NUMERICAL ANALYSIS OF THE CAHN-HILLIARD EQUATION AND APPROXIMATION FOR THE HELE-SHAW PROBLEM, PART I: ERROR ANALYSIS UNDER MINIMUM REGULARITIES

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1 NUMERICAL ANALYSIS OF THE CAHN-HILLIARD EQUATION AND APPROXIMATION FOR THE HELE-SHAW PROBLEM PART I: ERROR ANALYSIS UNDER MINIMUM REGULARITIES XIAOBING FENG AND ANDREAS PROHL Abstract. In this first part of a series we propose and analyze under minimum regularity assumptions a semi-discrete in time scheme and a fully discrete mixed finite element scheme for the Cahn-Hilliard equation u t + u fu = 0 arising from phase transition in materials science where is a small parameter known as an interaction length. The primary goal of this paper is to establish some useful a priori error estimates for the proposed numerical methods in particular by focusing on the dependence of the error bounds on. Quasi-optimal order error bounds are shown for the semi-discrete and fully discrete schemes under different constraints on the mesh size h and the local time step size k m of the stretched time grid and minimum regularity assumptions on the initial function u 0 and domain Ω. In particular all our error bounds depend on only in some lower polynomial order for small. The cruxes of the analysis are to establish stability estimates for the discrete solutions to use a spectrum estimate result of Alikakos and Fusco [3] and Chen [5] and to establish a discrete counterpart of it for a linearized Cahn-Hilliard operator to handle the nonlinear term on the stretched time grid. It is this polynomial dependency of the error bounds that paves the way for us to establish convergence of the numerical solution to the solution of the Hele-Shaw Mullins-Sekerka problem as 0 in Part II [6] of the series. Key words. Cahn-Hilliard equation Hele-Shaw Mullins-Sekerka problem phase transition semi-discrete and fully discrete schemes mixed finite element method AMS subject classifications. 65M60 65M 65M5 35B5 35K57 35Q99 53A0. Introduction. This paper is the first part of a series cf. [6] which devote to error analysis of a mixed finite element approximation for the Cahn-Hilliard equation and convergence analysis of the numerical solution to the solution of the Hele-Shaw Mullins-Sekerka problem. While Part II [6] of this series focuses on the approximation of the Hele-Shaw problem under some stronger regularity assumptions this paper mainly concerns the error analysis of the mixed finite element method for the Cahn-Hilliard equation under minimum regularity assumptions on the domain and the initial data. Specifically we shall propose and analyze a semi-discrete in time method and a fully discrete mixed finite element time-stepping method for the Cahn-Hilliard equation the super-index on u is suppressed for notational brevity u t + u fu = 0 in Ω T := Ω 0 T. u n = n u fu = 0 in Ω T := Ω 0 T. u = u 0 in Ω 0}.3 where Ω R N N = 3 is a bounded domain with a C boundary Ω. T > 0 is a fixed constant and f is the derivative of a smooth double equal well potential taking its global minimum value 0 at u = ±. A typical example of f is fu := F u and F u = u. submitted to SIAM J. Numer. Anal. on July 5 00 Department of Mathematics The University of Tennessee Knoxville TN U.S.A. xfeng@math.utk.edu. Mathematisches Seminar II Christian-Albrechts-Universität Kiel Ludewig-Meyn-Str. D-098 Kiel Germany. apr@numerik.uni-kiel.de.

2 Xiaobing Feng and Andreas Prohl The existence of bistable states suggests that a nonconvex energy functional is associated with the equation see the discussion below. In order to achieve broader applicability in this paper we shall consider more general potentials which satisfy some structural assumptions see Section and our analysis will be carried out based on these assumptions. We like to remark that nonsmooth potentials have also been considered in the literature for the Cahn-Hilliard equation for that we refer to [ 8 6 7] and the references therein. The equation. was originally introduced by Cahn and Hilliard [] to describe the complicated phase separation and coarsening phenomena in a melted alloy that is quenched to a temperature at which only two different concentration phases can exist stably. Note that the equation. differs from the original Cahn-Hilliard equation see [] in the scaling of the time so that t here called the fast time represents t in the original formulation. In the equation u represents the concentration of one of the two metallic components of the alloy mixture. The parameter is an interaction length which is small compared to the characteristic dimensions on the laboratory scale. The two boundary conditions in. the outward normal derivatives of u and u fu vanish on Ω imply that none of the mixture can pass through the walls of the container Ω; the first condition is the most natural way to ensure that the total free energy of the mixture decreases in time which is required by thermodynamics when there is no interaction between the alloy and the containing walls. The evolution of the concentration consists of two stages: the first stage rapid in time is known as phase separation and the second slow in time is known as phase coarsening. At the end of the first stage fine-scaled phase regions are formed which are separated by a thin region usually considered as a hypersurface called the interface. At the end of the second stage the solution will generically tend to a stable state which minimizes the energy functional associated with.. For more physical background derivation and discussion of the Cahn-Hilliard equation and related equations we refer to [ ] and the references therein. It is well-known that the Cahn-Hilliard equation. is a gradient flow with the Liapunov energy functional J u := φ u dx and φ u = u + F u.. Ω Here the energy density φ u is a nonconvex function. It is also known [] that the elliptic operator L CH u := u fu associated with the Cahn-Hilliard equation. is the representation of the Fréchet derivative J u of J u in the space H Ω. Another gradient flow for the same Liapunov energy functional in. is the Allen-Cahn equation u t u + fu = 0.5 which was originally introduced by Allen and Cahn [] to describe the motion of antiphase boundaries in crystalline solids see [ 3 5] and references therein. It is also known [] that the elliptic operator L AC u := u+ fu associated with the Allen-Cahn equation.5 is the representation of the Fréchet derivative J u in the space L Ω. On the other hand the Cahn-Hilliard equation is known to conserve the total mass because its solution satisfies d dt ux tdx = 0 but the Allen-Cahn Ω equation does not conserve the total mass.

