stability. Applying a different Gagliardo-Nirenberg inequality and using te reciprocal stability above, it follows tat To finis up, we argue as before ψ L C ψ 6 d 6 ψ d 6 H + C ψ C ψ 6 d 6 ψ d 6 + C ψ..55 ψ L ψ I ψ L + I ψ ψ L + ψ L C d/6 ψ I ψ + C 1 d/6 ψ + ψ L C d/6 ψ ψ + C d/6 ψ I ψ + C 1 d/6 ψ + ψ L C 1 d/6 ψ + C ψ 6 d 6 ψ d 6 + C ψ = C 1 d/6 ψ 6 d 6 ψ d 6 + C ψ 6 d 6 ψ d 6 + C ψ = C ψ 6 d 6 ψ d 6 + C ψ

11 Teorem.9. Let φ m, m, pm S S S be te unique solution of.1a.1c. Suppose tat E φ 0 < C0, independent of, and tat Ω is a convex polyedral. Te following estimate olds for any, τ > 0: M [ ] τ φ m + φ m 86 d d L C T + 1,.57 m=1 wit some constant C > 0 independent of, τ, and T. Proof. We first observe tat for any v S, v, v S, a v, v = v = v 1,..58 Taking ψ = φm in.1b, we ave ε φ m = ε 1 φ m, φm ε 1 φ m 1, φm m, φm = ε 1 φ m, φ m + ε 1 φ m 1, φ m + m, φ m = ε 1 φ m φ m + ε 1 φ m 1 φ m + m φ m Cε φ m + Cε φ m 1 + Cε 1 m + ε φ m Cε φ m L φ m + C + Cε 1 m + ε φ m Cε φ m L + C + Cε 1 m + ε φ m. Te first estimate follows upon summing and te result from.. To get te second estimate, we appeal to.: φ m L φ m φm L + φ m C φ m d 6 d φ m 5d 6 d + C φ m + φ m C + + C φ m d 6 d..59 φ 0 Hence, Summing gives te result. φ m 86 d d L C + C φ m..60 Error Estimates for te Fully Discrete Convex Splitting Sceme.1 Preliminary Estimates We utilize some notation to simplify te error analysis. To tis end, define te time lag operator L τ φt := φt τ, and te backward difference operator δ τ φt := φt Lτ φt τ. Define te approximation errors E φ a := φ R φ, E a := R,.1 σ φ := δ τ R φ t φ.. 11

12 Define te piecewise constant in time functions, for m = 1,... M and for t t m 1, t m ], ˆφt := φ m, ˆt := m, ût := um, ˆpt := pm, were φ m, m, um, and pm are te solutions of te fully discrete convex-splitting sceme.1a.1c. We take ˆφ0 = φ 0, et cetera, as is natural. Finally, let us define E φ := R φ ˆφ, E φ := φ ˆφ, E := R ˆ, E := ˆ.. Proposition.1. Te following key error equation olds for all t [τ, T ]: ε E + ε δ τ E φ + ετ δ τ E φ = σ φ, E + b φ, u, E b L τ ˆφ, û, E + E a, δ τ E φ + τ δ τ φ, δ τ E φ ε + ε 1 L τ E φ, δ τ E φ ε 1 φ ˆφ, δ τ E φ.. Proof. Weak solutions φ, wit te iger regularities.9.1 solve te following variational problem: t φ, ν + ε a, ν b φ, u, ν = 0, ν H 1 Ω,.5a, ψ ε a φ, ψ ε 1 φ φ, ψ ξ, ψ = 0, ψ H 1 Ω,.5b were u := Pγφ. By definition of te Ritz projection, for all ν, ψ S, we see tat δ τ R φ, ν + ε a R, ν = σ φ, ν + b φ, u, ν,.6a ε a R φ, ψ R, ψ = E a, ψ ε 1 φ L τ φ, ψ + τ ε δ τ φ, ψ..6b Tus, for τ t T, and all ν, ψ S, δ τ ˆφ, ν + ε a ˆ, ν = b L τ ˆφ, û, ν,.7a ε a ˆφ, ψ ˆ, ψ = ε 1 ˆφ L τ ˆφ, ψ,.7b were û = P γl τ ˆφ ˆ. Subtracting.7a.7b from.6a.6b, we ave, for all ν, ψ S, δ τ E φ, ν ε a + ε a E, ν = + b φ, u, ν b L τ ˆφ, û, ν σ φ, ν E φ, ψ E, ψ = E a, ψ + τ ε δ τ φ, ψ + ε 1 L τ E φ, ψ,.8a ε 1 φ ˆφ, ψ..8b Setting ν = E in.8a, ψ = δ τ E φ in.8b and summing te two equations, we ave te result. For te error estimates tat we pursue in tis section, we sall assume tat weak solutions ave te additional regularities: φ W 0, T ; W 1 Ω L 0, T ; W Ω 1 L 0, T ; W q+1 Ω,.9 L 0, T ; W6 1 Ω L 0, T ; W q+1 Ω,.10 u L 0, T ; H q Ω,.11 φ L 0, T ; H q Ω,.1 were q 1 is te spatial approximation order. We need some preliminary estimates, te proofs of wic can be found in [1]. 1

