• Aucun résultat trouvé

A priori error estimates of a finite‐element method for an isothermal phase‐field model related to the solidification process of a binary alloy

N/A
N/A
Protected

Academic year: 2021

Partager "A priori error estimates of a finite‐element method for an isothermal phase‐field model related to the solidification process of a binary alloy"

Copied!
25
0
0

Texte intégral

(1)IMA Journal of Numerical Analysis (2002) 22, 281–305. A priori error estimates of a finite-element method for an isothermal phase-field model related to the solidification process of a binary alloy D. K ESSLER† D´epartement de Math´ematiques, Ecole Polytechnique F´ed´erale, 1015 Lausanne, Switzerland AND. J.-F. S CHEID‡ Universit´e Henri Poincar´e Nancy 1, Institut Elie Cartan, B.P. 239, 54506 Vandoeuvre-les-Nancy C´edex, France [Received on 26 January 2001; revised on 9 October 2001]. We introduce a piecewise linear finite-element scheme with semi-implicit time discretization for an evolutionary phase field system modelling the isothermal solidification process of a binary alloy. This system can be written in a vectorial form as a nonlinear parabolic system. The convergence of the scheme with error estimate is then proved by introducing a generalized vectorial elliptic projector. Keywords: phase field model; parabolic system; finite element; a priori error estimates.. 1. Introduction The phase field model we consider describes the isothermal solidification process of a binary alloy. It involves the relative concentration c of one component with respect to the mixture and an order parameter φ called the phase field, which accounts for the solidification state of the alloy by taking values between 0 (in a pure solid phase) and 1 (in a pure liquid phase). The model we study is very similar to the Warren and Boettinger model, see Warren & Boettinger (1995). It has been already introduced in former publications (see e.g. Kessler 2001, Kessler et al. 1998, Kessler et al. 2000, Rappaz & Scheid 2000) and has successfully been used to simulate dendritic growth, see Kr¨uger et al. (to appear). A recapitulation of the modelling is beyond the scope of this paper, and we will immediately introduce the mathematical problem. Let Ω be an open subset of R2 with smooth boundary ∂Ω and a unit normal vector ν. For T > 0, the time evolution of φ = φ(x, t) and † Email: daniel.kessler@epfl.ch. ‡ Email: jean-francois.scheid@iecn.u-nancy.fr c The Institute of Mathematics and its Applications 2002 .

(2) 282. D . KESSLER AND J .- F. SCHEID. c = c(x, t) for x ∈ Ω and t ∈ [0, T ] is governed by the following equations:    ∂φ = M∆φ + F1 (φ) + cF2 (φ) in Ω × (0, T ),    ∂t      ∂c in Ω × (0, T ), = div D1 (φ)∇c + D2 (c, φ)∇φ (P) ∂t  ∂φ ∂c   = =0 on ∂Ω × (0, T ),    ∂ν ∂ν  φ(0) = φ , c(0) = c in Ω , 0 0 where M is a positive constant. The nonlinear functions Fi , Di , i = 1, 2 in Problem (P) satisfy the following assumptions: (H). · F1 , F2 and D1 , D2 are Lipschitz and bounded functions. · D1 is a positive function bounded below by a positive constant Ds .. The aim of this paper is to analyse a numerical scheme for (P) based on a finiteelement space discretization and a semi-implicit time scheme. For technical reasons, we restrict ourselves to the bidimensional space case. We obtain optimal error estimates for the scheme we consider and we prove that the scheme is unconditionaly convergent. Error estimates for finite-element methods have been performed in the past for thermal pure element phase field models (see Chen & Hoffmann (1994)), but the nonlinearities in such models are different from those of the solutal model we consider in this paper. We also mention Barrett & Blowey (1998) in which an isothermal phase separation model is described by a coupled system of Cahn–Hilliard equations. In order to study the convergence of the numerical scheme we first rewrite problem (P) in a convenient vectorial form. Problem (P) can be read as a uniformly parabolic system for the auxiliary vectorial variable u = (φ, αc) where the positive parameter α has to be chosen small enough. The idea to get optimal error estimates is mainly based on the introduction of a generalized vectorial projector related to the vectorial form of Problem (P). In Section 2, we introduce the vectorial form of (P) and we specify the mathematical framework. The numerical scheme we study is stated in Section 3. The main result of this paper is established in Section 4 where an optimal error estimate is derived. To this end, we first define in Section 4.1 a generalized vectorial elliptic projector for which some error bounds are obtained. This projector is used in Section 4.2 to prove the convergence result. Finally, we present in Section 5 some numerical results that confirm the theoretical prediction.. 2. Mathematical problem in vectorial form We transform Problem (P) to a vectorial form by defining the vectorial variable u = (φ, αc)T where α is an arbitrary positive parameter that will be fixed later. Then Problem.

(3) FINITE ELEMENTS FOR AN ISOTHERMAL PHASE - FIELD MODEL. (P) reads as a vectorial problem of the form: Find u (x, t) ∈ R2 such that  ∂ u. . u)  = div (D(. u )∇ u ) + F(. in Ω × (0, T ),    ∂t ∂ u. (PV ) =0 on ∂Ω × (0, T ),    ∂ν   in Ω , u (0) = u 0. 283. (1) (2) (3). where the 2 × 2 triangular matrix D is given by .  M 0 F1 (φ) + cF2 (φ). D(. u) = and F(. u) = α D2 (c, φ) D1 (φ) 0 . ∂ ∂ and where div (D(. u )∇ u ) := i=1,2 D(. u) u . ∂ xi ∂ xi Assumption (H) related to Problem (P) leads to the following assumptions for the vectorial problem (PV ): (A1) F is a 2-vector of Lipschitz bounded components. We call L F the maximum of the components’ Lipschitz constants. (A2) D is a 2×2 lower triangular matrix whose coefficients are given by d11 = M > 0, d12 = 0, d21 = α D2 (. u ) and d22 = D1 (. u )  Ds > 0. The functions D1 (. u) and D2 (. u ) are Lipschitz bounded functions. We call D M the maximum of the components’ absolute bounds and L D the maximum of the components’ Lipschitz constants. Since M > 0 and D1 (. u )  Ds > 0 for all u ∈ R2 , we can choose the parameter α small enough for D(. u ) to be uniformly positive definite. Indeed, if we choose α < 2(M Ds )1/2 / ||D2 ||∞ where ||D2 ||∞ = supu ∈R2 |D2 (. u )|, then it can be shown that v T D(. u ). v  min(M, Ds ). v T v for all u , v ∈ R2 . So in addition to (A1) and (A2), it is plainly justified to make an extra assumption on the positiveness of the matrix. (A3) The matrix D(. u ) is uniformly positive definite, i.e. there exists a constant Dm > 0 independent of u such that v T D(. u ). v  Dm v T v for all u , v ∈ R2 . Remark. As we pointed out previously, assumption (A3) is fulfilled if α is chosen small enough. Now, we recall some basic properties about vectorial calculus that will be useful later on. First, throughout this paper we denote by (:) the double scalar product in R2 ⊗ R2 , such that ∂vk ∂w j A jk , v , w. ∈ R2 . (4) A∇ v : ∇ w. = ∂ x ∂ x i i i=1,2 j,k=1,2 It is then clear that for all matrix A ∈ M2×2 (R) with bounded components, we have

(4)

(5) A∇ v : ∇ w. = AT ∇ w. : ∇ v , for all v , w. ∈ H 1 (Ω , R2 ). Ω. Ω. (5).

