Numerische Mathematik

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1 Numer. Mat. DOI /s Numerisce Matematik Error analysis of a mixed finite element metod for a Can Hilliard Hele Saw system Yuan Liu 1 Wenbin Cen 1 Ceng Wang Steven M. Wise Received: 6 October 015 / Revised: 9 February 016 Springer-Verlag Berlin Heidelberg 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 Feng and Wise SIAM J Numer Anal 50:10 14, 01, 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 isderived.tenonlinearanalysisisaccomplisedwitteelp 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,wereS is te finite element space. B Steven M. Wise swise1@utk.edu Yuan Liu @fudan.edu.cn Wenbin Cen wbcen@fudan.edu.cn Ceng Wang cwang1@umassd.edu 1 Scool of Matematical Sciences, Fudan University, Sangai 004, Cina Matematics Department, University of Massacusetts, Nort Dartmout, MA 0747, USA Matematics Department, University of Tennessee, Knoxville, TN 7996, USA 1

2 Y. Liu et al. Matematics Subject Classification 5K5 5K55 65K10 65M1 65M60 1 Introduction Let 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 µ φu, in T := 0, T, 1.1a µ = ε 1 φ φ ε φ, in T, 1.1b u + p = γφ µ, in T, 1.1c u = 0, in T, 1.1d n φ = n µ = 0, u n = 0, on 0, T, 1.1e wit initial data φ 0 = φ0, H 1. Weassumetattemodelparameters satisfy ε, γ > 0. We can reformulate te model by eliminating te velocity: t φ = ε µ + φ p + γφ µ, in T, 1.a µ = ε 1 φ φ ε φ, in T, 1.b p = γ φ µ, in T, 1.c n φ = n µ = n p = 0, on 0, T. 1.d If needed, te velocity may be back-calculated as u = p + γφ µ. Aweak formulation of te problem may be expressed as t φ, ν + ε µ, ν + p + γφ µ, φ ν = 0, ν H 1, 1.a ε 1 φ φ, ψ + ε φ, ψ µ, ψ = 0, ψ H 1, 1.b p + γφ µ, q = 0, q H 1, 1.c for almost every t 0, T. Wewillalsoconsideraweakformulationtatkeepste velocity as separate variable: 1 t φ, ν + εa µ, ν b φ, u,µ = 0, ν H 1, 1.4a ε 1 φ φ, ψ + εa φ, ψ µ, ψ = 0, ψ H 1, 1.4b u, v + c v, p γ b φ, v,µ = 0, v L, 1.4c c u, q = 0, q H 1, 1.4d

3 Error Analysis of a Mixed Finite Element Metod were a u,v := u, v, b ψ, v, ν := ψv, ν, c v, q := v, q. 1.5 We consider Eφ = 1 φ 1 + ε 4ε φ = 1 4ε φ4 L 4 1 ε φ + 4ε + ε φ, 1.6 wic is defined for all φ H := { φ H 1 φ φ 0, 1 = 0 }, were φ 0 = φ 0xdx. Fromnowon,wedenoteby te standard L norm, provided 1 tere is no ambiguity. Clearly, Eφ 0forallφ H. It is straigtforward to sow tat weak solutions dissipate te energy 1.6. In oter words, 1.1a 1.1e isa conserved gradient flow wit respect to te energy 1.6. Precisely, for any t [0, T ], we ave te energy law t Eφt γ us + ε µs ds = Eφ 0, 1.7 and, in addition, te following mass conservation law: φt,, 1 = φ 0, 1 = φ 0. Formally,onecanalsoeasilydemonstratetatµ in 1.1b istevariational derivative of E wit respect to φ. Insymbols,µ = δ φ E. Definition 1.1 Define W := { } u L u, q = 0, q H Te projection P : L W is defined via Pw = p + w, 1.9 were p H 1 := { φ H 1 φ, 1 = 0 } is te unique solution to p + w, q = 0, q H Clearly Pw W for any w L.Furtermore,weave Lemma 1. P is linear, and, given w L, itfollowstat Pw w, v = 0, v W In particular, since Pw W, Pw w, Pw = 0, 1.1 1

4 Y. Liu et al. and, consequently, for all w L. Pw w, 1.1 Wit te projection, we ave te following alternate weak formulation: t φ, ν + ε µ, ν + Pγφ µ, φ ν = 0, ν H 1, 1.14a ε 1 φ φ, ψ + ε φ, ψ µ, ψ = 0, ψ H b Equivalently, wit u = Pγφ µ, weave t φ, ν + ε µ, ν u, φ ν = 0, ν H 1, 1.15a ε 1 φ φ, ψ + ε φ, ψ µ, ψ = 0, ψ H b 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 [4,5,7,4,5,9], 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] andanoptimalrateconvergenceanalysiswasprovidedindetail.suca 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 1