13 Lemma.. Suppose tat φ, is a weak solution to.5a.5b, wit te additional regularities.9.1. Ten, for any, τ > 0, tere exists C > 0, independent of and τ, suc tat σ φ t C q + Cτ..1 Lemma.. Suppose tat φ, is a weak solution to.5a.5b, wit te additional regularities.9.1. Ten, for any, τ > 0, φ ˆφ E C ˆφ + 1 φ..1 L Proof. For t [0, T ], te following estimate is valid: φ ˆφ φ = φ ˆφ ˆφ φ φ ˆφ φ + ˆφ E φ φ L 6 φ + ˆφ E φ L + ˆφ E φ L 6 6 L E C ˆφ + 1 φ,.15 L were C > 0 is independent of t [0, T ]. Ten, using te unconditional a priori estimates in. and te assumption tat φ L 0, T ; H 1 Ω, te result follows. In our error analysis we need to make use of some non-standard approximation results for te Ritz projection. Te proof of te following can be gleaned from te material in [6, C. 8] and []. Teorem.. Let Ω R be a convex polyedral domain. Assume tat te solution u of te Neumann-Poisson equation au, v = f, v, v H 1 Ω, as regularity u W 1 p Ω, for some p [, ]. Ten tere are constants C > 0 and 0, suc tat te stability R u W 1 p C u W 1 p.16 olds, provided 0 < < 0. Furtermore, if u Wp q+1 Ω, u R u W 1 p C q u W q+1 p were q is te order of te polynomial approximation defining R.,.17 Remark.5. If Ω is a convex polyedral domain, it is proven in [] tat te following best approximation property olds for te omogeneous Diriclet-Poisson problem: u R u L C inf χ S u χ L,.18 were u H 1 0 W 1. It is expected to be straigtforward to prove suc a result for omogeneous Neumann-Poisson problem as well. Wit suc a result, te last teorem will follow. 1

14 Lemma.6. Suppose tat φ, is a weak solution to.5a.5b, wit te additional regularities.9.1. Ten, for any, τ > 0 and any arbitrary θ > 0, tere exists a constant C > 0, independent of and τ, but dependent upon θ, suc tat ε E + ε δ τ E φ + ετ δ τ E φ Cτ + C q + b φ, u, E b L τ ˆφ, û, E + C L τ E φ + θ δτ E φ L W 1 ε 1 R φ ˆφ, δ τ E φ..19 Proof. Using Lemma., te Caucy-Scwarz inequality, te Poincaré inequality, and te fact tat σ φ, 1 = 0, we get te following estimates: if E t is te spatial average of E t, for 0 < t T, ten σ φ, E = σ φ, E E An application of Teorem. implies tat As a consequence, we arrive at Now, it follows tat σ C φ E C σ φ ε + E C q + Cτ + ε E..0 E a W 1 = R W 1 C q W q+1 E a, δ τ E φ E a W 1 δ τ E φ W 1 C q + θ δ τ E φ τ δ τ φt L τ t t τ W 1 C q W q+1. δ τ E φ W 1..1 s φs L ds Cτ,. and, terefore, using a Poincaré-type inequality, for any θ > 0, τ δ τ φ, δ τ E φ ε C τ δτ φ L δ τ E φ Cτ + θ δ τ E φ W 1 W 1.. Wit similar steps, te next-to-last term in. is controlled by, ε 1 L τ E φ, δ τ E φ Lτ C E φ L δ τ E φ W 1 C q L τ φ W q+1 + C L τ E φ + θ δ τ E φ L W 1 C q + C L τ E φ + θ δ τ E φ L,. using Teorem. in te second step. Te last term in. can be divided into φ ˆφ, δ τ E φ = φ R φ, δ τ E φ W 1 R φ ˆφ, δ τ E φ..5 1

