NUMERICAL ANALYSIS OF THE CAHN-HILLIARD EQUATION AND APPROXIMATION FOR THE HELE-SHAW PROBLEM, PART II: ERROR ANALYSIS AND CONVERGENCE OF THE INTERFACE

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1 NUMERICAL ANALYSIS OF THE CAHN-HILLIARD EQUATION AND APPROXIMATION FOR THE HELE-SHAW PROBLEM, PART II: ERROR ANALYSIS AND CONVERGENCE OF THE INTERFACE XIAOBING FENG AND ANDREAS PROHL Abstract. In this second part of the series, we focus on approximating the Hele-Shaw problem via the Cahn-Hilliard equation u t + ε u ε fu)) = 0 as ε 0. The primary goal of this paper is to establish the convergence of the solution of the fully discrete mixed finite element scheme proposed in [ to the solution of the Hele-Shaw Mullins-Seera) problem, provided that the Hele- Shaw Mullins-Seera) problem has a global in time) classical solution. This is accomplished by establishing some improved a priori solution and error estimates, in particular, an L L )-error estimate, and maing full use of the convergence result of [. Lie in [0,, the cruxes of the analysis are to establish stability estimates for the discrete solutions, use a spectrum estimate result of Aliaos and Fusco [3 and Chen [, and establish a discrete counterpart of it for a linearized Cahn-Hilliard operator to handle the nonlinear term. Key words. Cahn-Hilliard equation, Hele-Shaw Mullins-Seera) problem, phase transition, biharmonic problem, fully discrete mixed finite element method, Ciarlet-Raviart element AMS subject classifications. 65M60, 65M, 65M5, 35B5, 35K57, 35Q99, 53A0. Introduction. In the first part [ of this series, we proposed and analyzed a semi-discrete in time) and a fully discrete mixed finite element method for the Cahn-Hilliard equation:.).).3) u t + ε u ε fu)) = 0 in Ω T := Ω 0, T ), u n = n ε u ε fu)) = 0 in Ω T := Ω 0, T ), u = u ε 0 in Ω 0}. Note that the super-index ε on the solution u ε is suppressed for notation brevity. We established a priori solution estimates and optimal and quasi-optimal error estimates under minimum assumptions on the domain Ω and the initial datum function u ε 0. Special attention was given to the dependence of the error bounds on ε. It was shown that all the error bounds depend on ε only in some low polynomial order for small ε. In this second part of the series, we are concerned with the second stage of the evolution of the concentration, that is, the motion of the interface. We focus on submitted to SIAM J. Numer. Anal. on July 5, 00 Department of Mathematics, The University of Tennessee, Knoxville, TN 37996, U.S.A. xfeng@math.ut.edu). Mathematisches Seminar II, Christian-Albrechts-Universität Kiel Ludewig-Meyn-Str. 4, D-4098 Kiel, Germany. apr@numeri.uni-iel.de).

2 Xiaobing Feng and Andreas Prohl approximating the Hele-Shaw Mullins-Seera) problem:.4) w = 0 in Ω \ Γ t, t [0, T,.5) w = 0 on Ω, t [0, T, n.6) w = σκ on Γt, t [0, T, V = [ w.7) on Γ t, t [0, T, n Γ t.8) Γ 0 = Γ 00 when t = 0 via the Cahn-Hilliard equation as ε 0. Here σ = F s) ds, and 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 to Ω + t and Ω t, the exterior and interior of Γ t in Ω cf. [, ). The main objective of this paper is to establish the convergence of the fully discrete mixed finite element method proposed in [ to the solution of the Hele-Shaw problem.4)-.8), provided that the Hele-Shaw problem has a global in time) classical solution. To our nowledge, such a numerical convergence result is not yet nown in the literature for the Cahn-Hilliard equation. We also note that the convergence of the Cahn-Hilliard equation to the Hele-Shaw model was established in [ under the same assumption. To show convergence, we need to establish stronger error estimates, in particular, a L J; L ) estimate. We are able to obtain the desired error estimates by first proving some improved a priori solution estimates, and then an improved discrete spectrum estimate under the assumption that the Hele-Shaw problem admits a global in time) classical solution. Lie in [, the cruxes of the analysis are to establish stability estimates for a discrete solution, use a spectrum estimate result of Aliaos and Fusco [3 and Chen [, and establish a discrete counterpart of it for a linearized Cahn-Hilliard operator to handle the nonlinear term. The analysis of this paper is carried out for a general class of admissible double equal well potentials and initial data u ε 0 H 4 which can be bounded in terms of negative powers of ε in H j Ω) norm for j =,, 3, 4; see the general assumptions GA )-GA 3 ) in Section and 3, and the fully space-time discretization of.)-.3) in Section 3. We exemplify our results for our fully discrete scheme for the case fu) = u 3 u. The fully discrete scheme, based on a mixed variational formulation for u and the chemical potential w := ε u + ε fu), is defined as dt U m ), ψ h + W m ).9), ψ h = 0 ψh S h,.0) ε U m ), v h + fu m ) ), v h = W m, v h ) v h S h, ε with some starting value U 0 S h. Here S h H Ω) denotes the finite element space of globally continuous, piecewise affine functions.

3 Numerical analysis of the Cahn-Hilliard Equation II: Convergence of the Interface 3 We assume there exist positive ε-independent constants m 0 and σ j for j =,,, 5 such that.) m 0 := u ε 0 x) dx, ), Ω Ω.) J ε u ε 0) := ε uε 0 L + ε F uε 0) L C ε σ,.3) w ε 0 H l := ε uε 0 + ε fuε 0 ) H l C ε σ +l, l = 0,,..4) lim u ts) L C ε σ5. s 0 + Our first main result for the case fu) = u 3 u is the following one: Theorem.. Let U m, W m )} M solve.9)-.0) on a quasi-uniform space mesh T h of size Oh), allowing for inverse inequalities and H -stability of the L - finite element projection, and a quasi-uniform time mesh J of size O). Suppose the Hele-Shaw problem.4)-.8) has a global in time) classical solution and u ε 0 be the ones constructed in [. Define for some ν > 0 µ := µn, ν) = min ν, 8 N 8 ρ ε, N) := ε ε maxσ+7,σ3+4} }, +ε 6 N maxσ+5,σ3+} maxσ+ 3,σ3+ 7,σ+4} + ε σ5), ρ ε) := ε maxσ+5,σ3+ 7,σ+ 5,σ4+}, ρ 4 ε) := ε maxσ+ 3,σ3+ 7,σ+4,σ4}, ρ 5 ε) := ε maxσ+9,σ3+6,σ+4,σ4+}, π ; ε, N, σ i ) := ρ ε, N) N) 6 8 N) ε 3σ +3)+8 N)σ ) 6 8 N) π, h; ε, N, σ i, ν) := ρ ε) [ h 8 N) 4 µ ε σ +)[6+8 N) 4N 6 8 N), 6 + h µ ρ 5 ε) ρ ε) N 8 + h µ) [ ρ ε) 6 N) 8 N + ρ4 ε) + h ν µ) ε σ+)}, rh, ; ε, N, σ i, ν) := π ; ε, N, σ i ) + h +µ) π, h; ε, N, σ i, ν). Then, under the following mesh and starting value constraints ). ε 3, ). h ln h ε ρ ε) 4 8 N, 48 [+ 3). rh, ; ε, N, σ i, ν) ε 8 N) ρ ε) N 8 N, 4). U 0, ) = u ε 0, ) 5). u ε 0 U 0 H C h +ν u ε 0 H, the solution of.9)-.0) converges to the solution of.)-.3) and satisfies for some positive constant C = Cu ε 0; C 0, T ; Ω) i) max ut m) U m H + d t ut m ) U m ) H ) C [ rh, ; ε, N, σ i, ν),