3 Numerical analysis of the Cahn-Hilliard Equation I: Error Analysis 3 In addition to the reason that the Cahn-Hilliard equation is widely accepted as a good model to describe the phase separation and coarsening phenomena in a melted alloy it has also been extensively studied in the past decade due to its connection to an interesting and complicated free boundary problem which is known as the Mullins- Sekerka problem arising from studying solidification/liquidation of materials of zero specific heat which is also known as the two-phase Hele-Shaw problem arising from the study of the pressure of immiscible fluids in the air [ ]. It was first formally shown by Pego [3] that as 0 the function w := u + fu known as the chemical potential tends to a limit which together with a free boundary Γ := 0 t T Γ t t} satisfies the following Hele-Shaw Mullins-Sekerka problem: Here w = 0 in Ω \ Γ t t [0 T ].6 w = 0 n on Ω t [0 T ].7 w = σκ on Γ t t [0 T ].8 V = [ w ] n Γ t on Γ t t [0 T ].9 Γ 0 = Γ 00 when t = 0..0 σ = F s ds. κ and V are respectively the mean curvature and the normal velocity of the interface Γ t n is the unit outward normal to either Ω or Γ t [ w n ] Γ t := w+ n w n and w + and w are respectively the restriction of w in Ω + t and Ω t the exterior and interior of Γ t in Ω. Also u ± in Ω ± t for all t [0 T ] as 0. The rigorous justification of this limit was successfully carried out by Alikakos Bates and Chen [] under the assumption that the above Hele-Shaw Mullins-Sekerka problem has a classical solution. Later Chen [6] formulated a weak solution to the Hele-Shaw Mullins-Sekerka problem and showed using an energy method that the solution of.-.3 approaches as 0 to a weak solution of the Hele-Shaw Mullins- Sekerka problem. Also using an energy method Stoth [35] established a global in time convergence result for the case of three-dimensional radial symmetry and Dirichlet boundary conditions. It is clear that the study of the Cahn-Hilliard equation. is of great value for understanding phase transition and for investigating the Hele-Shaw Mullins-Sekerka free boundary problem by taking advantage of the fact that the solution of the Cahn-Hilliard equation is known to exist for all times []. In particular this is attractive from the computational point of view. Due to the nonlinearity in the Cahn- Hilliard equation its solution only can be sought numerically. The primary numerical challenge for solving the Cahn-Hilliard equation results from the presence of the small parameter in front of the nonlinear term in the equation. Recall convergence of the Cahn-Hilliard equation to the Hele-Shaw Mullins-Sekerka model only when is small. On the other hand the equation becomes a singularly perturbed fourth order heat equation for small. To resolve the solution numerically one has to use small space mesh size h and time step size k which must be related to the parameter. Numerical approaches are often based on a mixed formulation of.-.3 which

4 Xiaobing Feng and Andreas Prohl involves the chemical potential w u t = w in Ω T. w = fu u in Ω T. u ν = w = 0 ν on Ω 0 T.3 ux 0 = u 0 x x Ω.. We refer to [ 9] and refererences therein for more discussions on well-posedness and regularities of the Cahn-Hilliard and the biharmonic problems. In the past fifteen years numerical approximations of the Cahn-Hilliard equation with a fixed have been developed and analyzed by many authors. Elliott and Zheng [] analyzed a continuous in time semi-discrete conforming finite element discretization in one space dimension. Numerical experiments of the method in one space dimension were reported in []. Elliott and French [3] proposed a continuous in time semi-discrete nonconforming finite element method based on the Morley nonconforming finite element method [0 7]. Optimal order error estimates were also established for the nonconforming method under the assumption that the solution is smooth. Elliott French and Milner [] proposed and analyzed a continuous in time semi-discrete splitting finite element method mixed finite element method which approximates simultaneously the concentration u and the chemical potential w. Optimal order error estimates were shown under the assumption that the finite element approximation u h of the concentration u is bounded in L. Later Du and Nicolaides [9] analyzed a fully discrete splitting finite element method in one space dimension under weaker regularity assumptions on the solution u of the Cahn-Hilliard equation and established optimal order error estimates by first proving the boundedness of u h in L. Copetti and Elliott [8] considered the Cahn-Hilliard equation with a nonsmooth logarithmic potential function. A fully discrete splitting finite element method was proposed and convergence of the method was also demonstrated. In one space dimension French and Jensen [7] analyzed the long time behavior of the continuous time semi-discrete conforming hp-finite element approximations. Recently extensive studies have been carried out by Barrett and Blowey and others on the finite element approximations of the Cahn-Hilliard system for multi-component alloys with constant or degenerate mobility we refer to [5 6 7] and the references therein for detailed expositions. We like to point out that the results of all papers cited above were established for the Cahn-Hilliard equation with a fixed interaction length. No special effort and attention were given to address issues such as how the mesh sizes h and k depend on and how the error bounds depend on. In fact since all those error estimates were derived using a Gronwall inequality type argument at the end of the derivations it is not hard to check that all error bounds contain a factor exp T which clearly is not very useful when is small. Unlike the numerical works mentioned above the focus of this series is on approximating the solution of the Cahn-Hilliard equation. for small which is the case for both applications we are interested in: simulating the second stage of the concentration evolution process for general regularities Part I and approximating the solution including the free boundary of the Hele-Shaw Mullins-Sekerka problem via the Cahn-Hilliard equation Part II. The primary goal of this paper is to develop a semi-discrete in time and a fully discrete approximation based on a mixed

5 Numerical analysis of the Cahn-Hilliard Equation I: Error Analysis 5 variational formulation for the initial-boundary value problem.-.3 and to establish useful error bounds for general regularities which show growth only in low polynomial order of for the proposed schemes under some reasonable constraints on mesh sizes h and k. To our knowledge such error estimates for the Cahn-Hilliard equation have not been known in the literature. In addition such error bounds serve as the basis for computing the solution of the Cahn-Hilliard equation and the solution of the Hele-Shaw Mullins-Sekerka problem. The subsequent analysis applies to a general class of admissible double equal well potentials and initial data u 0 H +l l = 0 that can be bounded in terms of negative powers of ; see the general assumptions GA -GA 3 in Sections and 3. Our fully discrete scheme based on a mixed variational formulation for u and the chemical potential w is defined as dt U m η h + W m η h = 0 ηh V h.5 U m v h + fu m W m v h = 0 vh V h.6 with some starting value U 0 V h. Here V h H Ω denotes the continuous piecewise linear finite element space. We consider this discrete system on the equidistant mesh J k and also on the stretched mesh J k := t m} M m=0 of local mesh sizes k m+ m + k 0 for 0 t m+ ˆt 0 γk 0 for t m+ ˆt 0.7 with the basic mesh size k 0 and some positive constants γ and ˆt 0 = O. Notice that both meshes require asymptotically the same amount of computation cost cf. Section 3. We assume that there exist positive constants m 0 and σ j for j = 3 such that m 0 := u 0 x dx.8 Ω Ω J u 0 := u 0 L + F u 0 L C σ.9 w 0 H l := u 0 + fu 0 H l C σ +l l = 0..0 We now summarize our main results in this paper. Let 0 < β < be an arbitrary number. On the equidistant time mesh Jk = t m} M m=0 and for u 0 H Ω we show a convergence rate Ok β for the implicit Euler semi-discretization see Theorem 3. which can be improved to Ok β 0 on the stretched time mesh Jk = t m} M m=0 see Theorem 3.6. Theorem.3 contains error estimates for the fully discrete approximation of.-.3 on Jk using the continuous piecewise linear mixed finite element. The results in Theorems 3. and 3.6 are obtained under general regularity assumptions for.-.3. Moreover mesh constraints which relate k 0 and h and under which the above convergence rates hold are explicitly formulated. The constraints indicate that small values of β severely restrict the size of k 0. In the case that u 0 H 3 Ω and either Ω is a convex polygonal domain for N = or the boundary Ω is of class C for N = 3 we show quasi-optimal order in k and optimal order in h convergence on the equidistant time mesh for the fully discrete mixed finite element approximation see Corollaries 3.5 and..