(6) 284. D . KESSLER AND J .- F. SCHEID. Furthermore, for any matrix A ∈ M2×2 (R) with bounded and Lipschitz components, the following Green formula:

(7)

(8)

(9) ∂ v. A∇ v : ∇ w. =− div (A∇ v ) · w. + A ·w. (6) ∂ν Ω Ω ∂Ω holds for all v ∈ H 2 (Ω , R2 ) and w. ∈ H 1 (Ω , R2 ). From the analytical point of view, the well-posedness of the original problem (P) has been investigated in Rappaz & Scheid (2000), under assumption (H). These results applied to the vectorial form lead in particular to an existence and uniqueness result for (PV ). Under assumptions (A1)–(A3) and if the initial data u 0 = (u 0,1 , u 0,2 )T ∈ H 2 (Ω ) × ∂u 0,1 H 1 (Ω ) satisfy = 0 on ∂Ω , then for any T > 0, there exists a unique solution ∂ν . T u = (u 1 , u 2 ) of Problem (PV ) such that u 1 ∈ C 0 [0, T ]; H 2 (Ω ) ∩ H 1 (Ω × (0, T )).   and u 2 ∈ C 0 [0, T ]; H 1 (Ω ) ∩ H 1 0, T ; L 2 (Ω ) . Finally, let us indicate that the solution u satisfies the following variational formulation:

(10)

(11)

(12) ∂ u. . u ) · v , ∀.  F(. v ∈ H 1 (Ω , R2 ), · v. + D(. u )∇ u. : ∇ v. =   Ω ∂t Ω Ω (PV ) a.e. t ∈ (0, T ) (7)     u (0) = u 0 . (8) This variational formulation will be useful for the expression of the numerical scheme in the next section and consequently for the error analysis. 3. Numerical scheme From now, we shall assume that the domain Ω is a convex polygonal subset of R2 . The case when Ω is a smooth convex domain can be treated as in Ciarlet (1978, 4.4) or Strang & Fix (1988), but we do not consider it for the sake of simplicity. We approximate Problem (PV ) by a P1 -finite element in space, semi-implicit in time discretization. To begin with, let us introduce some notation. We denote by Th a regular triangulation of the domain Ω (see Ciarlet (1978)), where h is the diameter of the biggest triangle in Th and we define the space Vh = {vh ∈ C 0 (Ω ); vh | K ∈ P1 (K ), ∀K ∈ Th } and Vh2 = Vh × Vh . For a given integer N  1, we denote by τ = T /N the time step and by t n = nτ , the current time for n = 0, . . . , N . We consider the approximation u nh of the exact solution u (t n ). For the rest of the paper, we choose an initial data u 0 belonging to H 2 (Ω ; R2 ) so that it is a continuous function. We denote by rh the Lagrange interpolation operator on Vh ×Vh and note that rh u 0 is well defined. Based on the variational formulation (PV ), we now introduce an approximate problem for u nh :  For n = 1, . . . , N , find u nh ∈ Vh2 such that for all v h ∈ Vh2 ,    

(13)

(14)

(15) u nh − u n−1 h θn n. u n−1 ) · v h , (Ph,τ ) F(. · v h + D(w. h )∇ u h : ∇ v h = (9) h  Ω τ  Ω Ω    0 u h = rh u 0 , (10).

(16) FINITE ELEMENTS FOR AN ISOTHERMAL PHASE - FIELD MODEL. where θ ∈ [0, 1] and the vector w. hθn is defined from u nh = (u n1h , u n2h )T and u n−1 by h.  n (1 − θ)u n−1 1h + θ u 1h w. hθn = . n−1 u 2h. 285. (11). It is easy to see that for all θ ∈ [0, 1], the discrete problem (Ph,τ ) has a unique solution. This is due to the fact that the matrix D(. u ) is lower triangular and that the component n−1 T n (D(. u ))11 = M is a constant. So, at first, from u n−1 = (u n−1 h 1h , u 2h ) we determine u 1h by solving (9) with v h = (vh , 0)T ; then since the second component of w. hθn does not n n T depend on u h at all, we determine u 2h by solving (9) with v h = (0, vh ) . Also, for any θ ∈ [0, 1] we do not have to solve nonlinear algebraic equations at each time step, while still granting inconditional convergence, as we will see in the next section. Lastly, note that in the approximate problem (Ph,τ ) we consider, no numerical integration is taken into account. 4. Convergence result The following theorem states the main result of this paper, concerning the convergence of the solution u nh of the discrete problem (Ph,τ ) to the exact solution u of the continuous problem (PV ). We need some extra assumptions on the triangulation of Ω . We assume that (A4) The triangulation Th verifies an inverse assumption, i.e. there exists a constant β such that ∀K ∈ Th , h/h K  β, where h K stands for the diameter of the triangle K. T HEOREM 4.1 Let assumptions (A1),(A2) and (A3) be fulfilled. Suppose that the triangulation Th satisfies the inverse assumption (A4). If the solution u of Problem (PV )  belongs to H 1 0, T ; H 2 (Ω , R2 ) ∩ W 1,∞ (Ω , R2 ) , then there exist two positive constants C and τ ∗ independent of h and τ such that for 0 < τ  τ ∗,   max u (t n ) − u nh  L 2 (Ω ,R2 )  C(h 2 + τ ). (12) 0n  N. The proof of Theorem 4.1 is given in Section 4.2. It is based on the introduction of a generalized vectorial elliptic projector, which is defined and studied in Section 4.1. Notice that we do not know if the solution of Problem (PV ) is in general sufficiently regular to satisfy the hypothesis of the above theorem. A general study of regularity has not been done and is beyond the scope of this paper. 4.1. A generalized vectorial elliptic projector. We will introduce a vectorial elliptic projector which is a generalization of the scalar elliptic projector used for instance by Thom´ee (1991). Through this section, we deal with a 2× 2 matrix which is not assumed to be triangular nor symmetric. In particular, the results of Section 4.1.2 about the properties of the vectorial projector are valid for a general 2×2 matrix..

(17) 286. D . KESSLER AND J .- F. SCHEID. D EFINITION 4.1 Let D(x) be a 2×2 matrix of bounded functions, positive definite uniformly with x ∈ Ω . We define the generalized vectorial elliptic projector (GVP) πh. H 1 (Ω , R2 )−→Vh2 u  −→πh u.

(18)

(19) by the relation D∇(. u − πh u ) : ∇ v h + (. u − πh u ) · v h = 0, Ω. :. Ω. ∀. vh ∈ Vh .. (13). The Lax–Milgram lemma ensures that πh is well-defined. Note that the second term in the left-hand side of equation (13) is necessary to account for Neumann boundary conditions for u in our problem. 4.1.1 Time-dependent GVP. We now consider a time-dependent matrix that will depend on both space x ∈ Ω and time t ∈ [0, T ], and we define a time-dependent generalized vectorial projector. We assume that (H1) D ∈ C 0 ([0, T ]; L ∞ (Ω , M2×2 (R))). (H2) D is uniformly positive definite, i.e. there exists a positive constant β independent of x and t such that ξ T D(x, t)ξ  β ξ T ξ for all ξ ∈ R2 and x ∈ Ω , t ∈ [0, T ]. We introduce a time-dependent bilinear form in H 1 (Ω , R2 ) defined for all t ∈ [0, T ] by

(20)

(21) u , v ∈ H 1 (Ω , R2 )  −→ at (. u , v ) = D(t)∇ u : ∇ v + u · v , (14) Ω. Ω. Under assumptions (H1), (H2) it is straightforward that the bilinear form at (·, ·) is coercive and continuous on H 1 (Ω , R2 ) uniformly with t, i.e. one can exhibit coercivity and continuity constants which are independent of t. Lax–Milgram’s lemma then allows us to generalize Definition 4.1 as the next definition. D EFINITION 4.2 Under assumptions (H1), (H2) we define the time-dependent GVP     πh : C 0 [0, T ]; H 1 (Ω , R2 ) −→L ∞ 0, T ; Vh2 u  −→πh u. by the relation. at (. u (t) − πh u (t), v h ) = 0,. ∀. vh ∈ Vh2 ,. ∀t ∈ [0, T ].. (15). 4.1.2 Properties of the time-dependent GVP. We will now give some properties for the time-dependent GVP. We derive error bounds that will be key ingredients for the a priori estimates in Section 4.2 for the proof of Theorem 4.1. P ROPOSITION 4.1 Under assumptions (H1), (H2), if in addition D.   L ∞ 0, T ; W 1,∞ (Ω , M2×2 (R)) and u ∈ C 0 [0, T ]; H 2 (Ω , R2 ) then πh u.  C 0 [0, T ]; Vh2 and there exists a positive constant C independent of h, such that ||. u − πh u || L ∞ (0,T ;L 2 (Ω ,R2 )) + h ||. u − πh u || L ∞ (0,T ;H 1 (Ω ,R2 ))  Ch 2 .. ∈ ∈. (16).