4 4 Xiaobing Feng and Andreas Prohl ii) iii) ut m ) U m L ) ut m ) U m ) L ) C h ε σ+ ) + ε [ rh, ; ε, N, σ i, ν) }, C h ε σ+ ) + ε [ rh, ; ε, N, σ i, ν) }. Moreover, if U 0 satisfies then there also hold iv) v) max ut m) U m L + C u ε 0 U 0 L C h u ε 0 H, d t ut m ) U m ) L ) h ε maxσ+ 5,σ3+} + 4 ε [ rh, ; ε, N, σ i, ν) }, max ut m) U m L C Furthermore, if = Oh q ) for some N 3 following additional estimates vi) vii) viii) h ln h ρ ε) 4 8 N + h N 4 ε [ rh, ; ε, N, σ i, ν) max U m L 3 C 0, max ut m) U m L + + ε C wt m ) W m ) L ) }. < q < 8 N) + 4µ, then there hold the d t ut m ) U m ) L ) h ε maxσ+ 5,σ3+,σ} [ + ε 7 rh, ; ε, N, σi, ν) }, max ut m) U m L C h ln h ρ ε) 4 4 N 8 N + h ε maxσ+ 5,σ3+,σ} +h [ N ε 7 rh, ; ε, N, σi, ν) }. In addition, for some β >, let W 0 be a value satisfying Then wt m ) W m also satisfies ix) max wt m) W m L + w ε 0 W 0 L C h β, d t wt m ) W m ) L ) C h ρ ε) + [ ε 5 rh, ; ε, N, σi, ν) + h β},

5 Numerical analysis of the Cahn-Hilliard Equation II: Convergence of the Interface 5 x) max wt m) W m L C h 4 N ln h 3 N ρ ε) + h [ N ε 5 rh, ; ε, N, σi, ν) + h β }. Remar: a). Theorem. is the combination of Theorem 3.3, Theorem 3.4 and Corollary 3.5 in the case of the special potential function F u) = 4 u ), or fu) = u 3 u. b). L J; H )- and L J; L )-estimates for U m and the L J; L )-estimate for W m all are optimal with respect to h and, and L J; H )- and L J; L )- estimates for U m are quasi-optimal. c). Both, L -projection Q h u ε 0 and the elliptic projection P hu ε 0 of uε 0 are valid choices for starting value U 0. Also, both Q h w0 ε and P h w0 ε are valid candidates for W 0. To establish the above error estimates, the following three ingredients play a crucial role in our analysis. To establish stability estimates for the solution of the fully discrete scheme. To handle the nonlinear) potential term in the error equation using a spectrum estimate result due to Aliaos and Fusco [3 and Chen [ for the linearized Cahn-Hilliard operator.5) L CH := ε ε f u)i), where I denotes the identity operator and u is a solution of the Cahn-Hilliard equation.), see Proposition.5 for details. To establish a discrete counterpart of above spectrum estimate, see Proposition 3. for details. The above L J; L Ω) ) error estimate combined with the convergence result on the Cahn-Hilliard equation to the Hele-Shaw problem proved in [ then immediately allows us to establish the convergence of the numerical solution of the fully discrete scheme to the solution of the Hele-Shaw problem. Our second main result for the case fu) = u 3 u is the following convergence theorem. Theorem.. Let Ω be a given smooth domain and Γ 00 be a smooth closed hypersurface in Ω. Suppose that the Hele-Shaw problem.4)-.8) starting from Γ 00 has a classical solution w, Γ := 0 t T Γ t t})) in the time interval [0, T such that Γ t Ω for all t [0, T. Let u ε 0x)} 0<ε be the family of smooth uniformly bounded functions as in Theorem 5. of [. Let U ε,h, x, t), W ε,h, x, t)) denote the piecewise linear interpolation in time) of the fully discrete solution U m, W m )} M. Also, let I and O stand for the inside and outside in Ω T ) of Γ. Then, under the mesh and starting value constraints of Theorem. with ν = and = Oh q ) for some < q < 8 N) + 4µ we have N 3 i) ii) U ε,h, x, t) ε 0 uniformly on compact subset of O, U ε,h, x, t) ε 0 uniformly on compact subset of I. Moreover, when N =, let = Oh q ) for some N < q < 4 N) + µ and choose W 0 such that w0 ε W 0 L C h β for some β > q, then we also have iii) W ε,h, x, t) ε 0 wx, t) uniformly on Ω T.

6 6 Xiaobing Feng and Andreas Prohl Remar: It can be shown that the assertion iii) also holds when N = 3 under a stronger constraint on the starting value U 0 and some higher regularity assumptions on u), see Section 4 for more discussions. Our third main result for the case fu) = u 3 u is the following convergence theorem for the numerical interface. Theorem.3. Let Γ ε,h, t := x Ω ; U ε,h, x, t) = 0 } denote the zero level set of U ε,h,. Then under the assumptions i) and ii) of Theorem., we have sup dist x, Γt ) ) ε 0 0 uniformly on [0, T. x Γ ε,h, t Our last main result for the case fu) = u 3 u is the following convergence rate estimate result for numerical interface. Theorem.4. Let δ be a positive constant such that dist Γ t, Ω) > δ for all t [0, T. Then, under the assumptions for i) and ii) of Theorem., there holds sup dist x, Γt ) ) δ x Γ ε,h, t uniformly on [0, T. We remar that using a similar approach a parallel study was carried out by the authors in [0 for the Allen-Cahn equation and the related curvature driven flows. On the other hand, unlie the Allen-Cahn equation which is a gradient flow in L, the Cahn-Hilliard equation is a gradient flow only in H, which maes the analysis for the Cahn-Hilliard equation in this paper much more delicate and complicated than that for the Allen-Cahn equation given in [0. The paper is organized as follows: In Section, we shall derive some improved a priori estimates for the solution of.)-.3) under the condition that the Hele- Shaw problem has a global in time) classical solution. Special attention is given to dependence of the solution on ε in various norms. In Section 3, we analyze the fully discrete mixed finite element method proposed in [ for the Cahn-Hilliard equation, which consists of the bacward Euler discretization in time and the lowest order Ciarlet-Raviart mixed finite element for the biharmonic operator) discretization in space. Optimal and quasi-optimal error estimates in stronger norms, including L J; L ) norm, are obtained for the fully discrete solution. It is shown that all the error bounds depend on ε only in low polynomial orders for small ε. Lie in [0,, the spectrum estimate and its discrete counterpart play a crucial role in the proofs. Finally, Section 4 is devoted to establishing the convergence of the fully discrete solution to the solution of the Hele-Shaw problem. Using the L J; L ) error estimate and the convergence result of [, we show that the fully discrete numerical solution converges to the solution including the free boundary) of the Hele-Shaw problem, provided that the latter admits a global in time) classical solution.. Energy estimates for the differential problem. In this section, we derive some energy estimates in various function spaces up to L J; H 4 Ω)) H J; H 3 Ω)) in terms of negative powers of ε for the solution u of the Cahn-Hilliard problem.)-.3) for given u ε 0 H 4 Ω). The basic estimates are derived under general minimum) regularities, while the improved estimates are established under the assumption that the Hele-Shaw problem admits a global in time) solution. The same notation as in Part I of the series [ are used throughout this paper, in particular, v and v

