A SPLITTING METHOD FOR THE CAHN-HILLIARD EQUATION WITH

A SPLITTING METHOD FOR THE CAHN-HILLIARD EQUATION WITH INERTIAL TERM MAURIZIO GRASSELLI AND MORGAN PIERRE Abstract. P. Galenko et al. proposed a Cahn-Hilliard model with inertial term in order to model spinodal decomposition caused by deep supercooling in certain glasses. Here we analyze a finite element space semidiscretization of their model, based on a scheme introduced by C. M. Elliott et al. for the Cahn-Hilliard equation. We prove that the semidiscrete solution converges weakly to the continuous solution as the discretization parameter tends to 0. We obtain optimal a priori error estimates, assuming enough regularity on the solution. We also show that the semidiscrete solution converges to an equilibrium as time goes to infinity and we give a simple finite difference version of the scheme.

Keywords: discrete negative norms, mixed finite elements, L Ã ojasiewicz inequality, error estimates. AMS Classification 65M12, 65M60, 65M15, 35Q99, 82C26. 1. Introduction In this paper we consider a space semidiscretization of the modified Cahn-Hilliard equation ²utt + νut + α∆2 u − ∆f (u) = 0,

in Ω × R+ ,

(1.1)

with periodic boundary conditions. Here Ω = Πdk=1 (0, Lk ) (Lk > 0 for k = 1, . . . , d), 1 ≤ d ≤ 3, f is the derivative of a nonconvex potential (typically, f (y) = y 3 − y), α > 0 is related to the interfacial energy, and the constant coefficients ², ν satisfy ² > 0 and ν ≥ 0. The unknown u is the relative concentration of one phase. When ² = 0 and ν = 1, we recover the well-known Cahn-Hilliard equation (see [4], cf. also [28] and references therein). On the other hand, P. Galenko et al. [9, 10, 11] have proposed to add the inertial term ²utt in order to model non-equilibrium decompositions caused by deep supercooling in certain glasses. Equation (1.1) shows a good agreement with experiments [12]. In comparison with the Cahn-Hilliard equation, equation (1.1) presents some mathematical difficulties because the solutions do not regularize in finite time anymore. The onedimensional case is rather well understood [2, 6, 13, 26, 27], but the two-dimensional [17] or the three-dimensional case [16, 30] is far more complicated, because the so-called energy bounded solutions (of H 1 -type) are no longer bounded in L∞ (see Theorem 2.1 below). We refer in particular the reader to the introduction of [16] for an analysis of these issues. In this paper we perform a numerical analysis of a space semidiscretization of (1.1) in view of computations. This analysis is justified because of the hyperbolic-like features which make it difficult to find fully discretized schemes which are stable. In this regard, a time semidiscretization and numerical simulations will possibly be presented in another paper. In the papers cited above it is always assumed that ν > 0 in order to have some dissipation (or simply ν = 1 by a change of the time scale). On the contrary, here we can also include the “conservative” case ν = 0 since we are mostly interested in finding error estimates on finite time intervals. Moreover, allowing ν = 0 can be useful for studying numerical schemes with small dissipation. 1

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MAURIZIO GRASSELLI AND MORGAN PIERRE

The space semidiscretization that we consider is a finite element scheme based on a splitting formulation introduced by C. M. Elliott et al. for the Cahn-Hilliard equation [7] which has already been extended to other Cahn-Hilliard type equations with regularizing properties (see for instance [1, 20]). We first prove that a solution of the semidiscrete scheme converges weakly to an energy solution, as the discretization parameter h tends to 0 (see Theorem 3.5). This result is refined in Theorem 4.6 where we prove optimal a priori error estimates, assuming enough regularity on the solution. In both cases, the main difficulty is to deal with the negative norms (H −1 or H −2 ). The paper is organized as follows. We first introduce in Section 2 some notation and and we report a number of results concerning the continuous problem. The splitting scheme and the weak convergence result are given in Section 3. Section 4 is concerned with the error estimates; we recall in particular some standard estimates for discrete negative seminorms. In the last section, we show how our finite element scheme can be understood as a natural finite difference semidiscretization. Notice that we have considered periodic boundary conditions, but similar results hold (to some extent) if, instead, we consider homogeneous Dirichlettype boundary conditions or homogeneous no-flux boundary conditions (see Remarks 4.9 and 4.10). 2. The continuous problem 2.1. Notation and assumptions. Let L2 (Ω) with scalar product (·, ·) and norm k · k0 . For each u ∈ L2 (Ω), we denote by m(u) its average 1 (u, 1), with |Ω| = Πdi=1 Li . (2.1) m(u) = |Ω| We define L˙ 2 (Ω) = {u ∈ L2 (Ω), m(u) = 0}, 2 and for every u ∈ L (Ω), we set u˙ = u − m(u) 1 We denote V = Hper (Ω) = H 1 (T) the L2 -Sobolev space on the torus T = Πdk=1 (R/Lk Z). Using the inclusions V ⊂ L2 (Ω) ⊂ V 0 , the semiscalar product a(u, v) = (∇u, ∇v),

∀u, v ∈ V

2

defines a linear operator A : D(A) → L (Ω) with domain 2 D(A) = Hper (Ω) = H 2 (T).

In fact, A is the Laplace operator with periodic boundary conditions. The operator A˙ : ˙ → L˙ 2 (Ω) is the restriction of A to L˙ 2 (Ω) with domain D(A) ˙ = D(A) ∩ L˙ 2 (Ω). D(A) The operator A is a nonnegative self-adjoint operator; it has an orthonormal basis of eigenvectors (ei )i∈N , associated to the eigenvalues (λi )i∈N , with λ0 = 0 < λ 1 ≤ λ2 ≤ · · · ≤ λi ≤ · · · , 1/2

λi → +∞ as i → +∞.

In particular, e0 = 1/|Ω| . For all i ≥ 0, the function ei belongs to D(T), the space of functions which are of class C ∞ on T. By spectral theory [19], it is possible to define the powers (A + I)s , for all s ∈ R, and the spaces ¡ ¢ Vs = D (A + I)s/2 , V−s = (Vs )0 , ∀s ≥ 0. The space Vs can be identified with the space (see for instance [23]) ½ ¾ ∞ X s 0 s 2 H (T) = u |u ∈ D (T), λi hu, ei iD0 (T),D(T) < +∞ . i=0

A SPLITTING METHOD FOR THE CAHN-HILLIARD EQUATION WITH INERTIAL TERM

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Similarly, A˙ is a positive self-adjoint operator and it is possible to define the powers A˙ s and the spaces V˙ s = D(A˙ s/2 ), V˙ −s = (V˙ s )0 , ∀s ≥ 0. In fact, when s ≥ 0, X X u= ui ei ∈ V˙ s ⇐⇒ λsi u2i < +∞, i≥1

and

A˙ s u =

X

i≥1

λsi ui ei ,

for all u =

i≥1

X

ui ei , ∈ V˙ 2s .

i≥1

For all s ∈ R, the operator As/2 maps V˙ s onto L˙ 2 (Ω), and the space V˙ s is endowed with the Euclidean norm |u|s = kA˙ s/2 uk0 . It is easily seen that the map u 7→ (m(u), u), ˙ where 1 m(u) = hu, 1iD0 (T),D(T) , u˙ = u − m(u) |Ω| defines an isomorphism from Vs onto R × V˙ s , so that any element u ∈ Vs is identified by (m(u), u) ˙ and that the norm on Vs is equivalent to the Euclidean norm kuks = (m(u)2 + |u| ˙ 2s )1/2 . By extension, for u ∈ Vs , we denote |u|s = |u| ˙s the seminorm on Vs . When necessary, we will denote P˙ : Vs → V˙ s the map u 7→ u. ˙ For all r < s, we have the compact inclusion V˙ s ⊂ V˙ r . We also have V˙ 0 = L˙ 2 (Ω), V1 = V and V−1 = V 0 . For all s ∈ R, the operator A˙ can be extended as an isomorphism from V˙ s onto V˙ s−2 and we will denote A˙ this extension. Similarly, the operator A operates from Vs into Vs−2 . The nonlinearity f is a polynomial of odd degree whose leading coefficient is positive and which vanishes at 0: 2p+1 X ai y i , ∀y ∈ R, a2p+1 > 0, (2.2) f (y) = i=1

with any p ∈ N if d = 1 or d = 2, and with p ∈ {0, 1, 2} if d = 3. We denote F the antiderivative of f which vanishes at 0, i.e., F (y) =

2p+1 X i=1

ai i+1 y , i+1

∀y ∈ R.