(22) FINITE ELEMENTS FOR AN ISOTHERMAL PHASE - FIELD MODEL. 287. P ROPOSITION 4.2 Under assumptions (H1), (H2), if in addition D ∈ . ∩H 1 (0, T ; L ∞ (Ω , M2×2 (R))) and u. ∈ L ∞ 0, T ; W 1,∞ (Ω , M2×2 (R)) .  H 1 0, T ; H 2 (Ω , R2 ) then πh u ∈ H 1 0, T ; Vh2 and there exists a positive constant C independent of h, such that      ∂    ∂       u − π u − π u. u. + h  Ch 2 . (17) (. (. ) ) h h  ∂t  2  2   ∂t 2 2 1 2 L (0,T ;L (Ω ,R )) L (0,T ;H (Ω ,R )) P ROPOSITION 4.3 Let assumptions (H1), (H2) be fulfilled. Suppose that the trian. gulation Th satisfies the inverse assumption (A4). If u ∈ H 1 0, T ; H 2 (Ω , R2 ) ∩.  L ∞ 0, T ; W 1,∞ (Ω , R2 ) then there exists a positive constant C independent of h, such that ||∇πh u || L ∞ (0,T ;L ∞ (Ω ,R2 ))  C.. (18). Remark. Propositions 4.1 and 4.2 are still valid in space dimension 3. However, Proposition 4.3 is not available in space dimension greater than 2. Indeed, the constant C in (18) depends on h 1−d/2 where d is the space dimension. Now we deal with the proofs of the three propositions. 4.1.3 Proof of the properties of the time-dependent GVP. We will need a lemma for proving the properties of the GVP. This result extends a regularity result from Grisvard (1985) from scalar elliptic problems to elliptic systems. L EMMA 4.1 Let A ∈ W 1,∞ (Ω , M2×2 (R)) be a uniformly positive definite matrix and. ∈ H 1 (Ω , R2 ) to the equation let f ∈ L 2 (Ω , R2 ). Then the solution w

(23)

(24)

(25) A∇ w. : ∇ v + w. · v = (19) f · v , ∀. v ∈ H 1 (Ω , R2 ), Ω. Ω. is actually in H 2 (Ω , R2 ) and satisfies constant C independent of f such that. Ω. ∂w. = 0 a.e. on ∂Ω . Furthermore, there exists a ∂ν.     ||w. || H 2 (Ω ,R2 )  C  f . L 2 (Ω ,R 2 ). .. (20). Proof of Lemma 4.1. Let f = ( f 1 , f 2 ) ∈ L 2 (Ω ) × L 2 (Ω ). According to Lax–Milgram’s lemma, there exists a unique solution w. = (w1 , w2 ) ∈ H 1 (Ω ) × H 1 (Ω ) to (19). We 1,∞ (Ω ). Since A is uniformly positive definite, there note A = (ai j )1i, j 2 with ai j ∈ W are three positive constants βi , i = 1, 2, 3, such that 0 < β1  a11  β2 and a11 a22 − a12 a21  β3 > 0. Under the lemma’s assumptions, Grisvard’s result (Grisvard, 1985, Theorem 3.2.1.3) tells us that there exists a unique w˜ 2 ∈ H 2 (Ω ) satisfying homogeneous Neumann boundary conditions such that a.e. in Ω . a12 − div a22 − a21 ∇ w˜ 2 + w˜ 2 a11.  (21) a12 a12 · (a11 ∇w1 + a21 ∇w2 ) . = f2 − ( f 1 − w1 ) + ∇ a11 a11.

(26) 288. D . KESSLER AND J .- F. SCHEID. For the same reasons, once w˜ 2 ∈ H 2 (Ω ) is given, there exists a unique w˜ 1 ∈ H 2 (Ω ) satisfying homogeneous Neumann boundary conditions such that a.e. in Ω − div (a11 ∇ w˜ 1 ) + w˜ 1 = f 1 + div (a21 ∇ w˜ 2 ) .. (22). Let v ∈ H 1 (Ω ). If we subtract a weak form of (21) with v as a test function, from (19) with v = (a12 /a11 v, −v), we find that.

(27)  a12 a22 − a21 ∇(w2 − w˜ 2 ) · ∇v + (w2 − w˜ 2 )v = 0, ∀v ∈ H 1 (Ω ). (23) a11 Ω If we subtract a weak form of (22) with v as a test function, from (19) with v = (v, 0), we find that

(28) a11 ∇(w1 − w˜ 1 ) · ∇v + a21 ∇(w2 − w˜ 2 ) · ∇v + (w1 − w˜ 1 )v = 0, ∀v ∈ H 1 (Ω ). Ω. (24) By first choosing v = w2 − w˜ 2 in (23) and then v = w1 − w˜ 1 in (24), we conclude that w1 ≡ w˜ 1 and w2 ≡ w˜ 2 . Therefore w. = (w1 , w2 ) ∈ H 2 (Ω , R2 ) and w. satisfies − div(A∇ w). +w. = f. ∂w. =0 ∂ν. a.e. in Ω ,. (25). a.e. on ∂Ω .. (26). Using the assumptions on the matrix A, it follows that there exists a constant Cˆ > 0 such that.      ˆ ||∆w. || L 2 (Ω ,R2 )  C ||w . (27). || H 1 (Ω ,R2 ) +  f  2 2 L (Ω ,R ). On the other hand, we obtain from (19) with v = w. that there exists a constant C¯ > 0 such that     ||w. || H 1 (Ω ,R2 )  C¯  f  2 (28) 2 L (Ω ,R ). Combining (27), (28) and Theorems 3.1.3.3 and 3.2.1.3 from Grisvard (1985), we find the estimate (20) stated in the lemma.  Remark. Note that this proof could be generalized to an elliptic system with more than two unknowns. Now, we are able to prove the three propositions. Proof of Proposition 4.1. We note rh : C 0 (Ω , R2 ) → Vh2 the P1 -Lagrange interpolation operator on Vh . It is well known (see e.g. Ciarlet 1978) that the interpolation error on H 1 norm can be estimated by ||w. − rh w. || L 2 (Ω ,R2 ) + h ||w. − rh w. || H 1 (Ω ,R2 )  C h 2 |w|. H 2 (Ω ,R2 ) ,. ∀w. ∈ H 2 (Ω , R2 ), (29).

(29) FINITE ELEMENTS FOR AN ISOTHERMAL PHASE - FIELD MODEL. 289. where |·| H 2 (Ω ,R2 ) denotes the H 2 semi-norm and C is a positive constant independent of w. and h. With the previously introduced notation, we can write that for all t ∈ [0, T ], using first the coercivity of at , then Definition 4.2 and finally the continuity of at , β ||. u (t) − πh u (t) ||2H 1at (. u (t) − πh u (t), u (t) − πh u (t)) at (. u (t) − πh u (t), u (t) − rh u (t)) η ||. u (t) − πh u (t) || H 1 ||. u (t) − rh u (t) || H 1 ,. (30) (31) (32). where β and η are positive constants. Using the interpolation error estimate (29), we find that ||. u (t) − πh u (t) || L ∞ (0,T ;H 1 (Ω ,R2 ))  C1 h,. (33). where C1 depends on ||. u || L ∞ (0,T ;H 2 (Ω ,R2 )) and is independent of h. 2 For the L -error estimate, we use Aubin–Nitsche’s technique, by introducing the dual problem to the definition of πh u (t). We define, for a fixed t ∈ [0, T ], the auxiliary function w. ∈ H 1 (Ω , R2 ) as the solution to the adjoint equation

(30) at (. v , w). = (34) u (t) − πh u (t)) · v , for all v ∈ H 1 (Ω , R2 ). (. Ω. Once again, Lax–Milgram’s lemma ensures that w. is well-defined. Using assumption (H2), the regularity of D and u , and Lemma 4.1 with A = D T , we obtain that w. ∈ H 2 (Ω , R2 ) and that there exists a constant C2 independent of u and h, such that |w|. H 2 (Ω ,R2 )  C2 ||. u − πh u || L 2 (Ω ,R2 ) .. (35). From (34), Definition 4.2 and the continuity of at , we find that ||. u (t) − πh u (t) ||2L 2 = at (. u (t) − πh u (t), w) = at (. u (t) − πh u (t), w. − rh w). − rh w  η ||. u (t) − πh u (t) || H 1 ||w. || H 1. (36) (37) (38). Using result (33), interpolation estimate (29) and the dual H 2 -bound (35), we find that there exists a positive constant C3 such that ||. u (t) − πh u (t) || L 2 (Ω ,R2 )  C3 h 2 ,. (39). and since this last inequality is valid for any fixed t ∈ [0, T ], we obtain ||. u − πh u || L ∞ (0,T ;L 2 (Ω ,R2 ))  C3 h 2 ,. (40). where C3 depends on ||. u || L ∞ (0,T ;H 2 (Ω ,R2 )) and is independent of h. Finally, using the coercivity of at (·, ·), Definition 4.2 for πh , H¨older’s inequality and the above estimates (33), it can be easily proved under the proposition’s assumptions that lim ||πh u (t) − πh u (s) || H 1 (Ω ,R2 ) = 0,. s→t.  i.e. that πh u ∈ C 0 [0, T ]; Vh2 .. ∀t ∈ [0, T ],. (41) .

