ANALYSIS OF A FULLY DISCRETE FINITE ELEMENT METHOD FOR THE PHASE FIELD MODEL AND APPROXIMATION OF ITS SHARP INTERFACE LIMITS

Transcription

1 ANALYSIS OF A FULLY DISCRETE FINITE ELEMENT METHOD FOR THE PHASE FIELD MODEL AND APPROXIMATION OF ITS SHARP INTERFACE LIMITS XIAOBING FENG AND ANDREAS PROHL Abstract. We propose and analyze a fully discrete finite eleent schee for the phase field odel describing the solidification process in aterials science. The priary goal of this paper is to establish soe useful a priori error estiates for the proposed nuerical ethod, in particular, by focusing on the dependence of the error bounds on the paraeter, known as the easure of the interface thickness. Optial order error bounds are shown for the fully discrete schee under soe reasonable constraints on the esh size h and the tie step size k. In particular, it is shown that all error bounds depend on 1 only in soe lower polynoial order for sall. The cruxes of the analysis are to establish stability estiates for the discrete solutions, to use a spectru estiate result of Chen 15 and to establish a discrete counterpart of it for a linearized phase field operator to handle the nonlinear effect. Finally, as a nontrivial byproduct, the error estiates are used to establish convergence of the solution of the fully discrete schee to solutions of the sharp interface liits of the phase field odel under different scaling in its coefficients. The sharp interface liits include the classical Stefan proble, the generalized Stefan probles with surface tension and surface kinetics, the otion by ean curvature flow, and the Hele-Shaw odel. 1. Introduction In this paper we shall propose and analyze a fully discrete finite eleent tiesplitting ethod for the phase field odel 1.1) 1.) 1.3) 1.) α)ϕ t ϕ + 1 fϕ) = s)u in Ω T := Ω 0, T ), c)u t u = ϕ t in Ω T, u n = ϕ n = 0 in Ω T := Ω 0, T ), ϕ = ϕ 0, u = u 0 in Ω 0, where Ω R N N =, 3) is a bounded doain with the sooth boundary Ω. T > 0 is a fixed constant, and f is the derivative of a sooth double equal well 1991 Matheatics Subject Classification. 65M60, 65M1, 65M15, 35B5, 35K57, 35Q99, 53A10. Key words and phrases. Phase field odel, Allen-Cahn equation, Cahn-Hilliard equation, Stefan proble, surface tension and kinetics, otion by ean curvature, Hele-Shaw odel, phase transition, fully discrete schee, finite eleent ethod. To be subitted to Math. Cop. 1

2 XIAOBING FENG AND ANDREAS PROHL potential taking its global iniu value 0 at ϕ = ±1. A typical exaple of f is 1.5) fϕ) := F ϕ) and F ϕ) = 1 ϕ 1). The existence of bistable states suggests that nonconvex energy is associated with the odel see the discussion below). We like to reark that nonsooth potentials have also been considered in the literature for the phase field odel, for that we refer to 7, 1 and the references therein. We also note that the super-index on the solution u, ϕ ) is suppressed for notation brevity. The phase field odel for solidification was introduced by Caginalp 9, Collins and Levine 1, Fix 9 and Langer 3 to treat phenoena such as crystal growth and the fusion and joining of aterials, which are not captured by the classical Stefan proble. The odel consists of a heat equation 1.) and a Ginzburg- Landau/Allen-Cahn equation 1.1) cf., ). Note that the original phase field odel consists of equations 1.1) and 1.), with α) = O1), s) = O1) and c) = O1). In the odel, u represents the teperature and ϕ is an order paraeter which will vary continuously but soehow describes the phase of the aterial. ϕ is scaled so that ϕ 1 represents the liquid phase and ϕ 1 the solid phase. α,, s and c are respectively the relaxation tie, a icroscopic scale, a surface tension scale, and the specific heat. We ephasize that the paraeter is usually sall copared to the characteristic diensions on the laboratory scale. The two boundary conditions in 1.3), the outward noral derivatives of u and ϕ vanish on Ω, iply no gain or loss of heat energy through the walls of the container Ω. For ore physical background, derivation, and discussion of the phase field odel and related equations, we refer to, 5, 6, 9, 1,, 9, 31, 3, 35, 37, 1 and the references therein. It is known that the phase field odel can be forulated as a gradient flow with the Liapunov energy functional 1.6) J ϕ, u) := φ ϕ, u) dx, where Ω φ ϕ, u) := 1 α) ϕ + 1 c)s) F ϕ) + α) α) u in the Hilbert space H 1 0 L, where H 1 0 denotes the ean-zero subspace of H 1, the dual of the Sobolev space H 1. Note that the energy density φ ϕ, u) is not convex in ϕ. In addition to the reason that the phase field odel for solidification is widely accepted as a good odel for treating phenoena which are not covered by the classical Stefan proble, it has also been used as a coputational) odel to copute a wide range of sharp interface probles, including the classical and generalized Stefan probles, the otion by ean curvature flow and the Hele-Shaw odel by taking advantage of the fact that the solution of the phase field odel exists at all ties and the singularities of the free boundaries do not pose either nuerical or theoretical difficulties. Furtherore, it could provide sufficient inforation for the possible extensions of these free boundary probles beyond any singularities. Indeed, the connection between the phase field odel and the sharp interface probles has been an extensively studied topic in recent years cf., 3, 7, 10, 11, 17, 16,, 39, 0 and the references therein). It was first forally shown by Caginalp 10 that, as

3 FINITE ELEMENT APPROXIMATIONS FOR THE PHASE FIELD MODEL 3 0, the function u tends to a liit u 0, which, together with a free boundary Γ := 0 t T Γ t t), satisfies the following free boundary proble: 1.7) 1.) 1.9) 1.10) 1.11) 1.1) c 0 u 0 t u0 = 0 in Ω T \ Γ, u 0 n = 0 on Ω T, V = 1 u 0 on Γ, n Γ u 0 = d 0 κ Γ α 0 V ) on Γ, u 0 = u 0 0 in Ω 0, Γ 0 = Γ 00 when t = 0, where c 0, α 0, c 0 are non-negative constants independent of, V is the noral velocity of the interface Γ positive when the otion is directed towards the liquid), κ Γ is the su of the principal curvatures of the interface in space), n is the unit outward noral to either Ω or Γ, u0 n Γ := u0 + n u0 n denotes the jup of the noral derivatives of u 0 across Γ. Also ϕ ±1 uniforly in every copact subset of Ω T \ Γ as 0. Later, the rigorous justification of this liit was successfully carried out by Caginalp and Chen 11, using a siilar ethodology as in 3, under the assuption that the above free boundary proble has a unique classical solution. Also, Soner 39 proved the weak convergence of solutions of a phase field odel with ϕ-dependent latent heat to the sharp interface liit in a very general setting that is applicable even when the sharp interface proble does not have a classical solution. We note that when s) = 0 and α) = 1, equation 1.1) decouples fro 1.) and becoes the Allen-Cahn equation ϕ t ϕ + 1 fϕ) = 0. Its connection to the otion by ean curvature flow was established by de Mottoni and Schatzan and to the generalized otion by ean curvature flow by Evans, Soner and Souganidis. When α) = c) = 0 and s) = 1, equations 1.1) and 1.) reduce to the Cahn-Hilliard equation 1 ϕ t + ϕ 1 fϕ)) = 0. The convergence of the Cahn-Hilliard equation to the Hele-Shaw odel 36 was recently carried out by Alikakos, Bates and Chen 3. It is clear that the study of the phase field odel 1.1)-1.) is of great value for understanding solidification processes, in particular, in presence of surface tension and surface kinetics, and for coputing a wide range of sharp interface probles by taking advantage of the fact that the solution of the phase field odel is known to exist for all tie cf. 3). As pointed out earlier, this is particularly attractive fro the coputational point of view. Due to the nonlinearity in the odel, its solution only can be sought nuerically. The priary nuerical challenge for solving the phase field odel results fro the presence of the -dependent coefficients in the equations of the odel. Recall that the phase field odel approxiates the free boundary proble only when becoes very sall. On the other hand, the