(2.3)

We have in particular the Sobolev injection V1 ,→ L2p+2 (Ω), which implies L(2p+2)/(2p+1) (Ω) ⊂ V10 . The functional u 7→ f (u) is Lipschitz continuous on bounded subsets of V1 with values into (V1 )0 = V−1 (see for instance [21]). Also, note that there exist constants c1 > 0 and c2 ≥ 0 such that F (y) ≥ c1 y 2p+2 − c2 , ∀y ∈ R. (2.4) 2.2. The continuous problem. The abstract formulation of the singular Cahn-Hilliard equation (1.1) with periodic boundary conditions reads: ²utt + νut + αA2 u + Af (u) = 0, u(0) = u0 ,

ut (0) = u1 ,

(2.5) (2.6)

where u0 and u1 are initial data. We recall two fundamental a priori estimates. We first take the scalar product of (2.5)(2.6) by the constant function 1/|Ω|; we obtain ²m(utt ) + νm(ut ) = 0,

(2.7)

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MAURIZIO GRASSELLI AND MORGAN PIERRE

with initial data m(u(0)) = m(u0 ) and m(ut (0)) = m(u1 ). Thus, if ν > 0, ²³ ν ´ ν m(u(t)) = m(u0 ) + m(u1 ) 1 − exp(− t) , m(ut (t)) = m(u1 ) exp(− t), ν ² ² and m(u(t)) is bounded on [0, +∞). If ν = 0, then m(u(t)) = m(u0 ) + tm(u1 ),

m(ut (t)) = m(u1 ),

(2.8)

(2.9)

and m(u(t)) is bounded on [0, T ] for all T > 0. Second, formally applying A˙ −1 P˙ to (2.5), and taking the scalar product by ut , we find that α d 2 ² d (2.10) |u˙ t |2−1 + ν|u˙ t |2−1 + |u| ˙ + (f (u), u˙ t ) = 0. 2 dt 2 dt 1 Moreover, we have Z d F (u) = (f (u), u˙ t ) + (f (u), m(ut )). (2.11) dt Ω In view of (2.10), we introduce the energy functional E : V1 × V−1 → R defined by Z ² α 2 E(u, v) = |u˙ t |2−1 + |u| ˙ 1+ F (u). 2 2 Ω If m(u1 ) = 0, then integrating (2.10) and (2.11) from 0 to t, and using (2.8)-(2.9), we get Z t (2.12) E(u(t), ut (t)) + ν |ut |2−1 ≤ E(u0 , u1 ), ∀t ≥ 0. 0

In particular, t 7→ E(u(t), ut (t)) is nonincreasing. In the general case m(u1 ) 6= 0, we use that, by the definitions (2.2) and (2.3) of f and F , there exist two constants c3 > 0, c4 ≥ 0 such that |f (s)| ≤ c3 F (s) + c4 , ∀s ∈ R, (2.13) so that µ Z ¶ Z |(f (u), m(ut ))| ≤ |m(ut )| |f (u)| ≤ km(ut )kL∞ (0,T ) c3 F (u) + c4 |Ω| . Ω

Ω

Thus, since km(ut )kL∞ (0,T ) = |m(u1 )|, we obtain µ Z ¶ Z ² d α d 2 d 2 2 |u˙ t |−1 + ν|u˙ t |−1 + |u| ˙ + F (u) ≤ |m(u1 )| c3 F (u) + c4 |Ω| , 2 dt 2 dt 1 dt Ω Ω and by Gronwall’s lemma, Z t c4 |Ω| c3 |m(u1 )|t E(u(t), ut (t)) + ν |ut |2−1 ≤ E(u0 , u1 )ec3 |m(u1 )|t + (e − 1), c3 0

(2.14)

∀t ≥ 0. (2.15)

Thus, if E(u0 , u1 ) < +∞, u ∈ L∞ (0, T ; V1 ),

ut ∈ L∞ (0, T ; V−1 ),

∀T > 0.

We use here that F is bounded from below (see (2.4)) Z F (v) ≥ −c2 |Ω|, ∀v ∈ V.

(2.16)

Ω

The computations above are valid if u is regular enough. In the general case, they can be justified by means of a Galerkin approximation. From these a priori estimates, we deduce Theorem 2.1. Assume that f is defined by (2.2) and let (u0 , u1 ) ∈ V1 × V−1 . Then, there exists a solution u of (2.5)-(2.6) such that u ∈ L∞ (0, T ; V1 ),

ut ∈ L∞ (0, T ; V−1 ),

∀T > 0,

(2.17)

and u satisfies the energy estimate (2.15). If d = 1 (with no restriction on p), or if d = 2 and p ∈ {0, 1}, then there exists at most one solution u of (2.5)-(2.6) with regularity (2.17).

A SPLITTING METHOD FOR THE CAHN-HILLIARD EQUATION WITH INERTIAL TERM

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Following [17], a solution u of (2.5)-(2.6) with regularity (2.17) and which satisfies the energy estimate (2.15) is called an energy solution. Proof. The proof of existence can be made by standard Galerkin approximation, using the basic a priori estimates (2.8) and (2.15). We refer the reader to [22] (see also [6] for the case d = 1 and [30] for the case d = 3 with Dirichlet-type boundary conditions). Also Theorem 3.5 below provides an energy solution. As far as uniqueness is concerned, the proof is standard in one dimension (see again [6]). If d = 2, uniqueness is proven in [17] by means of a more refined argument which takes advantage of the so-called Br´ezis-Gallouet inequality (cf. [3]) to show the whole Galerkin approximating sequence converge to (the) solution. ¤ Remark 2.2. By (2.2), the map u 7→ f (u) is Lipschitz continuous on bounded subsets of V with values into V 0 . Therefore, if u is a solution of (2.5) which has regularity (2.17), then f (u) ∈ L∞ (0, T ; V−1 ). Hence equation (2.5) has to be understood in D0 (0, T ; V−3 ), with utt ∈ L∞ (0, T ; V−3 ), and (u, ut ) belongs (at least) to C([0, T ]; V−1 × V−3 ) [22]. The initial conditions (2.6) make sense in V−1 × V−3 . Remark 2.3. Existence of an energy bounded solution can also be established in dimension one without any restriction on the growth of f , thanks to the embedding H 1 (Ω) ,→ L∞ (Ω) (see [13]). In dimension two or three, similarly to the semilinear wave equation (cf. [22]), the existence of an energy bounded solution holds when f is a continuous function of polynomially controlled growth satisfying a suitable coercivity condition. In particular, if d = 3 then one has to require that the initial datum u0 is such that F (u0 ) ∈ R, F being an antiderivative of f , unless this is ensured by u0 ∈ V1 . Note that existence holds also for negative times, i.e., with (0, T ) replaced by (−T, 0). Remark 2.4. The existence of more regular (global) solutions can be proven relatively easy in the case d = 1. This can also be done in dimension two for f with cubic controlled growth, though proofs are much more technical (see [17]). If d = 3, existence of smoother (global) solutions has been recently established in [16] for ² small enough and taking (smooth) initial data bounded by a constant which blows up as ² goes to 0. In this case, solutions are such that (at least) u ∈ L∞ (0, T ; V2 ) ,→ L∞ (0, T ; C(Ω)) and no restriction on the growth of f is required. Hence, uniqueness is straightforward [16]. On the contrary, in dimension three (no restriction on ²) for f of cubic controlled growth, uniqueness of energy bounded solutions is still an open problem as well as the existence of smoother solutions. A situation which reminds the well-known case of incompressible Navier-Stokes equations. 3. Weak convergence of the splitting method For the space semidiscretization of (1.1), we introduce (V h )h>0 , a sequence of finite dimensional subspaces of V such that for all h > 0, V h contains the constants; ∪h>0 V

h

is dense in V.

(3.1) (3.2)

The space V h is typically a space of conformal finite elements (see Section 4.1). By assumption (3.1), for every uh ∈ V h , the function u˙ h = u − m(u) belongs to V h . We will set V˙ h := {uh ∈ V h : m(uh ) = 0}. Assumptions (3.1)-(3.2) imply that ∪h>0 V˙ h is dense in V˙

(3.3)

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MAURIZIO GRASSELLI AND MORGAN PIERRE

The space discretization that we use is based on the following formulation of (1.1), which is (formally) equivalent to ²utt + νut w

= =

∆w, −α∆u + f (u).

The space semidiscretization reads: let (uh0 , uh1 ) ∈ V h × V h and find (uh , wh ) : [0, +∞] → V h × V h which satisfies ²(uhtt , ϕ) + ν(uht , ϕ) = h

(w , χ) = h

u (0) =

uh0 ,

−(∇wh , ∇ϕ) ∀ϕ ∈ V h , h

(3.4)

h

h

α(∇u , ∇χ) + (f (u ), χ) ∀χ ∈ V ,

(3.5)

uht (0)

(3.6)

=

uh1 .

Remark 3.1. If ² = 0 and ν = 1 in (3.4)-(3.5), we recover the splitting method introduced by C. M. Elliott et al. for the Cahn-Hilliard equation [7]. It will prove useful to have the matrix version of this scheme. Let (ehi )1≤i≤N h be an orthonormal basis of V h for the L2 (Ω) scalar product, with eh1 = 1/|Ω|1/2 . We look for PN h PN h uh (t) = i=1 ui (t)ehi and wh (t) = i=1 wi (t)ehi . Define the matrix A = (Aij )1≤i,j≤N h , with (A)ij and set



 u1   U =  ...  , uN h

= (∇ehi , ∇ehj ) 

 w1   W =  ...  , wN h

1 ≤ i, j ≤ N h , 

 (f (uh ), eh1 )   .. F h (U ) =  . . h h (f (u ), eN h )

The scheme (3.4)-(3.5) is equivalent to ²Utt + νUt W

= =

−AW,

(3.7) h

αAU + F (U ).