(31) 290. D . KESSLER AND J .- F. SCHEID. Proof of Proposition 4.2. We take advantage of the fact that Vh is a finite-dimensional space. Let us call πi (t) for i = 1, . . . , 2n h the coordinates of πh u (t) in a basis of Vh2 defined by a set of linearly independent elements {ψ 1 , . . . , ψ 2n h }, where n h is the dimension of Vh , i.e. πh u (t) =. 2n h. πi (t)ψ i. (42). i=1. Definition 4.2 can then be translated as. where. π(t). b(t). A(t)π(t). = b(t), ∀t ∈ [0, T ],.  = π j (t) 1 j 2n ,. 

(32) h

(33) = bk (t) = D(t)∇ u (t) : ∇ ψ k + u (t) · ψ k Ω. and. A(t). =. Ω. .

(34)

(35). ak j (t) = ψ j · ψk D(t)∇ ψ j : ∇ ψk + Ω. Ω. (43). 1k 2n h. ,. . 1k, j 2n h. By the existence and uniqueness of π (t) (Lax–Milgram’s lemma), we know that A(t) is invertible for all t ∈ [0, T ] and we get. π(t). = A−1 (t)b(t),. ∀t ∈ [0, T ].. (44). Since H 1 (0, T ) ⊂ C 0 ([0, T ]), and A(t) is invertible for all t ∈ [0, T ], it is clear that b ∈ H 1 (0, T ; R2n h ) and A−1 ∈ H 1 (0, T ; M2n h ×2n h (R)), and therefore π ∈ H 1 (0, T ; R2n h ). Thus we have that πh u ∈ H 1 (0, T ; Vh2 ). We now differentiate (15) with respect to t and obtain

(36)

(37) ∂ D  (t)∇ (. u (t) − πh u (t)) : ∇ v h + D(t)∇ u (t) − πh u (t)) : ∇ v h (. ∂t Ω Ω

(38) ∂ + for all v h ∈ Vh2 , a.e. in (0, T ), (45) u (t) − πh u (t)) · v h = 0, (. ∂t Ω where D  (t) stands for the matrix of the time derivatives of the components of D(t). With similar steps as (29)–(33), it is then easy to prove that    ∂    u. (46) u − π  C4 h, ) (. h  ∂t  2 L (0,T ;H 1 (Ω ,R 2 ))    ∂ u     where C4 depends on ||. u || L ∞ (0,T ;H 2 (Ω ,R2 )) ,   and ∂t  L 2 0, T ; H 2 (Ω , R2 )     D  2 but is independent of h. L (0,T ;L ∞ (Ω ,M2×2 )) 2 In order to get L -error estimate, again we use Aubin–Nitsche’s technique. This time we define w(t). ∈ H 1 (Ω , R2 ) as the solution to the adjoint equation

(39) ∂ at (. u (t) − πh u (t)) · v , ∀. v , w(t)). = v ∈ H 1 (Ω , R2 ), a.e. t ∈ (0, T ). (47) (. Ω ∂t.

(40) FINITE ELEMENTS FOR AN ISOTHERMAL PHASE - FIELD MODEL. 291. Thus w(t). is well-defined a.e. in (0, T ) and applying Lemma 4.1 we find that w(t). ∈. H 2 (Ω , R2 ) and ∂ w(t) = 0 a.e. on ∂Ω , for a.e. t ∈ (0, T ), and that there exists a constant ∂ν C5 > 0 such that    ∂    ||w(t). || H 2 (Ω ,R2 )  C5  , a.e. t ∈ (0, T ). (48) u (t) − πh u (t))  (. ∂t L 2 (Ω ,R 2 ) Using (47) and (45), we find that, a.e. in (0, T ),  2 .  ∂  ∂    u. (t)) = a u. (t)) , w(t). − r w(t). u (t) − π u (t) − π (. (. h t h h  ∂t  2 ∂t L (Ω ,R 2 )

(41) + D  (t)∇(. u (t) − πh u (t)) : ∇(w(t). − rh w(t)). Ω

(42) − D  (t)∇(. u (t) − πh u (t)) : ∇ w(t).. (49) Ω. Applying Green’s formula (6) with property (5) to the last term of the right-hand side, and using the continuity of at , we find that, a.e. in (0, T ),  2  ∂    ||w(t). − rh w(t). || H 1 (Ω ,R2 ) u (t) − πh u (t))   ∂t (. L 2 (Ω , R 2 )    ∂  u (t) − πh u (t))  × η  (. ∂t H 1 (Ω ,R 2 )     +  D (t)  L ∞ (Ω ,M ) ||. u (t) − πh u (t) || H 1 (Ω ,R2 ) 2×2

(43)   + ∇ D  (t)T ∇ w. + D  (t)T ∆w. . (. u (t) − πh u (t)) . Ω. (50) Integrating in t and using the Cauchy–Schwarz inequality, we find that  2  ∂   u − πh u )   ∂t (. L 2 (0,T ;L 2 (Ω ,R 2 ))    ∂   ||w. − rh w u − πh u ) . || L 2 (0,T ;H 1 (Ω ,R2 )) η  (. ∂t L 2 (0,T ;H 1 (Ω ,R 2 ))     +  D  L 2 (0,T ;L ∞ (Ω ,M )) ||. u − πh u || L ∞ (0,T ;H 1 (Ω ,R2 )) 2×2   u − πh u || L ∞ (0,T ;L 2 (Ω ,R2 )) . (51) + 2  D   L 2 (0,T ;W 1,∞ (Ω ,M )) ||w. || L 2 (0,T ;H 2 (Ω ,R2 )) ||. 2×2 Using then (46) and Proposition 4.1, we obtain that there exists a positive constant C6 which depends on ||D || H 1 (0,T ;W 1,∞ (Ω ,M2×2 )) but independent of h such that  2  ∂   u − πh u )   ∂t (. L 2 (0,T ;L 2 (Ω ,R 2 ))  . || L 2 (0,T ;H 1 (Ω ,R2 )) + h 2 ||w  C6 h ||w. || L 2 (0,T ;H 2 (Ω ,R2 )) .. − rh w. (52).

(44) 292. D . KESSLER AND J .- F. SCHEID. From interpolation estimate (29) and the H 2 -norm estimate (48) together with (52), we conclude that there exists a positive constant C7 independent of h such that    ∂   (53)  C7 h 2 . u − πh u )   ∂t (. L 2 (0,T ;L 2 (Ω ,R 2 ))  Proof of Proposition 4.3. Using assumption (A4), we can write the following inverse inequality in Vh (see Ciarlet 1978, p. 140): there exists a positive constant C independent of h such that ||∇vh || L ∞ (Ω ,R2 )  Ch −1 ||∇vh || L 2 (Ω ,R2 ) ,. ∀vh ∈ Vh .. (54). Therefore, since πh u (t) − rh u (t) ∈ Vh2 , we have for a.e. t ∈ (0, T ) −1 ||∇ (πh u (t) − rh u (t)) || L ∞ (Ω ,R2 ||∇(πh u (t)−rh u (t))|| L 2 (Ω ,R2 ) )Ch. Ch −1 ||∇ (πh u (t) − u (t)) || L 2 (Ω ,R2 ).  + ||∇ (. u (t) − rh u (t)) || L 2 (Ω ,R2 ) .. (55) (56). Then using interpolation estimate (29) and (33), we infer that for a.e. t ∈ (0, T ) ||∇ (πh u (t) − rh u (t)) || L ∞ (Ω ,R2 )  C8 ,. (57). where C8 is independent of h, and depends on ||. u || L ∞ (0,T ;H 2 (Ω ,R2 )) . 1,∞ On the other hand, we can estimate a W -interpolation error for u . For the Lagrange interpolation operator, we have that (see Ciarlet 1978, p. 121) there exists a constant C independent of h and u such that, for a.e. t ∈ (0, T ), ||. u (t) − rh u (t) ||W 1,∞ (Ω ,R2 )  C ||. u (t) ||W 1,∞ (Ω ,R2 ) ,. (58). ||∇rh u (t) || L ∞ (Ω ,R2 )  ||∇ (rh u (t) − u (t)) || L ∞ (Ω ,R2 ) + ||∇ u (t) || L ∞ (Ω ,R2 )  (1 + C) ||. u (t) ||W 1,∞ (Ω ,R2 ) .. (59). and therefore. Finally, using (57) and (59), we find that there exists a constant C9 independent of h such that for a.e. t ∈ (0, T ) ||∇πh u (t) || L ∞ (Ω ,R2 )  ||∇ (πh u (t) − rh u (t)) || L ∞ (Ω ,R2 ) + ||∇rh u (t) || L ∞ (Ω ,R2 )  C9 . (60) Proposition 4.3 is then proved.. . Remark. For the more general case of space dimension d  3, estimate (54) goes actually as h −d/2 , and then rather than inequality (60), we get an estimate depending on h 1−d/2 . So for d = 3, the constant is not bounded with h..