4 XIAOBING FENG AND ANDREAS PROHL equations of the odel becoe singularly perturbed heat equations for sall. To resolve the solution nuerically, one has to use sall space) esh size h and tie) step size k, which ust be related to the paraeter. In the past fifteen years, nuerical approxiations of the phase field odel with a fixed have been developed and analyzed by several authors. Caginalp and Lin 1 and Lin 3 also see 30) proposed an explicit finite difference schee and an iplicit Crank-Nicolson schee for the original phase field odel, the convergence and error estiates of the schees were shown under a restriction which is equivalent to α) c). The restriction excludes soe physically interesting cases. Chen and Hoffann 19 proposed a fully discrete finite eleent ethod which uses P 1 conforing finite eleent for space discretization and the backward Euler ethod for tie discretization. An optial order error estiate in L J; L ) was proved for the fully discrete ethod. A siilar fully discrete finite eleent ethod was analyzed later by Yue 3 for the case of nonsooth initial data. Leyk and Roberts 33 presented a Krylov subspace type solution) ethod for solving the algebraic proble resulted fro a finite difference discretization of the phase field odel. Caginalp and Socolovsky 13 proposed a coputational ethod which consists of soothing a sharp interface proble within the scaling of the distinguished liits of the phase field odel that preserve physically iportant paraeters. The coputations fro single-needle dendritic to faceted crystals are carried out continuously by adjusting the paraeters in the ethod. Very recently, Provatas, Goldenfeld and Dantzig 3 proposed an adaptive finite eleent algorith using dynaic data structures, which enables to siulate syste sizes corresponding to experiental conditions. Nuerical siulations were presented for two-diensional, tie-dependent dendritic evolution with and without surface tension anisotropy. We ephasize that the nuerical analyses of all papers cited above were developed for the phase field odel with a fixed. No special effort and attention were given to rigorously address issues such as how the esh sizes h and k depend on and how the error bounds depend on. In fact, since those error estiate results were derived using a Gronwall inequality type arguent at the end of the derivations cf. 19), it is not hard to check that all error bounds contain a factor exp T ), which clearly is not very useful when is sall. Unlike the nuerical works entioned above, the focus of this paper is on approxiating the solution of the phase field odel 1.1)-1.) for sall, which is the case for the applications we are interested in. The two priary goals of the paper are: i) to analyze a fully discrete finite eleent ethod for the initial-boundary value proble 1.1)-1.), and to establish useful error bounds, which show growth only in low polynoial orders of 1, for the proposed schee under soe reasonable constraints on esh sizes h and k; ii) to establish the convergence of the fully discrete finite eleent solution to the solutions of the sharp interface probles which are the distinguished liits of the phase field odel under different scaling in its coefficients. To our knowledge, such error estiates and convergence results for the phase field odel have not been known in the literature. On the other hand, such error bounds are valuable to have for coputing the solution of the phase field odel and the solution of its sharp interface liits. To avoid soe coplicated technicalities without coproising the ain ideas and results, the analysis of this paper will be carried out for the case of the quartic potential given in 1.5), although the subsequent analysis and results apply to a

5 FINITE ELEMENT APPROXIMATIONS FOR THE PHASE FIELD MODEL 5 general class of adissible double equal well potentials which satisfy soe structural assuptions as described in 5, 6, 7. In the rest of this section, we shall suarize the ain results of this paper. As in 19, the first fully discrete finite eleent ethod we propose for the phase field odel is the following schee P A k,h ): find Φ, U ) V h for = 1,,, M such that for all η h, v h ) V h α) d t Φ ), η h + Φ ) 1, η h + fφ ) ), η h = s) U ) 1.13), η h, c) dtu ), vh + U ), v h = dt Φ ) 1.1), v h, with soe starting value Φ 0, U 0 ) V h that approxiate ϕ 0, u 0 ). Here V h H 1 Ω) denotes the finite eleent space of continuous piecewise affine functions, k is the tie step and d t U := 1 k U U 1 ) denotes the backward difference quotient in tie. Clearly, a coupled nonlinear syste has to be solved at each tie step when using schee P A k,h ). Coputationally, this is not efficient, nor is it convenient. A ore practical schee would be one which coputes Φ and U in succession at each tie step; ore iportantly, this would allow one to use existing coputer codes for solving the Allen-Cahn/Ginzburg-Landau equation and for solving the heat equation. To that end, we propose our second fully discrete schee P B k,h ) which seeks Φ, U ) V h for = 1,,, M satisfying α) d t Φ ), η h + Φ ) 1, η h + fφ ) ), η h = s) U 1 ) 1.15), η h, c) dtu ), vh + U ), v h = dt Φ ) 1.16), v h, for all η h, v h ) V h, together with soe starting value Φ 0, U 0 ) V h. Reark 1.1. a). Evidently, schee P B k,h ) allows independent coputations of the iterates Φ and U at each tie step. As expected, the benefit of splitting in schee P B k,h ) is at an expense of stronger esh condition on k copared to the fully iplicit schee P A k,h ). See Rearks 3.1, 3., and 3.. b). Another way to decouple schee P A k,h ) is to evaluate the right-hand side of 1.1) in an explicit anner. We found soe theoretical indication that an additional ter kκ 1 κ d t Φ, κ 1, κ > 0 in 1.13) resp. 1.15)) ight help in those scenarios to stabilize the discrete schee. We assue there exist nonnegative -independent constants σ j for j = 1,.., such that 1.17) ϕ 0 1, in Ω, 1.1) J ϕ 0, u 0 ) C σ1, 1.19) 1.0) ϕ 0 1 fϕ 0) + s)u 0 L C σ, u 0 H l C σ 3+l for l = 0, 1. The subsequent analysis will be carried out for schee P B k,h ); a corresponding one for schee P A k,h ) follows easily using the sae arguents. Our first ain result is suarized in the following theore cf. Theores below).

6 6 XIAOBING FENG AND ANDREAS PROHL Theore 1.1. Let Φ, U ) M =0 solve 1.15)-1.16) on a regular tie esh J k := t M =0 of size Ok) and a quasi-unifor spatial esh T h of size Oh). Suppose 1.17)-1.0) hold and ϕ tt, u tt L J; L ). Also, assue that the free boundary proble 1.7)-1.1) has a unique classical solution. Then, if the esh sizes k and h satisfy soe sallness constraints with respect to > 0 see Theores for explicit forulas), there exists soe constant C = C T, Ω, σ i, ξ i ; α), c), s); C 0 ) > 0 see Theores for the explicit for), which grows in a low polynoial order in 1 as 0, such that there hold the following error estiates: i) ii) iii) iv) v) vi) vii) viii) ix) x) ax 0 M ax 0 M 3 α) k α) k ax 0 M α) ϕt ) Φ L C k + h ), c)α) ut ) U L C k + h ), M M ϕt ) Φ ) L ut ) U ) L 1 1 C k + h ), C k + h ), ϕt ) Φ ) L C k + h ), ax ut ) U ) ) L C k + h, 0 M M α) k d t ϕt ) Φ ) 1 L C k + h ), c) k α) M c)α) d t ut ) U ) L 1 C k + h ), ax ϕt ) Φ L 0 M C h N ln h 3 N + k + h h N ax 0 M ut ) U L C h N ln h 3 N + k + h h N. Reark 1.. We note that the above theore is a suary of the ain results of Theores To see detailed and precise stateents and other related estiates, we refer to Section 3. In order to establish the above error estiates, the following three ingredients play a crucial role: To establish stability estiates for the solution of the fully discrete ethod. To handle the nonlinear) potential ter in the error equation using a generalized) spectru estiate result due to Chen 15 for the linearized phase field operator L P F := 1 f ϕ)i Θ 1.1) I

