ENERGY STABLE AND CONVERGENT FINITE ELEMENT SCHEMES FOR THE MODIFIED PHASE FIELD CRYSTAL EQUATION

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ELEMENT SCHEMES FOR THE MODIFIED PHASE FIELD CRYSTAL EQUATION

Maurizio Grasselli, Morgan Pierre

To cite this version:

Maurizio Grasselli, Morgan Pierre. ENERGY STABLE AND CONVERGENT FINITE ELEMENT SCHEMES FOR THE MODIFIED PHASE FIELD CRYSTAL EQUATION. 2015. �hal-01118961�

MODIFIED PHASE FIELD CRYSTAL EQUATION

MAURIZIO GRASSELLI AND MORGAN PIERRE

Abstract. We propose a space semi-discrete and a fully discrete finite element scheme for the modified phase field crystal equation (MPFC). The space discretiza- tion is based on a splitting method and consists in a Galerkin approximation which allows low order (piecewise linear) finite elements. The time discretization is a second-order scheme which has been introduced by Gomez and Hughes for the Cahn-Hilliard equation. The fully discrete scheme is shown to be unconditionally energy stable and uniquely solvable for small time steps, with a smallness condi- tion independent of the space step. Using energy estimates, we prove that in both cases, the discrete solution converges to the unique energy solution of the MPFC equation as the discretization parameters tend to 0. Using a Lojasiewicz inequal- ity, we also establish that the discrete solution tends to a stationary solution as time goes to infinity. Numerical simulations illustrate the theoretical results.

Keywords: Finite elements, second-order schemes, gradient-like systems, Lojasiewicz inequality.

MSC 2010: 65M60, 65P40, 74N05, 82C26.

1. Introduction

In this paper, we analyze finite element discretizations of the modified phase field crystal (MPFC) equation

βu_tt+u_t= ∆[∆²u+ 2∆u+f(u)], in Ω×(0,+∞), (1.1) with periodic boundary conditions on the d-parallelepiped Ω = Π^d_k=1(0, L_k) (L_k >0 for k= 1, . . . , d) in space dimension d= 1, 2 or 3. Equation (1.1) is endowed with initial conditions

u(0, x) =u₀(x), u_t(0, x) =v₀(x), x∈Ω. (1.2) The unknown function u is the phase function andβ >0 is a relaxation parameter.

The nonlinearity f is the derivative of a polynomial potentialF (see (2.1)-(2.2) for a precise definition). A relevant example in applications is given by

f(s) =s³+ (1−ε)s, (1.3)

where ε∈R is constant.

When β = 0, equation (1.1) is known as the phase field crystal (PFC) equation:

it has been employed to model and simulate the dynamics of crystalline materials, including crystal growth in a supercooled liquid, dendritic and eutectic solidification, epitaxial growth [6, 7, 11, 12, 31, 33]. In the phase field approach, the number den- sity of atoms is approximated by the phase function u. The parameter ε in (1.3) is

1

proportional to the undercooling i.e. ε∼T_e−T,T_e being the equilibrium tempera- ture at which the phase transition occurs. The PFC equation is a gradient flow for the free energy

E(u) = Z

Ω

1

2|∆u|²− |∇u|²+F(u)

dx. (1.4)

It preserves the total mass and can be viewed as an analog of the Swift-Hohenberg equation [36].

The MPFC equation (1.1) (withβ >0) was recently proposed in [34] (cf. also [12, 15, 16, 35]) in order to account for elastic interactions. Equations like (1.1) have also been derived in [14] to take large deviations from thermodynamic equilibrium into account.The MPFC equation is no longer a gradient flow for (1.4), but it is possible to associate to a solution (u, ut) a “pseudo-energy”, obtained by adding to E a kinetic energy term (see (2.7)). This leads in a natural way to the notion of energy solution introduced by Grasselli and Wu [25] for the MPFC equation, following a terminology used for the modified Cahn-Hilliard equation [22, 23, 24]. Existence of a unique energy solution was proved by Grasselli and Wu in [25] as well as the convergence of single trajectories to single stationary states. The analysis of global dynamics (that is, existence of global and exponential attractors) for the MPFC equation was carried out in [25, 26].