5 Error Analysis of a Mixed Finite Element Metod error term turns out to be a Helmoltz projection of te nonlinear error associated wit γφ µ. Inturn,eventeigestorderdiffusiontermintestandardCan 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,,14 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 non-positive 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 [,17,4,9], 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, adiscretelaplacianoperator,,isintroducedforanyh 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 couldbeappliedsotatanunconditionalconvergencefortetimestep τ in terms of te spatial resolution isavailable,andaconstraintforbotτ 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 Sect..Tereinwerecallanunconditionalenergystabilityand afewoterrefinedstabilityestimates,andadiscretegagliardo Nirenberginequalityis establised in te finite element space. Subsequently, te detailed convergence analysis is given by Sect., wic results in an optimal rate error estimate. Finally, a useful discrete Gronwall inequality is restated in Appendix 1. 1

6 Y. Liu et al. 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 ], witτ = t i t i 1, i = 1,...,M. Suppose T = {K } is a conforming, sape-regular, globally quasi-uniform family of triangulations of.for r Z +,definemr := { v C 0 } v K P r K, K T H 1. Define L 0 := { φ L φ, 1 = 0 }.WesetS := Mq and S := S L 0,were q is a positive integer. Te mixed convex-splitting sceme is defined as follows [19]: for any 1 m M,givenφ m 1 S,findφ m,µm S and p m S suc tat δτ φ m, ν + ε a µ m, ν + φ m 1 p m + γφm 1 were µ m, ν = 0, ν S,.1a ε 1 φ m φ m 1, ψ + ε a φ m, ψ µ m, ψ = 0, ψ S,.1b p m + γφ m µm, ζ = 0, ζ S,.1c δ τ φ m Te operator R : H 1 S is te Ritz projection: := 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 Now we define a discrete projection. Definition.1 Define W := u m := pm γφm 1 µ m L..4 {u L u, q = 0, q S }..5 Observe tat W W.TeprojectionP : L W is defined via P w = p + w,.6 were p S is te unique solution to p + w, q = 0, q S..7 1

7 Error Analysis of a Mixed Finite Element Metod Clearly P W.Furtermore,weave Lemma. P is linear, and given any w L,itfollowstat P w w, v = 0, v W..8 In particular, since P w W, 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 wit te compatible boundary conditions w n = 0 on and p H q+1 were p = Pw w..11 Ten P w Pw C q p H q+1..1 Proof By definition, were p H 1 is te unique solution to and were p S is te unique solution to Tus Pw = p + w,.1 p + w, q = 0, q H 1,.14 P w = p + w,.15 p + w, q = 0, q S..16 P w Pw = p p C q p H q+1,.17 by a standard approximation estimate. We may re-express te sceme as δτ φ m, ν + ε a µ m, ν + b φ m 1, P γφ m 1 µ m, ν = 0, ν S,.18a 1

8 Y. Liu et al. ε 1 φ m φ m 1, ψ + ε a φ m, ψ µ m, ψ = 0, ψ S,.18b or equivalently, wit u m := P γφ m 1 µ m L,as δτ φ m, ν + ε a µ 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 µ 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,findφ m,µm S and u m X, p m Q suc tat δτ φ m, ν + ε a µ 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 µ m, ψ = 0, ψ S,.0b u m, v + c v, p + γ b φ m 1, v,µ m = 0, v X,.0c c u m, q = 0, q Q,.0d were X L and Q H 1 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.. Unconditional solvability and energy stability In tis subsection, we demonstrate some results from [1,19]tat are important forte 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,findT 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.,adiscreteanaloguetoteh 1 norm. Te following result can be found in [1,19]. 1

9 Error Analysis of a Mixed Finite Element Metod Lemma.4 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,andteinducednegativenorm satisfies ζ, χ ζ 1, := ζ, ζ 1, = sup χ.. 0 =χ S Consequently, for all χ S and all ζ S, Te following Poincaré-type estimate olds: ζ, χ ζ 1, χ..4 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: for some C > 0 tat is independent of. ζ C 1 ζ 1,, ζ S,.6 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,µm, pm S S S be te unique solution of.1a,.1b. Tentefollowingenergylawoldsforany, τ > 0: E φ l + τε l { ε + τ + 1 ε m=1 l µ m 1 + τ γ m=1 δ τ φ m 1 + 4ε l u m m=1 δ τ φ m φ m δ τ φ m + 1 } δτ φ m = E φ 0,.7 ε for all 0 l M. Te discrete energy law immediately implies te following uniform in and τ a priori estimates for φ m, µm,andum.notetat,fromtispoint,wewillnottrackte dependence of te estimates on te interface parameter ε > 0, toug tis may be of importance, especially if ε is made smaller [19]. 1