7 FINITE ELEMENT APPROXIMATIONS FOR THE PHASE FIELD MODEL 7 with respect to the operator N P F := α) 1.) Θ s) where I and Θ denote the identity and zero operators. Here, ϕ is the solution of the phase field odel 1.1)-1.), see Lea. for details. To establish a discrete counterpart of the above spectru estiate, we refer to Lea 3.3 for details. As a nontrivial byproduct, the above l J k ; L ) error estiate cobined with the convergence result on the phase field odel to its sharp interface liiting probles established in 11 iediately allows us to establish the convergence of the fully discrete finite eleent solution to the solution of the free boundary proble 1.7)-1.1). Our second ain result is the following convergence theore. Theore 1.. Let Ω be a given sooth doain and Γ 00 be a sooth closed hypersurface in Ω. Suppose that the free boundary proble 1.7)-1.1) starting fro Γ 00 has a classical solution u 0, Γ := 0 t T Γ t t) ) such that Γ t Ω for all t 0, T. Let ϕ 0, u 0 ) 0< 1 be the faily of sooth uniforly bounded functions as in Theore.1 and. of 11. Let Φ,h,k x, t), U,h,k x, t) ) denote the piecewise linear interpolation in tie) of the fully discrete solution Φ, U ) M =0 of 1.15)-1.16). Also, let I and O stand for the inside and outside in Ω T ) of Γ. Then, under the esh and starting value constraints of Theore 1.1 we have i) U,h,k u 0 C0 Ω T ) ii) iii) 0 0, c), Φ,h,k x, t) 0 1 uniforly on copact subset of O, Φ,h,k x, t) 0 1 uniforly on copact subset of I in each of six cases of different cobinations of c 0, α 0 and d 0 as described in Theores.1 and. of 11 also see Theore.1 below). Our third ain result is the following convergence theore for the nuerical interface. Theore 1.3. Let Γ,h,k t := x Ω ; Φ,h,k x, t) = 0 denote the zero level set of Φ,h,k. Then under the assuptions of Theore 1., there holds sup x Γ,h,k t distx, Γt ) ) 0 0 uniforly on 0, T. We reark that using a siilar approach parallel studies were carried out by the authors in 5 for the Allen-Cahn equation and the related curvature driven flows, and in 6, 7 for the Cahn-Hilliard equation and the Hele-Shaw proble. As pointed out earlier, the forer corresponds to s) = 0 and α) = 1 in the phase field odel and the latter is obtained when α) = c) = 0 and s) = 1. The success of the approach, based on a spectru estiate for the corresponding linearized operators, is due to the fact that it does not rely on the axiu and coparison principles, which are known not to hold for the Cahn-Hilliard equation and the phase field odel. On the other hand, the required spectru estiate does hold in each of the three cases, although the application of the estiate in the

8 XIAOBING FENG AND ANDREAS PROHL cases of the Cahn-Hilliard equation and the phase field odel is rather delicate and coplicated. The paper is organized as follows: In Section, we shall derive soe a priori estiates for the solution of 1.1)-1.). Special attention is given to the dependence of the solution on in various nors. In Section 3, we analyze the fully discrete finite eleent ethod 1.15)-1.16) for the phase field odel 1.1)-1.). The ethod consists of the sei-explicit) backward Euler discretization in tie and the P 1 conforing finite eleent discretization in space. Optial error estiates in energy nor and quasi-optial error estiate in l J k ; L ) nor are obtained for the fully discrete solution. It is shown that all the error bounds in Theore 1.1 see Theores ) depend on 1 only in low polynoial orders for sall. Like in 5, 6, 7, the spectru estiate cf. Lea.) and its discrete counterpart cf. Lea 3.3) play a crucial role in the proofs. Finally, Section is devoted to establishing the convergence of the fully discrete solution to the solution of the free boundary proble 1.7)-1.1) by showing Theores 1. and 1.3. Cobining the l J k ; L ) error estiate and the convergence result of 11, we show that the fully discrete nuerical solution converges to the solution including the free boundary) of the free boundary proble, provided that the latter adits a global in tie) classical solution.. Energy estiates for the differential proble In this section, we derive soe energy estiates in various function spaces up to H 1 J; H Ω) ) in ters of negative powers of for the solution ϕ, u ) to the phase field odel 1.1)-1.) for given ϕ 0, u 0 ) H Ω), where J = 0, T ). Throughout this paper, the standard space, nor and inner product notation are adopted. Their definitions can be found in, 0. In particular,, ) denotes the standard inner product on L Ω), and H k Ω) denotes the Sobolev space of the functions which and their up to kth order derivatives are L -integrable. Also, C and C are used to denote generic positive constants which are independent of and the tie and space esh sizes k and h. In order to trace dependence of the solution on the sall paraeter > 0, we assue that the initial function ϕ 0 and u 0 satisfy the following assuption. General Assuption GA) There exist positive constants σ j for j = 1,..,, such that 1.17)-1.0) hold. We now study dependence of the solution of 1.1)-1.) on the given data of the proble, in particular on > 0. The first lea is a corollary of Theore 3.1 of 11. It shows boundedness of the first coponent ϕ of the solution ϕ, u ) of the phase field odel, provided that the liiting free boundary proble 1.7)-1.1) has a global in tie) classical solution. This boundedness result enables us to establish iproved a priori estiates for the solution of the phase field odel. We reark that weaker estiates can be shown without using this boundedness result, hence, without assuing existence of a global in tie) classical solution for the phase field odel. Lea.1. Let 1.17) hold. Suppose that the free boundary proble 1.7)-1.1) has a unique global in tie) classical solution. Then there exists a faily of sooth initial data functions ϕ 0, u 0 ) 0< 1 and constants 0 0, 1 and C 0 > 0, such that for all 0, 0 ) the solution ϕ, u ) of the phase field odel 1.1)-1.) with

9 FINITE ELEMENT APPROXIMATIONS FOR THE PHASE FIELD MODEL 9 the above initial data u 0 satisfies.1) ϕ L Ω T ) 3 C 0. Proof. Using a atched asyptotic expansion technique, it was shown in 11 that there exists a faily of sooth approxiate solutions ϕ A, u A ) to the solution ϕ, u ) of 1.1)-1.) satisfying the assuptions of Theore 3.1 of 11 was constructed in Section of 11. One condition is ϕ A L Ω T ) C 0 for soe C 0 > 0. It was then proved in Theore 3.1 of 11 that ϕ A, u A ) is very close to u, w ) in L p Ω T ) for soe p > see 3.3) on page 7 of 11). Now,.1) follows fro a regularization arguent. The arguent goes as follows in three steps: i) odified f into f such that f = f in 3 C 0, 3 C 0) and f is linear for ϕ > C 0 ; ii) it is not hard to show that the solution ϕ, u) of the phase field odel with the new nonlinearity f satisfies the estiate.1) when 0, 0 ) for soe sall 0 0, 1; iii) it follows fro the uniqueness of the solution of the phase field odel that u u. Lea.. Suppose that ϕ 0, u 0 ) satisfies GA), and fϕ) = ϕ3 ϕ. Then, the solution of 1.1)-1.) satisfies the following estiates: i) ess sup 0, ϕ L + 1 F ϕ) L 1 + s)c) u L + α) ϕ t s) L + s) 0 us) L ds α) J ϕ 0, u 0), c) ii) ess sup c) u L + u t s) L 0, + us) L ds iii) iv) v) vi) vii) viii) ix) 0 0 J ϕ 0, u 0) + C 1 + c) σ3, α) ϕs) L ds + s)α) J ϕ 0, u 0 ), ess sup α) ϕ t L 0, ess sup ϕ L B, 0, 0 + ϕ tt s) H 1 ds B 3, ess sup c) u t L 0, ess sup u L B 5, 0, 0 u tt H 1 ds B 6, ϕ t s) L ds B 1, u t L ds B, where B i B i ; c), s), α)) for i = 1,.., are defined as follows: B 1 := ) + s) c) J ϕ 0, u 0 ) + C σ3 + C α) σ+1), B := C 1 + α)s) c) + s) c) + α) in σ, σ3,

10 10 XIAOBING FENG AND ANDREAS PROHL B 3 := J 3 ϕ 0, u 0) + B 1 α) + s) α) c) J ϕ 0, u 0) + C 1 + c) ) σ3, B := B 3 + J ϕ 0, u 0) c) + C1 + c) c) σ3 + 1 c) B 5 := c)b + B 1 α), B 6 := 1 c) B + B 3. In addition, if.) li ϕ t s) L C ξ1, s 0 + for soe ξ 1, ξ 0, then there also hold x) xi) where B 7 := B := 0 c) ϕ tt s) L ds + ess sup 0 0, σ + σ+1) α), li u t s) L C ξ, s 0 + ϕ t L + α) u tt s) L ds + ess sup u t L 0, c) u t s) L ds B, C α)j α) ϕ 0, 5 N u 0 ) Cs) J ϕ 0, u 0 ) + α)c) 1 c) B 7 + ξ. 0 B N 1 0 ϕ t s) L ds B 7, J ϕ 0, u ) 0 + C 1 + c) σ 3 + ξ1 α), Proof. i) The assertion is the iediate consequence of the basic energy law associated with the phase field odel d dt J ut)) = φ t L s) α) u L, where J, ) is defined by 1.6). ii) This assertion follows directly fro testing equation 1.) by u and using assertion i). iii) Multiplying 1.1) by ϕ we get α) d dt ϕ L + ϕ L + 1.3) fϕ), ϕ) = s) u, ϕ ). The assertion then follows fro cobining the above inequality with the following estiates 1 ) fϕ), ϕ = 1 f ϕ), ϕ ) 1 ϕ L, s) u, ϕ ) = s) u, ϕ ) 1 ϕ L + s) u L iv) Differentiating 1.1) with respect to tie gives.) α) ϕ tt ϕ t + 1 f ϕ)ϕ t = s) u t.