(3.8)

Using (3.8) in (3.7), we see that U satisfies ²Utt + νUt + αA2 U + AF h (U ) = 0. 2

(3.9) +

h

In particular, there exists a unique maximal solution U ∈ C ([0, T ), V ) which satisfies (3.9) and (3.6). We want to derive a priori estimates similar to (2.8) and (2.14). First, letting ϕ = 1/|Ω| in (3.4), we see that m(uh ) satisfies the differential equation ²m(uhtt ) + νm(uht ) = 0, so that m(uh (t)) = m(uh0 ) + m(uh1 )

ν ´ ²³ 1 − exp(− t) , ν ²

ν m(uht (t)) = m(uh1 ) exp(− t), ²

(3.10) (3.11)

if ν > 0, and m(uh (t)) = m(uh0 ) + tm(uh1 ), m(uht (t)) = m(uh1 ), (3.12) if ν = 0. The matrix A is not invertible, because of the constants. For this purpose, we define A˙ = (Aij )2≤i,j≤N h , (3.13) so that A˙ is a symmetric positive definite matrix. For a vector h

X = (x1 , . . . , xN h ) ∈ RN , h we denote X˙ = (x2 , . . . , xN h ) ∈ RN −1 ; in particular, we have

U = (u1 , U˙ ) and F˙ h (u1 , U˙ ) = (f (uh ), ehi )2≤i≤N h .

A SPLITTING METHOD FOR THE CAHN-HILLIARD EQUATION WITH INERTIAL TERM

7

Equation (3.9) implies that ²U˙ tt + ν U˙ t + αA˙ 2 U˙ + A˙ F˙ h (u1 , U˙ ) = 0.

(3.14) t ˙ −1 ˙ Multiplying (3.14) by Ut A , we find the identity ² d ˙ t ˙ −1 ˙ α d ˙t ˙ ˙ (U A Ut ) + ν U˙ tt A˙ −1 U˙ t + (U AU ) + U˙ tt F˙ h (u1 , U˙ ) = 0, (3.15) 2 dt t 2 dt which is a discrete version of (2.10). In view of this, we introduce the discrete energy E h : V h × V h → R by setting Z α ² F (uh ), E h (uh , v h ) = |u˙ h |21 + |v˙ h |2−1,h + (3.16) 2 2 Ω where | · |−1,h is the Euclidean norm on V˙ h defined by h

|v˙ h |2−1,h

:= V˙ t A˙ −1 V˙ ,

h

v =

N X

vi ehi ,

V˙ = (v2 , . . . , vN h )t .

(3.17)

i=2

Notice that |v˙ h |−1,h is a discrete version of |v˙ h |−1 but it is not equal, because we have not used a spectral approximation. We thus have the discrete version of Theorem 2.1: Proposition 3.2. Let (uh0 , uh1 ) ∈ V h × V h . There exists a unique solution (uh , wh ) of (3.4)(3.6) such that (uh , wh ) ∈ C 2 ([0, +∞); V h × V h ). Moreover, Z t h h h E (u (t), ut (t)) + ν |uht |2−1,h ≤ E h (uh0 , uh1 )ec3 |m(u1 )|t 0

+

c4 |Ω| c3 |m(u1 )|t (e − 1), c3

∀t ≥ 0.

(3.18)

Proof. By the Cauchy-Lipschitz Theorem, equation (3.9) has a unique maximal solution U ∈ C 2 ([0, T + ); V h ) which satisfies the initial conditions (3.6). The function wh is uniquely defined by (3.8). We have the a priori estimates (3.11) and (3.12) on m(uh ) and m(uht ), which show that (m(uh ), m(uht )) is bounded in L∞ (0, T ; R × R) for all T ∈ (0, T + ). Next, we notice that Z d (3.19) F (uh ) = (f (uh ), u˙ ht ) + (f (uh ), m(uht )) = U˙ tt F˙ h (u1 , U˙ ) + (f (uh ), m(uht )), dt Ω so that (3.15) reads d h h E (u (t), uht (t)) + ν|u˙ ht |2−1,h = (f (uh ), m(uht )), ∀t ∈ [0, T + ). (3.20) dt By (2.13) and (3.11)-(3.12), µ Z ¶ d h h h 2 h h h E (u (t), ut (t)) + ν|u˙ t |−1,h ≤ |m(u1 )| c3 F (u ) + c4 |Ω| , ∀t ∈ [0, T + ). dt Ω Gronwall’s lemma gives the estimate (3.18), for all t ∈ [0, T + ). This implies, since v 7→ R F (v) is bounded from below on V h (recall (2.16)), that (u˙ h , u˙ ht ) ∈ L∞ ([0, T ], V˙ h × V˙ h ), for Ω all T ∈ (0, T + ). Thus, (uh , uht ) ∈ L∞ (0, T ; V h × V h ), for all T ∈ (0, T + ), and T + = +∞. ¤ We can also state: Theorem 3.3. Let (uh0 , uh1 ) ∈ V h × V h , and let (uh , wh ) ∈ C 2 ([0, +∞); V h × V h ) be the solution of (3.4)-(3.6) given by Proposition 3.2. If ν > 0, then (uh (t), uht (t), wh (t)) → (uh,? , 0, w? ), where (u

h,?

, 0, w

h,?

h

as t → +∞,

h

) ∈ V × V × R is a steady state for (3.4)-(3.5), i.e., a solution of α(∇uh,? , ∇χ) + (f (uh,? ), χ) = (wh,? , χ),

∀χ ∈ V h .

(3.21)

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MAURIZIO GRASSELLI AND MORGAN PIERRE

Remark 3.4. Notice that when ν = 0, one cannot expect convergence to equilibrium, due to the energy conservation. Moreover, it is worth recalling that, because of the translation invariance of equation (1.1) and because of the periodic boundary conditions, the set of equilibria for the continuous problem (2.5)-(2.6) can be a continuum even in one dimension. For the space semidiscretized problem (3.4)-(3.6), the number of stationary states is expected to be very large in general. Thus the convergence to a single stationary state is not a trivial consequence of the dissipative nature of equation (1.1). Ã ojasiewicz inequality for analytic Proof. Convergence to equilibrium is a consequence of the L nonlinearities. When m(uh1 ) = 0, we can apply [18, Theorem 1.1] to the ODE (3.14), which is a second-order gradient-like system. In the general case m(uh1 ) 6= 0, the system is no longer gradient-like. Then we have to use the fact that m(uh (t)) converges exponentially fast to a constant as t → +∞ and arguing as in the proofs of [18, Theorem 1.1] and [15, Theorem 4.2], we obtain a similar result. For the reader’s convenience, we sketch the proof. We assume, just for the sake of simplicity, that ² = ν = α = 1. Equation (3.11) reads m(uh (t)) = m(uh0 ) + m(uh1 ) − m(uh1 ) exp(−t),

m(uht (t)) = m(uh1 ) exp(−t),

(3.22)

so that (3.20) and (2.13) imply

µ ¶ d h h h h 2 h h h h E (u (t), ut (t)) + |u˙ t |−1,h ≤ |m(u1 )| exp(−t) c3 E (u (t), ut (t)) + c4 |Ω| , dt

∀t ≥ 0.

(3.23) R∞ Using Gronwall’s lemma and the estimate 0 exp(−t)dt ≤ 1, we find that, for all t ≥ 0, Z t ¡ ¢ h h h E (u (t), ut (t)) + |u˙ ht (τ )|2−1,h dτ ≤ |m(uh1 )| c3 E h (uh0 , uh1 ) + c4 |Ω| exp(c3 |m(uh1 )|) . 0

(3.24) In particular, by (2.16) and (3.22), (uh (t), uht (t)) ∈ L∞ (0, +∞; V h × V h ). Moreover, we have uht (t) → 0 in V h , as t → +∞.

R +∞

(3.25)

Indeed, by (3.24) 0 |u˙ ht |2−1,h < +∞, and we know that t 7→ |u˙ ht (t)|2−1,h is uniformly continuous on [0, +∞), since d h2 |u˙ | (t) = 2U˙ ttt (t)A˙ −1 U˙ t (t), dt t −1,h

∀t ≥ 0,

and U˙ tt (t), defined by (3.14), is bounded in RN −1 , uniformly in t ∈ [0, +∞). Let now (tn )n≥0 be a sequence such that tn → +∞ and uh (tn ) → uh,? . Then, letting n → +∞, for the solution of the ODE (3.9) with initial conditions (uh (tn ), uht (tn )), and using (3.25), we find that uh,? is a steady state, i.e., there exists wh,? ∈ R such that (uh,? , wh,? ) satisfies (3.21). Now, we notice that m(uh (t)) → m(uh0 ) + m(uh1 ) (see (3.22)), so we introduce the useful change of variable Z y h h ˜ ˜ f (y) = f (m(u0 ) + m(u1 ) + y), and F (y) = f˜(σ)dσ. h

0

We also define the functional G(V˙ ) = and we observe that

1 ˙t ˙ ˙ V AV + 2

Z

h

Ω

N X F˜ ( vi ehi ),

∀V˙ = (v2 , . . . , vN h )t ∈ RN

i=2

∇V˙ G(V˙ ) = A˙ V˙ + F˜˙ h (V˙ ),

∀V˙ ∈ RN

h

−1

,

h

−1

,

(3.26)

A SPLITTING METHOD FOR THE CAHN-HILLIARD EQUATION WITH INERTIAL TERM

where

9

h

N X F˜˙ h (V˙ ) = (f˜( vj ehj ), ehi )2≤i≤N h . j=2

With these notations, equation (3.14) can be rewritten into the form ˙ U˙ tt + U˙ t + A˙ 2 U˙ + A˙ F˜˙ h (U˙ ) = AH(t), with

´ ³ ¢ ¡ H(t) = f˜(u˙ h (t)) − f˜ u˙ h (t) − m(uh1 ) exp(−t) , ehi

2≤i≤N h

(3.27) ,

∀t ≥ 0.