From the numerical analysis point of view, the MPFC equation has been studied in [3, 4, 17, 37, 38], while the literature on the PFC equation is more abundant (see, e.g., [2, 5, 10, 19, 28, 39]). In [3, 4, 38], the authors proposed unconditionally energy stable and unconditionally uniquely solvable finite difference schemes. The time discretization was based on a convex splitting of the pseudo-energy and was either first order or second order. A priori error estimates were proved assuming enough regularity on the solution. A time semi-discrete scheme was used in [37] to establish the existence of a weak solution and of a unique strong solution to the MPFC equation up to any positive final time T > 0. In [17], an unconditionally energy stable finite element scheme was derived, but no proof of convergence was given.

The main purpose of this paper is to derive and analyze a second order (in time) fully discrete finite element scheme for the MPFC equation. For the space discretiza- tion we use a splitting approach which is well known in the context of phase field models (cf., e.g., [9, 21]). This argument allows to consider low order (piecewise linear) finite elements, although the analysis is carried out in a more general setting, namely a Galerkin approximation. For the time discretization, we use a modified Crank-Nicolson scheme which was introduced by Gomez and Hughes for the Cahn- Hilliard equation [18]; their approach represents an interesting alternative to secant schemes (cf. the discussion in [40]).

We prove that our scheme is unconditionally energy stable, solvable for any time step, and uniquely solvable for small enough time steps, with a smallness assumption independent of the space step. Using the energy estimate, and assuming only some natural conditions on the initial conditions, we show that the solution of the fully discrete scheme converges to the energy solution of problem (1.1)-(1.2) as the time step τ and the space steph tend to 0. Finally, we prove that the discrete solution tends to a stationary solution as time goes to infinity. This last issue is not trivial

since the set of steady states can be very complicated (see [32] for an analysis of the one dimensional stationary problem, cf. also [8]).

For equations involving a second order time derivative such as (1.1), second order time discretizations are very interesting because they do not regularize in finite time, unlike first order schemes: a fundamental property of the continuous model is therefore reproduced at the discrete level. In contrast with the second order two-step scheme in [3, 4], we loose the unconditional unique solvability, but one advantage is that our one-step scheme can be used with variable time steps. Moreover, we do not assume ε < 1 in (1.3), and we do not have any restriction on the initial value ofu_t. Since energy solutions have a weaker regularity than the weak solutions, our convergence result as (h, τ) → (0,0) holds with the minimal regularity on the solution.

Our proofs are crucially based on the energy estimates. In order to establish the convergence to equilibrium, we also use the gradient-like structure of the problem and a Lojasiewicz inequality [30], as in the continuous case [25]. Related results have been obtained for first order schemes applied to second-order gradient-like equations in [1, 20], but the case of the second-order scheme here is more involved.

The paper is structured as follows. In Section 2 we introduce the functional setting and we recall useful results concerning the continuous problem. In Section 3 we consider the space semi-discrete problem. We show its well-posedness and establish energy estimates which allow us to prove the convergence of the semi-discrete solution to the energy solution of the MPFC problem (1.1)-(1.2). This gives a framework for the fully discrete problem which is treated in Section 4. Section 5 is concerned with the convergence to equilibrium for the fully discrete problem. In Section 6, numerical simulations in one and two space dimension illustrate the theoretical results.

2. The continuous problem

2.1. Notation and functional spaces. For a real Banach space V with dual V^⋆, we indicate byh·,·i_V^⋆_,V the duality product betweenV andV^⋆. We denote byH_per^m , m∈N, the space of functions that are inH_loc^m(R^d) and periodic with period Ω. For any m∈N,H_per^m is a Hilbert space for the scalar product

((u, v))_m= X

|κ|≤m

Z

Ω

D^κu(x)D^κv(x)dx

(κ being a multi-index) and its associated normkuk_m = ((u, u))^1/2_m .