7 Numerical analysis of the Cahn-Hilliard Equation II: Convergence of the Interface 7 are same as there. For the detailed definitions, we refer to Section of [. Again, C and C are used to denote generic positive constants which are independent of ε and the time and space mesh sizes and h. In this paper, we are mainly concerned with the second stage of the evolution of the concentration u, that is, the motion of the interface, and focus on approximating the Hele-Shaw problem via the Cahn-Hilliard equation discretized by a fully discrete mixed finite element method. For these purposes, we rewrite.)-.3) as.) u t = w in Ω T,.) w = ε fu) ε u in Ω T,.3) u ν = w = 0 ν on Ω 0, T ),.4) ux, 0) = u ε 0 x) x Ω, where w physically represents the chemical potential. We refer to [8, 7 and refererences therein for more discussions on well-posedness and regularities of the Cahn- Hilliard and the biharmonic problems. Lie in [0,, we consider the following general double equal-well potential function F : General Assumption GA ) ) f = F, for F C 4 R), such that F ±) = 0, and F > 0 elsewhere. ) f u) satisfies for some finite p > and positive numbers c i > 0, i = 0,.., 3, c u p c 0 f u) c u p + c 3. 3) There exist 0 < γ, γ > 0, δ > 0 and C 0 > 0 such that ) i) fa) fb), a b γ f a)a b), a b ) ii) af a) 0 for all a C 0. γ a b +δ for all a C 0, Remar: We note that the above GA ) differs slightly from those of [ in ) and 3). It is trivial to chec that GA ) implies.5) f u)v, v) c 0 v L, v L Ω), which will be utilized several times in the paper. Example: The potential function F u) = 4 u ), consequently, fu) = u 3 u, is often used in physical and geometrical applications [4, 9, 5,, 3. For convenience, we verify GA ) -GA ) 3. First, GA ) holds trivially. Since f u) = 3u, GA ) holds with c = c = 3 and c 0 = c 3 =. A direct calculation gives.6) fa) fb) = a b) [ f a) + a b) 3a b)a. Hence, GA ) 3 holds with γ =, γ = 3, δ = and any constant C 0 0. Also,.5) holds with c 0 =. In the rest of this section, we shall establish some basic and improved a priori estimates for the solution of the Cahn-Hilliard equation under the assumption that the Hele-Shaw problem.4)-.8) has a global in time) classical solution cf. [).

8 8 Xiaobing Feng and Andreas Prohl These improved a priori estimates are necessary for us to obtain error estimates in stronger norms in the next section. Lemma.. Suppose that f satisfies GA ), then the solution of.)-.3) satisfies the following estimates: i) ii) ε ess sup [0, ) u L + } ε F u) L ) ess sup u p L C + J p ε u ε 0 ) [0, ) u t H ds w L ds p as in GA ) ), = J εu ε 0 ), iii) ess sup u L C ε J εu ε 0 ). [0, ) Proof. The assertion i) is the immediate consequence of the basic energy law associated with the Cahn-Hilliard equation d u t t) dt J H,.7) εut)) = wt) L, where.8) J ε u) := [ε u + ε F u) dx t 0. Ω The conclusions of ii) and iii) are in turn the immediate corollaries of i). The next proposition is a corollary of Theorem. and.3 of [. It shows the boundedness of the solution of the Cahn-Hilliard equation, provided that the Hele- Shaw problem.4)-.8) has a global in time) classical solution. This boundedness result is the ey for us to be able to establish improved a priori estimates for the solution of the Cahn-Hilliard equation. We remar that the estimates in [ were obtained without assuming existence of a global in time) classical solution for the Hele-Shaw problem, hence, we could not be able to show the boundedness of the solution of the Cahn-Hilliard equation there. Proposition.. Suppose that f satisfies GA ), and the Hele-Shaw problem.4)-.8) has a global in time) classical solution. Then there exists a family of smooth initial datum functions u ε 0 } 0<ε and constants ε 0 0, and C 0 > 0, such that for all ε 0, ε 0 ) the solution u of the Cahn-Hilliard equation.)-.3) with the above initial data u ε 0 satisfies.9) u L Ω T ) 3 C 0. Proof. A proof of the assertion is buried in the middle of the proof of Theorem.3 of [. In fact, the assertion of Theorem.3 of [ was proved by establishing.9) first. Here we only setch the main idea of the proof. First, using a matched asymptotic expansion technique, a family of smooth approximate solutions u ε A, wε A ) to the solution u, w) of.)-.4) satisfying the assumption of Theorem. of [ was constructed in Section 4 of [. One condition is u ε A L Ω T ) C 0 for some C 0 > 0. Second, it was proved in Theorem. of [ that

9 Numerical analysis of the Cahn-Hilliard Equation II: Convergence of the Interface 9 u ε A, wε A ) is very close to u, w) in Lp Ω T ) for some p > see.7) on page 69 of [). Finally,.9) was proved using a regularization argument. The argument goes as follows in three steps: i) f is modified into f such that f = f in 3 C 0, 3 C 0) and f is linear for u > C 0 ; ii) It was shown that the solution u of the Cahn-Hilliard equation with the new nonlinearity f satisfies the estimate.9) when ε 0, ε 0 ) for some small ε 0 0, ; iii) It follows from the uniqueness of the solution of the Cahn-Hilliard equation that u u. Remar: As in [, the result of Proposition. is proved for a special family of initial data u ε 0x)} 0<ε. On the other hand, as explained in the introduction of [, this is not a serious restriction for the purpose of approximating the Hele-Shaw problem since i) at the end of the first stage of the evolution of the concentration, u has the required profile, and ii) the solution of the Hele-Shaw problem.4)-.8) depends only on Γ 00 and Ω. The next lemma states a Poincaré-Friedrichs type inequality for any function w which has the form.), it was first proved in Lemma 3.4 of [3. We note that u ε in the lemma does not have to be the solution of the Cahn-Hilliard equation. Lemma.3. Suppose that u ε satisfies.0) u ε t) dx = m 0, ) t 0, Ω Ω where m 0 is independent of ε. Let J ε u ε ) be defined by.8) and w ε be defined by.). Then there exist a large) positive constant C and a small) positive constant ε 0 such that for every ε 0, ε 0.) ) w ε, t) L C J ε u ε, t)) + w ε, t) L t 0. To derive a priori estimates in high norms we need to require u ε 0 satisfies the following conditions: General Assumption GA ) There exist positive ε-independent constants m 0 and σ j for j =,,, 4 such that ) m 0 := u ε 0 x) dx, ), Ω Ω ) J ε u ε 0) = ε uε 0 L + ε F uε 0) L C ε σ, 3) w ε 0 H l := ε u ε 0 ε fuε 0) H l C ε σ +l, l = 0,,. Lemma.4. Suppose f satisfies GA ), u ε 0 satisfies GA ), and Ω is of class C 3,. Assume the solution u of.)-.3) satisfies.9). Then u, w) satisfies the

10 0 Xiaobing Feng and Andreas Prohl following estimates: i) ut) dx = m 0, ) t 0, Ω ii) iii) iv) Ω 0 0 ess sup [0, ) u L ds C ε σ+3), u L ds C ε σ+5), u t H w L + ε 0 u t L ds C ε max σ+3,σ3}, v) vi) vii) viii) ess sup u L C ε max σ+ 5,σ3+}, [0, ) ess sup u L C ε max σ+ 5,σ3+}, [0, ) 0 0 ess sup [0, ) u t L ds w L ds + ess sup [0, ) ε u L C ε max σ+ 7,σ3+,σ+}, u t L + ε u t L ds C ε max σ+ 3,σ3+ 7,σ+4,σ4}. 0 Moreover, in addition to GA ) suppose that there exists σ 5 > 0 such that.) lim s 0 + u ts) L Cε σ5, then the solution of.)-.3) also satisfies the following estimates: for N =, 3 ix) ess sup [0, ) x) u t L + ε u t L ds 0 C ε maxσ+7,σ3+4} + ε 6 N maxσ+5,σ3+} maxσ+ 3,σ3+ 7,σ+4} 0 +ε σ5) C ρ 0 ε, N), u tt H ds C ερ 0ε, N) Cρ ε, N), xi) ess sup u L C ε maxσ+5,σ3+ 7,σ+ 5,σ4+} C ρ ε). [0, ) Proof. Since most of the proofs follow the same line as in the proofs of Lemma. and. of [, we only point out the main steps and differences, in particular, we lie to mention that.9) holds under the assumption that the Hele-Shaw problem.4)-.8) has a global in time) classical solution, it will be used below in the proofs of iii)-xi). The proof of i) is trivial. The assertion ii) is obtained from testing.) with u and applying.5) to the nonlinear term. To show iii), we write from.).3) u = ε fu) ε w.