(45) FINITE ELEMENTS FOR AN ISOTHERMAL PHASE - FIELD MODEL. 293. 4.2 Proof of the convergence result First of all, let us remark that the GVP assumptions (H1), (H2) as well as the regularity assumptions in Propositions 4.1–4.3, are implied by the assumptions (A1)–(A3) and the regularity assumption on the u of (PV ) in Theorem 4.1 with D = D(. u ). So exact solution  we can define πh u ∈ H 1 0, T ; Vh2 and use the three GVP properties for πh u given by Propositions 4.1–4.3. Also, from now on, ||· ||0 will denote the norm of L 2 (Ω , R2 ), Q n the  tn space–time domain (t n−1 , t n ) × Ω and g n = τ1 t n−1 g(t) dt the average of an integrable function g on [t n−1 , t n ]. Finally, let us define for n = 0, . . . N , δ u nh = πh u (t n ) − u nh .. (61). From the numerical scheme (9), for all v h ∈ Vh2 and for n = 1, . . . , N , we have

(46) 

(47)

(48)    θn n + τ D( w. )∇δ u. : ∇ v. = δ u nh − δ u n−1 · v. πh u (t n ) − πh u (t n−1 ) · v h h h h h h Ω Ω Ω

(49) +τ D(w. hθn )∇πh u (t n ) : ∇ v h Ω

(50). u n−1 ) · v h . F(. −τ (62) h Ω. Furthermore, since both u and πh u are in we have

(51) 

(52)  . πh u (t n ) − πh u (t n−1 ) · v h = πh u (t n ) − u (t n ) · v h Ω Ω

(53)   − πh u (t n−1 ) − u (t n−1 ) · v h

(54) Ω   u (t n ) − u (t n−1 ) · v h + H 1 (0, T ; L 2 (Ω , R2 )),.

(55) =τ. Ω.

(56) n n ∂ ∂ u. · v h . (63) (πh u − u ) · v h + τ Ω ∂t Ω ∂t. Now, using (7) of the exact problem (PV ), we deduce that

(57)

(58)

(59) n n ∂ u. n. F(. u ) · v h − · v h = D(. u )∇ u : ∇ v h . ∂t Ω Ω Ω. (64). Then from (63) and (64) together with (62), we obtain

(60) 

(61)  δ u nh − δ u n−1 · v. + τ D(w. hθn )∇δ u nh : ∇ v h h h Ω. Ω.

(62). =.

(63) n ∂ n (πh u − u ) · v h − τ D(. u )∇ u : ∇ v h Ω ∂t Ω

(64) 

(65)  n n−1. F(. u ) − F(. +τ u h ) · v h + τ D(w. hθn )∇πh u (t n ) : ∇ v h .. τ. Ω. Ω. (65).

(66) 294. D . KESSLER AND J .- F. SCHEID. Moreover, by the definition (15) of the GVP, we get for all v h ∈ Vh2

(67). n. Ω.

(68). D(. u )∇ u : ∇ v h. =. n. Ω.

(69). D(. u )∇πh u : ∇ v h +. n. Ω. (πh u (t) − u (t)) · v h .. (66). Then using (66) in (65), we obtain that

(70) 

(71)  n−1 n δ u h − δ u h · v h + τ D(w. hθn )∇δ u nh : ∇ v h Ω. Ω.

(72).

(73) n ∂ n =τ (. u − πh u ) · v h (πh u − u ) · v h + τ ∂t Ω Ω

(74)   n +τ u )∇πh u : ∇ v h D(w. hθn )∇πh u (t n ) − D(.

(75) Ω   n. u ) − F(. u n−1 ) · v h , F(. +τ for all v h ∈ Vh2 . h. (67). Ω. We may now choose v h = δ u nh in (67). Using assumption (A3) and applying Cauchy– Schwarz’ and Young’s inequalities five times to (67), we get the following inequality, valid for all ε1 , . . . , ε4 > 0 and for n = 1, . . . , N : . 1 2.  2   2 2    − (ε1 + ε2 + ε3 ) δ u nh 0 − 12 δ u n−1  + τ (Dm − ε4 ) ∇δ u nh 0 h 0   n 2      2 2 2 τ  ∂ τ  τ 2  n n 2 . u n−1 )  u ) − F(.  u − πh u  +  (πh u − u )  + .  F(. h  0 0 4ε1  ∂t 4ε2 4ε3 0     τ  n 2. hθn )∇πh u (t n ) − D(. + u )∇πh u  . (68) D(w 0 4ε4. We must now estimate the four terms of the right-hand side of inequality (68) before  2 using the discrete Gronwall lemma to obtain the final estimate of δ u nh 0 . (i) The first term in the right-hand side of (68) can be immediatly estimated using the Cauchy–Schwarz inequality. We get   n 2  ∂    (πh u − u )   ∂t . =. 0.  . 2

(76) 

(77) t n  ∂ 1  (πh u − u )dt  dx   Ω  τ t n−1 ∂t 2

(78)

(79) tn    1  ∂ (πh u − u ) dt dx  ∂t  n−1 τ Ω t 2  1  ∂ , (. u − πh u )    τ ∂t L 2 (Q n ). (69). where |·| stands for the vectorial norm. (ii) In a similar way, the second term can be estimated as   1 n 2  u − πh u ||2L 2 (Q n ) . u − πh u   ||. . 0 τ. (70).

(80) FINITE ELEMENTS FOR AN ISOTHERMAL PHASE - FIELD MODEL. 295. (iii) The third term of (68) can be read as    2

(81)  1

(82) t n   2  n   . u (t)) − F(. u n−1 ) dt  dx.. u n−1 )  = F(. u ) − F(.  F(.  h h   n−1 0 τ Ω t. (71). Then we use the Cauchy–Schwarz inequality and the Lipschitz assumption (A1) on F in order to get  2

(83) 1

(84) t n  2  n . u n−1 ) dt dx. u n−1 )   u ) − F(. F(. u (t)) − F(.  F(.  h h 0 Ω τ t n−1 2 L2

(85)

(86) t n    F u (t) − u n−1 . h  dt dx τ Ω t n−1 2 2

(87)

(88) t n  2L2 

(89)

(90) t n   n−1 n−1  n−1  F  u (t) − u (t u (t ) + ) − u h  . . . τ Ω t n−1 Ω t n−1 (72)  t ∂ u. Now, since we have the following relation u (t) − u (t n−1 ) = t n−1 ∂t (s) ds, for all t ∈ [t n−1 , t n ], it is easy to see that  

(91)

(92) tn  2  ∂ u 2    u (t) − u (t n−1 ) dt  τ 2  . (73) . ∂t  L 2 (Q n ) Ω t n−1 Then we deduce from (72) and (73) that   2 .  n n−1  2  ∂ u. u h )   2τ L F  u ) − F(.  F(. 0 ∂t. 2    2. L (Q n ).  2  n−1  u (t + 2L2F . ) − u n−1  . h 0. Finally, introducing the projector πh u (t n−1 ) in the above estimate, we obtain    2  2  n n−1  2  ∂ u . u h )   2τ L F  u ) − F(.  F(. 0 ∂t  L 2 (Q n )   2 2  n−1    u (t +4L2F . ) − πh u (t n−1 )  + 4L2F δ u n−1  . h 0. 0. (74). (75). (iv) The final term of (68) requires a little bit more work to estimate. Using Cauchy– Schwarz’ and Young’s inequalities, we can separate it in two terms as    2  n 2 n   . hθn )∇πh u (t n ) − D(. u )∇πh u   2  D(w. hθn ) − D(. u ) ∇πh u (t n )  D(w 0 0   n n 2  +2 D(. u ) ∇πh u (t n ) − D(. u )∇πh u  . 0. (76) We will start by estimating the first right-hand term of (76):  2  n  . hθn ) − D(. u ) ∇πh u (t n )   2 ||∇πh u ||2L ∞ (0,T ;L ∞ (Ω ,R2 ))  D(w 0. ×. 2. i, j=1.   n 2 . hθn ) − Di j (. u )  , Di j (w 0. (77).