10 Y. Liu et al. Lemma.6 Suppose tat is convex polyedral. Let φ m,µm, pm S S S be te unique solution of.1a.1c.assumetate φ 0 < C0,independentof. Ten for any 0 m M, φ m dx = φ 0 dx,.8 and tere is a constant C > 0 independent of and τ suc tat te following estimates old for any, τ > 0: max 0 m M max 0 m M τ M m=1 m=1 [ φ m + φ m ] 1 C,.9 [ φ m 4 L 4 + φ m ] + φ m H 1 C,.0 [ µ m + u m ] C,.1 M [ φ m φ φm 1 + m φ m 1 + φ m φm φm 1 τ τ + φ m φ m 1 ] C,. M m=1 m=1 [ φ m + µ m + φ m ] 46 d d L CT + 1,. M δτ φ m W 1 C,.4 for some constant C > 0 tat is independent of, τ, andt. We are able to prove te next set of aprioristability estimates witout any restrictions of and τ.beforewebegin,wewillneedtediscretelaplacian, : S S, wic is defined as follows: for any v S, v S denotes te unique solution to te problem v, χ = a v, χ, χ S..5 In particular, setting χ = v in.5, we obtain 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 1 v v 1/ v 1/,.6 v C v,.7

11 Error Analysis of a Mixed Finite Element Metod and v C v..8 Proof Te first inequality follows from.6 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,itisimpossibletodirectlyapplystandardGagliardo Nirenberg inequalities involving iger order norms, suc as H or H.Nowtatweavete definition of,wecanprovetefollowingdiscretegagliardo Nirenberginequality. Similar tecniques can be found in te existing works [5,8]forrelatedfiniteelement estimates involved wit iger order derivatives. Teorem.8 Suppose tat is convex and polyedral. Ten, for any ψ S d 4 d ψ L C ψ 6 d 6 d ψ + C ψ L 6 L 6, d =,,.40 ψ L C ψ d 6 ψ 6 d 6 + C ψ, d =,,.41 and, consequently, ψ ψ L C ψ 46 d d ψ 46 d 4 5d + C ψ, d =,,.4 ψ L C ψ d 1 ψ 1 d 1 + C ψ, d =,,.4 using te Poincaré inequality and estimate.6. Proof Define HN := { φ H 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 = According to te definitions of R in.andtediscretelaplacianin.5, ψ = R ψ.moreover, ψ = ψ in L. Terefore,tereisaconstantC > 0suc tat ψ H C ψ = C ψ..45 We summarize some standard inverse inequalities, wic can be found in [6,11]: 1

12 Y. Liu et al. ϕ W m q C d q d p l m ϕ W l p, ϕ S, 1 p q, 0 l m 1,.46 for some constant C > 0. By I : H S we denote te C 0, piecewisepolynomial nodal interpolation operator, and we recall te following approximation estimate from [6,11]: for any φ H, andany q, φ I φ W m q C d q d m φ H,.47 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 Terefore, by te triangle inequality, 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 C d ψ..48 ψ L 6 ψ L 6 + C d ψ..49 On te oter and, using.7 and.46, we ave 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 4 d 6 d ψ L C ψ L 6 4 d d 6 d ψ + C ψ H L 6 6 d C ψ L 6 ψ 6 d 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 1

13 Error Analysis of a Mixed Finite Element Metod C d ψ + ψ L C d ψ 4 d 6 d ψ 6 d d 4 d 6 d +C ψ L 6 ψ 6 d d + C ψ L 6 4 d 6 d C ψ L 6 ψ 6 d d + C ψ L 6,.5 were te inequality.50 isappliedintelaststep.teresult.40 isproven. 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,tetriangle inequality, a standard approximation estimate for te Ritz projection, and te inverse inequality.7, it follows tat ψ ψ φ + ψ ψ + ψ C ψ + ψ = C ψ,.54 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..56 Teorem.9 Let φ m,µm, pm S S S be te unique solution of.1a.1c. Suppose tat E φ 0 < C0, independent of, and tat is a convex polyedral. Te following estimate olds for any, τ > 0: τ M m=1 [ φ m + ] 86 d φ m d L C 4 T + 1,.57 wit some constant C 4 > 0 independent of, τ, andt. 1