15 Using te stability R φ W C φ 1 W and te non-standard approximation results from Teorem., and te assumed regularities of te PDE solution, te first term above can be bounded 1 as follows: for any θ > 0, φ R φ, δ τ E φ C φ R φ + θ δ τ E φ W 1 W 1 C φ + φr φ + R φ Ea φ φ + C φ R φ R φ + θ δ τ E φ L L W 1 C φ E φ φ L + C a + R φ E φ L a φ + C R φ E φ L a + θ δ τ E φ L W 1 C φ W 1 Ea φ E + C L φ W 1 φ φ L 6 + C a L φ 6 W 1 Ea φ + θ δ τ E φ L W 1 C q + θ δ τ E φ..6 W 1 Combining.0.6 leads to te result. Now, let us consider te error of te triple form in.. Define I := b φ, u, E b L τ ˆφ, û, E..7 Lemma.7. Suppose tat φ, is a weak solution to.5a.5b, wit te additional regularities.9.1. Ten, for any, τ > 0, tere exists a constant C > 0, independent of and τ, suc tat I γ P L τ ˆφ E + C ˆD0 τ + q + C ˆD Lτ 0 E φ + ε E,.8 were ˆD 0 := L τ ˆφ L Proof. By adding and subtracting appropriate terms, we ave I = b Ea φ, u, E + b L τ E φ, u, E + b τδ τ R φ, u, E Te last term is te only one tat will give us any concern. Recall tat te discrete and continuous velocities can be described as We obtain te following useful decomposition: γ 1 u û = Pφ P L τ ˆφ ˆ + b L τ ˆφ, u û, E..0 u = Pγφ, û = P γl τ ˆφ ˆ..1 = Pφ P φ + P φ P L τ ˆφ ˆ = Pφ P φ + P τδ τ φ + P L τ φ P L τ ˆφ ˆ = Pφ P φ + P τδ τ φ + P L τ E φ + P L τ ˆφ E.. 15

16 Let s deal wit all te above terms except for te last one. Define I 5 := Pφ P φ + P τδ τ φ + P L τ E φ.. Ten I 5 Pφ P φ + P τδ τ φ + P L τ E φ C q φ H q + Cτ L 6 tφ L + 6 Lτ Ea φ + 6 L τ E φ C q + τ + C q L 6 φ H q+1 + Lτ L 6 E φ L C q + τ + C L τ E φ.. From.0 we ave I = b Ea φ, u, E + b L τ E φ, u, E + b τδ τ R φ, u, E Ea φ u L 6 L E + L τ E φ u L 6 L E E φ Lτ + a + Ea φ + τδ τ φ u E b C q + ε E + C L τ E φ + ε E + Cτ + ε E + b L τ ˆφ, u û, E Now, using. we ave b L τ ˆφ, u û, E + b L τ ˆφ, u û, E L τ ˆφ, u û, E..5 = γb L τ ˆφ, I5, E γb L τ ˆφ, P L τ ˆφ E a, E γb L τ ˆφ, P L τ ˆφ E, E C L τ ˆφ I 5 + ε E L 16 + C q Lτ ˆφ L H q+1 + ε E 16 γb L τ ˆφ, P L τ ˆφ E, E C ˆD 0 τ + q + C ˆD Lτ 0 E φ + ε E 8 γ P L τ ˆφ E..6 To finis up, adding.5 and.6 leads to te result. Combining Lemmas.6 and.7, we get immediately te following result: Lemma.8. Suppose tat φ, is a weak solution to.5a.5b, wit te additional regularities.9.1. Ten, for any, τ > 0, and any arbitrary θ > 0, tere exists a constant C > 0, independent of and τ, but dependent on θ, suc tat ε E + ε δ τ E φ + ετ δ τ E φ + γ P L τ ˆφ E C ˆD 0 τ + q + C ˆD Lτ 0 E φ + C L τ E φ + θ δ τ E φ ε 1 R φ ˆφ, δ τ E φ W 1 L..7 16