11 FINITE ELEMENT APPROXIMATIONS FOR THE PHASE FIELD MODEL 11 Now we test.) with ϕ t to get Observing we have α) α) d dt ϕ t L + ϕ t L + 1 f ϕ), ϕ t ) = s) ) u t, ϕ t. 1 f ϕ), ϕ t ) 1 ϕ t L s) ) u t, ϕ t 1 ϕ t L + s) u t L d dt ϕ t L + ϕ t L ϕ t L + s) u t L. Integrating over 0, T in t gives the assertion. v) We proceed as in iii),.3) is replaced by ϕ L + 1 ϕ L α) ϕ t L + s) u L + 1 ϕ L. Then the assertion follows cobining the above estiate with estiates i), ii) and iv). vi) We start fro.) to obtain ϕ tt H 1 1 α) ϕ t L + 1 α) sup ψ H 1 f ϕ)ϕ t, ψ ) ψ H 1 1 α) ϕ t L + s) α) u t H α) + s) u t, ψ) sup α) ψ H ψ 1 H 1 f ϕ)ϕ t, ψ ) sup ψ H 1 ψ H 1 Since H 1 Ω) L 6 Ω) for N 3, the last ter above is bounded fro above by C α) f ϕ) L 3 ϕ t L C ) c ϕ L α) + c 3 ϕ t L. Hence, the assertion follows fro i) and iv). vii) Differentiating 1.) with respect to tie gives..5) c) u tt u t = ϕ tt. Testing.5) with u t, and integrating the resulting equation with respect to tie, the assertion follows fro vi) and ii). viii) The assertion follows directly fro equation 1.), and assertions vii) and iv). ix) Fro.5), we have u tt H 1 1 ) u t L + ϕ tt c) H 1. The assertion follows fro cobining the above inequality and the estiates vi) and vii).

12 1 XIAOBING FENG AND ANDREAS PROHL x) We test.) with ϕ tt and use an interpolation result which bounds the L Ω)-nor by W 1, Ω) and W, Ω) nors to get α) ϕ tt L + d dt ϕ t L 1 3 α) f ϕ) L ϕ t L + s) α) u t L C 3 α) ) ϕ 5 N L ϕ N 1 L + ϕ L ϕ t L + s) α) u t L. Integrating in t fro 0 to T gives the first part of the assertion. The second part follows fro ultiplying.) with ϕ t and using the first part estiate and estiates ii) and iv). xi) After differentiating 1.) with respect to t, we test the resulting equation with u tt to get c) u tt L + 1 d dt u t L 1 c) ϕ tt L. Then the first part of the assertion follows fro x) and the second part follows fro differentiating 1.) in t and testing the the resulting equation with u t. Lea.3. Under the assuptions of Lea.1 and., the estiates vi)-xi) of Lea. are iproved to new estiates which have the sae fors as previous ones while each B j is replaced by B j for j = 3,,,, respectively. The nubers B j j=3 are given by B 1 B 3 := α) + s) α) c) J ϕ 0, u 0 ) + C 1 + c) σ3 + 1 α) J ϕ 0, u 0), B := B 3 + J ϕ 0, u 0 ) c) + C1 + c) c) σ3 + 1 σ + σ+1) c) α), B 5 := c) B + B 1 α), B 6 := 1 B c) + B 3, B 7 := C J ϕ 0, u 0 ) s) + α) α)c) J ϕ 0, u 0) + C 1 + c) σ3, B := 1 c) B 7 + ξ. Proof. The proofs of the assertions are in the sae line as those of vi)-xi) of Lea.; the only difference is that now f ϕ) is uniforly bounded in for 0 < 0 since ϕ is uniforly bounded in. Hence, there is no need to use Sobolev ebedding to control f ϕ). We conclude this section by citing the following result of 15, 11 on a low bound estiate of the generalized) spectru of the linearized phase field operator L CH

13 FINITE ELEMENT APPROXIMATIONS FOR THE PHASE FIELD MODEL 13 in 1.1) with respect to the operator N CH in 1.). The estiate plays a crucial role in our error analysis. Lea.. Suppose that GA) holds. Let λ P F denote the sallest generalized) eigenvalue of the eigenvalue proble ) ) ψ ψ L CH = λ N w CH. w Then there exists a positive constant C 0 such that λ P F.6) Moreover, λ P F C 0. satisfies i) if α) c 0, s) 0, c) 0 for soe constant c 0 > 0, then.7) ψ L + 1 f ϕ)ψ, ψ) inf ψ H 1 Ω) α) ψ C 0 ; L ii) if α) 0, s) c 0, s) c) c 0 for soe constant c 0 > 0, then.) λ P F = ψ L + 1 f ϕ)ψ, ψ) + s) inf c) w ψ L ψ H 1 Ω) α) ψ w H L + s) w C 0, L Ω) where ϕ denotes the first coponent of the solution vector ϕ, u ) to the phase field odel 1.1)-1.). Reark.1. It is easy to see that.) also holds under the assuptions of i), and.7) is a stronger estiate than.) in this case. If c) = 0, it is assued that ψ = w, and the ter s) c) w ψ L is reoved in.). The above results were established in 11, 15 for any ϕ which satisfies soe special profile cf. page 7 of 11). It was shown that the first coponent ϕ of the solution vector ϕ, u ) to the phase field odel 1.1)-1.) indeed satisfies the required profile. 3. Error analysis for a fully discrete finite eleent approxiation In this section, we shall establish soe stability and convergence properties for schee P B k,h ). We reark that a corresponding analysis for schee PA k,h ) follows fro the sae arguents. Let T h be a quasi-unifor triangulation of Ω such that Ω = K T h K K T h are tetrahedrons in the case N = 3). Here h := ax K Th h K denotes the esh size of T h, see, 0 for further details. Let V h be the finite eleent subspace of H 1 Ω) associated with T h and consisting of continuous and piecewise linear functions on T h, that is, V h := v h CΩ) : v h K P 1 K), K T h. For the error analysis, we need to introduce the elliptic projection operator P h : H 1 Ω) V h, 3.1) 3.) ψ P h ψ, v h ) = 0 v h V h, ψ P h ψ, 1) = 0.