Since uh ∈ L∞ (0, +∞; V h ) and f˜ is locally Lipschitz continuous, we have |H(t)| ≤ c5 exp(−t),

∀t ≥ 0,

(3.28)

for some constant c5 > 0 (here, |·| denotes the Euclidean norm of a vector). Following [15, 18], we define, for all t ≥ 0, 1 M(t) = U˙ t (t)t A˙ −1 U˙ t (t) + G(U˙ (t)) − G(U˙ ? ) + β[A˙ −1 U˙ t (t)]t [A˙ U˙ (t) + F˜˙ h (U˙ (t))] + c6 exp(−t), 2 with β > 0 small and c6 > 0 large, to be chosen later on. By computation, we find M0 (t) = U˙ t (t)A˙ −1 U˙ (t) + U˙ t (t)[A˙ U˙ (t) + F˜˙ h (U˙ (t))] + β[A˙ −1 U˙ (t)]t [A˙ U˙ (t) + F˜˙ h (U˙ (t))] tt

t

tt

t

+β[A˙ −1 U˙ t (t)]t [A˙ U˙ t (t) + ∇F˜˙ h (U˙ (t)) · U˙ t (t)] − c6 exp(−t). Replacing U˙ tt by its value from (3.27), we obtain M0 (t) =

−U˙ tt (t)A˙ −1 U˙ t (t) + U˙ tt (t)H(t) − β|A˙ U˙ (t) + F˜˙ h (U˙ (t))|2 −β[A˙ −1 U˙ (t)]t [A˙ U˙ (t) + F˜˙ h (U˙ (t))] + βH t (t)[A˙ U˙ (t) + F˜˙ h (U˙ (t))] t

+β[A˙ −1 U˙ t (t)]t [A˙ U˙ t (t) + ∇F˜˙ h (U˙ (t)) · U˙ t (t)] − c6 exp(−t). Now, we use the fact that U˙ (t) and U˙ t (t) are uniformly bounded on [0, +∞), the estimate (3.28) and Young’s inequality, in order to obtain, for c6 > 0 large enough, −M0 (t) ≥

β β U˙ tt (t)A˙ −1 U˙ t (t)) + |A˙ U˙ (t) + F˜˙ h (U˙ (t))|2 − |A˙ −1 U˙ t (t)|2 2µ 2 ¶ ˙ h ˙ −1 ˙ ˙ ˙ ˜ ˙ −β|A Ut (t)||Ut (t)| kAk + sup k∇F (U (t))k . t≥0

Since all norms are equivalent in finite dimension, for β > 0 small enough, we have β −M0 (t) ≥ N 2 (t), ∀t ≥ 0. (3.29) 2 where N 2 (t) := |U˙ t (t)|2 + |A˙ U˙ (t) + F˜˙ h (U˙ (t))|2 , ∀t ≥ 0. h Let U ? = (u?1 , U˙ ? ) ∈ RN denote the vector associated to the limiting state uh,? . By (3.25), M(tn ) → 0 as n → +∞. By (3.29), t 7→ M(t) is nonincreasing, so M(t) ≥ 0 for all t ≥ 0. If M(t¯) = 0 for some t¯ ≥ 0, then M(t) = 0 for all t ≥ t¯, and U˙ is constant, by (3.29). Thus, we may assume that M(t) > 0 for all t ≥ 0. Now, we notice that the function G is a polynomial of the variables (v2 , . . . , vN h ), so that we can apply the L Ã ojasiewicz inequality in its original form [24, 25]: there exist ρ ∈ (0, 1/2) and η > 0 such that h ∀V˙ ∈ RN −1 , |V˙ − U˙ ? | ≤ η ⇒ |G(V˙ ) − G(U˙ ? )|1−ρ ≤ |A˙ V˙ + F˜˙ h (V˙ )|. This inequality implies that if t ≥ 0 satisfies |U˙ (t) − U˙ ? | ≤ η, |U˙ t (t)| ≤ 1

and

N (t) > exp(−(1 − ρ)t),

10

MAURIZIO GRASSELLI AND MORGAN PIERRE

then M(t)1−ρ ≤ c7 N (t), for some constant c7 > 0 independent of t. As a consequence, we deduce −

d ρβ M(t)ρ = −ρM0 (t)M(t)ρ−1 ≥ N (t). dt 2c7

Arguing as in [15, p. 55], we infer from this estimate that, for n large enough, Z +∞ Z +∞ ˙ |Ut (t)|dt ≤ N (t)dt < +∞. tn

tn

Thus, U˙ (t) has a limit as t → +∞, which is necessarily U˙ ? , and this concludes the proof. ¤ The energy estimate (3.18) implies convergence of the discrete solution to a solution of the continuous problem, as h → 0. Theorem 3.5. Let (uh0 , uh1 )h>0 in V h × V h such that ¢ ¡ (m(uh0 ), m(uh1 ))h>0 and E h (uh0 , uh1 ) )h>0 are bounded, and let uh ∈ C 2 ([0, +∞); V h ) be the solution of (3.4)-(3.6) given by Proposition 3.2. Then, up to a subsequence, uh → u weakly ? in L∞ (0, T ; V ), uh → u strongly in C([0, T ]; H),

∀T > 0, ∀T > 0,

where u is an energy solution of (2.5)-(2.6). ¡ ¢ Proof. Let T > 0. By (3.11) or (3.12), (m(uh ))h>0 and (m(uht ) h>0 are bounded in L∞ (0, T ). By (3.18), (u˙ h )h>0 is bounded in L∞ (0, T ; V ), and (|u˙ ht |−1,h )h>0 is bounded in L∞ (0, T ). Let z˙ h : [0, T ] → V˙ h be defined by (∇z˙ h (t), ∇ϕh ) = (uht , ϕh ),

∀ϕh ∈ V˙ h .

(3.30)

With the notations introduced above, we have ˙ Z(t) = A˙ −1 U˙ t (t), with z˙ h (t) =

PN h

h i=2 zi (t)ei

t ≥ 0,

and Z˙ = (z2 , . . . , zN h ). In particular, we have |z˙ h |21 = Z˙ t A˙ Z˙ = U˙ tt A˙ −1 U˙ t = |u˙ h |2−1,h ,

and (z˙ h )h>0 is bounded in L∞ (0, T ; V ). This implies that (uh )h>0 is precompact in the space C([0, T ]; H). Indeed, (uh )h>0 is uniformly bounded from [0, T ] with values in V1 , and V1 is precompact in H. Moreover, for all 0 ≤ τ ≤ t ≤ T , Z t |u˙ h (t) − u˙ h (τ )|20 = 2 (u˙ ht (σ), u˙ h (σ) − u˙ h (τ ))dσ, τ Z t = 2 (∇z˙ h (σ), ∇[u˙ h (σ) − u˙ h (τ )]dσ, τ

≤

4kz˙ h kL∞ (0,T ;V ) ku˙ h kL∞ (0,T ;V ) |t − τ |.

We also have, by (3.11)-(3.12), |m(uh (t)) − m(uh (τ ))| ≤ |m(uh1 )||t − τ |.

A SPLITTING METHOD FOR THE CAHN-HILLIARD EQUATION WITH INERTIAL TERM

11

Thus, the sequence (uh )h>0 is uniformly equicontinuous from [0, T ] with values in H. By Ascoli-Arzel`a’s Theorem, (uh )h>0 is precompact in C([0, T ]; H), as claimed. Thus, up to a subsequence, we have uh → u uh → u f (uh ) → f (u) f (uh ) → f (u) z˙ h → z˙

weakly ? in L∞ (0, T ; V ), strongly in C([0, T ]; H), a.e. in Ω × (0, T ), weakly in Lq (0, T ; Lq (Ω)), weakly ? in L∞ (0, T ; V ),

where q = (2p+2)/(2p+1) > 1. Now, let ϕ˙ ∈ V˙ , and let ϕ˙ h ∈ V h such that ϕ˙ h → ϕ˙ strongly ˙ = (ϕ2 , . . . , ϕN h ) in V˙ (this is possible by assumptions (3.1)-(3.2)). For every h > 0, let Φ h P N ˙ t A˙ −1 , we find with ϕ˙ h = i=2 ϕi ehi . Multiplying (3.14) to the left by Φ ˙ t A˙ −1 U˙ tt + ν Φ˙ t A˙ −1 U˙ t + αΦ ˙ t A˙ U˙ + Φ˙ t F˙ h (u1 , U˙ ) = 0, ²Φ or, equivalently, ²(z˙th , ϕ˙ h ) + ν(z˙ h , ϕ˙ h ) + α(∇uh , ∇ϕ˙ h ) + (f (uh ), ϕ˙ h ) = 0.