14 Y. Liu et al. Proof We first observe tat for any v S, v, v S, a v, v = v = v..58 1, Taking ψ = φm in.1b, we ave ε φ m = ε 1 φ m, φ m ε 1 φ m 1, φm µ m, φm = ε 1 φ m, φ m + ε 1 φ m 1, φ m + µ m, φ m = ε 1 φ m φ m + ε 1 φ m 1 φ m + µ m φ m Cε φ m + Cε φ m 1 + Cε 1 µ m + ε φ m Cε φ m 4 φ m L + C + Cε 1 µ m ε + Cε φ m 4 L + C + Cε 1 µ m + ε φ m Te first estimate follows upon summing and te result from.. To get te second estimate, we appeal to.4: Hence, φ m L φ m φm L + φ m C φ m 46 d d φ m φ m. 4 5d 46 d +C φ m φ + m C + + C φ m 46 d d..59 φ 0 Summing gives te result. φ m 86 d d L C + C φ m..60 Errorestimatesfortefullydiscreteconvexsplittingsceme.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 := 1

15 Error Analysis of a Mixed Finite Element Metod φt L τ φt τ.defineteapproximationerrors E φ a := φ R φ, E µ a := µ R µ,.1 σ φ := δ τ R φ t φ.. Define te piecewise constant in time functions, for m = 1,...M and for t t m 1, t m ], ˆφt := φ m, ˆµt := µm, ût := um, ˆpt := pm, were φ m, µm, um,andpm are te solutions of te fully discrete convex-splitting sceme.1a.1c. We take ˆφ0 = φ 0,etcetera,asisnatural.Finally,letus define E φ := R φ ˆφ, E φ := φ ˆφ, E µ := R µ ˆµ, E µ := µ ˆµ.. Proposition.1 Te following key error equation olds for all t [τ, T ]: ε µ E + ε δ τ E φ + ετ = σ φ, E µ µ + b φ, u, E + ε 1 L τ E φ, δ τ E φ δ τ E φ b L τ ˆφ, û, E µ ε 1 φ ˆφ, δ τ E φ + + τ δ τ φ, δ τ E φ ε E a µ, δ τ E φ..4 Proof Weak solutions φ,µ wit te iger regularities.9.1 solve te following variational problem: t φ, ν + ε a µ, ν b φ, u, ν = 0, ν H 1,.5a µ, ψ ε a φ, ψ ε 1 φ φ, ψ ξ, ψ = 0, ψ H 1,.5b were u := Pγφ µ. BydefinitionofteRitzprojection,forallν, ψ S,we see tat δ τ R φ, ν + ε a R µ, ν = σ φ, ν + b φ, u, ν,.6a ε a R φ, ψ R µ, ψ = E a µ, ψ ε 1 φ L τ φ, ψ + τ ε δ τ φ, ψ. Tus, for τ t T,andallν, ψ S, δ τ ˆφ, ν ε a ˆφ, ψ + ε a ˆµ, ν = b L τ ˆφ, û, ν ˆµ, ψ = ε 1 ˆφ L τ ˆφ, ψ.6b,.7a,.7b 1

16 Y. Liu et al. were û = P γ L τ ˆφ ˆµ.Subtracting.7a,.7bfrom.6a,.6b, we ave, for all ν, ψ S, δ τ E φ, ν + ε a E µ, ν = σ φ, ν + b φ, u, ν b L τ ˆφ, û, ν,.8a E φ, ψ E µ, ψ = E a µ, ψ + τ ε δ τ φ, ψ + ε 1 L τ E φ, ψ ε a ε 1 φ ˆφ, ψ..8b Setting ν = E µ ave te result. in.8a, ψ = δ τ E φ in.8b andsummingtetwoequations,we 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 4 L 0, T ; W 1 L 0, T ; W q+1,.9 µ L 0, T ; W 1 6 L 0, T ; W q+1,.10 u L 0, T ; H q,.11 φ µ L 0, T ; H q,.1 were q 1istespatialapproximationorder. We need some preliminary estimates, te proofs of wic can be found in [1]. Lemma. Suppose tat φ,µ is a weak solution to.5a,.5b, witteaddi- tional regularities.9.1.ten,forany,τ > 0,tereexistsC > 0,independent of and τ, suctat σ φ t C q + Cτ..1 Lemma. Suppose tat φ,µ is a weak solution to.5a,.5b, wit te additional regularities.9.1. Ten,forany,τ > 0, φ ˆφ C ˆφ + 1 E φ..14 L Proof For t [0, T ],tefollowingestimateisvalid: φ ˆφ φ = φ ˆφ ˆφ φ φ ˆφ φ + ˆφ E φ φ L 6 φ + ˆφ E φ L 6 L 6 + ˆφ E φ L C ˆφ E + 1 φ,.15 L were C > 0isindependentoft [0, T ]. Ten,usingteunconditionalapriori estimates in.andteassumptiontatφ L 0, T ; H 1, te result follows. 1