6 6 Xiaobing Feng and Andreas Prohl The analyses to be given below study the effects of temporal and spatial discretization independently for given initial data u 0 H l Ω l = 3; the complexity of initial data captured in terms of the parameters σ i for i =.. growth p > and degree of the nonmonotonicity δ > 0 of f and the value > 0 are all taken into account here to draw conclusions for a robust numerical scheme which necessarily relates the different scales and k 0 h under the premise to derive error bounds that depend only polynomially on. This scenario not only gives practically relevant error bounds for quantities of interest in materials science i.e. concentration but also paves the way to approximate the Hele-Shaw Mullins-Sekerka problem via the Cahn-Hilliard equation in the second part [6] of this series. The main result for.5-.6 and general f is given in Theorem.3 here we present it in a simplified form. Theorem.. Let U m W m } M m=0 solve.5-.6 on J +l k := t m } M m=0 and for u 0 H 3 l Ω l = 0. Suppose that T h is a quasi-uniform triangulation of Ω allowing for inverse inequalities and H -stability of the L -projection in the continuous linear finite element space. For any fixed 0 < β < if the mesh sizes k 0 h and the starting value U 0 satisfy some appropriate constraints see Theorem.3 and Corollary. for the precise descriptions then there hold i ii max 0 m M ut m U m H M + M k m ut m U m L C k m ut m U m H C k β 0 ν + h ν } k β 0 ν + h ν } where C = CT Ω σ i p δ β; ln k 0 is homogeneous in the last argument and νj = ν j σ i p δ β for j =. Here it is understood that k 0 = k for J k. To establish the above error estimates the following three ingredients play a crucial role in our analysis. To establish stability estimates for the discrete solutions of the semi-discrete in time and the fully discrete schemes. To handle the nonlinear potential term in the error equation using a spectrum estimate result due to Alikakos and Fusco [3] and Chen [5] for the linearized Cahn-Hilliard operator L CH := f ui. where I denotes the identity operator and u is a solution of the Cahn-Hilliard equation.; see Proposition. for details. To establish a discrete counterpart of above spectrum estimate. We remark that using a similar approach a parallel study was also carried out by the authors in [5] for the Allen-Cahn equation and the related curvature driven flows. On the other hand unlike the Allen-Cahn equation which is a gradient flow in L the Cahn-Hilliard equation is a gradient flow only in H which makes the analysis for the Cahn-Hilliard equation in this paper more delicate and complicated than that for the Allen-Cahn equation given in [5].

7 Numerical analysis of the Cahn-Hilliard Equation I: Error Analysis 7 The paper is organized as follows: In Section we shall derive some a priori estimates for the solution of.-.3 where special attention is given to the dependence of the solution on in various norms. In Section 3 we consider the backward Euler semi-discrete in time scheme for the Cahn-Hilliard equation and establish some stability estimates for the semi-discrete solution. We then obtain a sub-optimal error bound which depends on only in a low polynomial order for small as is summarized in Theorems The spectrum estimate plays a crucial role in the proof. In Section we propose a fully discrete approximation obtained by discretizing the semi-discrete scheme of Section 3 in space using the lowest order Ciarlet-Raviart mixed finite element method. Optimal order error bounds depending on only in a low polynomial order are shown for the fully discrete method in Theorem.3. The main ideas are to establish some stability estimates for the fully discrete solutions and more importantly to prove a discrete counterpart of the spectrum estimate of [3 5].. Energy estimates for the differential problem. In this section we derive some energy estimates in various function spaces in terms of negative powers of for the solution u the Cahn-Hilliard equation. for given u 0 H +l Ω l = 0. Here J = 0 T and H k Ω denotes the standard Sobolev space of the functions which and their up to kth order derivatives are L -integrable. Throughout this paper the standard space norm and inner product notation are adopted. Their definitions can be found in [0 7]. In particular denotes the standard inner product on L Ω. Also c c j C C C j are generic positive constants which are independent of and the time and space mesh sizes k k 0 and h. In addition define for r 0 H r Ω := H r Ω H r 0 Ω := w H r Ω; < w > r = 0 } where < > r stands for the dual product between H r Ω and H r Ω; we denote L 0Ω H 0 0 Ω. For v L 0Ω let v := v H Ω L 0Ω be the solution to and define v as v = v in Ω v = 0 n on Ω v := v = v. We make the following general assumptions on the derivative f of the potential function F : General Assumption GA f = F for F C R such that F ± = 0 and F > 0 elsewhere. f u satisfies for some finite < p N N and positive numbers c i > 0 i = c u p c 0 f u c u p + c 3. 3 There exist 0 < γ γ > 0 and δ > 0 such that for all a fa fb a b γ f aa b a b γ a b +δ.

8 8 Xiaobing Feng and Andreas Prohl Remark: It is trivial to check that GA implies f uv v c 0 v L v L Ω. which will be utilized several times in the paper. Example: The potential function F u = u consequently fu = u 3 u is often used in physical and geometrical applications [ 8 6]. For readers convenience we verify GA -GA 3 for the case in the following. First GA holds trivially. Since f u = 3u GA holds with c = c = 3 and c 0 = c 3 =. A direct calculation gives fa fb = a b [ f a + a b 3a ba ].. Hence GA 3 holds with γ = γ = 3 and δ =. Also. holds with c 0 =. In order to trace dependence of the solution on the small parameter > 0 we assume that the initial function u 0 satisfies the following conditions: General Assumption GA There exist positive -independent constants m 0 and σ j j = 3 such that.8-.0 hold. Lemma.. Suppose that f satisfies GA and u 0 H Ω satisfies.8-.9 in GA. Then the following estimates hold for the solution u w of.-.: i ii iii iv v vi vii viii ix where u dx = m 0 t 0 Ω Ω ess sup [0 ] u L + } F u L u t s H ds ws L ds = J u 0 us ds C σ+3 0 ess sup u t L + 0 u t s L ds [0 ] 0 ws C maxσp +p+σ} L ds ess sup u L C ρ [0 ] ess sup τt u t H + τs u t L ds C σ+3 [0 ] 0 ess sup τt u t L + τs u t L ds C maxσp +p+σ+} [0 ] 0 0 u tt s L ds C ρ τs u tt s H ds C ρ 0 ρ := maxσp +p+3σ+3σ+} ρ := maxσp 3+p+σ++σp +p}

9 Numerical analysis of the Cahn-Hilliard Equation I: Error Analysis 9 and τ τt = mint t 0 } for any fixed small number 0 < t 0. Proof. i The assertion follows immediately from integrating. over Ω and using the boundary condition.. ii This assertion is the immediate consequence of the basic energy law associated with the Cahn-Hilliard equation where d dt J ut = Ω u t t H wt L.3 J u := [ u + ] F u dx t 0.. iii Multiply. by u. by u and add these equations. Integration by parts on the nonlinear term and. lead to d dt u L + u L f u u c 0 u L. The assertion then follows from ii. iv We formally differentiate.-. in time u tt w t = 0.5 w t = f uu t u t..6 Testing.5 with u t.6 with u t and using Young s inequality we get d dt u t L + u t L = f uu t u t.7 f u L 3 u t L u t L 6 u t L + C 3 f u L 3 u t L 6. The last term in.7 can be bounded by 3 f u L 3 ut L 3 c u p + c L 3p 3 ut L..8 Coming back to.7 and integrating over [0 then gives the result. v We multiply. by u. by u and use. to get u L u t L 8 f u u u.9 u t L + 8 c 0 u L. The assertion follows from ii and iv. vi Testing.5 with τ u t.6 with τu t and using Young s inequality give d τ u t H + τ u t L ut L τ f u u t dt ut L + c 0τ u t L τ u t L + + c 0 τ 3 u t H.