(3.31)

Letting h → 0 in (3.31), we find that ²(z˙t , ϕ) ˙ + ν(z, ˙ ϕ) ˙ + α(∇u, ˙ ∇ϕ) ˙ + (f (u), ϕ) ˙ = 0, in D0 (0, T ), i.e., in the sense of distributions in (0, T ). Letting h → 0 in (3.30), we see that (∇z, ˙ ∇ϕ) ˙ = (u˙ t , ϕ) ˙ in D0 (0, T ). Thus, u˙ t = A˙ z˙ belongs to L∞ (0, T ; V˙ −1 ), and ²(A˙ −1 u˙ tt , ϕ) ˙ + ν(A˙ −1 u˙ t , ϕ) ˙ + α(∇u, ˙ ∇ϕ) ˙ + (f (u), ϕ) ˙ = 0, in D0 (0, T ). This is true for all ϕ˙ ∈ V˙ , so ²A˙ −1 u˙ tt + ν A˙ −1 u˙ t + αA˙ u˙ + P˙ (f (u)) = 0,

in D0 (0, T ).

˙ we find Applying A, ²u˙ tt + ν u˙ t + αA˙ 2 u˙ + A˙ ◦ P˙ (f (u)) = 0,

in D0 (0, T ).

Moreover, letting h → 0 in (3.10), we obtain ²m(utt ) + νm(ut ) = 0,

in D0 (0, T ).

These two relations show that the function u is a solution of (2.5). The energy estimate (2.15) is obtained by letting h → 0 in (3.18). ¤ Remark 3.6. If the energy solution u with initial data (u0 , u1 ) ∈ V × V 0 is unique, uh0 → u0 weakly in V, (m(uh1 ), |uh1 |−1,h )h>0 is bounded in R2 and (uh1 , ϕh ) → (u1 , ϕ) when ϕh → ϕ strongly in V, then in Theorem 3.5, the whole sequence uh tends to u as h → 0. 4. Error estimates We have seen that a solution uh of the semidiscrete scheme (3.4)-(3.5) converges to an energy solution u of (3.4), in the sense specified in Theorem 2.1. In this section, we want to be more specific by proving error estimates when u is assumed to be regular enough.

12

MAURIZIO GRASSELLI AND MORGAN PIERRE

4.1. Further notation and some useful estimates. For this purpose, we let {T h }h>0 be a quasiuniform family of conforming polygonal decompositions of Ω. Every decomposition T h is composed of d-simplices uniquely (i.e., triangles if d = 2 and tetrahedrons if d = 3) or of d-parallelepipeds uniquely (i.e., rectangles if d = 2 and parallelepipeds if d = 3) (see [5, 29, 8] for details); the decomposition takes into account the periodic boundary conditions, so that T h is in fact a triangulation of T. We associate to T h = ∪T ∈T h T the conformal finite element space of lowest order, P1 or Q1 (see for instance [8]): if the reference element of T h is a d-simplex, then n o h ∈ P1 (T ) ∀T ∈ T h V h := v h ∈ C 0 (T) : v|T and if the reference element of T h is a d-parallelepiped, n h V h := v h ∈ C 0 (T) : v|T ∈ Q1 (T )

o ∀T ∈ T h .

If d = 1, T is simply a subdivision of [0, L] and the notions P1 and Q1 coincide. It is well-known that V h has finite dimension, that V h ⊂ V and that V h satisfies assumption (3.1)-(3.2) [5]. Since the family {T h }h>0 is quasiuniform, we have the inverse estimates kv h k1 h

kv kL∞ (Ω)

≤

C0 h−1 kv h k0 ,

∀v h ∈ V h ,

≤

−d/2

h

C0 h

h

kv k0 ,

(4.1) h

∀v ∈ V ,

(4.2)

where C0 is independent of h. In fact, C0 depends only on the family {T h }h>0 . We now introduce the operator of orthogonal projection P h : L2 (Ω) → V h

(4.3)

for the L2 (Ω) scalar product, and the operator of orthogonal projection Πh : V1 → V h h

for the V1 scalar product, i.e., for u ∈ V , Π u ∈ V

h

(4.4)

is uniquely defined by

m(Πh u) = m(u) and (∇Πh u, ∇ϕh ) = (∇u, ∇ϕh ),

∀ϕh ∈ V h .

(4.5)

Obviously, we have kP h vk0

≤ kvk0 ,

∀v ∈ V0 ,

(4.6)

h

≤ kvk1 ,

∀v ∈ V1 .

(4.7)

kΠ vk1

The standard L2 and H 1 error estimates for Πh (see, e.g., [5]) state that there exists a constant C1 > 0 which depends only on the family {T h }h>0 such that, for r ∈ {1, 2}, kΠh u − uk0 + hkΠh u − uk1 ≤ C1 hr |u|r ,

∀u ∈ Vr .

(4.8)

Estimates (4.8) are obtained via Cea’s Lemma and duality arguments from similar estimates for the nodal interpolate I h u of u. More precisely, for every u ∈ C(Ω), let I h u denote the unique function in V h such that I h u(xi ) = u(xi ) for every node xi of the triangulation T h . Since 1 ≤ d ≤ 3, the Sobolev inequality yields V2 ⊂ C(T) and there exists a constant C10 which depends only on the family {T h }h>0 such that ku − I h uk0 + hku − I h uk1 ≤ C10 h2 kuk2 ,

∀u ∈ V2 .

0,γ

(4.9) 0,γ

We will also use the inclusion V2 ⊂ C (T) for some γ ∈ (0, 1], where C (T) denotes the usual H¨older space. We can choose γ = 1 if d = 1, γ ∈ (0, 1) if d = 2, and γ = 1/2 if d = 3. In particular, we have ku − I h ukL∞ (Ω) ≤ C2 hγ kuk2 , for some constant C2 = C2 (Ω, d).

∀u ∈ V2 ,

(4.10)

A SPLITTING METHOD FOR THE CAHN-HILLIARD EQUATION WITH INERTIAL TERM

13

4.2. Discrete negative seminorms. It will be useful to have a more general definition of the negative norm used in Section 3. Following [32], let T h : L˙ 2 (Ω) → V˙ h be the linear operator defined by T h f = uh where, for f ∈ L˙ 2 (Ω), uh ∈ V˙ h solves (∇uh , ∇v h ) = (f, v h ), h

∀v h ∈ V˙ h . h

˙ −1

(4.11) h

In terms of the operators introduced above, T = Π A so that T is a discrete version of ∆−1 . We note that T h is selfadjoint and positive semidefinite on L˙ 2 (Ω) since (g, T h f ) = (∇T h g, ∇T h f ) = (f, T h g),

∀f, g ∈ L˙ 2 (Ω),

and

(f, T h f ) = |∇T h f |20 ≥ 0, ∀f ∈ L˙ 2 (Ω). Moreover, T h is positive definite on V˙ h for the L2 -scalar product: if (ehi )1≤i≤N h is the orthonormal basis of V h introduced above, with eh1 a constant, then A˙ −1 (recall (3.13)) is the matrix of T h h in the basis (eh )2≤i≤N h of V˙ h . i

|V˙

By spectral theory, it is possible to define, for any s > 0, the selfadjoint operator (T h )s . We define for s ∈ N the discrete negative seminorm |v|−s,h = |(T h )s/2 v|20 = ((T h )s v, v)1/2 , ∀v ∈ L˙ 2 (Ω), (4.12) which is an Euclidean norm on V˙ h (for s = 0, (T h )0 = I and |v|−s,h = |v|0 ). In fact, if PN h vi eh , and V˙ = (v2 , . . . , vN h )t , then v˙ h ∈ V˙ h , with v˙ h = i=2

i

|v˙ h |2−s,h = V˙ t A˙ −s V˙ . For s = 1, we recover the norm | · |−1,h defined previously (see (3.17)). As a useful shortcut, we also define the seminorm ¡ ¢1/2 kvk−s,h = m(v)2 + |v|2−s,h , ∀v = (m(v), v) ˙ ∈ L2 (Ω),

(4.13)

h

which is an Euclidean norm on V . The discrete seminorm k · k−s,h is equivalent to the corresponding continuous negative norm k · k−s , modulo a small error: Lemma 4.1. There exists a constant C3 independent of h such that, for s ∈ {1, 2}, kvk−s,h ≤ C3 (kvk−s + hs kvk0 ) and kvk−s ≤ C3 (kvk−s,h + hs kvk0 ) ,

∀v ∈ L2 (Ω).

Proof. We follow the proof of Lemma 5.3 in [32]. It is sufficient to prove the assertion for ˙ all v ∈ L(Ω). Let T = A˙ −1 . For s = 1 and v ∈ L˙ 2 (Ω), we have |v|2−1,h = (T h v, v) = (T v, v) + (T h v − T v, v) ≤ |v|2−1 + C1 h2 |T v|2 |v|0 , using T h v = Πh T v and (4.8). By elliptic regularity, |T v|2 ≤ C4 |v|2 for some constant C4 independent of v, so |v|2−1,h ≤ |v|2−1 + C1 C4 h2 |v|20 . This proves the first inequality for s = 1. For s = 2 and for v ∈ L˙ 2 (Ω), we have |v|−2,h = |T h v|0 ≤ |T v|0 + |T h v − T v|0 ≤ |v|−2 + C1 C4 h2 |v|0 . The second inequality is obtained by interchanging the roles of T h and T .