14 1 XIAOBING FENG AND ANDREAS PROHL It is well-known that P h has the following approxiation properties, 0, : 3.3) 3.) 3.5) 3.6) ψ P h ψ L + h ψ P h ψ) L Ch ψ H ψ H Ω), ψ P h ψ L Ch N ln h 3 N ψ H ψ H Ω), ψ P h ψ) t L J;L ) Ch ψ t L J;H ) ψ H 1 J; H ), ψ P h ψ) t L J;H 1 ) Ch ψ ψ W, where W = w ; w H 1 J; H 1 ), w <, w = w H 1 J;H 1 ) + N i,j=1 xj xi w t L J;H 1 ) We note that a short proof of 3.) can be found in 5. To be used in later analysis, we also define the discrete negative) Laplace operator h : V h H 1 Ω) V h by ) h ψ, η h ) = ψ, η h ) η h V h H 1 Ω). We now state a basic stability result for the discrete schee P B k,h ) Lea 3.1. Let fϕ) = ϕ 3 ϕ, and c) c 0 > 0. The solution Φ, U ) M =0 of 1.15)-1.16) satisfies for values k < inα) c), s), ax Φ L M F Φ ) L 1 + s)c) U L M +k =0 k α) ) dt Φ L + s) U L + k d tφ L + k d t Φ 1 ) L + ks)c) s) k d t U L α)j ϕ 0, u 0 ). Proof. We first rewrite fφ ) as follows fφ ) = 1 Φ 1 ) Φ + Φ 1 + kd t Φ ). Multiplying the equation by d t Φ gives 3.) 1 fφ ), d t Φ ) = 1 Φ 1, d t Φ 1) ) + k Φ 1, d t Φ ) 1 Φ 1 ± Φ 1 1), d t Φ 1) ) k d tφ L 1 d t Φ 1 L + k d t Φ 1) L k d tφ L.

15 FINITE ELEMENT APPROXIMATIONS FOR THE PHASE FIELD MODEL 15 We now test 1.15) with d t Φ and 1.16) with s)u. Adding the resulting equations leads to α) d t Φ L + d t Φ L + k d tφ L 3.9) + 1 d t Φ 1 L + k d t Φ 1) L + c)s) d t U L + kc)s) d t U L + s) U L k d tφ L s)k d tu, d t Φ ) k d tφ L + k 3 s) d t U L. The assertion follows fro suing 3.9) over fro 0 to M. Reark 3.1. The esh condition guarantees that all coefficients on the right-hand side of the stability estiate are positive. In case of schee 1.13)-1.1) the constraint weakens to k α). Lea 3.. Let fϕ) = ϕ 3 ϕ. Under the esh constraint of Lea 3.1, the solution U M =0 of 1.15)-1.16) also satisfies the following stability estiate: M α)k ax α) d t Φ L + s) U 1 M L + k d t Φ L + d t Φ L + c)s) d tu L + s)k d tu L α) 1 + s) 3 C J ϕ 0, u 0) + s) σ3. Proof. First, take η h = d t Φ after applying the difference operator d t to 1.15) and v h = s)d t U in 1.16), then add the resulting equations. The assertion iediately follows fro taking suation over, using Lea 3.1 and the following inequalities dt fφ ), d t Φ ) = f ξ), d t Φ ) d t Φ L, s)k d t U, d t Φ ) α)k d t U L + s) k α) d tφ L. In order to establish error bounds that depend on low order polynoials of 1, we need discrete versions of the spectral estiates of Lea.. To that end, we define 3.10) C 1 = ax f ξ), ξ C 0 and C is the sallest positive -independent constant such that 3.11) ess sup ϕ P h ϕ L C h N ln h 3 N ess sup ϕ H J J C h N ln h 3 N B 1.

16 16 XIAOBING FENG AND ANDREAS PROHL Lea 3.3. Suppose that the assuptions of Lea.1-. hold, and C 0 and 0 are sae as there. Then for 0, 0 there hold the following estiates: i) if α) c 0, s) 0, c) 0 for soe constant c 0 > 0, then 3.1) λ h P F ψ L + 1 inf f P h ϕ)ψ, ψ) 0 ψ H 1 Ω) α) ψ C 0 ; L ii) if α) 0, s) c 0, s) c) c 0 for soe constant c 0 > 0, then λ h ψ L P F inf ψ H 1 Ω) α) ψ w H L + s) w L Ω) 1 + f P h ϕ)ψ, ψ) + s) c) w ψ L 3.13) α) ψ L + s) w C 0, L provided that k and h satisfy 3.1) h N ln h 3 N C 1 C B 1 ) 1 C0 α). In the above, ϕ denotes the first coponent of the solution vector ϕ, u ) to the phase field odel 1.1)-1.). Proof. Fro the definition of C 1 and C, we iediately have ess sup P h ϕ L ess sup ϕ L + ϕ P h ϕ L J J provided that h satisfies 3.1). By Mean Value Theore, 3.15) 3 ess sup ϕ L C 0. J ess sup f P h ϕ ) f ϕ ) L sup f ξ) ess sup ϕ P h ϕ L J ξ C 0 J C 1 C h N ln h 3 N B 1 C 0 α). Using the inequality a b a b and 3.15), we get 3.16) f P h ϕ) f ϕ) f P h ϕ) f ϕ) f ϕ) C 0 α). Substituting 3.16) into the definition of λ h P F, in case i) we get λ h ψ L + 1 P F inf f ϕ)ψ, ψ ) 0 ψ H 1 Ω) α) ψ C 0 L C 0. In case ii) we have ψ λ h L + 1 P F inf f ϕ)ψ, ψ ) + s) c) w ψ L ψ H 1 Ω) α) ψ w H L + s) w L Ω) C 0 α) ψ L α) ψ L + s) w L C 0.

17 FINITE ELEMENT APPROXIMATIONS FOR THE PHASE FIELD MODEL 17 The proof is copleted. We now are ready to state the first ain theore, which establishes error estiates for the schee 1.15)-1.16) in the case i) as defined in Lea.. Theore 3.1. Let Φ, U ) M =0 solve 1.15)-1.16) on a quasi-unifor tie esh J k := t M =0 of size Ok) and a quasi-unifor space esh T h of size Oh). Suppose GA) holds and the free boundary proble 1.7)-1.1) has a unique classical solution. Also, assue that α) c 0, s) 0, c) 0 for soe constant c 0 > 0. Then under the following esh and starting value constraints 1). k in1, α) in, c) α) s), N+1 ). h N ln h 3 N C 0 α) C 1 )C B 1 ) 1 3). µ)k + π)h 1 α) N 1 ). Φ 0 ϕ 0 L C h ϕ 0 H, 5). U 0 u 0 L Ch u 0 H, ax 0 M N α) B 1 + B N J ϕ α) 0, u 0) N B N N, N for N =, 3, the solution of 1.15)-1.16) satisfies the error estiates i) α) ϕt ) Φ L Cρ 1 ) 1 h + π) 1 h + µ) 1 k, ii) iii) iv) v) c)s) k ax 0 M 3 α)k M =0 1 ut ) U L C ρ ) 1 h + π) 1 h + µ) 1 k, k ut ) U ) j=0 M L C ρ 3 ) 1 1 h + π) h + µ) 1 k, ϕt ) Φ ) 1 L C ρ ) 1 1 h + π) h + µ) 1 k, α) ax ϕt ) Φ L 0 M C where ρ i ), µ) and π) are defined by 3.17) ρ 1 ) 1 h N ln h 3 N + π) 1 h + µ) 1 k h N ρ 1 ) = α) B, ρ ) = c)s) J ϕ 0, u 0 ) c) σ, ρ 3 ) = J ϕ 0, u 0 ) c) σ, α) ρ ) = 3 α) + s)c) J ϕ 0, u 0), µ) = α)1 + s)1 + c) + B 3 c) +s)c) 1 + c) B6 + s) k J ϕ 0, u 0) c) σ3 α) c),,

18 1 XIAOBING FENG AND ANDREAS PROHL 3.1) s) 3 J ϕ 0, u 0) + s) σ3+1 ; c)s) 1 + s) π) = + s) α) α) c) + s)α) J ϕ 0, u 0) +s)c) J ϕ 0, u 0 ) c) σ3 + α) 1 + B 3 + s)c) σ + s) c) σ + B B + B N 1 B N h N. α) Proof. The proof is divided into four steps. The first step estiates the consistency error of the schee; the second and third steps deal with the error due to the nonlinear ter fϕ) and how to bound it in ters of soe low order polynoial in 1, with the help of the stability estiates of Lea 3.1 and 3. and the spectral estiates of Lea 3.3; the final step eploys an inductive arguent to handle the super-quadratic cubic) ter in fa) fb), a b). Step 1: We decopose the global errors Eϕ := ϕt ) Φ and Eu := ut ) U into 3.19) E ϕ := Θ ϕ + Υ ϕ, E u := Θ u + Υ u, where Θ ϕ := ϕt ) P h ϕt ), Υ ϕ = P h ϕt ) Φ, Θ u := ut ) P h ut ), Υ u = P h ut ) U. Then the error equations are given by α) d t Υ ) ϕ, η h + Υ ϕ, η h ) + 1 fph ϕt )) fφ ) ), η h 3.0) = α) d t Θ ϕ, η h ) + s) Θu 1 + Υ u, η h ) +ks)d t ut ), η h ) s)kd t Υ u, η h ) 3.1) 1 ) fϕt )) fp h ϕt )), η h + α)r ϕ, η h ). c) d t Υ u, v ) ) h + Υ u, v h = d t Υ ϕ + d tθ ϕ, v h) c) dt Θ u, v h) + c)r u + R ϕ, v h), for all η h, v h ) V h. Where 3.) R u = 1 k t+1 t s t )u tt s) ds, R ϕ = 1 k Using Schwartz inequality, we have for r = 1, 0 3.3) k R ϕ H 1 r k =0 =0 t+1 t s t )ϕ tt s) ds. t +1 s t ) t +1 ds ϕ tt s) H ds r t t C k ϕ tt L J,H r ) C k B3, if r = 1, C k B7, if r = 0.