10 0 Xiaobing Feng and Andreas Prohl We then obtain vi from ii. vii Multiplying.5 by τu t.6 by τ u t leads to d dt τ u t L + τ u t L = u t L + τ f uu t u t.0 u t L + Cτ f u L 3 u t L u t L u t L + Cτ 3 f u L 3 u t L + τ u t L. Using an argument similar to.8 the fact that H Ω L 3p Ω and vi we get the assertion vii. viii Testing.5 with u tt and.6 with 3 u tt leads to u tt L u t L + f uu t 3 u tt u t L + f u L u t L 3 u tt L C c u p L p + c 3 u t L + u tt L. The assertion viii then follows from H Ω L p Ω and ii iv. ix Multiplying.5 by τ 3 u tt.6 by τ u tt gives τ 3 utt L + d τ ut L = τ dt f uu t u tt + ut L C f u L 3 u t L + τ 3 utt L + ut L.. Then the above inequality iv and ii imply the assertion. The above estimates are derived under the minimum regularity assumption u 0 H Ω. They show the strong dependency of the solution on negative powers of in high norms. On the other hand we show in the following that the estimates will improve drastically if the initial data u 0 H 3 Ω and the boundary Ω C are considered. Alternatively the subsequent results also hold for convex polygonal domains in the case N =. Lemma.. Suppose that f satisfies GA and u 0 H 3 Ω satisfies GA and Ω is of class C. Then the solution of.-. satisfies the following estimates: i u dx = m 0 t 0 Ω Ω ii ess sup u t H + u t L ds C maxσ+3σ3} [0 ] 0 iii ess sup u L + us H ds C 3 σp +p+} where iv [0 ] 0 u tt H ds C ρ 0 ρ := maxσp 3+p+σp +σ+p+σ3 }.

11 Numerical analysis of the Cahn-Hilliard Equation I: Error Analysis Proof. i The proof of assertion i is trivial. ii It is same as step vi in the proof of Lemma. except for omitting the time weight. iii We multiply. by u. by u and integrate by parts. d dt u L + u L = f u u u 3 f u L 3 u L 6 + u L. The assertion follows from integration over times 0 s < and applying iii of Lemma.. iv This estimate follows directly from.. We conclude this section by citing the following result of [3 5] on a low bound estimate of the spectrum of the linearized Cahn-Hilliard operator L CH in.. The estimate plays an important role in our error analysis. Proposition.3. Suppose that GA holds. Then there exists a positive constant C 0 such that the principle eigenvalue of the linearized Cahn-Hilliard operator L CH in. satisfies for small > 0 or equivalently λ CH λ CH ψ L + inf f uψ ψ C 0 ψ H Ω ψ 0 L ψ L + inf f uψ ψ 0 ψ H Ω w C 0. w=ψ L 3. Error analysis for a semi-discrete in time approximation. We start this section with a weak formulation of.-.: Find ut wt [H Ω] such that for almost every t 0 T u t η + w η = 0 η H Ω 3. u v + fu v = w v v H Ω 3. ux 0 = u 0 x x Ω. 3.3 Note that u t = 0 that is the mass ut = u 0 is conserved for all t 0. A semi-discrete mixed formulation via implicit Euler method on the time mesh Jk := t m} M m=0 reads: Find u m w m } M [H Ω] such that for every 0 m M d t u m+ η + w m+ η = 0 η H Ω 3. u m+ v + fum+ v = w m+ v v H Ω 3.5 with u 0 = u 0. Here Jk := t m} M m=0 is a quasi-uniform partition of [0 T ] of mesh size k := T M. Also d tu m+ := u m+ u m /k.

12 Xiaobing Feng and Andreas Prohl It turns out from the subsequent analysis that this scheme on the time mesh Jk only performs sub-optimal see Theorem 3. for general regularities see Lemma. and quasi-optimal see Corollary 3.5 under additional assumptions on regularity of the problem see Lemma.. The reason for the sub-optimal convergence in the case of general regularities is the lack of an estimate for u tt in L J H Ω. In order to construct an optimally convergent time discretization scheme for.-. in the case u 0 H Ω we suggest to compute iterates u m+ of on a stretched mesh Jk := t m} M m=0 of local mesh sizes k m+ m + k 0 for 0 t m+ ˆt 0 γk 0 for t m+ ˆt with the basic mesh size k 0 and some positive constants γ and ˆt 0 = O; see Chapter 0 of [33]. Obviously this grid structure is very fine near the origin with increasing mesh size at increasing times and requires Ok0 iteration steps to overcome the critical time interval [0 ˆt 0 ]. It will be proved in Theorem 3.6 that the benefit of using the stretched mesh Jk is that it results in quasi-optimal error bounds. For equidistant meshes the scheme has been used mostly in the literature see [5 9] and the reference therein. However a verification of an estimate that corresponds to ii of Lemma. for the semi-discrete solution is not immediate since GA has no evident discrete analogy. The necessity of this result will be clear in the subsequent error analysis for that we make the final general assumption on f which applies to both meshes Jk and J k : General Assumption 3 GA 3 Suppose that there exists α < γ 3 < and c > 0 such that f satisfies for any 0 < k m α0 and any set of discrete in time functions φ m } M m=0 H Ω γ 3 k m d t φ m H + k m d t φ m L + k m fφ m d t φ m + c J φ 0 c F φl L l M. 3.7 A direct consequence of 3.7 are the following stability estimates for the scheme to be valid on both meshes Jk and J k. Moreover additional estimates in strong norms are shown for the stretched mesh which indicates its stabilizing effect. Lemma 3.. For k m α0 and u 0 H Ω the solution of the scheme satisfies the following estimates on both meshes Jk and J k. i ii iii u m dx = m 0 m 0 Ω Ω max 0 m M + u m L + } F um L k m w m L + d tu m H + k m d t u m L } c J u 0 k m u m L C σ+3. m=0