¤

We point out some other inequalities related to these negative seminorms. First, recall the Poincar´e inequality: |v| ˙ 0 ≤ cP |v| ˙ 1 , ∀v˙ ∈ V˙ , (4.14) −1/2

where cP = λ1

. By induction, we have

Lemma 4.2. Let s ∈ N. Then |v| ˙ −s−1,h ≤ cP |v| ˙ −s,h ,

∀v˙ ∈ L˙ 2 (Ω).

(4.15)

14

MAURIZIO GRASSELLI AND MORGAN PIERRE

These estimates imply in particular that kvk−1,h ≤ c0P kvk0 where

c0P

and

kvk−2,h ≤ (c0P )2 kvk0 ,

∀v ∈ L2 (Ω),

(4.16)

= max(1, cP ).

˙ Then, by definition (4.11) of T h and by the Poincar´e inequality (4.14), Proof. Let f ∈ L(Ω). h 2 (T h f, f ) = (∇T h f, ∇T h f ) = |T h f |21 ≥ c−2 P |T f |0 .

On the other hand, (T h f, f ) ≤ |T h f |0 |f |0 so that eliminating |T h f |0 , we have |T h f |0 = |f |−2,h ≤ c2P |f |0 .

(4.17)

Similarly, we get |f |2−1,h = |(T h )1/2 f |20 = (T h f, f ) ≤ |T h f |0 |f |0 . Using (4.17), we obtain |f |−1,h ≤ cP |f |0 , (4.18) 2 h s/2 ˙ which is estimate (4.15) for s = 0. Now, let v˙ ∈ L (Ω), and choose f = (T ) v˙ in (4.18). This reads |(T h )s/2 v| ˙ −1,h = |v| ˙ −s−1,h ≤ cP |(T h )s/2 v| ˙ 0 = cP |v| ˙ −s,h , and the proof is complete. ¤ Analogously, we can deduce from the inverse estimate (4.1) some inverse estimates for the discrete negative seminorms. Lemma 4.3. Let s ∈ N. Then there holds kv h k−s,h ≤ C0 h−1 kv h k−s−1,h ,

∀v h ∈ V h .

Proof. Let (ehi )1≤i≤N h denote as previously an orthonormal basis of V h for the L2 (Ω) scalar PN product, with eh1 a constant, and let A˙ be as previously. Let v˙ ∈ V˙ h , v˙ = i=2 vi ehi and V˙ = (v2 , . . . , vN h )t . The inverse estimate (4.1) reads V˙ t A˙ V˙ ≤ C02 h−2 V˙ t V. Replacing V˙ by A˙ −(s+1)/2 V˙ in this estimate, we have V˙ t A˙ −s/2 V˙ ≤ C02 h−2 V˙ t A˙ −(s+1) V˙ , that is, |v| ˙ −s,h ≤ C0 h−1 |v| ˙ −s−1,h . The result follows immediately.

¤

4.3. Error estimates. We are now ready to derive error estimates on a finite time interval [0, T ] (T > 0). Assume that u is a solution of (2.5)-(2.6) on [0, T ] which is regular enough, and let uh be a solution of the discretized scheme (3.4)-(3.6) on [0, T ]. In order to estimate the error kuh − uk, we define, following [7, 32] uh − u = θu + ρu , with θu = uh − Πh u, h

w −w u

w

w

w

h

h

= θ + ρ , with θ = w − Π w,

ρu = Πh − u, w

(4.19)

h

ρ = Π w − w.

w

(4.20) u

The estimates on ρ and ρ follow from (4.8). In the next lemma, we estimate θ and θw . Lemma 4.4. Let u be a solution of (2.5) such that u ∈ L∞ (0, T ; L∞ (Ω)),

u ∈ L2 (0, T ; V3 ),

ut , utt ∈ L2 (0, T ; V1 ),

(4.21)

and let uh ∈ C 2 ([0, T ]; V h ) be a solution of (3.4)-(3.5). Then the following inequality holds ´ d ³ ˙u 2 ²|θt |−2,h + α|θ˙u |20 + ν|θ˙tu |2−2,h ≤ 3|θ˙tu |2−2,h + ²2 |ρ˙ utt |2−2,h + ν|ρ˙ ut |2−2,h dt +kf (uh ) − f (u)k20 + |ρ˙ w |20 , (4.22)

A SPLITTING METHOD FOR THE CAHN-HILLIARD EQUATION WITH INERTIAL TERM

15

in D0 (0, T ), and u ²m(θtt ) + νm(θtu ) = 0,

in D0 (0, T ).

(4.23)

Proof. Let u be a solution of (2.5) which is regular enough. Then, introducing the so-called “chemical potential” w = αAu + f (u),

(4.24)

= −(∇w, ∇ϕ), ∀ϕ ∈ V, = α(∇u, ∇χ) + (f (u), χ),

(4.25) (4.26)

we have ²(utt , ϕ) + ν(ut , ϕ) (w, χ)

∀χ ∈ V.

Subtracting (4.25) from (3.4), and using the definitions (4.19) and (4.20), we obtain u , ϕh ) + ν(θtu , ϕh ) = −(∇θw , ∇ϕh ) − ²(ρutt , ϕh ) − ν(ρut , ϕh ), ²(θtt

∀ϕh ∈ V h .

(4.27)

Choosing ϕh = 1 in (4.27), and using (ρu (t), 1) = 0 for all t ≥ 0, we obtain (4.23). Subtracting (4.26) from (3.5), we obtain similarly (θw , χh ) = α(∇θu , ∇χh ) + (f (uh ) − f (u), χh ) − (ρw , χh ),

∀χh ∈ V h .

(4.28)

Definition (4.3) of P h shows that, for all ϕh , χh ∈ V h (ρu , ϕh ) = (P h (ρu ), ϕh ) and (f (uh ) − f (u), χh ) = (P h (f (uh ) − f (u)), χh ). Using the matrix notation introduced in Section 3, equations (4.27)-(4.28) are equivalent to ²Θutt + νΘut Θw

= =

u −AΘw − ²Rtt − νRtu , αAΘu + N Lu − Rw ,

where Θu and Θw are the vectors associated with θu and θw , respectively, Ru is the vector associated with P h (ρu ), Rw is the vector associated with P h (ρw ), and N Lu is the vector associated with P h (f (uh ) − f (u)). Keeping only the components corresponding to eh2 , . . . , ehN h , this reads ˙ utt + ν Θ ˙ ut ²Θ ˙w Θ

= =

u ˙ w − ²R˙ tt −A˙ Θ − ν R˙ tu , ˙ u + NL ˙ u − R˙ w , αA˙ Θ

(4.29) (4.30)

Next, we multiply (4.30) by −A˙ and we add (4.30), so that u ˙ utt + ν Θ ˙ ut + αA˙ 2 Θ ˙ u = −²R˙ tt ˙ u + A˙ R˙ w . ²Θ − ν R˙ tu − A˙ N L

(4.31)

˙ ut )t A˙ −2 and we use the Cauchy-Schwarz inequality. This yields We multiply (4.31) by (Θ µ ¶ 1 d u 2 u 2 ˙ ˙ ²|θt |−2,h + α|θ |0 + ν|θ˙tu |2−2,h ≤ ²|θ˙tu |−2,h |P h (ρ˙ utt )|−2,h + ν|θ˙tu |−2,h |P h (ρ˙ ut )|−2,h 2 dt + |θ˙u |−2,h |P˙ h (f (uh ) − f (u))|0 + |θ˙u |−2,h |P h (ρ˙ w )|0 . t

t

Finally, we note that |P h v| ˙ −2,h = |v| ˙ −2,h ,

∀v˙ ∈ L˙ 2 (Ω),

since T h (P h v) ˙ = T h v, ˙ we use (4.6) and we apply the Cauchy-Schwarz inequality. This gives the expected estimate (4.22). The regularity (4.21) that is assumed on u implies that the computations used above are valid. In particular, we have f (u) ∈ L∞ (0, T ; V1 ), so that, by definition (4.24), w ∈ L∞ (0, T ; V1 ). In fact, by (4.25) and by elliptic regularity, w ∈ L2 (0, T ; V3 ); using (4.24) and elliptic regularity again [14], we even have u ∈ L2 (0, T ; V5 ). The proof is complete. ¤

16

MAURIZIO GRASSELLI AND MORGAN PIERRE

Remark 4.5. Let (u0 , u1 ) ∈ V2 × V0 . Then, taking advantage of the improved regularity, it is easily seen (see [16]) that there exists at most one solution u of (2.5)-(2.6) such that u ∈ C 0 ([0, T ]; V2 ),

ut ∈ C 0 ([0, T ]; V0 ).