17 Error Analysis of a Mixed Finite Element Metod 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.4 Let R be a convex polyedral domain. Assume tat te solution uofteneumann Poissonequation au,v= f,v, v H 1, as regularity u Wp 1,forsomep [, ].TentereareconstantsC > 0 and 0,suctattestability R u W 1 p C u W 1 p.16 olds, provided 0 < < 0.Furtermore,ifu Wp q+1, 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 is a convex polyedral domain, it is proven in []tattefollowing 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.Itisexpectedtobestraigtforwardtoprovesucaresultfor omogeneous Neumann Poisson problem as well. Wit suc a result, te last teorem will follow. Lemma.6 Suppose tat φ,µ is a weak solution to.5a,.5b, wit te additional regularities.9.1.ten,forany,τ > 0 and any arbitrary θ > 0,tere exists a constant C > 0, independentofandτ,butdependentuponθ,suctat ε µ E + ε δ τ E φ + ετ δ τ E φ Cτ + C q + b φ, u, E µ b L τ ˆφ, û, E µ + C L τ E φ + θ δτ E φ L ε 1 R φ ˆφ, δ τ E φ W Proof Using Lemma., tecaucy Scwarzinequality,tePoincaréinequality, and te fact tat σ φ, 1 = 0, we get te following estimates: if E µ t is te spatial average of E µ t,for0< t T,ten σ φ, E µ = σ φ, E µ Eµ C σ φ E µ C σ φ + ε C q + Cτ + ε E µ E µ..0 1

18 Y. Liu et al. An application of Teorem.4 implies tat As a consequence, we arrive at E a µ W 1 = R µ µ W 1 C q µ q+1 W. E a µ, δ τ E φ E a µ W 1 Now, it follows tat δτ E φ C q + θ δ τ E φ 4 τ δ τ φt L τ 4 t t τ W 1 W 1 C q µ W q+1 and, terefore, using a Poincaré-type inequality, for any θ > 0, τ ε δ τ φ, δ τ E φ C τ δτ φ L δ τ E φ δ τ E φ W 1..1 s φs L ds Cτ,. W 1 Cτ + θ δ τ E φ 4 Wit similar steps, te next-to-last term in.4 iscontrolledby, W 1.. ε 1 L τ E φ, δ τ E φ C Lτ 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 4 W 1 C q + C L τ E φ + θ δ τ E φ L,.4 4 using Teorem.4 in te second step. Te last term in.4 canbedividedinto W 1 φ ˆφ, δ τ E φ = φ R φ, δ τ E φ R φ ˆφ, δ τ E φ..5 Using te stability R φ W 1 C φ W 1 and te non-standard approximation results from Teorem.4, and te assumed regularities of te PDE solution,te first term above can be bounded as follows: for any θ > 0, 1 φ R φ, δ τ E φ C φ R φ + θ δ τ E φ W 1 4 C φ + φ R φ + R φ Ea φ W 1 L

19 Error Analysis of a Mixed Finite Element Metod + C φ φ R φ R φ + θ δ τ E φ L 4 W 1 C φ 4 E φ L a L + C φ + R φ Ea φ φ L + C R φ Ea φ + θ δ τ E φ L 4 W 1 C φ 4 W E φ 1 a L + C φ W 1 φ E φ L 6 a L 6 + C φ E φ W 1 a L + θ δ τ E φ 4 W 1 C q + θ δ τ E φ..6 4 W 1 Combining.0.6 leadstoteresult. Now, let us consider te error of te triple form in.4. Define I 4 := b φ, u, E µ b L τ ˆφ, û, E µ..7 Lemma.7 Suppose tat φ,µ is a weak solution to.5a,.5b, wit te additional regularities.9.1.ten,forany,τ > 0,tereexistsaconstantC > 0, independent of and τ,suctat I 4 γ P L τ ˆφ E µ + C ˆD 0 τ + q + C ˆD 0 Lτ E φ + ε 4 E µ,.8 were ˆD 0 := L τ ˆφ L Proof By adding and subtracting appropriate terms, we ave I 4 =b E φ a, u, Eµ +b L τ E φ, u, Eµ +b τδ τ R φ, u, E µ +b L τ ˆφ, u û, E µ..0 Te last term is te only one tat will give us any concern. Recall tat te discrete and continuous velocities can be described as u = Pγφ µ, û = P γ L τ ˆφ ˆµ..1 We obtain te following useful decomposition: γ 1 u û = Pφ µ P L τ ˆφ ˆµ 1