17 Te next step is to prove tat te dual norm δ τ E φ W 1 can be bounded in a convenient way. Lemma.9. Suppose tat φ, is a weak solution to.5a.5b, wit te additional regularities.9.1. Ten, for any, τ > 0, δ τ E φ Cε E Lτ + Cγ P L τ ˆφ E + C E φ + C ˆD 1 0 q + τ,.8 W 1 were C > 0 is independent of and τ. Proof. Here we follow te ideas in [19]. Let Q be te standard L projection into S. For any ν W 1, Ω, denote ν = Q ν in.8a. Recall te estimate for σ φ from Lemma., δ τ E φ, ν = δ τ E φ, ν = ε a E, ν + σ φ, ν + b φ, u, ν b L τ ˆφ, û, ν ε E ν + σ φ ν + b φ, u, ν b L τ ˆφ, û, ν C ε E + q + τ ν W 1 + b φ, u, ν b L τ ˆφ, û, ν..9 For te last two terms above, we repeat te tecniques used to analyze I in.0. Define I 6 := b φ, u, ν b L τ ˆφ, û, ν..0 Recalling te estimates in.,.5, and.6, we can estimate I 6 as follows: I 6 = b E a φ, u, ν + b L τ E φ, u, ν + b τδ τ R φ, u, ν + b L τ ˆφ, u û, ν Ea φ u L 6 L ν + L τ E φ u L 6 L ν + τδ τ R φ u ν b L τ ˆφ, u û, ν C q + τ + L τ E φ ν + γb L τ ˆφ, I5, ν + γb L τ ˆφ, P L τ ˆφ E a, ν + γb L τ ˆφ, P L τ ˆφ E, ν C q + τ + L τ E φ ν + γ L τ ˆφ I 5 ν L 6 L + C q Lτ ˆφ L H q+1 ν L + γ L τ ˆφ P L 6 L τ ˆφ E ν L C ˆD 1 0 q + τ + γ P L τ ˆφ E Lτ + E φ ν L..1 Combining.9 and.1, we get δ τ E φ, ν C ˆD 1 0 q + τ + ε E + γ P L τ ˆφ E Lτ + E φ ν W 1 = C ˆD 1 0 q + τ + ε E + γ P L τ ˆφ E Lτ + E φ ν W 1.. Te last estimate is due to te W 1 stability of te L projection into te finite element space. See, for example, [7]. 17

18 Now, if we coose θ in.7 sufficiently small, and apply Lemma.9, te following result could be easily obtained: Lemma.10. Suppose tat φ, is a weak solution to.5a.5b, wit te additional regularities.9.1. Ten, for any, τ > 0, tere exists a constant C > 0, independent of and τ, suc tat ε E 8 + ε δ τ E φ + γ P L τ ˆφ E C ˆD 0 τ + q + C ˆD Lτ 0 E φ + C L τ E φ L ε 1 R φ ˆφ, δ τ E φ... Estimates for te Cubic Nonlinear Error Term Now tat all te preliminary estimates ave been done, we will ten elaborate ow to deduce te stability for te error function.. Te result.7 is not enoug to get wat we want, since te last term of te rigt side as not been estimated yet. If it is estimated in te normal way, suc as using te Caucy-Scwarz inequality directly and summing every step, wat we get is at most a stable inequality coupled wit an implicit term like τĉ E φ on te rigt side wit Ĉ is dependent on some norm of te numerical solution ˆφ. In tis case, τ needs to be small enoug in order to be absorbed by te left side. In addition, te ig nonlinearity of te last term in.7 is anoter difficulty to be overcome. If we do not use dual norm estimates, wat we get from.7 is a discrete nonlinear Gronwall inequality wic leads us to te sub-optimal convergence rate. Te main result is demonstrated below. Lemma.11. Suppose tat φ, is a weak solution to.5a.5b, wit te additional regularities.9.1. Ten, for any, τ > 0, tere exists a constant C > 0, independent of and τ, suc tat ε E φ t m + ετ Cτ E t j + γτ ˆD j 0 τ + q + C were ˆD j 0 := ˆD 0 t j and A j := τ j+1 ˆD 0 + τδ τ φ j+1 + τδ τ φ j+1 P m 1 W 1 E t j φ j 1 A j E φ t j + ε τ 8 + τ 1 7 τδ τ φ j W 1 m 1 E φ t j,. + τ 1 τδ τ φ j+1 W 1..5 Proof. Our starting point is estimate.. Te last term of. can be rewritten as R φt m φ m, δ τ E φ t m = ζ m E φ t m, δ τ E φ t m,.6 were ζ m := R φt m + φ m R φt m + φ m