13 Numerical analysis of the Cahn-Hilliard Equation I: Error Analysis 3 Moreover there hold for J k iv v max m M d t u m L + k M d t u m L C maxσp +p+σ} max 0 m M um L C ρ and for J k vi vii max d } t u m H + m M um L + k 0 d t um H + k 0 d } t u m L + k m d t u m L C σ+3 + ln σp +p+ 3 }} k 0 max d } t u m L + m M um L + k 0 + d t u m L + d } t u m L k m d t u m L C ρ where d t ϕ m+ := k 0 ϕ m+ ϕ m} and ρ = max ln σ+p + σ+3 + ln σp +p+ 3 } k 0 k 0 +C σ+σ+ + σp +σ+p+} } ρ. In addition under the assumptions of Lemma. there also holds for the mesh J k viii ix max d tu m H + k M d m M t u m L C maxσ+3σ3} 3.8 max 0 m M um L + k M u m H C 3 σp +p+}. 3.9 m=0 Proof. The proof of i is trivial setting η = in 3.. To see ii we choose η v = w m+ d t u m+ in The assertion then follows from GA 3 and the inequality d t u m+ H w m+ L. The proof of iii is similar to that of iii in Lemma.. We choose η v = u m+ u m+ in and arrive at d t u m+ L + k m+ d t u m+ L + um+ L = f u m+ u m+ The assertion then follows from ii. c 0 um+ L.

14 Xiaobing Feng and Andreas Prohl To show assertion iv we first apply the difference operator d t to d t um+ η + d t w m+ η = 0 η H Ω 3.0 d t u m+ v + d tfu m+ v = d t w m+ v v H Ω. 3. For η v = d t u m+ d t w m+ using the Mean Value Theorem on d t fu m+ leads to d t d t u m+ L + k d t um+ L + d tu m+ L 3. = f ξd t u m+ d t u m+ d tu m+ L + C 3 f ξ L 3 d t u m+ L 6. Here ξ is a value between u m and u m+. For the following step we introduce u H Ω such that Ω Ω u dx = m 0 as the solution of d t u 0 ϕ = u 0 + fu0 ϕ 3.3 for all ϕ χ H Ω; ϕ = 0 }. Then summation over 0 m M and Cauchy s inequality together with ii imply the result. v We test by u m+ u m+ u m+ L d t u m+ L f u m+ u m+ the assertion follows from. ii and iv. Finally the estimates viii-ix can be shown using similar arguments to those in the proof of ii-iii of Lemma. with some straightforward modifications. We now analyze solutions u m} M m=0 which are obtained from the mesh J k. The estimates i-iii remain valid for the mesh Jk. Instead of iv and v we find the stronger results vi and vii. Rewrite 3. as vi We apply d t to this equation and find d t u m+ m + k 0 w m+ = d t um+ mk 0 d t w m+ w m+ = 0. We test the above equation with d t u m+ and find the counterpart of 3. d t dt u m+ L + k 0 d t u m+ L + mk 0 d t u m+ L d t u m+ L + k 0 d t u m+ L = mk 0 f ξ d t u m+ + f u m+ u m+ dt u m+.

15 Numerical analysis of the Cahn-Hilliard Equation I: Error Analysis 5 Here ξ is a value between u m and u m+. Using. the last line can be bounded by c 0 mk 0 d t u m+ L + C f u m+ L 3 u m+ L dt u m+ L 6 mk 0 d t u m+ L + Cmk 0 3 dt u m+ L + f u m+ L 3 u m+ L dt u m+ L 6. Let δ > 0. Using.8 the last contribution is bounded by Cδ c 5/ u m+ p + c L 3p 3 u m+ L + δ d t u m+ L Cδ σ+p +σ+ 7 } + mk 0 d t u m+ L + C k 0 mδ dt u m+ L. We insert this into 3.5 and multiply by k 0 finally sum over m from to l M. Note that M m ln M < and k m dt u m L = M k m k0 m dt u m L k m dt u m L. From i and discrete Gronwall s inequality we find for the choice δ = ln k 0 dt u l L + k 0 d t u m L + k m d t u m L + ul L + k 0 C σ+3 + ln σp +p+ 3 }} + k 0 d t u m+ L 3.6 dt u L + u L }. By ii the last two terms can be bounded by σ. We benefit at this point again from the scaling of the stretched mesh J k u u 0 L k k 0 dt u L C σ. 0 vii We test 3. with d t u m+. In the sequel ξ is a value between u m and u m+. d t d t u m+ L + k 0 d t u m+ L + mk 0 d t u m+ L 3.7 = mk 0 + d t u m+ L + k 0 d t u m+ L f ξ d t u m+ d t u m+ f u m+ u m+ d t u m+.

16 6 Xiaobing Feng and Andreas Prohl We multiply by k 0 sum over m from to M use the upper bound from.8 and the estimate f u m L 3 u m L d t u m L 6 C 3 f u m L mk 3 um L + mk 0 0 d t u m L C 3 c u m p + c mk L 3p 3 u m L + mk 0 0 d t u m L C σ+p 3 + mk 0 mk 0 d t u m L to get from 3.7 max m M d } t u m L + um L + k 0 m= d t um L d } t u m L + k m d t u m L m= d } t u L + u L + C σp +p+} ln k 0 + C M 3 k m f ξ L d 3 t u m L. 6 m= The logarithmic term comes again from the bound M m ln M <. The last term in 3.8 is estimated by C 3 k m c u m p L 3p + c 3 d t u m L 3.9 C ln σ+p +3 σ+3 + ln σp +p+ 3 }. k 0 k 0 The last inequality follows from an elementary calculation. The first two terms on the right hand side of 3.8 are bounded because of the structure of Jk. Therefore we come back to 3. taking m = 0 and testing the equation with d t u lead to d t u L + k 0 d t u L = k 0 f u u d t u c 0 u L + f u L 3 u L u 0 L 6. Similar to.8 and.9 using GA the above inequality is continued by c 0 u L + c 0 f u L 3 u0 L c 0 u L + u p + c c 0 L 3p 3 u 0 L C σ+ + σp +p } σ+ + σ+}.

17 Numerical analysis of the Cahn-Hilliard Equation I: Error Analysis 7 Using 3.0 and 3.9 we find the following upper bound for 3.8 C ln σ+p +3 σ+3 + ln σp +p+ 3 } k 0 k 0 +C σ+σ+ + σp +σ+p+}. This concludes the proof. 3.. Verification of GA 3 for the case fu = u 3 u. For the reader s convenience we verify GA 3 and show the following estimates for this specific case. The results are valid for both meshes J k and J k. Note that k m := k on the mesh J k. Lemma 3.. For k 3 resp. k m 3 for J k the solution um w m } M m=0 of satisfies for both meshes J k and J k Then max u m L + } 0 m M F um L + k m + 3 k m dt u m L k m d t u m L + k m d t u m } L J u 0. Proof. Rewrite fu m+ as follows fu m+ = u m+ [u m + u m+ ] + k m d t u m+. 3. fum+ d t u m+ = u m+ d t u m+ u m+ d t u m+ + k m u m+ ± u m d t u m+ k m d tu m+ L d t u m+ L + k m d t u m+ L k m d tu m+ L. The last contribution is decomposed as follows k m d tu m+ L k d tu m+ L dt u m+ L dt u m+ L + k m d tu m+ L. 3. Multiplying 3.7 by k m and summing over m from 0 to M we obtain 3.7 with α 0 = 3 γ 3 = and c =. The proof is completed by testing 3. with w m+ 3.5 with d t u m+ and applying 3.. The above derivation also suggests to consider the following variant scheme of : d t u m+ η + w m+ η = u m+ v + u m + u m+ } u m + u m+} v = w m+ v 3.