For m = 0, H_per⁰ = L²(Ω), the inner product as well as the norm on L²(Ω) are simply indicated by (·,·) and k · k, respectively. For sake of simplicity, we assume that R

Ω1dx=|Ω|= 1. The mean value of any functionu∈L²(Ω) is denoted by hui=

Z

Ω

udx,

and we set ˙u=u− hui.

The dual space of H_per^m is denoted byH_per^−m, and it is equipped with the operator norm given by

kT k_−m= sup

kukm=1, u∈H_per^m

|T(u)|.

For an operator u∈H_per^−m, we denote hui =hu,1i_H⁻m

per,H^m_per and we set ˙u =u− hui.

We denote by ˙H_per^m ={u∈H_per^m : hui= 0}(m∈Z) the Sobolev spaces for functions with zero mean. We will frequently use the fact thatH_per^m is isomorphic toR×H˙_per^m (m∈Z) through the decomposition u=hui+ ˙u.

Using the dense and continuous inclusions H_per¹ ⊂L²(Ω)⊂H_per⁻¹, the semi-scalar product on H_per¹ ,

(u, v)7→(∇u,∇v),

defines a linear operator A = −∆ : D(A) → L²(Ω) with domain D(A) = H_per² . We denote ˙A = −∆ : D( ˙A) → L²(Ω) the restriction of A to ˙L²(Ω), with domain D( ˙A) = ˙H_per² . We observe that ˙A is a positive self-adjoint operator with compact resolvent so that its powers ˙A^s (s∈ R) are well defined and it is possible to prove that ˙H_per^m =D( ˙A^m/2) (m∈Z).

Form=−1, we introduce an equivalent and more convenient norm| · |₋₁ on ˙H_per⁻¹ associated with the inner product

( ˙u,v)˙ ₋₁= ( ˙A^−1/2u,˙ A˙^−1/2v),˙ so that for any ˙u∈H˙_per⁻¹, we have

|u|˙ ₋₁=kA˙^−1/2uk˙ =k∇A˙⁻¹uk.˙

Similarly, form= 1, we will sometimes use the equivalent norm|·|₁inH_per¹ associated with the inner product

(u, v)₁ =huihvi+ (∇u,∇v).

Moreover, ˙A defines a continuous bijection from ˙H_per^m onto ˙H_per^m−2. In particular, fors=−1,

( ˙u,v)˙ ₋₁ =hA⁻¹u,˙ vi˙ _H1

per,Hper⁻¹ =hu,˙ A⁻¹vi˙ _H⁻¹

per,H_per¹ = (A^−1/2u,˙ A^−1/2v).˙ 2.2. Energy solutions. The nonlinearity f is a polynomial of odd degree whose leading coefficient is positive and which vanishes at 0:

f(s) =

2p+1X

i=1

a_isⁱ ∀s∈R (a_2p+1>0), (2.1) with p ∈ N^⋆ if d = 1 or d = 2 and with p ∈ {1,2} if d = 3. We denote F the antiderivative of f which vanishes at 0, i.e.,

F(s) =

2p+2X

i=2

a_i−1

i sⁱ ∀s∈R. (2.2)

We will make use of the Sobolev injection H_per¹ ⊂L^2p+2(Ω). In particular, there is a constant C_S=C_S(Ω) such that

kuk_L^2p+2_(Ω)≤C_S|u|1, ∀u∈H_per¹ , (2.3) and the map v 7→f(v) is Lipschitz continuous on bounded sets of H_per¹ with values intoL(2p+2)/(2p+1)(Ω)⊂H_per⁻¹. We also haveH_per² ⊂C⁰(Ω) with continuous injection.

Finally, we note that there exist constants c₁ ≥0,c₂ >0 and c₃ ≥0 such that

F(s)≥2s²−c₁ ∀s∈R, (2.4)

|f(s)| ≤c₂F(s) +c₃ ∀s∈R. (2.5)

We point out that the expression (2.2) includes the standard quartic potential (ob- tained with p= 1)

F(s) = s⁴

4 +(1−ε)

2 s². (2.6)

In contrast to some authors, we do not assume that ε <1 (we simply haveε∈R).