(93) 296. D . KESSLER AND J .- F. SCHEID. where Di j stands for the components of matrix D. We now introduce an auxiliary function.  (1 − θ)u 1 (t − τ ) + θ u 1 (t) θ w. (t) = , u 2 (t − τ ). for t  τ .. (78). For all combinations of i, j = 1, 2 and for n  1, we have    2 n 2 . hθn ) − Di j (. u )   2  Di j (w. hθn ) − Di j (w. θ (t n )) 0 Di j (w 0    n 2  θ n + Di j (w. (t )) − Di j (. u )  . 0. (79). Let us estimate the first term in the right-hand side of (79). By the use of Lipschitz assumption (A2) on the matrix D, we have  2 2  θn  Di j (w. hθn ) − Di j (w. θ (t n )) 0  L2D w. θ (t n ) 0 .. h − w Moreover, definition (78) for w. θ leads to  θn 2  n−1 w. h − w. θ (t n ) = (1 − θ)(u n−1 ) 1h − u 1 (t 2  2 n−1 +θ(u n1h − u 1 (t n )) + u n−1 ) 2h − u 2 (t.  2    n−1 n−1  2 n n 2 ,  2 . u h − u (t ) + θ u h − u (t ). (80). (81). from which we deduce that  2  2  θn 2    n−1 n−1  w. h − w. θ (t n ) 0  4 δ u n−1 + u (t ) − π u. (t )    .  h h 0 0      n 2  n 2 2  n 2 +θ δ u h 0 + θ u (t ) − πh u (t ) 0 .. (82). Thus from (80) and (82), and since 0  θ  1 we obtain  2    2   θn θ n 2 2  Di j (w. h ) − Di j (w. (t )) 0  4L D δ u n−1  + θ 2 δ u nh 0 h 0.  2  n−1  + . u (t ) − πh u (t n−1 )  0  n 2  n + u (t ) − πh u (t )  . 0. (83). Now, we estimate the second term of the right-hand side of (79). First we have

(94)   n 2 . θ (t n )) − Di j (. u )  = Di j (w 0. 

(95) n 2  1 t θ n Di j (w. (t )) − Di j (. u (t)) dt dx. Ω τ t n−1. (84).

(96) FINITE ELEMENTS FOR AN ISOTHERMAL PHASE - FIELD MODEL. 297. Then using the Cauchy–Schwarz inequality and the Lipschitz assumption (A2) on the matrix D, we get   2 L2  θ n n 2 . θ (t n )) − Di j (. u )   D w. (t ) − u  L 2 (Q ) . Di j (w n 0 τ. (85). Furthermore, we have for all t ∈ [t n−1 , t n ] 2  2  θ n 2  w. (t ) − u (t) = (1 − θ )u 1 (t n−1 ) + θ u 1 (t n ) − u 1 (t) + u 2 (t n−1 ) − u 2 (t) . (86)  t n−1 ∂u 1 Thus, remarking that (1 − θ )u 1 (t n−1 ) + θu 1 (t n ) − u 1 (t) = (1 − θ) t ∂t (s) ds +  t n ∂u 1  t n−1 ∂u 2 n−1 θ t ∂t (s) ds and u 2 (t ) − u 2 (t) = t ∂t (s) ds, we deduce by the Cauchy– Schwarz inequality that, since 0  θ  1,  θ n 2 w. (t ) − u (t)  τ.

(97).   ∂ u.   t n−1 ∂t tn. 2   dt, . for all t ∈ [t n−1 , t n ].. (87). Then from (85) and (87), we obtain     ∂ u. n 2 . θ (t n )) − Di j (. u )   τ L2D  Di j (w 0 ∂t. 2    2. .. (88). L (Q n ). The second right-hand term of (76) is estimated as follows. We have   n n 2  u ) ∇πh u (t n ) − D(. u )∇πh u  D(. 0 2

(98) 

(99) t n .  1  = D(. u (t)) ∇πh u (t n ) − ∇πh u (t) dt  dx.    n−1 τ Ω t. (89). By the Cauchy–Schwarz inequality and the boundedness assumption of D in (A2), we obtain that   2 D 2  n n 2  u ) ∇πh u (t n ) − D(. u )∇πh u   M ∇πh u (t n ) − ∇πh u  L 2 (Q ) . D(. n 0 τ Since for all t ∈ [t n−1 , t n ] we have ∇πh u (t n ) − ∇πh u (t) = using the Cauchy–Schwarz inequality and estimate (90), that.  tn t. (90). ∇ ∂π∂th u (s) ds, we deduce,.    . n n 2  n 2  ∂πh u u ) ∇π u. (t ) − D(. u )∇π u.  τ D D(.  h h M ∇ 0 ∂t. 2    2. L (Q n ). .. (91).

(100) 298. D . KESSLER AND J .- F. SCHEID. Grouping steps (76)–(91), we find that   n 2 . hθn )∇πh u (t n ) − D(. u )∇πh u  D(w.  2   n−1  ) − πh u (t n−1 )  u (t . 0 2      2     + ||. u (t n ) − πh u (t n ) ||20 + δ u n−1  + θ 2 δ u nh 0 h 2 0     ∂ u.  +32τ L2D ||∇πh u ||2L ∞ (0,T ;L ∞ (Ω ,R2 ))  ∂t  L 2 (Q n )    ∂πh u 2  +2τ D 2M ∇ . ∂t  2 0. 128L2D. ||∇πh u ||2L ∞ (0,T ;L ∞ (Ω ,R2 )). L (Q n ). (92) Let us denote. . K 1 = 128L2D ||∇πh u ||2L ∞ (0,T ;L ∞ (Ω ,R2 )). and. K 2 = max. K1 , 2D 2M . 4. (93). Then we have   n 2 . hθn )∇πh u (t n ) − D(. u )∇πh u  D(w 0 .      n−1 2 2 2  n 2 u − πh u || L ∞ (0,T ;L 2 (Ω ,R2 )) + δ u h  + θ δ u h 0  K 1 2 ||. 0   2 2   ∂ u   ∂πh u    +K 2 τ  . (94) + ∇   ∂t L 2 (Q n ) ∂t  L 2 (Q n ) We can now go back to inequality (68). We choose ε1 = ε2 = ε3 = τ/3 and ε4 = Dm . Then from estimates (69), (70), (75) together with inequality (68), we get for n = 1, . . . , N  2  2   ( 12 − τ ) δ u nh 0 − 12 δ u n−1  h 0     2 3 K2 3 2 2 2  ∂ u  τ   ||. u − πh u || H 1 (t n−1 ,t n ;L 2 (Ω ,R2 )) + L + 4 2 F 4Dm ∂t  L 2 (Q n )  .  K 2 2  ∂πh u 2 K1 2 τ ||. u − πh u ||2L ∞ (0,T ;L 2 (Ω ,R2 )) + τ ∇ + 3L F + 4Dm  ∂t  L 2 (Q n ) 2Dm   2 K1 K 1  n 2   + 3L2F + τ δ u n−1 τ δ u h 0 . (95)  + θ 2 h 0 4Dm 4Dm Thus we have 2   2   ( 12 − µ1 τ ) δ u nh 0 − ( 12 + µ2 τ ) δ u n−1  h 0.  . 3 2 2  ∂ u u − πh u || H 1 (t n−1 ,t n ;L 2 (Ω ,R2 )) + K 3 τ   ||. 4 ∂t +K 4 τ ||. u. − πh u ||2L ∞ (0,T ;L 2 (Ω ,R2 )) ,. 2    2 L.   ∂πh u. + ∇ ∂t (Q n ). 2    2. . L (Q n ). (96).

(101) FINITE ELEMENTS FOR AN ISOTHERMAL PHASE - FIELD MODEL. 299. where we have put µ1 = 1 +. θ 2 K1 , 4Dm. µ2 = 3L2F +. K1 4Dm. (97). and K3 =. 3 2 K2 L + , 2 F 4Dm. K1 . 2Dm. K 4 = 3L2F +. (98). Now, let us define τ∗ =. 1 > 0. 4µ1. (99). Note that τ ∗ depends on ||∇πh u || L ∞ (0,T ;L ∞ (Ω ,R2 )) but thanks to Proposition 4.3 the constant τ ∗ is independent of h. In that way, for all 0 < τ  τ ∗ , we have 1 2. − µ1 τ  14 .. (100). In addition, it is straightforward to prove that for all 0 < τ . τ ∗,. we have. ( 12 + µ2 τ )  ( 12 − µ1 τ )(1 + µτ ). (101). µ = 4(µ1 + µ2 ).. (102). where. Note also that µ does not depend on h and τ . Then using (100) and (101), we deduce from (96) that for all n = 1, . . . , N and 0 < τ  τ ∗, 2   n 2 δ u  − (1 + µτ ) δ u n−1   λn (103) h 0 h 0. where λn = 3 ||. u. − πh u ||2H 1 (t n−1 ,t n ;L 2 (Ω ,R2 )).   ∂ u. + 4K 3 τ  ∂t 2. 2    2 L.   ∂πh u. + ∇ ∂t (Q n ). + 4K 4 τ ||. u − πh u ||2L ∞ (0,T ;L 2 (Ω ,R2 )) .. 2    2. . L (Q n ). (104). Now, we sum inequality (103) over n, in order to get n n  .  n 2  0 2  k−1 2 δ u   δ u  + u. λ + µτ  δ  , k h h 0 h 0. k=1. k=1. 0. (105). for all 1  n  N and 0 < τ  τ ∗ . We can then use the discrete Gronwall lemma (see for instance Quarteroni & Valli 1991, Section 1.4) on inequality (105) and find that, for n = 1, . . . , N ,   n  2  n 2 δ u   δ u 0  + (106) λk exp(µT ). h h 0 0. k=1.