20 Y. Liu et al. = Pφ µ P φ µ + P φ µ P L τ ˆφ ˆµ = Pφ µ P φ µ + P τδ τ φ µ + P L τ φ µ P L τ ˆφ ˆµ = Pφ µ P φ µ + P τδ τ φ µ + P L τ E φ µ + P L τ ˆφ E µ.. Let s deal wit all te above terms except for te last one. Define Ten I 5 := Pφ µ P φ µ + P τδ τ φ µ + P L τ E φ µ.. I 5 Pφ µ P φ µ + P τδ τ φ µ + P L τ E φ µ C q φ µ H q + Cτ µ L 6 t φ L + 6 Lτ Ea φ µ + 6 L τ E φ µ C q + τ + C q µ L 6 φ H q+1 + µ Lτ L 6 E φ L C q + τ + C L τ E φ..4 From.0 we ave I 4 = b Ea φ, u, Eµ + b L τ E φ, u, Eµ + b τδ τ R φ, u, E µ E φ L a 6 u L µ E Lτ + E φ u L 6 L E µ + E a φ + L τ Ea φ + τδ τ φ u E µ b C q + ε µ E + C L τ E φ + ε µ E Cτ + ε µ E + b L τ ˆφ, u û, E µ 4 Now, using.4 weave b L τ ˆφ, u û, E µ 1 = γ b L τ ˆφ, I 5, E µ + b L τ ˆφ, u û, E µ L τ ˆφ, u û, E µ..5 γ b L τ ˆφ, P L τ ˆφ E a µ, Eµ γ b L τ ˆφ, P L τ ˆφ E µ, Eµ C L τ ˆφ I 5 + ε E µ L +C q Lτ ˆφ 4 16 L µ H q+1 + ε µ E γ b L τ ˆφ, P L τ ˆφ E µ 16, Eµ C ˆD 0 τ + q Lτ + C ˆD 0 E φ + ε µ E 8 γ P L τ ˆφ E µ..6

21 Error Analysis of a Mixed Finite Element Metod To finis up, adding.5 and.6 leadstoteresult. Combining Lemmas.6 and.7, wegetimmediatelytefollowingresult: Lemma.8 Suppose tat φ,µ is a weak solution to.5a,.5b, wit te additional regularities.9.1.ten,forany,τ > 0,and any arbitraryθ > 0,tere exists a constant C > 0, independentofandτ,butdependentonθ,suctat ε µ E + ε 4 δ τ E φ + ετ δ τ E φ + γ P L τ ˆφ E µ C ˆD 0 τ + q + C ˆD 0 Lτ E φ + C L τ E φ L + θ δ τ E φ ε 1 R φ ˆφ, δ τ E φ W 1 Te next step is to prove tat te dual norm δ τ E φ convenient way...7 W 1 can be bounded in a Lemma.9 Suppose tat φ,µ is a weak solution to.5a,.5b, wit te additional regularities.9.1. Ten,forany,τ > 0, δ τ E φ W 1 Cε µ E P + Cγ L τ ˆφ E µ were C > 0 is independent of and τ. + C Lτ E φ + C ˆD 1 0 q + τ,.8 Proof Here we follow te ideas in [19]. Let Q be te standard L projection into S. For any ν W 1,, denoteν = 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 µ, ν + σ φ, ν + b φ, u, ν b L τ ˆφ, û, ν ε E µ ν + σ φ ν + b φ, u, ν b L τ ˆφ, û, ν C ε µ 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 4 in.0. Define I 6 := b φ, u, ν b L τ ˆφ, û, ν..40 1

22 Y. Liu et al. Recalling te estimates in.4.6, we can estimate I 6 as follows: I 6 = b Ea φ, u, ν + b L τ E φ, u, ν E φ L a 6 u L ν + L τ E φ L τ ˆφ, u û, ν + b τδ τ R φ, u, ν + b L τ ˆφ, u û, ν u L 6 L ν + τδ τ R φu ν b C q + τ + L τ E φ ν + γ b L τ ˆφ, I 5, ν + γ b L τ ˆφ, P L τ ˆφ E a µ, ν + γ b L τ ˆφ, P L τ ˆφ E µ, ν C q + τ + L τ E φ ν + γ L τ ˆφ I 5 ν L 6 L + C q Lτ ˆφ µ L H q+1 ν L + γ L τ ˆφ P L 6 L τ ˆφ E µ ν L C ˆD 1 0 q + τ + γ P L τ ˆφ E µ Lτ + E φ ν L..41 Combining.9 and.41, we get δ τ E φ, ν C ˆD 1 0 q +τ +ε µ E P +γ L τ ˆφ E µ + Lτ E φ ν W 1 = C ˆD 1 0 q +τ +ε E µ +γ P L τ ˆφ E µ + Lτ E φ ν W 1..4 Te last estimate is due to te W 1 stability of te L projection into te finite element space. See, for example [7]. 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,µis a weak solution to.5a,.5b, wit te additional regularities.9.1.ten,forany,τ > 0,tereexistsaconstantC > 0, independent of and τ,suctat ε µ E + ε 8 δ τ E φ + γ P L τ ˆφ E µ C ˆD 0 τ + q + C ˆD 0 Lτ E φ + C L τ E φ L ε 1 R φ ˆφ, δ τ E φ..4. 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.4. Te result.7 is not enoug to 1