11 Numerical analysis of the Cahn-Hilliard Equation II: Convergence of the Interface The assertion follows from applying the operator to.3), assumption GA ), triangle inequality,.9) and i) of Lemma.. To derive the remaining estimates, we differentiate.) with respect to t.4) u tt + ε u t ε f u)u t ) = 0. The assertion iv) results from testing.4) with u t and using.5) as well as interpolating u t L by u t H and u t H. Notice that w L = u t H in view of.) and the fact that u t, ) = 0. Assertion v) follows from.3) and using the triangle inequality,.9),.), Lemma., i) and the assertion iv). Then vi) is obtained similarly after first applying the operator to.3). Testing.) with u t, and using i) of Lemma. and the assertion iv) give vii). Now viii) follows from testing.4) with u t, integrating by parts on the nonlinear term and using the assertion vii). To show ix), we multiply.4) by u t to get d dt u t L + ε u t L = ε f u)u t ), u t ) Using the following Gagliardo-Nirenberg inequality [,.5) u L C u = ε f u) uu t + f u) u t, u t ) ε u t L + C ε 3 u L u t L + u t L ). 6 N L u 4 N 6 N L + u L ) we have.6) d dt u t L + ε u t L C [ ) ε 3 u t L + u 4 6 N L + u t L. The assertion follows from.6) and iv), vi) and vii). From.4) we have < u tt, φ > u tt H = sup 0 φ H φ H ε u t L + ε f u)u t ) L. Hence u tt H ε u t L + ε u L u t L + u t L ). Statement x) then follows from integrating the above inequality over 0, T ) and using.5) and the assertion ix). Finally, we rewrite.) as u = ε fu) ε u t = ε [ f u) u + f u) u ε u t.

12 Xiaobing Feng and Andreas Prohl The assertion xi) follows from using the triangle inequality,.9), and the assertions v), vi) and viii). The proof is complete. Remar: From the construction of u ε A, wε A ) in Section 4 of [ we now that u ε 0 } 0<ε obtained in Proposition. satisfy GA ). In addition, the corresponding solution u satisfies.9), see Proposition., provided that the Hele-Shaw problem.4)-.8) has a global in time) classical solution. We conclude this section by citing the following result of [3, on low bound estimate of the spectrum of the linearized Cahn-Hilliard operator L CH in.5). The estimate plays an important role in our error analysis. Proposition.5. 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.5) satisfies for small ε > 0 or equivalently λ CH λ CH ε ψ L + inf ε f u)ψ, ψ) C 0 ψ H Ω) ψ 0, L ε ψ L + inf ε f u)ψ, ψ) 0 ψ H Ω) w C 0. w=ψ L 3. Error analysis for a fully discrete mixed finite element approximation. In this section we analyze the fully discrete mixed finite element method proposed in [ for.)-.4) under the condition that the Hele-Shaw problem has a global in time) classical solution cf. [). Under this assumption, we establish stronger error bounds than those of [, which were shown under general minimum) regularity assumptions, for the solution of the fully discrete mixed finite element method. In particular, we obtain L J; L )-error estimate, which is necessary for us to establish the convergence to the solution of the fully discrete mixed finite element scheme to the solution of the Hele-Shaw problem in the next section. We recall that the wea formulation of.)-.4) is defined as: Find u, w) [H Ω) such that for almost every t 0, T ) 3.) 3.) 3.3) u t, η) + w, η) = 0 η H Ω), ε u, v) + ε fu), v) = w, v) v H Ω), ux, 0) = u ε 0 x) x Ω. Note that u t, ) = 0, that is, the mass ut), ) = u ε 0, ) is conserved for all t 0. We also recall that the fully discrete mixed finite element discretization of 3.)- 3.3) is defined as: Find U m, W m ) } M [ S h such that 3.4) 3.5) d t U m+, η h ) + W m+, η h ) = 0 η h S h, ε U m+, v h ) + ε fu m+ ), v h ) = W m+, v h ) v h S h, with some suitable starting value U 0. Here J := t m } M is a quasi-uniform partition of [0, T of mesh size := T M and T h be a quasi-uniform triangulation of Ω. Also,

13 Numerical analysis of the Cahn-Hilliard Equation II: Convergence of the Interface 3 d t U m+ := U m+ U m )/ and 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 spaces nown as the Ciarlet-Raviart mixed finite elements for the biharmonic problem cf. [4, 4). In fact, it is not hard to chec that the following inf-sup condition holds 3.6) 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 3.7) Q h v v, η h ) = 0 η S h, and the elliptic projection P h : H Ω) S h by 3.8) 3.9) [P h v v, η h ) = 0 η h S h, P h v v, ) = 0. We refer to Section 4 of [0 for a list of approximation properties of Q h and P h. In the sequel, we confine to the meshes T h that result in H -stability of the Q h, see [0 and reference therein for the details. We also introduce space notation } S h := v h S h ; v h, ) = 0, } L 0 v Ω) := L Ω) ; v, ) = 0, and define the discrete inverse Laplace operator: h : L 0 Ω) S h such that 3.0) h v), η h) = v, ηh ) η h S h. To establish stability estimates for the solution of the fully discrete scheme 3.4)- 3.5) for general potential function F u), we mae the last structural assumption on fu). General Assumption 3 GA 3 ) Suppose that there exists α 0 0, 0 < γ 3 <, and c 4 > 0 such that f satisfies for any 0 < ε α0 and any set of discrete in time) functions φ m } M H Ω) 3.) ) γ 3 d t φ m H + ε d tφ m L + ε fφ m ), d t φ m) + c 4 J ε φ 0 ) c 4 ε F φl ) L l M.

14 4 Xiaobing Feng and Andreas Prohl We remar that the validity of GA 3 ) was proved in [ for the case of the quartic potential F u) = 4 u ) 4 with α 0 = 3, γ 3 = 4 and c 4 =. With help of GA 3 ) we are able to show that the solution of 3.4)-3.5) satisfies the following stability estimates. Lemma 3.. The solution U m, W m ) } M i) Ω Ω of 3.4)-3.5) satisfies U m dx = U 0 dx, m =,,, M, Ω Ω ii) d t U m H C W m L, m =,,, M, iii) iv) v) vi) vii) max ε U m L + } ε F U m ) L ) + W m L + ε d tu m L d t U m H CJ εu 0 ), max U m L p C ) + J ε U 0 ) p as in GA ) ), ess sup W m L ) + d t W m L + ε d tu m L CJ ε U 0 ), C ε 3 J ε U 0 ), ess sup d t U m L + M d t U m L C ε) ε 3 J ε U 0 ). Proof. The assertion i) is an immediate consequence of setting η h = in 3.4). For any φ H Ω), from 3.4), 3.7), and the stability of Q h in H Ω) cf. [0 and references therein) we have 3.) 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 3.4) and v h = d t U m+ in 3.5) and adding the resulting equations give 3.3) W m+ L + ε d dt U m+ L + ε d t U m+ L + ε fu m+ ), d t U m+ ) = 0. The statement then follows from GA 3 ) and ii) after multiplying 3.3) by and taing sum over m from 0 to l M ).