(102) 300. D . KESSLER AND J .- F. SCHEID. Furthermore, using the definition (104) of λk , we have that for all 1  n  N n. k=1. λk  3 ||. u − πh u ||2H 1 (0,T ;L 2 (Ω ,R2 ))   ∂ u. + 4K 3 τ  ∂t 2. 2    2 L.   ∂ u. + ∇ ∂t (0,T ;L 2 (Ω ,R2 )).   ∂πh u. ∂ u. + ∇ −∇ ∂t ∂t. 2    2 L. 2    2 L. (0,T ;L 2 (Ω ,R2 )) . (0,T ;L 2 (Ω ,R2 )). + 4K 4 T ||. u − πh u ||2L ∞ (0,T ;L 2 (Ω ,R2 )) .. (107). Then it is plain, using Propositions 4.1 and 4.2 for the properties of the time-dependent GVP, that there exists a positive constant C1 independent of h and τ , such that for all 1nN n. λk  C1 (h 4 + τ 2 ).. (108). k=1. Finally, using (10), (16) and (29), we find that there exists a constant C2 independent of h and τ such that    0  (109) δ u h   C2 h 2 . 0. Therefore, using inequalities (108) and (109) together in inequality (106), we find that there exists a constant C3 independent of h and τ such that, for any 0 < τ  τ ∗ ,  n  δ u   C3 (h 2 + τ ), for n = 1, . . . , N . (110) h 0 We complete the proof of the convergence result by writing  n      u (t ) − u n   u (t n ) − πh u (t n )  + δ u n  h 0 h 0 0. (111). for all 1  n  N . Then we use Proposition 4.1 and estimate (110) to conclude that there exists a constant C independent of h and τ such that  n  u (t ) − u n   C(h 2 + τ ), (112) h 0 for all 0 < τ  τ ∗ and n = 1, . . . , N . The theorem is then proved.. . 5. Numerical tests Numerical tests have been performed on an adimensional problem using data from Kessler et al. (1998) and Warren & Boettinger (1995). We refer to them for a complete physical description. We define the nonlinear terms F1 , F2 and D1 , D2 in Problem (P), for φ and c in the interval [0, 1]. Outside this interval, all the terms are truncated to constant values..

(103) FINITE ELEMENTS FOR AN ISOTHERMAL PHASE - FIELD MODEL. 301. TABLE 1 Values of the physical parameters M 63.5. α1 −3.23 · 1011. α2 1.20·106. β1 −1.64 · 109. β2 3.17·109. γ −8.85 · 10−11. D 10−5. As we shall see, this gives Lipschitz and bounded nonlinear terms and so we are in the framework of the previous analysis. We choose (for φ and c in [0, 1]) F1 (φ) = α1 g  (φ) + β1 p  (φ). and. F2 (φ) = α2 g  (φ) + β2 p  (φ),. where αi , βi are model parameters linked to physical characteristics of the binary alloy we will consider and g be the polynomial double-well type function defined by g(φ) = φ 1 φ 2 (1 − φ 2 ) and the related polynomial p = 0 g(s)ds/ 0 g(s)ds = φ 3 (6φ 2 − 15φ 4 + 10). On the other hand, we choose D1 (φ) = D + p(φ)(1 − D). and. D2 (c, φ) = γ c(1 − c)D1 (φ)F2 (φ),. where D stands for the ratio of the diffusive coefficients in the pure solid and liquid phases. The parameter γ is linked to properties of the materials. Notice that with those definitions, the terms Fi vanish for φ = 0 and φ = 1. Moreover, these terms are taken to be zero when φ is outside of interval [0, 1]. In that way, we obtain Lipschitz and bounded functions. The same process is applied for D1 which is then always positive, and for D2 . In fact, a maximum principle holds (see Rappaz & Scheid 2000) which guarantees that if initial data belong to [0, 1] then the same holds for the solution at any time and then the truncation procedure is justified. The physical example we consider is a Ni–Cu alloy. The numerical values of physical parameters are given in Kessler et al. (1998) and Warren & Boettinger (1995) and we report them in Table 1, for a problem adimensionalized in space relative to a domain characteristic length l = 2 · 10−4 cm and in time relative to the liquid diffusion characteristic time l/Dl , where Dl = 10−5 cm2 s−1 is the physical diffusivity coefficient in the liquid phase. These parameters have been derived from physical characteristics of the Ni–Cu alloy, assuming that the interface thickness is of order δ = 10−5 cm, which is higher than what would be physically expected, but allows for reasonable calculation meshes and time steps (the αi parameters go as 1/δ 2 and would become extremely high with a smaller value for δ). We choose θ = 1 in the scheme (9) of (Ph,τ ). First we present numerical tests. The adimensional problem is defined in the unit square, and we fix a final adimensional time t f = 10−5 (higher times can create stability problems due to the stiffness of the source terms, i.e. the high values of their Lipshitz constants). We then construct an exact and explicit solution. We add right-hand sides to equations in Problem (P) so that given functions φe (x, y, t) and ce (x, y, t) (defined for (x, y) in [0, 1] × [0, 1] and t ∈ [0, t f ]) are then solutions. For a first test we choose the infinitely differentiable functions      t t 1 φe (x, y, t) = ce (x, y, t) = sin 2π y + . 1 + sin 2π x + 2 tf tf We choose to relate the time step τ to the mesh size h of a regular mesh by the.

(104) 302. D . KESSLER AND J .- F. SCHEID. TABLE 2 Errors and convergence order for very regular test functions j 1 2 3 4 5 6 7 8 9. hj 5·000E−2 2·500E−2 1·667E−2 1·250E−2 1·000E−2 8·333E−3 7·143E−3 6·250E−3 5·556E−3. eh j 3·700E−1 9·332E−2 4·158E−2 2·343E−2 1·500E−2 1·043E−2 7·662E−3 5·868E−3 4·637E−3. relationship τ = 40h 2 . Let us denote by eh =. sj 1·988 1·994 1·994 1·996 1·997 1·998 1·998 1·998.   max u (t n ) − u nh  L 2 (Ω ,R2 ) the error. 1n  N. between the exact solution u = (φe , ce ) and the computed solution u h . We are interested in the local slope of the error with respect to h in logarithmic scale, which we define by ln(eh j ) − ln(eh j−1 ) sj = , ln(h j ) − ln(h j−1 ) where h j−1 and h j are choices of mesh sizes for two consecutive calculations, and eh j and eh j−1 the corresponding computed errors. The results of these tests are given in Table 2. Note that the slopes s j take values very close to 2. This simple test therefore confirms our theoretical result of convergence order h 2 + τ , with a very regular test function. Nevertheless, on physical simulations, the solutions are not as regular as the product of two sines, and their main feature is that their values change very fast on regions of length scale δ (recall that δ was one of the model parameters, discussed earlier in this section). For this reason, we now present a second numerical test, with test functions reproducing the features of the physical solutions, yet regular enough to be in the scope of our convergence theorem. We define ρ(t) = 0·15 + 0·5 ttf and we choose  0  if r (. x ) < ρ(t),     r (. x ) − ρ(t) φe (. x , t) = 0·5 1 − cos π if ρ(t) < r (. x ) < ρ(t) + 2δ,  2δ   1 if ρ(t) + 2δ < r (. x ), and.  0 if r (. x ) < ρ(t),       r (. x ) − ρ(t)   π if ρ(t) < r (. x ) < ρ(t) + δ, 0·3 + 0·2 1 − cos δ   ce (. x , t) = r (. x ) − ρ(t) − δ   π if ρ(t) < r (. x ) + δ < ρ(t) + 2δ, 0·7 − 0·1 1 − cos   δ   1 if ρ(t) + 2δ < r (. x ),.