In [25], a notion of energy solution was introduced. This is based on the following pseudo-energy:

E(u, v) =E(u) +β

2|v|˙ ²₋₁, (2.7)

which is well defined for any (u, v)∈H_per² ×H_per⁻¹.

Definition 2.1. A pair (u, u_t) is called anenergy solution to problem (1.1)-(1.2) if (u, u_t)∈L^∞(R₊;H_per² ×H_per⁻¹), u_tt∈L^∞(R₊;H_per⁻⁴), (2.8) E(u, u_t)∈L^∞(R₊), (2.9) and the following relations hold:

hβutt+uti= 0, (2.10)

A˙⁻¹(βu_tt+u_t) +A²u−2Au+f(u)− hf(u)i= 0,

in ˙H_per⁻², a.e. in R₊, (2.11) u(0) =u₀ in H_per² , u_t(0) =v₀ inH_per⁻¹. (2.12) Equation (2.10) can be interpreted as a conservation law for the mass. With this definition, we have

Theorem 2.2 ([25]). For any intial data(u₀, v₀)∈H_per² ×H_per⁻¹, problem (1.1)-(1.2) has a unique global energy solution (u, u_t). Moreover, any energy solution satifies the strong time continuity property

u∈C²(R₊;H_per⁻⁴)∩C¹(R₊;H_per⁻¹)∩C(R₊;H_per² ), as well as the following energy identity, for all s, t∈R₊ with s < t,

E(u(t), ut(t)) =E(u(s), ut(s))− Z _t

s

|u˙t(τ)|²₋₁dτ+ Z _t

s

hv0ie^{−τ /β} Z

Ω

f(u(τ))dxdτ.

(2.13) In particular, whenever hv0i= 0 then the pseudo-energy is nonincreasing.

3. The space semi-discrete problem

3.1. The space semi-discrete scheme. Our space discretization is based on two ideas: first, in view of the time discretization, we write the PDE (1.1) as a first order system; second, in order to use low order finite elements, we split the tri-Laplacian into three terms, in the spirit of a well-known splitting approach in Cahn-Hilliard type equations (see, e.g., [9, 21, 17]). We obtain the following system, which is (formally) equivalent to (1.1):









 u_t=v

βv_t=−v+ ∆w z=−∆u

w=−∆z+ 2∆u+f(u).

Now, let V_h denote a finite-dimensional subspace of H_per¹ which contains the con- stants. In applications, V_h will be a space of conforming finite elements (see Sec- tion 6). The space V_h can also be obtained with a spectral basis.

The space semi-discrete scheme reads: find u_h, v_h, z_h, w_h:R₊→V_h such that











(∂_tu_h, ϕ_h) = (v_h, ϕ_h)

β(∂_tv_h, ψ_h) =−(v_h, ψ_h)−(∇w_h,∇ψ_h) (z_h, ζ_h) = (∇u_h,∇ζ_h)

(w_h, ξ) = (∇z_h,∇ξ_h)−2(∇u_h,∇ξ_h) + (f(u_h), ξ_h),

(3.1)

for all ϕ_h, ψ_h, ζ_h, ξ_h inV_h. This problem is completed with initial conditions

u_h(0) =u⁰_h, v_h(0) =v⁰_h, (3.2) where u⁰_h andv_h⁰ are given inV_h.

It will be convenient to work with an appropriate basis ofV_h. For this purpose, let (eⁱ_h)1≤i≤Nhdenote an orthonormal basis ofVhfor theL²(Ω) scalar product, such that e¹_h ≡1. The integerN_h is the dimension of V_h. To every function r_h =PNh

i=1r_ieⁱ_h ∈ V_h corresponds a unique (column) vector R = (r1, . . . , r_Nh)^t, represented by the corresponding capital letter. We seek

u_h(t) =

Nh

X

i=1

ui(t)eⁱ_h≃(u1(t), . . . , u_Nh(t))^t=U(t), v_h ≃V, z_h≃Z, w_h ≃W.

Define A= (A_ij)_1≤i,j≤Nh, where

A_ij = (∇eⁱ_h,∇e^j_h), 1≤i, j≤N_h, (3.3) and let

F_h(U) = (F(u_h),1), so that ∇F_h(U) =

(f(u_h), e¹_h), . . . ,(f(u_h), e^N_h^h)t

.