19 FINITE ELEMENT APPROXIMATIONS FOR THE PHASE FIELD MODEL 19 Siilarly, 3.) k =0 R u H r C k u tt L J,H r ) C k B6, if r = 1, Ck c) B, if r = 0. Where B j for j = 3, 6, 7, are defined in Lea.3. Step : In the sequel, it turns out that the ost crucial ter to handle is s)υ u, η h ) in 3.0). Dealing with it requires a preparatory step: first, replacing the super-index in 3.1) by j, then suing the resulting equation over 1 j yields c) Υ u, v h ) + G u, v h ) = Θ ϕ + Υ ϕ, v h ) c) Θ u, v h ) + c) E 0 u, v h ) 3.5) +Eϕ, 0 v h ) + k ) c)r j u + R j ϕ, vh, j=1 where G 0 u = 0, and G u = k Υ j u. j=1 Now taking v h = s)υ u in 3.5) and η h = Υ ϕ in 3.0), and taking suation over fro 1 to l M) after adding the resulting equations we obtain 3.6) α) Υ l ϕ L + s) α)k Gl u L + k d t Υ ϕ L + s)k Υ u L + Υ ϕ L + c)s) Υ u L + 1 fph ϕt )) fφ ), Υ ) ϕ = k α) R ϕ, Υ ) ϕ + s)k ) c)r j u R j ϕ, Υ u j=1 α) d t Θ ϕ, ) Υ ϕ + s) Θ 1 u, Υ ) ϕ s) Θ ϕ, Υ ) u s)c) Θ u, ) Υ u + c)s) E 0 u, Υ ) u + s) E 0 ϕ, Υ ) u s)k d t ut ), Υ ) ϕ s)k dt Υ u, Υ ) ϕ k fϕt )) fp h ϕt )), Υ ) ϕ. Here we have used the fact that d t G u = Υ u in the first ter on the second line. We now bound ters on the right-hand side of 3.6) as follows. First, using the following suation by parts forula d t f, g ) = 1 f l, g l ) f 0, g 0 ) k f 1, d t g ),

20 0 XIAOBING FENG AND ANDREAS PROHL we have k s)k ) c)r j u R j ϕ, Υ u = s)k c)r j u R j ) ϕ, dt G u j=1 j=1 ) = s)k c)r j u R j ϕ, G l u j=1 c)r u R ϕ where we have used the fact that G 0 u = 0. Hence, k 3.7) s)k s)k c)r j u R j ) ϕ, Υ u j=1 c) R u H 1 + R ϕ H 1 G l u H 1 + G 1 s) Gl u L + s)k +s)c) 1 + c) k G u L + s)c) k ), G 1 u, u H 1 Υ u L R u s) 1 + c) H 1 + k c) Second, α) R ϕ, Υ ) ϕ α) R 3.) ϕ H 1 Υ ϕ H 1 α) R ϕ H. 1 Υ ϕ L + 3 α) Υ ϕ L + C α)1 + R ϕ H 1 Third, s)k d t ut ), Υ ) ϕ + s)k dt Υ u, Υ ) 3.9) ϕ s)k dtut) L + dtυ u L Υ ϕ L s) k d t ut ) L α) + d tp h ut ) L + d tu L + α) Υ ϕ L. Fourth, 3.30) k fϕt )) fp h ϕt )), Υ ϕ ) k = f ξ)θ ϕ, ) Υ ϕ α) Υ ϕ L + C 3 α) Θ ϕ L, where we have used the fact that ϕ and P h ϕ are bounded cf. Lea.1 and the first line of the proof of Lea 3.3).

21 FINITE ELEMENT APPROXIMATIONS FOR THE PHASE FIELD MODEL 1 Finally, the reaining ters on the right-hand side of 3.6), denoted by S 1, can be bounded together by 3.31) S 1 α) Υ ϕ L + 3 α) +C Υ ϕ L + c)s) Υ u L α)1 + d t Θ ϕ s) H 1 + α) Θ 1 u L +s)c) Θ u L + s E0 u L + Θ c) ϕ L + E0 ϕ L. Substituting 3.)-3.31) into 3.6), and using Leas.,.3, 3.1, 3. and estiates 3.3)-3.6) we get α) Υ l ϕ L + s) α)k Gl u L + k d t Υ ϕ L + s)k Υ u L + 1 α) Υ ϕ L + c)s) Υ u L + 1 fph ϕt )) fφ ), Υ ) 3.3) ϕ k k α) Υ l ϕ L + s) G u L + C s)c) 1 + c) R u H 1 + α)1 + s)1 + c) + ) R ϕ c) H 1 + C s) k 3 d t ut ) L α) + d tp h ut ) L + d tu L +C k 1 3 α) Θ ϕ L + α)1 + d t Θ ϕ H 1 + s) α) Θ 1 u L + s)c) Θ u L + E0 u L + s) Θ c) ϕ L + E0 ϕ L α) Υ l ϕ L + s) G u L + C µ)k + π 1 )h, where µ) is defined by 3.17) and 3.33) 1 + s) π 1 ) = 3 + s) α) α) c) + s)α) J ϕ 0, u 0) +s)c) J ϕ 0, u 0 ) σ3 + α) 1 + B 3 + s)c) σ + s) c) σ, where B 3 and B 6 are defined in Lea.3.

22 XIAOBING FENG AND ANDREAS PROHL Step 3: It reains to bound the last ter on the left-hand side of 3.3). This will be done using the discrete spectru estiate 3.1). First, using the identity we get 3.3) fa) fb) = a b) f a) + a b) 3a b)a, a, b R, fph ϕt )) fφ ), Υ ) ϕ Substituting 3.3) into 3.3) yields 3.35) f P h ϕt )), Υ ϕ ) ) + Υ ϕ L C Υ ϕ 3 L 3. α) Υ l ϕ L + s) α)k Gl u L + k d t Υ ϕ L + s)k Υ u L + c)s) Υ u L + α)3 Υ ϕ L + 1 Υ ϕ L +1 α) Υ ϕ L + 1 f P h ϕt )), Υ ϕ )) C µ)k + π 1 )h + k +s)k G u L + α)k µ)k + π 1 )h + C k + Ck Υ ϕ 3 L. 3 α) Υ l ϕ L + Ck Υ ϕ 3 L 3 f P h ϕt )), Υ ϕ ) ) α) Υ l ϕ L + s)k G u L Here we have used the fact that P h ϕt ) L C to get the last inequality. Then, using the discrete spectru estiate 3.1), the last ter on the left-hand side of 3.35) is bounded fro below as follows 1 α) Υ ϕ L ) f P h ϕt )), Υ ϕ ) ) C 0 α) Υ ϕ L, which can be absorbed into the second ter on the right-hand side of 3.35). Finally, we need to bound the last ter on the right-hand side of 3.35). This will be done using a spatial-teporal decoposition technique cf. 5, 7). We ake a shift in the super-index and use the triangle inequality to get 3.37) Eϕ 3 L 3 K T h k 3 d t E ϕ 3 L 3 K) + E 1 ϕ 3 L 3 K) For each ter of the second su on the right-hand side of 3.37), we interpolate L 3 K) between L K) and H K), ) Eϕ 1 3 L 3 K) C Eϕ 1 N L K) E 1 ϕ 1 N L K) + E 1 ϕ 3 L K) 3.3) C E 1 ϕ C E 1 ϕ 1 N L K) 1 N L K) B N. E 1 ϕ. N L K) + E 1 ϕ N L K) )