(105) FINITE ELEMENTS FOR AN ISOTHERMAL PHASE - FIELD MODEL. 303. TABLE 3 Errors and convergence order for test functions similar to physical solutions j 1 2 3 4 5 6 7 8 9. hj 5·000E−2 2·500E−2 1·667E−2 1·250E−2 1·000E−2 8·333E−3 7·143E−3 6·250E−3 5·556E−3. eh j 2·561E−1 5·071E−2 2·282E−2 1·295E−2 8·355E−3 5·810E−3 4·281E−3 3·284E−3 2·596E−3. sj 2·337 1·969 1·969 1·965 1·992 1·981 1·987 1·995. where r (. x ) is the distance between x and the centre of Ω . The isovalues of the solution are expanding concentric circles with a boundary layer of width 2δ. We follow the same procedure as for the previous tests. The results are given in Table 3. Again this test confirms the theoretical result of convergence order h 2 + τ . Finally, we illustrate the behaviour of physical solutions of (Ph,τ ) by showing graphs of functions φ and c computed on the original problem (without extra artificial source terms) for initial conditions defined as follows:   if r (. x ) < ρ0 ,  0    r (. x )−ρ0 φ0 (. x ) = 0·5 1 − cos π if ρ0 < r (. x ) < ρ0 + δ, (113) δ   1 if ρ + δ < r (. x ), 0. and   cs 0 0 c0 (. x) = c + (c0 − cl 0 ) r ( x )−ρ δ  l0 c0. if r (. x ) < ρ0 , if ρ0 < r (. x ) < ρ0 + δ, if ρ0 + 2δ < r (. x ),. (114). with radius ρ0 = 0·1 adimensional units. The computation is performed on a square domain of side 2 adimensional units (i.e. 4 · 10−6 m) with a roughly 600×600 unstructured mesh, and a final time 0.1 adimensional units (i.e. 4 · 10−4 s), achieved after 5000 time steps. Graphs of resulting functions are presented in Fig. 1. Notice that they are isotropic, as anisotropy has not been taken into account in the model presented in this paper. We also present radial profiles of the solutions in Fig. 2. Notice that the behaviour of the profile of c through the solid–liquid interface corresponds to what is expected from a sharp-interface limit asymptotic analysis of the model (see Kessler 2001). 6. Conclusion In this paper we have obtained error estimates of a finite-element method applied to a coupled system of nonlinear evolution equations. These equations are related to a phase.

(106) 304. D . KESSLER AND J .- F. SCHEID. 1 0·9 0·8 0·7 0·6 0·5 0·4 0·3 0·2 0·1 0. 0·5433 0·537 0·5307 0·5244 0·5182 0·5119 0·5056 0·4993 0·493 0·4868 0·4805 0·4742 0·4679. F IG . 1. φ(x) and c(x) at final time for a Ni–Cu alloy.. 1·1. 0·55. 1. 0·54 0·53. 0·8 0·52. 0·7 0·6. C1. Fraction of Solid. 0·9. 0·5 0·4. 1. 0·48. 0·2 0. 0·5 0·49. 0·3 0·1. 0·51. 0·47. 1. 0·46. 0 0·3 0·6 0·9 1·2 1·5 1·8 2·1 2·4 2·7 3. 0 0·3 0·6 0·9 1·2 1·5 1·8 2·1 2·4 2·7 3. Distance. Distance. F IG . 2. Profiles of φ(x) and c(x) at final time for a Ni–Cu alloy.. field model for the solidification of a binary alloy. The main idea is to reduce the equations to a nonlinear parabolic system and then introduce a generalized vectorial projector based on the elliptic part of the system operator. We derive projection errors in several norms. Error estimates of the piecewise linear finite-element method we used are obtained by comparing the approximate solution with the generalized projection of the exact solution at every time step in the L 2 norm. It is shown that error estimates are of order 2 in the space mesh size h and of order 1 in the time step τ . In addition, there is no condition connecting h and τ . Numerical tests supply results which are in good agreement with the theoretical predictions. Phase-field models are characterized by a small thickness region where an order parameter goes from 0 to 1 (the transition layer liquid/solid). Unfortunately, in the error estimates the constants depend on the inverse of this thickness so that constants become large when the thickness decreases. So in practice we need to have a small mesh size and time step in the transition region. Let us mention that a posteriori error estimates have also been performed on this model (see Kr¨uger et al. to appear) and provided a criteria for the.

(107) FINITE ELEMENTS FOR AN ISOTHERMAL PHASE - FIELD MODEL. 305. refinement of the mesh, thus allowing numerical calculations to be precise enough in the transition layer without refining the mesh in the whole domain, which would require too large calculation times. Finally, let us remark that our analysis should be applicable to more general nonlinear parabolic systems in Rn provided that the n×n-matrix of the system is triangular and can be reduced (as we did with our 2×2 matrix) to a definite positive matrix. Acknowledgements We thank Professor J. Rappaz for initiating and managing the phase field research group at the Department of Mathematics of EPFL. We acknowledge the Swiss National Science Foundation for financially supporting this project. We also thank Professor M. Rappaz and the Department of Materials Science, EPFL, for an ongoing collaboration with our research work. R EFERENCES BARRETT , J. W. & B LOWEY , J. F. (1998) Finite element approximation of a model for phase separation of a multi-component alloy with a concentration-dependent mobility matrix. IMA J. Numer. Anal., 18, 287–328. C HEN , Z. & H OFFMANN , K.-H. (1994) An error estimate for a finite-element scheme for a phase field model. IMA J. Numer. Anal., 14, 243–255. C IARLET , P. G. (1978) The finite element method for elliptic problems. Amsterdam: North-Holland. G RISVARD , P. (1985) Elliptic problems in nonsmooth domains. Boston, MA: Pitman. K ESSLER , D. (2001) Sharp interface limits of a thermodynamically consistentsolutal phase-field model. J. Crystal Growth, 224, 175–186. ¨ K ESSLER , D., K R UGER , O., R APPAZ , J. & S CHEID , J.-F. (2000) A phase-field model for the isothermal solidificationprocess of a binary alloy. Computer Assisted Mechanics and Engineering Sciences, 7, 279–288. ¨ K ESSLER , D., K R UGER , O. & S CHEID , J.-F. (1998) Construction d’un mod`ele de champ de phase a` temp´erature homog`ene pour la solidification d’unalliage binaire. DMA-EPFL Internal Report. ¨ K R UGER , O., P ICASSO , M. & S CHEID , J.-F. A posteriori error estimates and adaptive finite elements for a nonlinear parabolic problem related to solidification. Comp. Math. Appl. Mech. Eng., submitted. Q UARTERONI , A. & VALLI , A. (1991) Numerical approximation of partial differential equations. Berlin: Springer. R APPAZ , J. & S CHEID , J.-F. (2000) Existence of solutions to a phase-field model for the solidification process of a binary alloy. Math. Methods Appl. Sci., 23, 491–513. S TRANG , G. & F IX , G. J. (1988) An analysis of the finite element method. Cambridge: Cambridge University Press. T HOM E´ E , V. (1991) Galerkin Finite Element Methods for Parabolic Problems, 2nd edn. Berlin: Springer. WARREN , J. A. & B OETTINGER , W. J. (1995) Prediction of dendritic growth and microsegregation patterns in a binary alloy using the phase-field model. Acta Metall. Mater., 43, 689–703..

(108)

Références

Documents relatifs

Crouzeix-Raviart element, nonconforming method, stabilized method, nonlocking, a posteriori error estimates.. AMS

This paper concerns one of the simplest cases: the two-dimensional problem (which corresponds to a nonlinearity holding on a boundary of dimension one) writ- ten as a

Erik Burman, Daniel Kessler, Jacques Rappaz Convergence of the finite element method applied to an anisotropic phase-field model.. Volume 11, n o 1

Meddahi, Analysis of the coupling of primal and dual-mixed finite element methods for a two-dimensional fluid-solid interaction problem.. Meddahi, A new dual-mixed finite element

We prove higher regularity of the control and develop a priori error analysis for the finite element discretization of the shape optimization problem under consideration.. The derived

In order to suppress the round-off error and to avoid blow-up, a regularized logarithmic Schr¨ odinger equation (RLogSE) is proposed with a small regularization parameter 0 &lt; ε ≪

Convergence rates for FEMs with numerical quadrature are thus essential in the analysis of numerical homogenization methods and the a priori error bounds derived in this paper allow

Finally in section 6, we present new benchmark solutions for flows through mixed fluid–porous domains and for convective solidification: corner flows in a fluid overlying a porous