15 Numerical analysis of the Cahn-Hilliard Equation II: Convergence of the Interface 5 The assertions iv) and v) are immediate consequences of ii) and iii). To show vi), choose η h = d t W m+ in 3.4) and v h = d t U m+ in 3.5) after applying the difference operator d t to 3.5), and add the resulting equations we get 3.4) d t W m+ L + d t W m+ L + ε d tu m+ L + ε d tfu m+ ), d t U m+ ) = 0. By the Mean Value Theorem and.5) we bound the last term on the left hand side of 3.4) as follows, 3.5) ε d tfu m+ ), d t U m+ ) = f ξ), d t U m+ ) ε c 0 ε d tu m+ L ε d tu m+ L C ε 3 d tu m+ H. The assertion then follows from 3.4)-3.5) and the assertion iv) after multiplying 3.4) by and taing sum over m from 0 to l M ). Finally, to show statement vii), we first apply the difference operator d t to both sides of 3.4), then tae η h = d t U m+ in the resulting equation to have 3.6) d t d t U m+ L + d t U m+ L = d t W m+, d t U m+) ε d t W m+ L + ε d tu m+ L ). Multiplying 3.6) by and taing sum over m from 0 to l M ), the assertion then follows from vi). Remar: In view of i) of Lemma.4 and i) of Lemma 3., in order for the scheme 3.4)-3.5) to conserve the mass of the underlying physical problem, it is necessary to require U 0 u ε 0, ) = 0 for the starting value U 0. This condition will be assumed in the rest of this section. As is shown in [, in order to establish error bounds that depend on low order polynomials of ε, the crucial idea is to utilize the spectrum estimate result of Proposition.5 for the linearized Cahn-Hilliard operator. In the following we show that the spectrum estimate still holds if the function u, which is the solution of.)-.3), is replaced by its elliptic projection P h u and the nonlinear term is scaled by a factor ε, provided that the mesh size h is small enough. As expected, this result plays a critical role in our error analysis for the fully discrete finite element discretization. For u the solution of.)-.3), let C 0 be same as in.9) and define 3.7) C = max v C 0 f v), and C be the smallest positive ε-independent constant such that cf. Chapter 7 of [8) 3.8) u P h u L J;L ) C h ln h u L J;W, ) C h ln h ρ 3 ε, N)

16 6 Xiaobing Feng and Andreas Prohl for some low order) polynomial function ρ 3 ε, N) in ε. We remar that the existence of C and ρ 3 ε, N) follows easily from xi) of Lemma.4 and the following Gagliardo- Nirenberg inequality [ u W, C D 4 u 4 8 N L u 4 N 8 N L + u L ), N =, 3. In fact, the above inequality,.9) and xi) of Lemma.4 imply 3.9) ρ 3 ε, N) ρ ε) 4 8 N, where ρ ε) is defined in xi) of Lemma.4. Proposition 3.. Let the assumptions of Proposition.5 hold and C 0 be same as there. Let u be the solution of.)-.) satisfying.9) and P h u be its elliptic projection. Then there holds for small ε > 0 3.0) λ h CH inf 0 ψ L 0 Ω) w=ψ, w n =0 ε ψ L + ε ε f P h u)ψ, ψ) w L ε)c 0 + ), provided that h satisfies the constraint 3.) h ln h C C ρ 3 ε, N)) ε. Proof. From the definitions of C and C, we immediately have P h u L J;L ) u L J;L ) + P h u u L J;L ) 4 3 u L J;L ) C 0 if h satisfies 3.). It then follows from the Mean Value Theorem that 3.) f P h u) f u) L J;L ) max ξ C 0 f ξ) P h u u L J;L ) C C h ln h ρε, N) ε. Using the inequality a b a b and 3.) we get 3.3) f P h u) f u) f P h u) f u) L J;L ) f u) ε. 3.4) In addition, we have ψ L = ψ, w ) ε ε ψ L + ε ) w L. ε Substituting 3.3)-3.4) into the definition of λ h CH we get λ h CH inf ε) [ ε ψ L + ε f u)ψ, ψ) 0 ψ L 0 Ω) w ε). L w=ψ, w n =0

17 Numerical analysis of the Cahn-Hilliard Equation II: Convergence of the Interface 7 The proof is completed by applying Proposition.5. The first main result of this section is stated in the following theorem. Theorem 3.3. Let U m, W m ) } M solve 3.4)-3.5) on a quasi-uniform space mesh T h of size Oh), allowing for inverse inequalities and H -stability of Q h, and a quasi-uniform time mesh J of size O). Suppose the assumptions of Lemma.4 and 3., Proposition.5 and 3. hold, in particular, α 0, σ i, ρ i ε, N) are the same as there. Let 0 < δ < 6 8 N for N =, 3, and define for some ν > 0 µ := µn, δ, ν) = min } 3.5) δ, ν, 8 N 8, 3 maxσ+ ρ 4 ε) := ε 3.6),σ3+ 7,σ+4,σ4}, ρ 5 ε) := ε maxσ+9,σ3+6,σ+4,σ4+}, π ; ε, N, δ, σ i ) := ρ ε, N) N)δ 6 8 N)δ ε 3σ +3)+δ8 N)σ ) 6 8 N)δ ρ ε) 6 8 N)δ, [ π, h; ε, N, δ, σ i, ν) := h 8 N)δ 4 µ ε σ 3.7) +)[6+8 N)δ 6 + h µ ρ 5 ε) 4Nδ ρ ε) δn 8 [ + h δ µ) ρ ε) 4+δ) N 8 N + ρ 4 ε) ) + h ν µ) ε σ+), 3.8) rh, ; ε, N, δ, σ i, ν) := π ; ε, N, δ, σ i ) + h +µ) π, h; ε, N, δ, σ i, ν). Then, under the following mesh and starting value constraints ). ε α0, ). h ln h ε ρ ε) 4 8 N, 48 [+ 3). rh, ; ε, N, δ, σ i, ν) ε 8 N)δ ρ ε) N 8 N, 4). U 0, ) = u ε 0, ), 5). u ε 0 U 0 H C h +ν u ε 0 H, the solution of 3.4)-3.5) converges to the solution of.)-.4) and satisfies i) max ut m) U m H + d t ut m ) U m ) H ) C [ rh, ; ε, N, δ, σ i, ν), ii) iii) ut m ) U m L ) ut m ) U m ) L ) C h ε σ+ ) + ε [ rh, ; ε, N, δ, σ i, ν) }, C h ε σ+ ) + ε [ rh, ; ε, N, δ, σ i, ν) } for some positive constant C = Cu ε 0 ; γ, γ, C 0, T ; Ω).

18 8 Xiaobing Feng and Andreas Prohl Proof. The proof is divided into four steps. Step one deals with consistency error due to the time discretization. Steps two and three use Proposition 3. and stability estimates of Lemma.4 and 3. to avoid exponential blow-up in ε of the error constants. In the final step, an inductive argument is used to handle the difficulty caused by the super-quadratic term in GA ) 3. Step : Let E m := ut m ) U m and G m := wt m ) W m. Subtracting 3.4)-3.5) from 3.)-3.), respectively, we get the error equations 3.9) 3.30) d t E m, η h ) + G m, η h ) = Ru tt ; m), η h ), ε E m, v h ) + ε futm )) fu m ), v h ) = G m, v h ), where 3.3) Ru tt ; m) = tm t m s t m ) u tt s)ds. It is easy to chec that 3.3) Ru tt ; m) H [ t m [ t m s t m ) ds t m u tt H ds t m C ρ ε, N), where ρ ε, N) is defined in x) of Lemma.4. Step : Introduce the decompositions: E m := Θ m + Φ m and G m := Λ m + Ψ m, where Θ m := ut m ) P h ut m ), Φ m := P h ut m ) U m, Λ m := wt m ) P h wt m ), Ψ m := P h wt m ) W m. Then from the definition of P h in 3.8)-3.9) we can rewrite 3.9)-3.30) as follows 3.33) 3.34) d t Φ m, η h ) + Ψ m, η h ) = Ru tt ; m), η h ) d t Θ m, η h ), ε Φ m, v h ) + ε fph ut m )) fu m ), v h ) = Ψ m, v h ) + Λ m, v h ) ε futm )) fp h ut m )), v h ). Since E m, Φ m L 0 Ω) for 0 m M, setting η h = h Φm in 3.33) and v h = Φ m in 3.34) and taing summation over m from to l M), after adding the