By choosing the test functions ϕ_h, ψ_h, ζ_h, ξ_h in (3.1) as the basis functions eⁱ_h, we obtain the following equivalent system:









 U_t=V

βV_t=−V −AW Z =AU

W =AZ−2AU +∇F_h(U).

(3.4)

Eliminating V,Z andW, we see that (3.4) is equivalent to

βUtt+Ut=−A[A²U −2AU +∇F_h(U)], t≥0. (3.5) SinceAis a discretization of −∆, this is natural space semi-discrete version of (1.1).

LetU denote a solution of (3.5). We notice that the first line and the first column ofAare filled with zeros (recalle¹_h ≡0, so that∇e¹_h ≡0). Thus, the first component of U,u₁(t) = (u_h(t),1), satisfies

β∂_ttu₁+∂_tu₁ = 0, t≥0. (3.6) Solving (3.6) with initial conditionsu₁(0) = (u⁰_h,1) =:u⁰₁ and∂_tu₁(0) = (v_h⁰,1) =:v₁⁰ yields

∂_tu₁(t) =v₁⁰e^−t/β =:a(t) and u₁(t) =M −βa(t), withM =βv₁⁰+u⁰₁. (3.7)

For every vectorR= (r1, . . . , rNh)^t∈R^Nh, we denote ˙R= (r2, . . . , rNh)^t∈R^Nh−1. Then ˙U satisfies

βU˙_tt+ ˙U_t=−A[ ˙˙ A²U˙ −2 ˙AU˙ + ˙∇F_h(U)], t≥0, (3.8) where ˙Ais the submatrix ˙A= (A_ij)_2≤i,j≤Nh, and

∇F˙ _h(U) =

(f(u_h), e²_h), . . . ,(f(u_h), e^N_h^h)t

.

We can also write ˙∇F_h(U) = ˙P(∇F_h(u1(t),U˙)), where ˙P : R^Nh → R^Nh−1 is the projection on the components 2, . . . , N_h. This shows that ˙∇F_h(U) is a “non au- tonomous” function of ˙U (recall thatu₁(t) is determined by (3.7)). For later purpose, we note that by (3.3), ˙Ais symmetric positive definite: in particular, ˙Ais invertible.

Conversely, any solutionU of (3.7)-(3.8) satisfies (3.4), i.e. that the second equa- tion of (3.4) is satisfied with V,Z and W given by the three other equations of the system (3.4).

3.2. Existence, uniqueness, and discrete energy estimate. The standard Eu- clidean norm inR^NhorR^Nh−1will be denoted|·|. We also use the following quadratic norm:

|R|˙ −1 =

R˙^tA˙⁻¹R˙1/2

, (3.9)

defined for all ˙R∈R^Nh−1. Notice that |A^sU|=|A˙^sU˙|(s >0,U ∈R^Nh). We set E_h(U) = 1

2|AU|²− |A^1/2U|²+F_h(U), (3.10) E_h(U, V) = E_h(U) +β

2|V˙|²₋₁. (3.11)

As a shortcut, for a solution (U, U_t) of (3.5), we will write E_h(t) =E_h(U(t), Ut(t)).

Notice that by the Cauchy-Schwarz inequality we have

|A^1/2U|²=U^tAU ≤ 1

4|AU|²+|U|². (3.12) Then, using also (2.4), we find that

E_h(U)≥ 1

4|AU|²+|U|²−c₁. (3.13) We first prove the following

Lemma 3.1. Any solutionU ∈C²([0, T);R^Nh) of (3.5)satisfies the energy equality d

dtE_h(t) +|U˙_t|²₋₁=v₁⁰e^−t/β(f(u_h),1), (3.14) and the energy estimate

E_h(t) + Z t

0

|U˙_t(s)|²₋₁ds≤ E_h(0)e^2c2|v01|β+ (c₁c₂+c₃)|v₁⁰|βe^2c2|v01|β, (3.15) for all t∈[0, T), where c₁, c₂ and c₃ depend only on f (see (2.4)-(2.5)), and where v₁⁰ =∂tu1(0).