18 8 Xiaobing Feng and Andreas Prohl for all tuple η v [ H Ω ]. It turns out that this new scheme has better stability properties than the scheme does as shown by the next lemma. Lemma 3.3. The solution u m } M m=0 of 3.3 satisfies for any k > 0 k 0 > 0 k m dt u m L + k m d tu m L } + max 0 m M um L + F um L } = J u 0. Proof. After testing 3.3 with w m+ 3. with d t u m+ the only term which needs special care is A u m + u m+ }u m + u m+ } d t u m+ appearing on the left hand side of the equation. We apply a binomial formula twice to reformulate this term as A = u m + u m+ } d t u m+ = u m + u m+ [ } d t u m+ ] = d t u m+ L. The proof then is completed by taking summation over m. Remark: Note that Lemma 3.3 holds under no constraint onto choices of k. However despite of this advantage of scheme we prefer scheme for its simpler structure and generality for different f. On the other hand it would be interesting to also analyze the scheme and compare the two schemes numerically. 3.. Error estimates for the scheme In this subsection we present the error analysis for under the assumptions GA -GA 3 starting with the mesh Jk. As is shown in Subsection 3. the stability result of the timediscrete scheme imposes some constraint on the time step size k. In fact in order to establish convergence of the discrete scheme this constraint needs to be strengthened according to the following convergence theorem which is the first of two main theorems in this subsection. Theorem 3.. Let u m w m } M m=0 solve on an equidistant mesh Jk = t m} M m=0 of mesh size Ok and u 0 H Ω. Suppose GA -GA 3 hold and < δ < 8 N. Let ρ and ρ be same as in Lemma. and ρ 3 = ρ [ ρ ] + β + ρ N β = Nδ 8 Nδ σ + Nδ +σ + 8 Nδ Nδ 8 Nδ ρ 5 N = [ ρ ] N [ ρ ] N [ ρ 6 N β = δ Nδ β8 + Nδ].

19 Numerical analysis of the Cahn-Hilliard Equation I: Error Analysis 9 For fixed positive values 0 < β < let k satisfy the following constraint k C min 3 β α 0 [ ρ 3 N ] ρ Nβ [ ρ5 N ] } ρ 6Nβ. 3.5 Then there exists a positive constant C = Cu 0 ; γ γ C 0 T Ω such that the solution of satisfies the following error estimate max ut m u m H + k 0 m M k d t ut m u m H +k β ut m u m L } C k β [ ρ ]. Proof. The proof is split into four steps: the first step deals with consistency error and shows the relevancy of the condition GA 3 imposed on f. Steps two and three use Proposition.3 and stability properties of the implicit Euler-method to avoid exponential blow-up in of the error constant. In the last step an inductive argument is used to handle the difficult caused by the super-quadratic term in GA 3. Step : Let e m := ut m u m L 0Ω and g m := wt m w m denote the error functions. Subtracting from respectively we obtain the error equations d t e m+ η + g m+ η = Ru tt ; m η 3.6 e m+ v + futm fu m+ v = g m+ v 3.7 which are valid for all η v [ H Ω ] and Ru tt ; m = k tm We choose η v = e m+ e m+ and find t m s t m u tt s ds. 3.8 d t e m+ L + k dt e m+ L + em+ L futm+ fu m+ e m+ = Ru tt ; m e m+. From ix of Lemma. k m=0 k Ru tt ; m H 3.30 [ t m+ s t m t m τs m=0 C k ρ. ][ t m+ ] ds τs u tt s H ds t m To control the last term on the left hand side of 3.9 we use GA 3 fut m+ fu m+ e m+ 3.3 γ f ut m+ e m+ e m+ γ em+ +δ L +δ.

20 0 Xiaobing Feng and Andreas Prohl Step : We want to use the following spectrum estimate result see Proposition.3 to bound from below the first term on the right hand side of 3.3 φ L + f uφ φ C 0 φ L φ H Ω 3.3 where C 0 > 0 is independent of. At the same time we want to make use of the H Ω norm of Ru tt ; m in order to keep the power of as low as possible in the error constant. The latter requires to keep portions of e m+ L on the left hand side of the error equation 3.9. To this end we apply 3.3 with a scaling factor γ kβ which together with 3.3 and 3.9 gives d t e m+ L + k dt e m+ L + [ γ kβ ] e m+ L C 0 γ kβ e m+ L γ k β f ut m+ e m+ e m C em+ +δ L +δ + C k β Rutt ; m H. From. the second term on the right hand side can be bounded as γ k β f ut m+ e m+ e m+ γ k β c 0 em+ L 3.3 γ k β c e m+ L + γ k β e m+ L. Then we obtain from 3.30 and 3.3 after summing 3.33 over m from 0 to l M k e l+ L + k dt e m+ L + } [ γ kβ ] e m+ L m=0 C 0 γ + c 0 γ k β 3 k e m+ L + C k β ρ 3.35 m=0 + Ck m=0 e m+ +δ L +δ. Note that k = O 3 β in the coefficient of the first term on the right hand side in order to avoid exponential growth in of the stability constraint arising from discrete Gronwall s inequality. Step 3: We now need to bound the super-quadratic term at the end of the inequality First a shift in the super-index leads to em+ +δ C e m +δ + k +δ d L +δ L +δ t e m+ +δ L +δ For the first term on the right hand side we interpolate L +δ between L and H and using v of Lemma. and v of Lemma 3. we infer e m Nδ L 3.37 em +δ C L +δ C e m 8+ Nδ L e m 8+ Nδ L + e m +δ L e m Nδ C ρ Nδ 8 e m 8+ Nδ 8 L. L + e m Nδ L