19 By Lemma A., τ ε ζ m E φ t m, δ τ E φ t m = τ ε 1 ε δ τ ζ m, E φ t m 1 τ ε ζ m, ζ m, δ τ E φ t m E φ t m..8 Observe tat te last two terms on te rigt-and-side of te last identity are non-positive and can be dropped in te analysis. For any 1 m M, summation of. implies tat ε E φ t m + ετ E t j + γτ Cτ + q τ + τ ε P m 1 ˆD j 0 + Cτ E t j φ j 1 ˆD j+1 0 E φ t j + C E φ t j L δ τ ζ m, E φ t m 1,.9 were we ave dropped te indicated non-positive terms from te rigt-and-side. definition of ζ j, Due to te ζ j+1 ζ j = τδ τ R φt j+1 R φt j+1 + R φt j + τδ τ R φt j+1 φ j + τδ τ φ j+1 + τδ τ φ j+1 R φt j+1 + R φt j E φ t j..50 Ten for every step t j, te following estimate is available: ζ j+1 ζ j, E φ t j C φt j+1 + φt j W 1 τδ τ φt j+1 L E φ t j L + C φ j τδ τ φt L j+1 L E φ t j L + C τδ τ φ j+1 E φ L t j + C τδ τ φ j+1 L, E φ t j + C τδ τ φ j+1, E φ t j Cτ E φ t j + C τδ τ φ j+1 φ E t j E + C τδ τ φ j+1 φ t j W + E φ 1 t j W W 1 Now define I 7 := E φ t j W + E φ 1 t j W. 1 19

20 We observe tat I 7 can be analyzed as E φ I 7 C t j L E + φ t j E φ t E j + φ L t j L E + φ t j E φ t j L E E φ C t E j + 1 φ L t j E + C φ L 6 t E j + φ L t j φ L t j L C E φ t j + C E φ 1 t j E φ t j + C E φ 1 t j E φ 7 t j + C E φ 1 t j E φ 5 t j C E φ t j + C E φ 1 t j E φ 7 t j + C E φ 1 t j E φ t j..5 Here we reduce te power of E φ t j in some terms above according to te L H 1 bound of E φ. We also appeal to te discrete Gagliardo-Nirenberg inequality. and.. Tis is ten fed into.51 to obtain ζ j+1 ζ j, E φ t j Cτ E φ t j + C + C τδ τ φ j+1 + C τδ τ φ j+1 W 1 W 1 Cτ E φ t j + C + Cτ 1 7 τδ τ φ j+1 + Cτ 1 τδ τ φ j+1 τδτ φ j+1 E φ t j E φ t j 8 7 τδτ φ j+1 W 1 W 1 Due to te definition of A j from.5, we arrive at + τδ τ φ j+1 1 E φ t j 7 W 1 E φ t j 1 E φ t j E + τδ τ φ j+1 φ W 1 t j E φ t j + ε τ E φ t j E φ t j + ε τ E φ t j..5 ζ j+1 ζ j, E φ t j CA j E φ t j + ε τ E φ 16 t j..5 For te term τ E φ t j in.9, we apply te discrete Gagliardo-Nirenberg inequality and L Young s inequality again: τ E φ t j Cτ E φ 1 L t j E φ 7 t j + Cτ E φ t j Cτ E φ t j + ε τ E φ 16 t j..55 Combining.9.55, we finis te proof. Te following lemma demonstrates an approac to deal wit te term τ E φ t j on te rigt-and-side in.. 0