23 Error Analysis of a Mixed Finite Element Metod 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.7isanoterdifficultytobeovercome.ifwedonotusedualnormestimates,wat we get from.7 isadiscretenonlineargronwallinequalitywicleadsustote sub-optimal convergence rate. Te main result is demonstrated below. Lemma.11 Suppose tat φ,µis a weak solution to.5a,.5b, wit te additional regularities.9.1.ten,forany,τ > 0,tereexistsaconstantC > 0, independent of and τ,suctat ε E φ t m + ετ 4 Cτ E µ t j + γτ j=0 P φ j 1 E µ t j ˆD j m 1 0 τ + q + C A j φ E t j + ε m 1 τ E φ 8 t j, were ˆD j 0 := ˆD 0 t j and A j := τ ˆD j τ 1 7 τδ τ φ j+1 τδ τ φ j+1 + τδ τ φ j W 1 j=0 W τ 1 τδ τ φ j+1 W Proof Our starting point is estimate.4. Te last term of.4 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,.46 were By Lemma 4., ζ m := R φt m + φ m R φt m + φ m τ ε = τ ε 1 ε ζ m E φ t m, δ τ E φ t m 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..48 1

24 Y. Liu et al. Observe tat te last two terms on te rigt-and-side of te last identity are nonpositive and can be dropped in te analysis. For any 1 m M, summationof.4 impliestat ε E φ t m + ετ 4 Cτ + q τ + τ ε µ E t j + γτ ˆD j m Cτ j=0 ˆD j+1 0 P φ j 1 E µ t j E φ t j + C E φ t j L δ τ ζ m, E φ t m 1,.49 were we ave dropped te indicated non-positive terms from te rigt-and-side. Due to te definition of ζ j, 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,tefollowingestimateisavailable: ζ j+1 ζ j,e φ t j C φt j+1 + φt j τδτ W 1 φt j+1 L E φ t j L + C φ j τδ L 4 τ φt j+1 L 4 E φ t j L 4 + C τδ τ φ j+1 E φ L 4 t j + C τδ τ φ j+1 L 4,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 W 1 t j W + E φ 1 t j W Now define I 7 := E φ t j W + E φ 1 t j W. 1 1

25 Error Analysis of a Mixed Finite Element Metod We observe tat I 7 can be analyzed as E φ I 7 C t j E L + 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 C E φ t j + C E φ 1 t 4 φ j E t j + C E φ 1 t 4 φ j E t j + C E φ 1 t 5 φ j E t j C E φ t j + C E φ 1 t 7 4 φ j E t j 4 + C E φ 1 t φ j E t j L Here we reduce te power of E φ t j in some terms above according to te L H 1 bound of E φ.wealsoappealtotediscretegagliardo Nirenberginequality.4 and.4. Tis is ten fed into.51 toobtain ζ j+1 ζ j, E φ t j Cτ E φ t j τδτ + C φ j+1 E + τδ τ φ j+1 φ W 1 t j + C τδ τ φ j+1 + C τδ τ φ j+1 W 1 W 1 E φ t j E φ t j Cτ E φ t j τδτ + C φ j+1 + τδ τ φ j+1 + Cτ 1 7 τδ τ φ j+1 + Cτ 1 τδ τ φ j W 1 4 W φ E t j 1 φ E t j W E φ t j E φ t j + ε τ E φ t j E φ t j + ε τ E φ t j..5 Due to te definition of A j from.45, we arrive at ζ j+1 ζ j, E φ t j CA j φ E t j + ε τ E φ 16 t j..54 1

26 Y. Liu et al. For te term τ E φ t j in.49, we apply te discrete Gagliardo Nirenberg L inequality and Young s inequality again:.τ E φ t j Cτ E φ 1 L t 7 4 φ j E t j 4 + Cτ E φ t j Cτ E φ t j + ε τ E φ 16 t j.55 Combining.49.55, 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.44. Lemma.1 Suppose tat φ,µis a weak solution to.5a,.5b, wit te additional regularities.9.1. Ten,forany,τ > 0, Proof Since ε E φ t j µ E t j +C 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 µ E t j + E µ t j + ε E φ t j,.57 wat we need to estimate is te last term above. To bound E µ + ε E φ, set ψ = E µ + ε Eφ E µ + ε E φ in.8b, wic in turn implies tat = E a µ, E µ + ε Eφ + τ δ τ φ, E µ ε + ε Eφ +ε 1 L τ E φ, E µ + ε Eφ ε 1 φ ˆφ, E µ + ε Eφ E µ C a + τ δ τ φ + φ ˆφ + Lτ E φ + 1 E µ + ε E φ..58 Using tecniques from Lemmas. and.6, teabovenormcanbecontrolledas E µ t j + ε E φ t j Cτ + C ˆφt j q + C E φ L t j 1 1 E + C ˆφt j φ L t j