21 Numerical analysis of the Cahn-Hilliard Equation I: Error Analysis Since Ω em dx = 0 the above inequality is continued by [ em +δ C L +δ γ ρ Nδ 8 e m 8+ Nδ 8 L ] [ γ e m 8+ Nδ 8 L ] 3.38 for some γ > 0 to be fixed in the sequel. The subsequent analysis deals with 0 < δ < 8 N which is the more involved case. It is crucial to recover a super-quadratic exponent for e m L in Step. We come back to 3.38 and to look for α > 0 such that which implies and then set γ α e m 8+ Nδ 8 α L 8 + Nδ 8 k [ β ] γ e m L α = or α = Nδ γ = C α k β α. We use these choices in 3.38 together with Young s inequality to find after a short calculation that 3.38 is continued by em +δ C [ k β] 8+ Nδ 8 Nδ ρ L +δ Nδ 8 Nδ e m + Nδ 8 Nδ + [ γ kβ 8 ] e m L. L 3.39 Similarly the second term on the right hand side of 3.36 can be bounded as k +δ d t e m+ +δ L +δ Ck+δ Ck+δ d t e m+ Nδ L d t e m+ 8+ Nδ L + d t e m+ +δ L d t e m+ 8+ Nδ L d t e m+ Nδ L Nδ + C k ρ Nδ 8 dt e m+ 8+ Nδ 8 L [ Nδ C k+ ρ Nδ 8 dt e m+ 8+ Nδ 8 γ We look for α > 0 such that This implies L ] [ + d t e m+ Nδ L γ dt e m+ 8+ Nδ 8 γ α dt e m+ 8+ Nδ 8 α L k dt e m+ L. L ] 3.0. and 8 + Nδ 8 α = or α = γ = k 8+ Nδ Nδ

22 Xiaobing Feng and Andreas Prohl Hence an upper bound for 3.0 is k dt e m+ L + 8+ Nδ Ck 8 Nδ Nδ Nδ ρ 8 Nδ dt e m+ 8+ Nδ 8 Nδ L = C k 3 8+ Nδ Nδ 8 Nδ ρ 8 Nδ dt e m+ 8+ Nδ 8 Nδ L + k dt e m+ L C k 3 8+ Nδ C k 3+ 8 Nδ max i=0 Nδ 8 Nδ ρ e m+i Nδ Nδ 8 Nδ σ + Nδ 8 Nδ Nδ 8 Nδ 8 Nδ L } d t e m+ L + k dt e m+ L Nδ ρ 8 Nδ dt e m+ L + k dt e m+ L. Finally combining these results with 3.35 and using vi of Lemma. and ii of 3. we get k e l+ L + k dt e m+ L + } 8[ γ kβ ] e m+ L m=0 C 0 γ + c 0 γ k β 3 k e m+ L + C k β ρ 3. m=0 +C [ k β] 8+ Nδ 8 Nδ ρ Nδ + +C k Nδ 8 Nδ k 8 Nδ σ + Nδ 8 Nδ e m + m=0 +σ +} ρ L Nδ 8 Nδ. Nδ 8 Nδ Step : We now conclude the proof by the following induction argument which is based on the results from Steps to 3. Suppose that for sufficiently small time steps satisfying k C min 3 β α 0 [ ρ 3 ] ρ Nβ [ ρ5 N ] } ρ 6Nβ 3. and 0 < β < there exist two positive constants c = c t l Ω u 0 σ i p c = c t l Ω u 0 σ i p; C 0 independent of k and such that the following inequality holds k max 0 m l e m L + k dt e m L + γ k β e m L c k β ρ exp c t l. 3.3 The last two constraints in 3. arise as follows. controlling the last error term in 3. The first of them comes from k + Nδ 8 Nδ σ + Nδ +σ [ 8 Nδ +} ρ ] Nδ 8 Nδ c k β ρ exp c t l.

23 Numerical analysis of the Cahn-Hilliard Equation I: Error Analysis 3 Note that the exponent in the second sum on the right hand side of 3. is bigger than hence we can recover 3.3 at the l + th time step by using the discrete Gronwall s inequality provided that [ k β ] 8+ Nδ 8 Nδ ρ Nδ 8 Nδ [ k β ρ ] + Nδ 8 Nδ c k β ρ exp c t l+ which gives the last constraint in 3.. The proof is complete. Remark: a. Theorem 3. is stated for 0 < δ < 8 N which covers assumption GA for the case N = 3. For N = the error estimate is valid for all 0 < δ < the above proof can be adapted and simplified for the case δ > 8 N. Note that in this case the crucial requirement of super-quadratical growth is already met in 3.38 then we can immediately jump to Step to proceed. Finally the case δ = 8 N is again easy to take care thanks to Lemma. and 3.. b. In addition to the spectrum estimate of Proposition.3 the stability estimate v of Lemma 3. is critical to the analysis. c. It is natural to estimate the error of in the norm of l Jk ; H Ω l Jk ; H Ω its structure allows to test with e m+ which then interferes with limited regularity property of u tt and cuts convergence rate in 3.30 down to sub-optimal order. As to be demonstrated in the sequel using stretched meshes Jk will help to attain a quasi-optimal order for the Euler method d. A straightforward idea to benefit from the damping property of. is to multiply 3.9 by a time-weight τ m+ := mint m+ } before summation; this would give an optimal convergence rate Ck ρ in 3.30 thanks to ix of Lemma.. On the other hand this would require to control the error e m } M m=0 in the norm of l Jk ; H Ω by using a parabolic duality argument. This strategy does not seem to be successful in the present analysis where we focus on non-exponential dependencies on ɛ of involved stability constants. e. It is clear that the smaller β the better the error bound since the exponent of k is closer to. Small values of β however restrict admissible time steps to small sizes. f. The proof of Theorem 3. suggests the following numerical stabilization technique for the Cahn-Hilliard equation d t u m+ η + w m+ η = 0 η H Ω 3. + kζ um+ v + fu m+ v = w m+ v v H Ω 3.5 ζ where ζ i 0 for i =. We will not go into further discussion of these methods here. For given more regular initial data u 0 H 3 Ω and domains Ω see assumptions in Lemma. the convergence rate can be improved to Ck β ρ. The key ingredient for proving that is to use v of Lemma. to improve on the estimate Corollary 3.5. Let u m w m } M m=0 solve on an equidistant mesh Jk = t m} M m=0 of mesh size Ok for u 0 H 3 Ω and Ω of class C or a convex polygonal domain when N =. Suppose GA -GA 3 hold and < δ < 8 N. Let ρ j be same as in Theorem 3.. For fixed positive values 0 < β < let k satisfy the following constraint k C min 3 β α 0 [ ρ 3 N ] ρ Nβ [ ρ5 N ] } ρ 6Nβ 3.6