23 FINITE ELEMENT APPROXIMATIONS FOR THE PHASE FIELD MODEL 3 The last step follows fro v) of Lea. and Lea 3.1. Siilarly, we can bound the first su on the right-hand side of 3.37) as k 3 d t Eϕ 3 L 3 K) Ck 3 d t Eϕ N L K) d teϕ 1 N 3.39) L K) + d teϕ 3 L K) ) Ck 3 d t Eϕ 1 N L K) d t Eϕ N L K) + d teϕ N L K) C B + B 1 α) N k 1 N d t Eϕ 1 N L K) Suing 3.3) and 3.39) over all K T h and using the convexity of the function gs) = s r for r > 1 and s 0 then leads to ) 3.0) E ϕ 3 L 3 C E 1 ϕ 1 N L B N + C B + B 1 α) N k 1 N d t Eϕ 1 N L. Because of 3.19) and Lea., the above estiate leads to Υ ϕ 3 L C Υ 1 3 ϕ + Θϕ 1 + Θ ϕ 3 L 3 C 1 N L B N + B + B 1 α) 1 N L B N + B + B 1 α) Θϕ 1 1 N L B N +C B + B 1 α) +C B + B 1 α) N k 1 N N k 1 N d t Υ ϕ 1 N L d t Θ ϕ 1 N L 1 N L B N + Θ ϕ 3 L + 3 Υ 1 ϕ N k Υ ϕ L + ) N Υ 1 ϕ L d t Υ ϕ L N Θ ϕ 1 N L + Θ 1 ϕ 1 N L Suing over fro 1 to l and using 3.3)-3.6), Leas. and 3.1 we get 3.1) k Υ ϕ 3 L C B B 3 + B N 1 B N h 1 N α) +C B N k Υ 1 ϕ 1 N L + C B + B N 1 α)j ϕ α) 0, u 0) N k 3 d t Υ ϕ L. The last ter of 3.1) can be absorbed by the corresponding ter on the left-hand side of 3.35) if k satisfies 3.) C B + B N 1 α)j ϕ α) 0, u 0) N k α).

24 XIAOBING FENG AND ANDREAS PROHL Substituting 3.36) and 3.1) into 3.35) leads to the following inequality 3.3) α) Υ l ϕ L + s) α)k Gl u L + k + s)k Υ u L + c)s) C + C 0 ) k +C B N k d t Υ ϕ L Υ u L + 3 α) Υ ϕ L + 1 Υ ϕ L G u L α) Υ l ϕ L + s) Υ 1 ϕ 1 N L + C µ)k + π)h, where µ) and π) is defined by 3.1) and 3.17), respectively. We note that the super-quadratic power in the last ter allows to control this error contribution by the subsequent inductive arguent. Step : We now finish the proof by the following inductive arguent. Suppose there exist two positive constants c 1 = c 1 T, Ω, σ i ), c = c T, Ω, σ i ; C 0 ), which are independent of, h and k such that there holds inequality α) ax Υ l ϕ 0 l L + s) α)k Gl u L + k d t Υ ϕ L + s)k Υ u L + c)s) Υ u L + α)3 Υ ϕ L ) Υ ϕ L c 1 µ)k + π)h expc t l ) for sufficiently sall h and k satisfying esh conditions 1)-3) stated in the theore. Because of 3.3), we can choose c 1 =, c = C + C 0. Since the exponent in the first ter on the last line of 3.3) is bigger than, hence, we can recover 3.) at the l + 1)th step by applying the discrete Grownwall s inequality, provided that k satisfies B N µ)k + π)h 1 N N 1 α) which iplies that 3.5) µ)k + π)h 1 α) N 1 c 1 µ)k + π)h expc t l+1 ), N B N N. Finally, the assertions i)-ii) follows fro 3.), 3.3), 3.5), and applying the triangle inequality to E u = Θ u + Υ u and E ϕ = Θ ϕ + Υ ϕ. The assertion iv) follows fro applying the inverse inequality bounding L nor in ters of L nor, using 3.) and 3.).

25 FINITE ELEMENT APPROXIMATIONS FOR THE PHASE FIELD MODEL 5 Reark 3.. a). The estiates in i)-iv) are optial in both h and k, and the one in v) is quasi-optial. b). The proof clearly shows how the three esh conditions arise. The condition 1) is for the stability of the fully discrete schee, the condition ) is required for having the discrete spectru estiate see Lea 3.3), finally the condition 3) is caused by the super-quadratic nonlinearity of f see Step 3 of the proof). c). In case of schee 1.13)-1.1), the esh condition 1) weakens to k in1, α) in, N+1 N α) B 1 + B N α) N J ϕ 0, u 0 ) d). The estiates of this theore are established under the iniu assuptions that ϕ tt L J; H 1 ) and u tt L J; H 1 ). If ϕ tt L J; L ) and u tt L J; L ) are assued, analysis can be siplified a little bit in 3.7) and 3.31), and error bounds on d t ut ) U ), d t ϕt ) Φ ) and ut ) U ) can also be obtained see Theore 3. below). e). Superconvergence in h) is seen for P h ϕt ) Φ ) in the L -nor. f). Regarding the choices of the starting values Φ 0 and U 0, clearly, both L - projections Φ 0 = Q h ϕ 0 and U 0 = Q h u 0, and both elliptic projections Φ0 = P h ϕ 0, U 0 = P h u 0 are valid choices, on the other hand, the L -projections have the advantage of being cheaper to be coputed. Theore 3.. Under the assuptions and esh constraints of Theore 3.1, there exists h 0 > 0 and k 0 > 0 or there exists 1 > 0) such that the following error estiates hold for h < h 0 and k < k 0 or for < 1 ). i) ii) iii) iv) ax 0 M Φ L C 0, ax 0 M α) k c)α) ut ) U L Cρ 5 ) 1 h + ˆπ) 1 h + ˆµ) 1 k, M ut ) U ) L 1 c)s) ax ut ) U L 0 M C C ρ 6 ) 1 1 h + ˆπ) h + ˆµ) 1 k, ρ 5 ) 1 h N ln h 3 N + ˆπ) 1 h + ˆµ) 1 k h N, where 3.6) 3.7) ρ 5 ) = c)α) B 5, ρ 6 ) = α) J ϕ 0, u 0 ) c) σ3, ˆπ) = B 6 + ˆµ) = 1 + c) B 6 + π) α) + s)π) + ρ ) c) α) µ) α) + s) c) α) + π) + ρ 1) c)α), µ) + J ϕ 0, u 0 ) c) σ3 µ) c) + c)α).

26 6 XIAOBING FENG AND ANDREAS PROHL In addition, if ϕ tt L J; L ), then we also have v) vi) vii) where 3.) 3.9) ax 0 M α) k k M ϕt ) Φ ) L Cρ 7 ) 1 h + ˆπ) 1 h + ˆµ) 1 k, M d t ϕt ) Φ ) L 1 k d t ϕt ) Φ ) L 1 ρ 7 ) = B, ˆπ) = α) ρ ) = α) B7 B 7 + s) ρ ) + π) c)α) C ρ ) 1 h + ˆπ) 1 h + ˆµ) 1 k, C ρ 9 ) 1 h + ˆπ) 1 h + ˆµ) 1 k,, ρ 9 ) = α)k B 7, + ρ 1) + π) 3 α) ˆµ) = α) B 7 + J ϕ 0, u 0 ) c) σ3 c) + µ) 3 α) + s) µ) c)α). Furtherore, if u tt L J; L ), then there also hold viii) ax ut ) U ) L C ρ 10 ) 1 h + ˆπ) 1 h + ˆµ) 1 k, 0 M M ix) c) k d t ut ) U ) 1 L C ρ 11 ) 1 h + ˆπ) 1 h + ˆµ) 1 k, x) where k M k d t ut ) U ) 1 L C ρ 1 ) 1 h + ˆπ) 1 h + ˆµ) 1 k, ρ 10 ) = B 5, ρ 11 ) = c) B 1 + c), ρ 1) = k B 1 + c). Proof. The estiate iv) of Theore 3.1 iplies that there exist h 0 > 0 and k 0 > 0 equivalently, there exists 1 > 0), such that 3.50) ax ϕt ) Φ L C 0 0 M for h < h 0 and k < k 0 or < 1 ). Where C 0 > 0 is defined in Lea.1. The assertion i) then follows iediately fro.1) and 3.50). To show the assertions ii) and iii), taking η h = Υ u and v h = α)υ u in 3.0) and 3.1), respectively, and adding the resulting equations yield c)α) d t Υ u L + k d tυ u L + α) Υ u L = α)c) R u, ) Υ u dt Θ u, ) Υ u + Υ ϕ, ) 3.51) Υ u s) Eu 1, Υ ) u s)k dt ut ), Υ u ) 1 fϕt )) fφ ), Υ ) u.,