21 Lemma.1. Suppose tat φ, is a weak solution to.5a.5b, wit te additional regularities.9.1. Ten, for any, τ > 0, ε E φ t j E t j +C Proof. Since ˆD j+1 0 E φ t j +C E φ t j 1 +C ˆD j+1 0 τ + q..56 ε E φ t j E t j + E t j + ε E φ t j,.57 wat we need to estimate is te last term above. To bound E + ε E φ, set ψ = E + ε Eφ in.8b, wic in turn implies tat E + ε E φ = E a, E + ε Eφ + τ δ τ φ, E ε + ε Eφ +ε 1 L τ E φ, E + ε Eφ ε 1 φ ˆφ, E + ε Eφ C E a + τ δ τ φ + φ ˆφ Lτ + E φ + 1 E + ε E φ..58 Using tecniques from Lemmas. and.6, te above norm can be controlled as E t j + ε E φ t j Cτ + C ˆφtj + 1 q + C E φ L t j 1 E + C ˆφtj + 1 φ L t j j+1 C ˆD 0 τ + q j+1 + C ˆD 0 E φ t j +C E φ t j A combination of Lemmas.1 and.11 yields te following teorem. Teorem.1. Suppose tat φ, is a weak solution to.5a.5b, wit te additional regularities.9.1. Ten, for any, τ > 0, tere exists a constant C > 0, independent of and τ, suc tat ε E φ t m + ετ 8 Cτ E t j + γτ ˆDj ˆD j 0 P τ + q + C E t j φ j 1 m 1 A j E φ t j..60 Te summability of te sequence A j is ten essential to apply te discrete Gronwall inequality. We ave te following lemma: 1

22 Lemma.1. Suppose tat φ, is a weak solution to.5a.5b, wit te additional regularities.9.1. Ten, for any 1 m M and any, τ > 0, tere exists a constant C > 0, independent of and τ, suc tat A j C..61 Proof. Recalling.5 for te definition of A j j+1, τ ˆD 0 is summable due to Teorem.9. τδ τ φ j+1 and τδ τ φ j+1 are summable due to. and. respectively. For te last two terms in W 1.5, it can be estimated due to te Caucy-Scwarz inequality τ 1 7 τδ τ φ j W Cτ 7 C τ 1 τ 1 τδ τ φ j+1 τδ τ φ j+1 τ 1 τδτ φ j+1 W 1 W 1 7 W C,.6 and τ 1 τδ τ φ j+1 W Cτ 1 C τ 1 τ 1 τδ τ φ j+1 τδ τ φ j+1 τ 1 τδτ φ j+1 W 1 W 1 7 W 1 C..6. Main Convergence Result Applying te discrete Gronwall inequality to.60, we get te optimal convergence rate for te numerical sceme. Teorem.15. Suppose tat φ, is a weak solution to.5a.5b, wit te additional regularities.9.1. Ten, for any, τ > 0, tere exists a constant C > 0, independent of and τ, suc tat ε E φ t m + ετ 8 E t j + γτ P φ j 1 E t j Cτ + q..6

23 Remark.16. A combination of.59 and.6 yields tat Acknowledgment τ E φ t j Cτ + q..65 Tis work is supported in part by te grants NSF DMS C. Wang, NSFC C. Wang, NSF DMS S. Wise, NSFC , and W. Cen, and te fund by Cina Scolarsip Council Y. Liu. Y. Liu tanks University of California-San Diego for support during is visit. C. Wang also tanks Sangai Key Laboratory for Contemporary Applied Matematics, Fudan University, for support during is visit. A Discrete Gronwall Inequality We need te following discrete Gronwall inequality, cited in [6, 0]: Lemma A.1. Fix T > 0, and suppose {a m } M m=1, {b m} M m=1 and {c m} M 1 m=1 are non-negative sequences suc tat τ M 1 m=1 c m C 1, were C 1 is independent of τ and M, and M τ = T. Suppose tat, for all τ > 0, a M + τ M b m C + τ m=1 M 1 m=1 a m c m, A.1 were C > 0 is a constant independent of τ and M. Ten, for all τ > 0, M M 1 a M + τ b m C exp τ c m C expc 1. A. m=1 m=1 Note tat te sum on te rigt-and-side of A.1 must be explicit. Lemma A.. Suppose {a m } M m=1 and {b m} M m=0 are sequences suc tat b 0 = 0. Define, for any integer m, 1 m M, I m := a j b j b j b j 1. A. Ten te following identity is valid: I m = 1 a j a j 1 b j a j b j b j a mb m. A. References [1] L. Baňas and R. Nürnberg. Adaptive finite element metods for te Can-Hilliard equations. J. Comput. Appl. Mat., 181: 11, 008. [] L. Baňas and R. Nürnberg. A posteriori estimates for te Can Hilliard equation wit obstacle free energy. MAN Mat. Model. Numer. Anal., 5: , 009.

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