19 Numerical analysis of the Cahn-Hilliard Equation II: Convergence of the Interface 9 equations we conclude 3.35) 3.36) h Φl L + = + h d tφ m L [ ε Φ m L + fph ut m )) fu m ), Φ m) ε [ Ru tt ; m), h Φm ) d t Θ m, h Φm ) + Λ m, Φ m ) + ε futm )) fp h ut m )), Φ m) + h Φ0 L. The first sum on the right hand side can be bounded as follows [ Ru tt ; m), h Φm ) d t Θ m, h Φm ) + Λ m, Φ m ) C [ Ru tt ; m) H + d tθ m H + ε Λ m H + h Φm L + ε ε γ } ) Φ m L ε) h Φm L + ε ε γ } ) Φ m L ε) [ +C ρ ε, N) + h 6 ρ 4 ε), where ρ ε, N) is defined in x) of Lemma.4 and ρ 4 ε) := ε maxσ+ 3,σ3+ 7,σ+4,σ4}. Here we have used the following approximation properties in H projection P h cf. [6) of the elliptic u P h u H C h 3 u H, u P h u) t H C h 3 u t H, w P h w H C h 3 w H. In view of.9) and the inequality at the beginning of the proof of Proposition 3., the second sum on the right hand side of 3.35) can be bounded by 3.37) ε futm )) fp h ut m )), Φ m) = ε f ξ)θ m, Φ m) [ ε ε γ ) Φ m L ε) + C ε 3 Θ m H ε ε γ ) Φ m L ε) + C h6 ε maxσ+8,σ3+3}.

20 0 Xiaobing Feng and Andreas Prohl By GA ) 3, the last term on the left hand side of 3.35) is bounded from below by 3.38) 3.39) fph ut m )) fu m ), Φ m) ε ε [ γ f P h ut m ))Φ m, Φ m) γ Φ m +δ Substituting 3.36)-3.38) into 3.35) we arrive at h Φl L + + γ ε h d tφ m L L +δ [ ε Φ m L + ε f P h ut m ))Φ m, Φ m) ε [ C ρ ε, N) + h 6 ρ 4 ε) + + γ ε Φ m +δ L +δ. h Φm L We could bound the last term on the left hand side from below using Proposition 3., however, this will consume all the contribution of ε Φ m L on the left hand side. On the other hand, in order to bound the super-quadratic term on the right hand side in Step 3 below, we do need some small) help from this ε Φ m L. For that reason, we only apply Proposition 3. with a scaling factor ε 3 we remar that any scaling of form ε γ γ 3) should wor, but γ = 3 seems to give the best constant), that is, [ ε 3 ) ε Φ m L + ε f P h ut m ))Φ m, Φ m) 3.40) ε ε 3 ) ε)c 0 + ) Φ m L, C 0 + ) Φ m L. We eep the leftover term γε4 ε Φm L on the left hand side, and move the leftover term γ ε f P h ut m ))Φ m, Φ m) to the right side to bound it from above by 3.4) 3.4) γ ε f P h ut m ))Φ m, Φ m) γ ε 4 ε) Φm L + γ ε) h Φm L. Combining 3.39)-3.4) we finally get h Φl L + [ h d tφ m L + γ ε 4 Φm L [ C ρ ε, N) + h 6 ρ 4 ε) + C 0 + γ + 3 ) + γ ε Φ m +δ L +δ, h Φm L.

21 Numerical analysis of the Cahn-Hilliard Equation II: Convergence of the Interface where we have used the fact that 0 < ε < and v h L = h v h L for any v h S h. Step 3: It remains to bound the super-quadratic term at the end of 3.4). Since Φ m = E m Θ m, the triangle inequality implies 3.43) 3.44) To bound E m +δ L +δ Φ m +δ L +δ C E m +δ L +δ + Θ m +δ L +δ ). E m +δ L +δ K T h in 3.43), we first mae a shift in the super-index to get [ E m +δ L +δ K) + +δ d t E m +δ. L +δ K) For each term in the first sum on the right hand side of 3.44), we use the Gagliardo- Nirenberg inequality [ which interpolates L +δ K) between L K) and H 4 K). [ E m +δ L +δ K) C D 4 E m δn 8 L K) Em 6+8 N)δ ) L K) + E m +δ L K), C E m 8 N)δ [ + 8 L K) D 4 ut m ) δn 8 L K) + Em δn 8 L K) C E m 8 N)δ + 8 L K) ρ ε) δn 8. Here we have used the estimates xi) of Lemma.4, ii) of Lemma. and v) of Lemma 3. to obtain the last inequality. Summing 3.45) over all K T h and using the convexity of the function gs) = s r for r > and s 0 we have 3.46) 3.47) E m +δ C ρ L +δ ε) δn 8 E m 8 N)δ + 8 L. Similarly, we can bound the second sum on the right hand side of 3.44). +δ d t E m +δ L +δ K) C +δ[ D 4 d t E m δn 8 L K) d te m 6+8 N)δ 8 L K) + d t E m +δ L K) C +δ d t E m 8 N)δ [ + 8 L K) D 4 d t ut m ) δn 8 L K) + d te m δn 8 L K) C +δ d t E m 8 N)δ [ + 8 L K) Nδ 8 D 4 ut m ) δn 8 L K) + d te m δn 8 8 N)δ + C 8 d t E m 8 N)δ + 8 L K) ρ ε) δn 8. L K) Here we have used the estimates viii) and xi) of Lemma.4, and vii) of Lemma 3. to obtain the last inequality. Taing sum on both sides of 3.47) over all K T h we get 3.48) +δ d t E m +δ 8 N)δ + C L +δ 8 ρ ε) δn 8 dt E m 8 N)δ + 8 L. It now follows from the triangle inequality E m L Θ m L + Φ m L, the inequalities 3.43), 3.44), 3.46) and 3.48) that 3.49) Φ m +δ L +δ C ρ ε) δn 8 + Θ m Φ m 8 N)δ + 8 L 8 N)δ + 8 L 8 N)δ N)δ + 8 L 8 N)δ ) + 8 L 8 N)δ d t Φ m 8 d t Θ m + C Θ m +δ L +δ.