On the other hand, if u is a solution of (2.5) with regularity (4.21), then standard regularity results [22, 31] show that u ∈ C 0 ([0, T ]; V2 ) (recall that V3 b V2 b V1 ) and ut ∈ C 0 ([0, T ]; V1 ). Thus, uniqueness is not an issue in Lemma 4.4 (cf. also Remark 2.4). In the following, C denotes a constant which is independent of h (but which may depend on the other parameters of the problem). Theorem 4.6. Let u be a solution of (2.5) such that u ∈ L2 (0, T ; V3 ), h

2

ut , utt ∈ L2 (0, T ; V2 ),

(4.32)

h

and let u ∈ C ([0, T ]; V ) be a solution of (3.4)-(3.5). If kuh (0) − u(0)k0 ≤ Ch2

and

kuht (0) − ut (0)k−2,h ≤ Ch2 ,

(4.33)

then kuh − ukL2 (0,T ;V0 ) + kuh − ukL∞ (0,T ;V−2 ) ≤ Ch2 , h

ku − ukL∞ (0,T ;V1 ) +

kuht

− ut kL∞ (0,T ;V−1 ) ≤ Ch.

(4.34) (4.35)

Remark 4.7. Note that the existence of a (unique) solution satisfying (4.32) can be guaranteed (at least in the cases d = 1, 2) taking (u0 , u1 ) ∈ V6 × V4 . If d = 3, then ² must be taken small enough and the norm of the initial data should be bounded by a constant depending on ² as specified in [16]. Proof. The regularity assumptions imply that u ∈ C 1 ([0, T ]; V2 ), and in particular, u ∈ L∞ (0, T ; L∞ (Ω)), since V2 ⊂ L∞ (Ω). Define M = kukL∞ (0,T ;L∞ (Ω)) < +∞. We have kuh (0) − u(0)kL∞

≤ kuh (0) − I h u(0)kL∞ + kI h u(0) − u(0)kL∞ , ≤ C0 h−d/2 kuh (0) − I h u(0)k0 + C2 hγ ku(0)k2 , ¡ ¢ ≤ C0 h−d/2 kuh (0) − u(0)k0 + ku(0) − I h u(0)k0 + C2 hγ ku(0)k2 ,

where, in the second line, we have used the estimates (4.2) and (4.10). In particular, by assumption (4.33) and estimate (4.9), we find kuh (0) − u(0)kL∞ (Ω) ≤ Ch2−d/2 + Chγ . h

h

(4.36)

Thus, for h > 0 small enough, ku (0) − u(0)kL∞ (Ω) ≤ 1/2 and ku (0)k ≤ M + 1/2. Now, let T h ∈ (0, T ] be the maximal time such that kuh (t)kL∞ (Ω) ≤ M + 1 for all t ∈ [0, T h ], and let Lf denote the Lipschitz constant of f on the interval [−(M + 1), M + 1]. Then, we have kf (uh (t)) − f (u(t))k0 ≤ Lf kuh (t) − u(t)k0 , Thus, using (4.22) and (4.15), we find ´ d ³ ˙u 2 ²|θt |−2,h + α|θ˙u |20 + ν|θ˙tu |2−2,h dt

∀t ∈ [0, T h ].

≤ 3|θ˙tu |2−2,h + ²2 c4P |ρ˙ utt |20 + νc4P |ρ˙ ut |20 +L2f kuh − uk20 + |ρ˙ w |20 ,

in D0 (0, T h ). Using now

´ ³ ku − uh k20 ≤ 2 |θ˙u |20 + |m(θu )|2 + 2kρu k20 ,

A SPLITTING METHOD FOR THE CAHN-HILLIARD EQUATION WITH INERTIAL TERM

17

together with the standard estimate (4.8), we obtain ´ d ³ ˙u 2 ²|θt |−2,h + α|θ˙u |20 ≤ 3|θ˙tu |2−2,h + 2L2f |θ˙u |20 + ²2 c4P C12 h4 |utt |22 + νc4P C12 h4 |ut |42 dt +2L2f |m(θu )|2 + 2Lf C12 h4 |u|22 + C12 h4 |w|22 , still in D0 (0, T h ). Gronwall’s lemma yields ³ ´ |θ˙tu (t)|2−2,h + |θ˙u (t)|20 ≤ C |θ˙tu (0)|2−2,h + |θ˙u (0)|20 + Ch4 kuk2W 2,2 (0,T ;V2 ) Z t +C |m(θu )|2 ,

(4.37)

0

for all t ∈ [0, T h ], for some constant C which depends on T , α, ², ν, C1 , Lf and cP , but which is independent of h and u. Here, we have denoted kuk2W 2,2 (0,T ;V2 ) = kuk2L2 (0,T ;V2 ) + kut k2L2 (0,T ;V2 ) + kutt k2L2 (0,T ;V2 ) . Notice that, using elliptic regularity and equation (4.25), the regularity which is assumed on u implies that w˙ ∈ L2 (0, T ; V4 ), with ¡ ¢ kwk ˙ L2 (0,T ;V4 ) ≤ C kut kL2 (0,T ;V2 ) + kutt kL2 (0,T ;V2 ) . Assumptions (4.33) on the initial condition, and the regularity u(0), ut (0) ∈ V2 imply kθu (0)k0 ≤ Ch2

and

kθtu (0)k−2,h ≤ Ch2 .

(4.38)

Indeed, we have kθu (0)k0 ≤ kuh (0) − u(0)k + ku(0) − Πh u(0)k ≤ Ch2 , by (4.8), and kθtu (0)k−2,h ≤ kuht (0) − ut (0)k−2,h + kut (0) − Πht (0)k−2,h ≤ Ch2 , by (4.16) and (4.8). We now solve (4.23) (see (2.8)-(2.9)) and we use ²³ ν ´ 1 − exp(− t) ≤ t, ν ² to see that |m(θu (t))| ≤ |m(θu (0))| + t|m(θtu (0))|,

∀t ≥ 0, |m(θtu (t))| ≤ |m(θtu (0))|.

(4.39)

Thanks to (4.38), this gives |m(θu (t))| ≤ Ch2 and |m(θtu (t))| ≤ Ch2 ,

∀t ∈ [0, T ].

(4.40)

Thus, (4.37) becomes |θ˙tu (t)|2−2,h + |θ˙u (t)|20 ≤ Ch4 ,

∀t ∈ [0, T h ],

(4.41)

for some constant independent of h. By an argument similar to the one which gave (4.36), estimates (4.40) and (4.41) imply that kuh (t) − u(t)kL∞ ≤ Ch2−d/2 + Chγ ,

∀t ∈ [0, T h ],

so that for h > 0 small enough, kuh (t) − u(t)kL∞ ≤ 1, and T h = 1. Summing up, we have proved that, for h > 0 small enough, sup kθu (t)k0 + sup kθtu (t)k−2,h ≤ Ch2 .

0≤t≤T

0≤t≤T

We now notice that Lemma 4.1 implies µ ¶ sup kθtu (t)k−2 ≤ C3 sup kθtu (t)k−2,h + h2 sup kθtu (t)k0 , 0≤t≤T

0≤t≤T

0≤t≤T

(4.42)

18

MAURIZIO GRASSELLI AND MORGAN PIERRE

and that the inverse estimate of Lemma 4.3 implies kθtu (t)k0 ≤ C02 h−2 kθtu (t)k−2,h ≤ C,

∀ t ∈ [0, T ],

so that (4.42) implies sup kθtu (t)k−2 ≤ Ch2 .

(4.43)

0≤t≤T

On the other hand, the standard estimate (4.8) yields ¡ ¢ sup kρu (t)k0 + sup kρut (t)k0 ≤ C1 h2 kukL∞ (0,T ;V2 ) + kut kL∞ (0,T ;V2 ) . 0≤t≤T

(4.44)

0≤t≤T

Observe that the regularity assumptions on u imply that kukL∞ (0,T ;V2 ) and kut kL∞ (0,T ;V2 ) are bounded, because the space W 1,2 (0, T ; V2 ), to which both u and ut belong, is continuously embedded in C 0 (Ω×[0, T ]). Estimate (4.34) is now a consequence of the triangular inequality, estimates (4.42), (4.43) and (4.16). In order to obtain estimate (4.35), we apply the inverse estimate (4.1) and Lemma 4.3 to (4.42), which reads sup kθu (t)k1 + sup kθtu (t)k−1,h ≤ Ch.

0≤t≤T

(4.45)

0≤t≤T

From this, we proceed as previously with a triangular inequality to obtain (4.35). The proof is complete. ¤ Remark 4.8. The L2 and H 1 estimates (4.34)-(4.35) for u − uh are optimal in the sense that the error ku − I h uk for the interpolate I h u of u has the same order. Similarly, the H −2 error for ut − uht is not better than O(h2 ) in general; in contrast, the H −1 estimate for ut −I h ut is O(h2 ), so that one could expect kut −uht k−1 = O(h2 ) instead of O(h). But in our problem, we work with the product space H 1 × H −1 , so that the H −1 estimate is dominated by the H 1 estimate. We can say that the error estimates in Theorem 4.6 are optimal for the L2 × H −2 and H 1 × H −1 norms. Remark 4.9 (Homogeneous Dirichlet boundary conditions). The weak convergence result of Theorem 3.5 and the error estimates in Theorem 4.6 can be adapted for the singularly perturbed Cahn-Hilliard equation (1.1) with homogeneous Dirichlet boundary conditions on a convex bounded domain of Rd (1 ≤ d ≤ 3) with smooth boundary. In this case, V1 = H01 (Ω), and since −∆ is positive definite on V1 , there is no need to introduce V˙ 1 , so that proofs are slightly easier. Choosing Ω as a convex domain and P 1 conforming elements allows to consider a triangulation Ωh such that Ωh ⊂ Ω, and consequently to build V h ⊂ V1 with optimal error estimates [32]. However, it is more difficult in this case to find solutions u which satisfy regularity assumptions similar to (4.32), because of the compatibility conditions on the boundary. Remark 4.10 (Homogeneous no-flux boundary conditions). For equation (1.1) endowed with homogeneous no-flux boundary conditions, the situation is more delicate. The space 1 V = Hper (Ω) is replaced by V = H 1 (Ω), where Ω is a bounded domain of Rd with smooth boundary (1 ≤ d ≤ 3), and the spaces Vs , V˙ s are defined accordingly. The weak convergence stated in Theorem 4.6 still holds, but it is more difficult in this case to build a finite element space V h ⊂ V which can actually be computed. In fact, since V h 6⊂ V in general, nonconforming methods or other approaches (see [32]) should be considered in order to adapt the error estimates of Theorem 4.6 in this case. 5. A finite difference version of the splitting scheme Equation (1.1) can easily be discretized in space by a finite difference method; this is important for numerical simulations because it allows the use of the Fast Fourier Transform, which in turn can make 3d computations possible. Our purpose here is to explain how the finite