27 Error Analysis of a Mixed Finite Element Metod C ˆD j+1 0 τ + q + C ˆD j+1 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,µis a weak solution to.5a,.5b, wit te additional regularities.9.1.ten,forany,τ > 0,tereexistsaconstantC > 0, independent of and τ,suctat ε E φ t m + ετ 8 Cτ E µ t j + γτ P φ j 1 E µ t j ˆD j ˆD j m 1 0 τ + q + C A j φ E t j..60 Te summability of te sequence A j is ten essential to apply te discrete Gronwall inequality. We ave te following lemma: Lemma.14 Suppose tat φ,µis a weak solution to.5a,.5b, wit te additional regularities.9.1. Ten,forany1 m Mandany,τ > 0, tere exists a constant C > 0, independentofandτ,suctat j=0 A j C..61 j=0 Proof Recalling.45 fortedefinitionofa j, τ ˆD j+1 0 is summable due to Teo- rem.9. τδ τ φ j+1 and τδ τ φ j+1 are summable due to. and.4 W 1 respectively. For te last two terms in.45, it can be estimated due to te Caucy Scwarz inequality j=0 τ 1 7 τδ τ φ j W j=0 Cτ 7 j=0 j=0 τ 1 4 τ 1 4 τδ τ φ j+1 τδ τ φ j+1 C τ 1 τδτ φ j+1 j=0 W 1 W W C,.6 1

28 Y. Liu et al. and j=0 τ 1 τδ τ φ j+1 4 W j=0 Cτ 1 j=0 j=0 τ 1 τ 1 τδ τ φ j+1 τδ τ φ j+1 C τ 1 τδτ φ j+1 j=0 W 1 W 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,µis a weak solution to.5a,.5b, wit te additional regularities.9.1.ten,forany,τ > 0,tereexistsaconstantC > 0, independent of and τ,suctat ε E φ t m + ετ 8 + γτ µ E t j P φ j 1 E µ j t Cτ + q..64 Remark.16 A combination of.59 and.64 yieldstat τ E φ t j Cτ + q..65 Acknowledgments 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. Appendix 1: Discrete Gronwall inequality We need te following discrete Gronwall inequality, cited in [6,0]: 1

29 Error Analysis of a Mixed Finite Element Metod Lemma 4.1 Fix T > 0, andsuppose{a m } m=1 M, {b m} m=1 M and {c m} m=1 M 1 are nonnegative sequences suc tat τ M 1 m=1 c m C 1,wereC 1 is independent of τ and M, and M τ = T.Supposetat,forallτ > 0, a M + τ M b m C + τ m=1 M 1 m=1 a m c m, 4.1 were C > 0 is a constant independent of τ and M. Ten, for all τ > 0, a M + τ M b m C exp τ m=1 M 1 m=1 Note tat te sum on te rigt-and-side of 4.1 must be explicit. c m C expc Lemma 4. 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 Ten te following identity is valid: I m = 1 a j a j 1 b j a j b j b j a mbm. 4.4 References 1. Baňas, L., Nürnberg, R.: Adaptive finite element metods for te Can Hilliard equations. J. Comput. Appl. Mat. 181, Baňas, L., Nürnberg, R.: A posteriori estimates for te Can Hilliard equation wit obstacle. MAN Mat. Model. Numer. Anal. 45, Barrett, J.W., Blowey, J.F.: An optimal error bound for a finite element approximation of a model for pase separation of a multi-component alloy wit non-smoot free energy. MAN. Mat. Model. Numer. Anal. 5, Baskaran, A., Hu, Z., Lowengrub, J., Wang, C., Wise, S.M., Zou, P.: Energy stable and efficient finite-difference nonlinear multigrid scemes for te modified pase field crystal equation. J. Comput. Pys. 50, Baskaran, A., Lowengrub, J., Wang, C., Wise, S.: Convergence analysis of a second order convex splitting sceme for te modified pase field crystal equation. SIAM J. Numer. Anal. 51, Brenner, S.C., Scott, L.R.: Te Matematical Teory of Finite Element Metods, rd edn. Springer, Berlin Cen, C.M., Huang, Y.Q.: Hig Accuracy Teory in Finite Element Metods, in Cinese. Hunan Science and Tecnology Press, Cangsa Cen, W., Conde, S., Wang, C., Wang, X., Wise, S.M.: A linear energy stable sceme for a tin film model witout slope selection. J. Sci. Comput. 5, Cen, W., Liu, Y., Wang, C., Wise, S.M.: Convergence analysis of a fully discrete finite difference sceme for Can-Hilliard Hele Saw equation. Mat. Comput doi: /mcom05 1

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