22 Xiaobing Feng and Andreas Prohl From [8, 4, 5 we now that 3.50) 3.5) 3.5) u P h u L C h u H, u P h u) t L C h u t H, u P h u L +δ C h u W,+δ C h 4+δ) N 8 N)+δ) u L. To control two terms which involve Φ m on the right hand side of 3.49), we only consider the case δ < 6 8 N because i) it covers most useful ranges of δ, ii) the analysis for the case δ > 6 8 N From the definition of h is easier to carry out since 6+8 N)δ 8 > 4 in this case. in 3.0) and Young s inequality we have 3.53) Φ m 6+8 N)δ 8 L Φ m 6+8 N)δ 6 L C [ ε 5 ρ ε) δn 8 h 6+8 N)δ 6 8 N)δ h + γ ε 5 ρ ε) δn 8 Φ m L 4γ. Φm 6+8 N)δ 6 L [6+8 N)δ Φm 6 8 N)δ L Similarly, d t Φ m 6+8 N)δ 8 L d t Φ m 6+8 N)δ 6 L h C [ ε 8+8 N)δ 8 ρ ε) δn N)δ ε 8 ρ ε) δn 8 + 4γ d tφ m 6+8 N)δ N)δ L 6 8 N)δ d t Φ m d t h Φm L. [6+8 N)δ 6 8 N)δ L Using d t Φ m [6+8 N)δ 6 8 N)δ L = d t Φ m L d tφ m 48 N)δ 6 8 N)δ ε σ + ) 48 N)δ 6 8 N)δ L 48 N)δ 6 8 N)δ dt Φ m L, we get 3.54) d t Φ m 6+8 N)δ 8 L 8+8 N)δ ε 8 ρ ε) δn 8 4γ d t h Φm L +C ε [+ 4δ8 N)σ +) 6 8 N)δ ρ ε) δn[6+8 N)δ 8[6 8 N)δ 8 N) δ + 8[6 8 N)δ dt Φ m L. Now, substituting 3.50)-3.54) into 3.49) summing over m from to l M) after multiplying 3.49) by γ ε and using iv) of Lemma.4 and vi) of Lemma 3.

23 Numerical analysis of the Cahn-Hilliard Equation II: Convergence of the Interface 3 lead to the following estimate 3.55) 3.56) γ ε Φ m +δ L +δ +C ε 4[4+8 N)δ 6 8 N)δ [ ε 4 ρ ε) 4 Φm L + 4 h d tφ m L 4Nδ 6 8 N)δ 8 N)δ 3+ +C 6 8 N)δ ρ ε) [ +C ρ ε) δn 8 ε σ +)[6+8 N)δ 8 N)δ 6 4+ h N)δ ε [+ 4δ8 N)σ +) h Φm L + 48 N)δ 6 8 N)δ 4Nδ 6 8 N)δ ε σ +) + h 4 ε maxσ+9,σ3+6,σ+4,σ+} + C h +δ) ρ ε) 4+δ) N 8 N Finally, substituting 3.55) into 3.4) we get h Φl L + [ 4 h d tφ m L + γ ε 4 4 Φm L C rh, ; ε, N, δ, σ i, ν) + C 0 + γ + 3) +C sε, N, δ, σ i ) l where rh, ; ε, N, δ, σ i, ν) is defined in 3.8) and 3.57) sε, N, δ, σ i ) = ε 4[4+8 N)δ 6 8 N)δ It is easy to chec from 3.8) that for a fixed ε 3.58) h Φm L 48 N)δ + h Φm 6 8 N)δ L, ρ ε) 4Nδ 6 8 N)δ. rh, ; ε, N, δ, σ i, ν) C + h +µ) ), provided that h. Step 4: We now conclude the proof by the following induction argument. Suppose there exist two positive constants c = c t l, Ω, u ε 0, σ i ), c = c t l, Ω, u ε 0, σ i ; C 0 ), independent of and ε, such that there holds 3.59) [ max 0 m l h Φm L + 4 h d tφ m L + γ ε 4 4 Φm L In view of xi) of Lemma.4 and 3.56), we can choose c rh, ; ε, N, δ, σ i, ν) expc t l ). c =, c = C 0 + γ + 3). Since the exponent in the last term of 3.56) is bigger than, we can recover 3.59) at the l + )th time step by using the discrete Gronwall s inequality, provided that h, satisfy 8 N)δ sε, N, δ, σ i ) rh, ; ε, N, δ, σ i, ν) + c 6 8 N)δ rh, ; ε, N, δ, σ i, ν) expc t l+ ).

24 4 Xiaobing Feng and Andreas Prohl That is, sε, N, δ, σ i ) rh, ; ε, N, δ, σ i, ν) 8 N)δ 6 8 N)δ C, which gives the mesh condition 3) in the theorem. Hence, we have shown that 3.60) [ max h Φm L + h d tφ m L + ε4 Φ m L C rh, ; ε, N, δ, σ i, ν). Finally, the assertion i) follows from 3.60) and applying the triangle inequality on E m = Θ m + Φ m. The assertions ii) and iii) follow in the same way. Note that we need to use Poincaré inequality on Φ m to show ii), and since Φ m S h, we can bound Φ m L by Φ m L. The proof is complete. Remar: a). The L J; H )-estimate is optimal with respect to h and, and L J; H )-estimate is quasi-optimal. b). The proof clearly shows how the three mesh conditions arise. The condition ) is for the stability of the time discretization see GA 3 )), the condition ) is to ensure the discrete spectrum estimate see Proposition 3.), finally, the condition 3) is caused by the super-quadratic nonlinearity of f see GA ) 3 ). Also notice that only smallness of and h with respect to ε is required but no restriction is imposed on the ratio between and h in the L J; H )- and L J; H )-norm estimates. c). It is well-nown [5 that the finite element solutions of all linear and some nonlinear parabolic problems exhibit superconvergence property in h) when compared with the elliptic projections of the solutions of underlying problems, it is worth pointing out that this superconvergence property also hold for the Cahn-Hilliard equation as showed by the inequality 3.60). d). Regarding the choices of the starting value U 0, clearly, both U 0 = Q h u ε 0 and U 0 = P h u ε 0 satisfy the conditions 4) and 5) with ν = in view of 3.7) and 3.9). In fact, they also satisfy a stronger inequality, see 3.6). The L -projection Q h u ε 0 has the advantage of being cheaper to be computed compared to the elliptic projection P h u ε 0. Note that the condition 4) is necessary in order for the scheme 3.4)-3.5) to conserve the mass. In the next theorem we derive error estimates in stronger norms under a slightly stronger requirement on the starting value U 0, which nevertheless is satisfied by both, L -projection U 0 = Q h u ε 0 and the elliptic projection U 0 = P h u ε 0. In addition, a mild constraint on admissible choices of, h) is required to assure their validity. Theorem 3.4. In addition to the assumptions of Theorem 3.3, if 3.6) u ε 0 U 0 L C h u ε 0 H, then the solution of 3.4)-3.5) also satisfies the following error estimates i) ii) max ut m) U m L + C d t ut m ) U m ) L ) h ε maxσ+ 5,σ3+ } + 4 ε [ rh, ; ε, N, δ, σ i, ν) }, max ut m) U m L C h ln h ρ ε) 4 8 N + h N 4 ε [ rh, ; ε, N, δ, σ i, ν) }.

25 Numerical analysis of the Cahn-Hilliard Equation II: Convergence of the Interface 5 Moreover, if = Oh q ) for some N 3 < q < 8 N) + 4µ, then there also hold iii) iv) v) max U m L 3 C 0, max ut m) U m L + + ε C h ε maxσ+ 7,σ3+ } + ε 7 d t ut m ) U m ) L ) wt m ) W m L ) [ rh, ; ε, N, δ, σi, ν) }, max ut m) U m L C h ln h ρ ε) 4 4 N 8 N + h ε maxσ+ 7,σ3+ } +h N ε 7 [ rh, ; ε, N, δ, σi, ν) }. for some positive constant C = Cu ε 0; γ, γ, C 0, T ; Ω). Proof. Since E m = Θ m + Φ m, G m = Λ m + Ψ m, it suffices to show the assertions i), ii), iv), v) hold for Φ m and Ψ m without the first term on the right hand side of each inequality. Notice that Φ m and Ψ m satisfy 3.33)-3.34). Using the identity the definition of h 3.6) Φl L + = d t Φ m, Φ m ) = d t Φ m L + d tφ m L, in 3.0) and the estimate 3.60) we have d tφ m L h d tφ m ), Φ m) + Φ0 L ε [ h d tφ m L + ε4 Φ m L + Φ0 L ε rh, ; ε, N, δ, σ, ν) + Φ0 L. The assertion i) then follows from 3.6) and 3.6). The assertion ii) is an immediate consequence of i), the inverse inequality bounding L by L -norm, and the L -estimate of Θ m see Chapter 7 of [8). To show iii), notice that under the mesh conditions of Theorem 3.3 and the assumption that = Oh q ) for some N 3 < q < 8 N) + 4µ, there holds for sufficiently small ε 3.63) max ut m) U m L 3 C 0,