A SPLITTING METHOD FOR THE CAHN-HILLIARD EQUATION WITH INERTIAL TERM

19

difference scheme can be derived from our finite element scheme (3.4)-(3.5) via a quadrature formula. For this purpose, let h = (hk )1≤k≤d with hk = Lk /Nk where Nk ∈ N? for 1 ≤ k ≤ d. We work with the grid G h = Πdk=1 (Z/Nk Z); in particular, an index (i1 , . . . , id ) ∈ G h can always be chosen such that 1 ≤ i1 < N1 , . . . , 1 ≤ id ≤ Nd . The value u(xi1 , . . . , xid ) of u at a grid point xi1 ,...,id = (i1 h1 , . . . , id hd ) with (i1 , . . . , id ) ∈ G h is approximated by ui1 ,...,id , so that h u is approximated by U = (ui1 ,...,id )(i1 ,...,id )∈G h ∈ RG (for sake of simplicity, we omit the h h index h on U ). The discrete finite difference Laplace operator ∆h : RG → RG is defined as h usually for U = (ui1 ,...,id )(i1 ,...,id )∈G h ∈ RG by the 2d + 1 stencil (∆h U )i1 ,...,id =

d X ui

1 ,...,ik +1,...,id

k=1

− 2ui1 ,...,id + ui1 ,...,ik −1,...,id , h2k

∀(i1 , . . . , id ) ∈ G h .

The most natural finite difference semidiscretization of equation (1.1) with periodic boundary conditions reads ²(ui1 ,...,id )tt + ν(ui1 ,...,id )t + α((∆h )2 U )i1 ,...,id − (∆h (f (U )))i1 ,...,id = 0, h

(5.1) h

for all (i1 , . . . , id ) ∈ G h , where U : [0, T ] → RG is the unknown function, and f : RG → RG is obviously defined by (f (U ))i1 ,...,id = f (ui1 ,...,id ),

h

h

∀ U = (ui1 ,...,id )(i1 ,...,id )∈G h ∈ RG .

Equation (5.1) is a second order ordinary differential system which is completed with initial h h conditions U (0) ∈ RG and Ut (0) ∈ RG . By the Cauchy-Lipschitz Theorem, given initial h conditions U (0) and Ut (0), there exists a unique maximal solution U : [0, T + ) → RG which solves (5.1). Proceeding as in the proof of Proposition 3.2, and replacing the L2 -scalar product (·, ·) by the approximate L2 -scalar product (·, ·)h with trapezoidal rule (see below), it is easily seen that the discrete energy of a solution remains bounded on finite time intervals, and consequently that T + = +∞. If ν > 0 , then it is possible to prove, as in Theorem 3.3, that U (t) tends to a discrete steady state as t tends to +∞. On the other hand, the grid G h gives a decomposition T h of T into N1 × . . . × Nd dparallelepipeds. Let V h be the conforming Q1 finite element space associated to this decomposition T h , as described in Section 4.1. We can consider the finite element scheme (3.4)-(3.5) for this space V h . For the actual computation of (3.4)-(3.5), we introduce the nodal basis (ϕi1 ,...,id )(i1 ,...,id )∈G h associated to G h ' T h , i.e., for all (i1 , . . . , id ) ∈ G h , ϕi1 ,...,id is the unique function in V h such that ( 1 if (j1 , . . . , jd ) = (i1 , . . . , id ), ϕi1 ,...,id (xj1 , . . . , xjd ) = ∀(j1 , . . . , jd ) ∈ G h . 0 otherwise, P P Letting uh = i∈G h ui ϕi , wh = i∈G h wi ϕi , the scheme (3.4)-(3.5) reads ²M h Utt + νM h Ut h

M W

=

−Ah W,

=

h

(5.2) h

αA U + F (U ),

(5.3)

where, for i = (i1 , . . . , id ) ∈ G h and j = (j1 , . . . , jd ) ∈ G h , Ahij = (∇ϕi , ∇ϕj ),

h Mij = (ϕi , ϕj )

and (F h (U ))i = (f (uh ), ϕi ).

(5.4)

Now, assume that the integrals in (5.4) are computed with the trapezoidal rule, and let ˜ h , A˜h denote the resulting matrices and F˜ h (U ) the resulting vector. Recall that for a M parallelepiped P of T h with sides h1 , . . . , hd , and for a function g : T → R, the trapezoidal rule reads Z Πd hk X g(x1 , . . . , xd )dx1 . . . dxd ≈ k=1 g(S), (5.5) 2d P

20

MAURIZIO GRASSELLI AND MORGAN PIERRE

where the sum is over all the 2d vertices S of P . The integral on Ω is computed through Z X Z g(x1 , . . . , xd )dx1 . . . dxd = g(x1 , . . . , xd ), Ω

P ∈T h

P

where the integral on every d-parallelepiped P is approximated by (5.5). For two functions u, v ∈ L2 (Ω), the L2 -scalar with trapezoidal rule (u, v)h is obtained by replacing g(x) with u(x)v(x) in the integral above, and using the trapezoidal rule on each parallelepiped. Then, a straightforward computation shows that ˜ ij = (Πdk=1 hk )δij and (F˜ h (U ))i = (Πdk=1 hk )f (ui ), M for all i, j ∈ G h , and δij is the Kronecker symbol. Another computation entails that A˜h ' −(Πdk=1 hk )∆h . The scheme (5.2)-(5.3) becomes then ²Utt + νUt

= ∆h W,

(5.6)

W

h

(5.7)

= −α∆ + f (U ).

Eliminating W , this system is equivalent to ²Utt + νUt + α(∆h )2 U − ∆h (f (U )) = 0,

(5.8)

and we recover the finite difference scheme (5.1), as claimed. Notice that, because of the quadrature formula, it is not clear whether the convergence result of Theorem 3.5 or whether the error estimates of Theorem 4.6 still apply to the scheme (5.8). This is an interesting open question related to the numerical analysis of nonconforming methods for this equation (see Remark 4.10). Acknowledgment This work was partially supported by the Partenariat Hubert Curien Galil´ee “Mod`eles math´ematiques en sciences des mat´eriaux”. References [1] F. Bai, C. M. Elliott, A. Gardiner, A. Spence, and A. M. Stuart. The viscous Cahn-Hilliard equation. I. Computations. Nonlinearity, 8(2):131–160, 1995. [2] A. Bonfoh, M. Grasselli, and A. Miranville. Singularly perturbed 1D Cahn-Hilliard equation revisited. Submitted. [3] H. Brezis and T. Gallouet. Nonlinear Schr¨ odinger evolution equations. Nonlinear Anal., 4(4):677–681, 1980. [4] J. W. Cahn and J. E. Hilliard. Free energy of a non-uniform system I. Interfacial free energy. J. Chem. Phys., 2: 258–267, 1958. [5] P. G. Ciarlet. The finite element method for elliptic problems. North-Holland Publishing Co., Amsterdam, 1978. Studies in Mathematics and its Applications, Vol. 4. [6] A. Debussche. A singular perturbation of the Cahn-Hilliard equation. Asymptotic Anal., 4(2):161–185, 1991. [7] C. M. Elliott, D. A. French, and F. A. Milner. A second order splitting method for the Cahn-Hilliard equation. Numer. Math., 54(5):575–590, 1989. ´ ements finis: th´ [8] A. Ern and J.-L. Guermond. El´ eorie, applications, mise en œuvre, volume 36 of Math´ ematiques & Applications (Berlin). Springer-Verlag, Berlin, 2002. [9] P. Galenko and D. Jou. Diffuse-interface model for rapid phase transformations in non-equilibrium systems. Phys. Rev. E, 71:046125 (13 pages), 2005. [10] P. Galenko and V. Lebedev. Analysis of the dispersion relation in spinodal decomposition of a binary system. Philos. Mag. Lett., 87:821–827, 2007. [11] P. Galenko and V. Lebedev. Local nonequilibrium effect on spinodal decomposition in a binary system. Int. J. Thermodyn., 11:21–28, 2008. [12] P. Galenko and V. Lebedev. Nonequilibrium effects in spinodal decomposition of a binary system. Phys. Lett. A, 372:985–989, 2008.

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