27 FINITE ELEMENT APPROXIMATIONS FOR THE PHASE FIELD MODEL 7 Each ter on the right-hand side of 3.50) can be bounded as follows: R u, ) Υ u dt Θ u, ) Υ u R u H 1 + d tθ u H 1 Υ 3.5) u H 1 1 c) Υ u L + 1 Υ u L 3.53) 3.5) Υ ϕ, Υ u +C1 + c) R u H 1 + d tθ u H 1. ) s) E 1 u, Υ ) ) u s)k dt ut ), Υ u Υ u L + c)α) Υ u L + α) + Cs) c)α) 1 fϕt )) fφ ), Υ u = 1 f ξ )E ϕ, Υ u ) Eu 1 L + k d t ut ) L. ) c)α) Υ u L + α) Υ ϕ L C 3 c)α) E ϕ L, where we have used the Mean Value Theore on f, and.1) and the assertion i). Substituting 3.5)-3.5) into 3.51) and taking su over fro 1 to l M) gives 3.55) c)α) C k Υ l u L + k M c)α)k M + s) c)α) 1 + c) +c)α)k d t Υ u L + α) Υ u L R u H 1 + d tθ u H 1 E 1 u L + k d t ut ) L + M Υ u L C ˆπ)h + ˆµ)k + c)α) k M Υ u L, + α) Υ ϕ L 1 3 c)α) E ϕ L where ˆπ) and ˆµ) are defined by 3.6) and 3.7), respectively. Assertions ii) and iii) follow fro 3.), 3.3), 3.5), 3.6), and i), ii), iv) of Theore 3.1 after applying the Grownwall s inequality to 3.55). The assertion iv) is obtained by using above estiate, 3.) and the inverse inequality bounding L -nor in ters of L -nor. To show the assertion v)-vii), we set η h = d t Υ ϕ in 3.0) to get 3.56) α) d t Υ ϕ L + d t Υ ϕ L + k d t Υ ϕ L = α) R ϕ, d t Υ ) ϕ dt Θ ϕ, d t Υ ) ϕ + s) Θ 1 u +s)k d t ut ), d t Υ ) ϕ + dt Υ u, d tυ ) ϕ + 1 fϕt )) fυ ϕ ), d t Υ ϕ ). + Υ u, d t Υ ) ϕ

28 XIAOBING FENG AND ANDREAS PROHL Its right-hand side, denoted by S, can be bounded together by S α) d t Υ ϕ L + C α) R ϕ L + d tθ ϕ 3.57) L + s) α) E 1 u L + Υ u L + k d t ut ) L α) E ϕ L, where we have bounded the last ter in the sae way as in 3.5). Substituting 3.57) into 3.56), taking su over fro 1 to l M) and applying Grownwall s inequality lead to 3.5) α) ax 0 l Υ ϕ L + k d t Υ ϕ L + k d t Υ ϕ L C ˆπ)h + ˆµ)k, where ˆπ) and ˆµ) are given by 3.) and 3.9). Note that we have used i) and ii) of Theore 3.1 and the assertion ii) above to bound the right-hand side of 3.57). The assertions v)-vii) follows fro applying triangle inequality on Eϕ = Θ ϕ + Υ ϕ and using 3.3), 3.5), 3.), 3.5) and 3.). Finally, we set v h = d t Υ u in the error equation 3.1) to get c) dt Υ u L + k d t Υ 3.59) u L d tυ u L C c) d t E ϕ L + R ϕ L + c) d t Θ u L + R u L. Then, the assertions viii)-x) follows iediately fro 3.59), 3.5), the assertion vi) above, 3.3) and 3.). Reark 3.3. a). We ephasize that the assertions i)-iv) are shown under the regularity assuptions ϕ tt L J; H 1 ) and u tt L J; H 1 ), the proof of each stateent is based on the one which precedes it, hence, the order of the stateents are iportant. On the other hand, it is not hard to see that under the assuptions ϕ tt L J; L ) and u tt L J; L ), the assertions ii)-x) can be proved siultaneously, and the proofs of ii)-iv) can be shortened a little bit. Finally, it is also not hard to check that a factor k 1 will appear in the right-hand sides of the assertions v)-x) if the assuption.) is reoved. b). With help of the inverse inequality bounding L -nor in ters of H 1 -nor, it is easy to see fro the above proof that the estiate v) of Theore 3.1 and the estiate iv) of Theore 3. can be iproved to the following ones, 3.60) ax ϕt ) Φ L 0 M C ϕ L J;W, ) h + ˆπ) 1 h + ˆµ) 1 3 N k ln h, 3.61) ax ut ) U L 0 M C u L J;W, ) h + ˆπ) 1 h + ˆµ) 1 k ln h 3 N,

29 FINITE ELEMENT APPROXIMATIONS FOR THE PHASE FIELD MODEL 9 provided that ϕ L J; W, ) and u L J; W, ). So far we have derived error bounds for the fully discrete schee 1.15)-1.16) in the case i) defined in Leas. and 3.3, that is, α) c 0, s) 0 and c) 0 for soe constant c 0 > 0. In the reaining part of this section, we will extend the above error estiate results to the case ii) which corresponds to α) 0, c) c 0 and s) c) c 0 for soe constant c 0 > 0. As expected, the analysis for the case ii) is ore delicate and coplicate than that of the case i) since the weaker) spectral estiate 3.13), instead of the stronger estiate 3.1), has to be used to avoid exponential growth of the error bounds in 1, which requires the use of soe nonstandard test functions in the error analysis to be given next. To reduce soe technicalities, we will only present the derivation of error bounds for the case ϕ tt L J; L ) and u tt L J; L ), and leave the derivation for the case ϕ tt L J; H 1 ) and u tt L J; H 1 ) to interested readers. Theore 3.3. Let Φ, U ) M =0 solve 1.15)-1.16) on a quasi-unifor tie esh J k := t M =0 of size Ok) and a quasi-unifor space esh T h of size Oh). Suppose GA) and.) hold, and the free boundary proble 1.7)-1.1) has a unique classical solution. Also, assue that α) 0, c) c 0 and s) c) c 0 for soe constant c 0 > 0. Then under the following esh and starting value constraints 1). k in1, α) in, c) α)s), N+1 N α) B 1 + B N α) ). h N ln h 3 N C 0 α) C 1 )C B 1 ) 1 3). ζ)k + η)h 1 α) N 1 ). Φ 0 ϕ 0 L C h ϕ 0 H, 5). U 0 u 0 L Ch u 0 H, N B N N, for N =, 3, the solution of 1.15)-1.16) satisfies the error estiates i) ii) iii) iv) v) N J ϕ 0, u 0 ) ax α) ϕt ) Φ L Cρ 1 ) 1 h + η) 1 h + ζ) 1 k, 0 M M c)s) k ax 0 M 3 α)k =0 1 ut ) U L C ρ ) 1 h + η) 1 h + ζ) 1 k, k ut ) U ) j=0 M L C ρ 3 ) 1 1 h + η) h + ζ) 1 k, ϕt ) Φ ) 1 L C ρ ) 1 1 h + η) h + ζ) 1 k, α) ax ϕt ) Φ L 0 M C ρ 1 ) 1 h N ln h 3 N + η) 1 h + ζ) 1 k h N,,