26 6 Xiaobing Feng and Andreas Prohl which together with.9) then implies 3.64) max U m [ L max utm ) L + ut m ) U m L 3 C0. Hence the assertion iii) holds. Now, taing η h = Φ m in 3.33) and v h = ε Ψm in 3.34) and adding the resulting equations we get 3.65) d t Φ m L + d tφ m L + ε Ψm L = Ru tt ; m), Φ m ) d t Θ m, Φ m ) ε Λm, Ψ m ) + ε futm )) fu m ), Ψ m). The first three terms on the right hand side can be bounded similarly as in 3.36), and the last term can be bounded as follows. By the Mean Value Theorem and Schwarz inequality we obtain 3.66) ε futm )) fu m ), Ψ m) = ε f ξ)e m, Ψ m) ε Ψm L + C ε 3 Em L. The assertion iv) follows from multiplying 3.65) by, summing it over m from to l M) and using 3.66) and the assertion ii) of Theorem 3.3. Finally, the assertion v) is a refinement of ii), it is based on the estimate iv), instead of i). The proof is complete. Remar: a). The estimate in i) is optimal in h and suboptimal in due to the factor 4 in the second term on the right hand side of the inequality. However, this estimate is important for establishing the L J; L )-estimate in ii), which then leads to the proof of the boundedness of U m in 3.64), and the improved estimates iv) and v). b). Optimal estimates in stronger norms also can be obtained for both E m and G m under stronger regularity assumptions on the solution u e.g. u tt L J; L )) of the Cahn-Hilliard equation and on the starting value U 0. These estimates include statements for E m in L J; H ) and H J; L ), and in L J; L ) and L J; H ) for G m. For more expositions in this direction, we refer to [7 also see [5), where a continuous in time) semi-discrete splitting finite element method was analyzed for a fixed ε > 0 under the assumption that the semi-discrete finite element approximate solution for u is bounded in L. Note that here we have indeed showed in iii) that our fully discrete solution U m is bounded in L. Corollary 3.5. Let the assumptions of Theorem 3.4 be valid, and W 0 be a value satisfying for some β > 3.67) P h w0) W 0 L C h β.

27 Numerical analysis of the Cahn-Hilliard Equation II: Convergence of the Interface 7 Then there hold the following estimates i) ii) max wt m) W m L + C h ρ ε) + max wt m) W m L C h 4 N ln h 3 N ρ ε) d t wt m ) W m ) L ) h ε maxσ+3,σ3} + ε 3 [ rh, ; ε, N, δ, σ i, ν) } + h β}, +h [ N h ε maxσ+3,σ3} + ε 3 [ rh, ; ε, N, δ, σ i, ν) } + h β }. Proof. First, from [8, 5 we now that 3.68) max Λm L + d t Λ m L ) C h ρ ε). Next, using the identity just above 3.6) we get 3.69) Ψl L + d tψ m L = d t Ψ m, Ψ m ) + Ψ0 L [ 4 d tψ m L + 4 Ψ m L + Ψ0 L. The first term on the right hand side can be absorbed by the second term on the left, a desired bound for the second term on the right has been obtained in the proof of iv) of Theorem 3.4. Hence, assertion i) follows from combining 3.68) and 3.69). The assertion ii) comes from applying the triangle inequality to G m = Λ m +Ψ m, the following estimate cf. Section 4 of [0) Λ m L C h 4 N ln h 3 N w H C h 4 N ln h 3 N ρ ε), and the inverse inequality bounding Ψ m L in terms of Ψ m L. The proof is complete. Remar: a). Clearly, both Q h w0 ε and P h w0 ε are valid candidates for W 0. b). Both estimates are not optimal due to the factor in the second term on the right hand side of each inequality. It can be shown that the estimates will be improved to optimal order first order in and second order in h) under some stronger regularity assumptions and starting value constraint. See b) of the remar after the proof of Theorem 3.4. We conclude this section by giving a very short proof of Theorem.. Proof of Theorem.. The assertions follows immediately from setting δ =, α 0 = 3, γ = and γ = 3 in Theorem 3.3 and Theorem 3.4.

28 8 Xiaobing Feng and Andreas Prohl 4. Approximation for the Hele-Shaw problem. The goal of this section is to establish the convergence of the solution U m, W m )} M of the fully discrete mixed finite element scheme 3.4)-3.5) to the solution of the Hele-Shaw problem.4)-.8), provided that the Hele-Shaw problem has a global in time) classical solution. It is shown that the fully discrete solution W m, as h, 0, converges to the solution w of the Hele-Shaw problem uniformly in Ω T. In addition, the fully discrete solution U m converges to ± uniformly on every compact subset of the outside and inside of the free boundary Γ of the Hele-Shaw problem, respectively. Hence, the zero level set of U m converges to the free boundary Γ. Our main ideas are to mae fully use of the convergence result that the Hele-Shaw problem is the distinguished limit, as ε 0, of the Cahn-Hilliard equation proved by Aliaos, Bates and Chen in [, and to exploit the closeness between the solution u of the Cahn-Hilliard equation and its fully discrete approximation U m, which is demonstrated by the error estimates in the previous section. We remar that as in [, our numerical convergence result is also established under the assumption that the Hele-Shaw problem has a global in time) classical solution. We refer to [, 3 and references therein for further expositions on this assumption and related theoretical wors on the Hele-Shaw problem. Although the results of this section hold for general potential F u) satisfying GA ), for the sae of clarity of the presentation, we only consider the quartic potential F u) = 4 u ) in this section. Let u ε denote the solution of the Cahn-Hilliard problem.)-.3). Note that we put bac the super index ε on the solution in this section. Let U ε,h, x, t), W ε,h, x, t)) denote the piecewise linear interpolation in time) of the fully discrete solution U m, W m ), that is, 4.) 4.) U ε,h,, t) := t t m W ε,h,, t) := t t m U m+ ) + t m+ t U m ), W m+ ) + t m+ t W m ), for t m t t m+ and 0 m M. Note that W 0 is defined in Corollary 3.5, and U ε,h, x, t) and W ε,h, x, t) are a continuous piecewise linear functions in space and time. Let Γ 00 Ω be a smooth closed hypersurface and let w, Γ := 0 t T Γ t t})) be a smooth solution of the Hele-Shaw problem.4)-.8) starting from Γ 00 such that Γ Ω [0, T. Let dx, t) denote the signed distance function to Γ t such that dx, t) < 0 in I t, the inside of Γ t, and dx, t) > 0 in O t := Ω \ Γ t I t ), the outside of Γ t. We also define the inside I and the outside O of Γ as follows I := x, t) Ω [0, T ; dx, t) < 0}, O := x, t) Ω [0, T ; dx, t) > 0}. For the numerical solution U ε,h, x, t), we denote its zero level set at time t by Γ ε,h, t, that is, 4.3) Γ ε,h, t := x Ω ; U ε,h, x, t) = 0 }. Before we give a proof of Theorem., we need to recall the following convergence result of [, see Theorem 5. of [, which proved that the Hele-Shaw problem is the distinguished limit, as ε 0, of the Cahn-Hilliard equation. Theorem 4.. Let Ω be a given smooth domain and Γ 00 be a smooth closed hypersurface in Ω. Suppose that the Hele-Shaw problem.4)-.8) starting from Γ 00 has a smooth solution w, Γ := 0 t T Γ t t})) in the time interval [0, T such that