24 Xiaobing Feng and Andreas Prohl Then there exists a positive constant C = Cu 0 ; γ γ C 0 T Ω such that the solution of satisfies the following error estimate max ut m u m H + k 0 m M k d t ut m u m H +k β ut m u m L } C k β [ ρ ]. For u 0 H Ω the error bound given in Theorem 3. is not optimal the crucial step where we loose accuracy by one order of magnitude on the time step k is 3.30 since we are only provided with a bound for τ u tt L J; H Ω ; see ix of Lemma.. The following result reflects the stabilizing effect of the mesh Jk in this respect. Note that the proof of Theorem 3. only requires iv-v of Lemma 3. which are replaced by vii in the case of the mesh Jk. Theorem 3.6. Suppose that the assumptions and shorthand notation of Theorem 3. hold. Define ρ N β = [ β + Nδ ] 8 Nδ [ ρ 6 N β = δ Nδ β6 + 3 Nδ]. For fixed positive values 0 < β < let k 0 satisfy the following constraint k 0 C min 3 α 0 [ β ρ3 N ] ρ Nβ [ ρ5 N ] ρ } 6Nβ. 3.7 Let u m w m } M m=0 be the solution to on the mesh J k defined in 3.6. Then there holds the following improved error estimate max ut M m u m H + 0 m M k m k m d t ut m u m } H +k β m ut m u m} L } C k β 0 [ ρ ]. Proof. The proof follows the steps of that of Theorem 3.. We only sketch the necessary modifications in the following. Step : On the stretched mesh J k the residual Ru tt m can be bounded as follows k m+ Ru tt m H = M k m+ C max 0 m M tm+ t m k m+ k m+ tm+ s s t m ds t m tm+ t m s s t m ds tm+ t m s t m u tt s ds τs u tt s H ds } ρ km C max ρ m M t m Ck0 m + max ρ C k0 ρ m M t m H

25 Numerical analysis of the Cahn-Hilliard Equation I: Error Analysis 5 thanks to ix of Lemma.. This improved upper bound for the residual replaces 3.30 in the proof of Theorem 3.. Step : This step is the same. Step 3: We use vii of Lemma 3. instead of v to bound max 0 m M u m L. Then 3.38 and 3.39 are replaced by em +δ C [ k β ] 8+ Nδ Nδ 8 Nδ L +δ m ρ 8 Nδ e m + Nδ 8 Nδ L [ γ kβ m 8 ] e m L. Again the argument applies for values δ < 8 N. Instead of 3.0 we now have km+ +δ d t e m+ +δ L +δ Ck+δ m+ Ck+δ m+ C k 3+ Finally 3. is replaced by e l L + d t e m+ Nδ L d t e m+ 8+ Nδ L Nδ σ 8 Nδ + Nδ m+ 8 Nδ d t e m+ 8+ Nδ L + d t e m+ +δ L d t e m+ Nδ Nδ L + d t e m+ Nδ L ρ 8 Nδ dt e m+ L + k m+ k m dt e m L + k m 8 C 0 γ + c 0 γ k β 0 3 d + C [ ρ ] Nδ 8 Nδ dt e m+ L. [ γ kβ m ] e m L } k m e m L + C k β 0 ρ 3.9 [ ] 8+ Nδ k m k β 8 Nδ m e m Nδ + σ 8 Nδ + Nδ +σ +C k0 [ ρ 8 Nδ +} ] Nδ 8 Nδ. Nδ + 8 Nδ L Step : The inductive argument now reads: Suppose that for sufficiently small basic time step k 0 satisfying k 0 C min 3 β α 0 [ ρ 3 ] ρ Nβ [ ρ5 N ] ρ } 6Nβ 3.50 and 0 < β < there exist two positive constants c = c tl Ω u 0 σ i p c = c tl Ω u 0 σ i p; C 0 independent of k 0 and such that the following inequality holds max 0 m l e m L + k m km dt e m L + γ k β m c k β 0 ρ exp c t l. e m L 3.5

26 6 Xiaobing Feng and Andreas Prohl We employ the following necessary criterion k + Nδ σ 8 Nδ + Nδ 0 +σ [ ρ 8 Nδ +} ] Nδ 8 Nδ c which implies the third condition in 3.7. Finally we need to make sure that 0 ρ 3.5 k β [ ] 8+ Nδ k m k β [ ρ 8 Nδ m ] Nδ [ 8 Nδ k β 0 ρ ] + Nδ c k β 0 ρ exp c t l+. This completes the induction argument and the proof too. 8 Nδ. Error analysis for a fully discrete mixed finite element approximation. In this section we propose and analyze a fully discrete mixed finite element method for on the meshes Jk and J k. The lowest order Ciarlet-Raviart mixed finite element method cf. Chapter 7 of [7] and [3] for the biharmonic problem is used for spatial discretization. Like in Section 3 special emphasis is given to analyze the dependence of the error bounds on. Throughout this section we assume that u 0 H Ω and Ω is of class C and that GA -GA 3 are satisfied. Sub-optimal error estimates for the fully discrete scheme on Jk and improved quasioptimal estimates for N = ; with slightly deteriorated rate for N = 3 on Jk are established see Theorem.3 and Corollary.. We recall that the fully discrete mixed finite element discretization of is defined as: Find U m W m } M [ ] S h such that d t U m+ η h + W m+ η h = 0 η h S h. U m+ v h + fu m+ v h = W m+ v h v h S h. with some suitable starting value U 0 and a quasi-uniform triangulation T h of Ω. Where S h denotes the P conforming finite element space defined by } S h := v h CΩ ; v h K P K K T h. The mixed finite element space S h S h is the lowest order element among a family of stable mixed finite element spaces known as the Ciarlet-Raviart mixed finite elements for the biharmonic problem cf. [7 3]. In fact it is not hard to check that the following inf-sup condition holds ψ h η h inf sup 0 η h S h 0 ψ h S h ψ h H η h H c 0 for some c 0 > 0. Also we note that d t U m+ = 0 which implies that U m+ = U 0 for m = 0 M. Hence the mass is also conserved by the fully discrete solution at each time step. We define the L Ω-projection Q h : L Ω S h by Q h v v η h = 0 η S h.3

27 Numerical analysis of the Cahn-Hilliard Equation I: Error Analysis 7 and the elliptic projection P h : H Ω S h by [P h v v] η h = 0 η h S h. P h v v = 0..5 We refer to Section of [5] for a list of approximation properties of P h. In the sequel we confine to meshes T h that allow for H -stability of Q h see [] for their further characterization. We also introduce space notation S h := v h S h ; v h = 0 and define the discrete inverse Laplace operator: h : L 0 Ω S h such that h v η h = v ηh η h S h..6 } Lemma.. For J k = Jk or J k the solution U m W m } M satisfies i U m dx = U 0 dx m = M Ω Ω Ω Ω ii d t U m H C W m L m = M m=0 of.-. iii iv max U m L + } 0 m M F U m L + k m W m L + k m d t U m L CJ U 0 k m d t U m H CJ U 0. Proof. The assertion i is an immediate consequence of setting η h = in.. For any φ H Ω from..3 and the stability of Q h in H Ω cf. [] and references therein we have d t U m φ = d t U m Q h φ + d t U m φ Q h φ Assertion ii then follows from = W m Q h φ C W m L φ L. d t U m d t U m φ H = sup C W m L. 0 φ H φ H To show assertion iii setting η h = W m+ in. and v h = d t U m+ in. and adding the resulting equations give W m+ L + d dt U m+ L + k m+ d t U m+ L.7 + fu m+ d t U m+ = 0.