Proof. Recall that ∂tu1(t) = v₁⁰e^−t/β (see (3.7)). On multiplying (3.8) by ˙U_t^tA˙⁻¹, and using

d

dt[F_h(U(t))] = (∇F_h(U), U_t(t)) =

Nh

X

i=1

∂_tu_i(t)(f(u_h), eⁱ_h), we find the energy equality (3.14). Estimate (2.5) yields

d

dtE_h(t) +|U˙_t|²₋₁ ≤ |v⁰₁|e^−t/β(c₂F_h(U) +c₃).

By the Cauchy-Schwarz inequality, we have

2|A^1/2U|²= 2U^tAU ≤ |AU|²+|U|². Thus we get

2E_h(U) = (|AU|²−2|A^1/2U|²+|U|²) +F_h(U) + (F_h(U)− |U|²)≥F_h(U)−c1, so that

F_h(U)≤2E_h(U, V) +c1. (3.16) Therefore we obtain

d

dtE_h(t) +|U˙t|²₋₁ ≤ |v₁⁰|e^−t/β(2c2E_h(t) +c1c2+c3). Letting η(t) =Rt

02c2|v⁰₁|e^−t/β and c^′₃ =c1c2+c3, Gronwall’s lemma yields E_h(t) +

Z t

0

|U˙_t(s)|²₋₁e^{η(t)−η(s)}ds≤ E_h(0)e^η(t)+ Z t

0

c^′₃|v₁⁰|e^−s/βe^{η(t)−η(s)}ds.

Sinceη(t) = 2c₂|v₁⁰|β(1−e^−t/β)≤2c₂|v⁰₁|β, we deduce the energy estimate (3.15).

Theorem 3.2. For every U⁰, V⁰ in R^Nh, there exists a unique solution U ∈ C²(R₊,R^Nh) of (3.5) such thatU(0) =U⁰ and Ut(0) =V⁰.

Proof. By the Cauchy-Lipschitz theorem, there exists a unique maximal solution U ∈ C²([0, T⁺);R^Nh) of (3.5) satisfying the given initial conditions. The energy estimate (3.15) shows that E_h is uniformly bounded for t ≥0. By (3.13), |U| and

|U˙t|−1 are uniformly bounded for t ≥ 0. This, together with the estimate (3.7) on

the mass, implies that T⁺= +∞.

3.3. Some additional notation. We assume now that (V_h)_h>0 is a family of sub- spaces of H_per¹ such that:

(H1) for all h >0,V_h has finite dimension and contains all the constants;

(H2) for any ϕ∈H_per¹ , there existsϕ_h ∈V_h such thatϕ_h →ϕ(strongly) inH_per¹ , ash tends to 0.

For the convergence result ash→0, it will be useful to haveh-dependent operators and norms. We denote A_h:V_h →V_h the linear operator such that for any q_h∈V_h, A_hq_h solves

(A_hq_h, ζ_h) = (∇q_h,∇ζ_h), ∀ζ_h∈V_h. (3.17) The operator A_h is a discrete Laplacian, A_h ≃ −∆_h. Notice that if q_h is constant, then A_hq_h = 0 so thatA_h is not invertible. In order to define a discrete version of A˙⁻¹, we introduce the subspace

V˙_h ={ϕ_h∈V_h : hϕ_hi= 0}.

The bilinear form (∇·,∇·) is symmetric positive definite on ˙V_h. We can define the operator ˙S_h : ˙V_h→V˙_h such that for any ˙r_h∈V˙_h, ˙S_hr˙_h is the unique solution of

(∇S˙_hr˙_h,∇ϕ˙_h) = ( ˙r_h,ϕ˙_h), ∀ϕ˙_h ∈V˙_h. (3.18) By choosing ζ_h ≡ 1 in (3.17), we see that A_h(V_h) ⊂ V˙_h, so that the restriction A˙_h : ˙V_h → V˙_h of A_h is well defined. Using (3.17)-(3.18), it is easily seen that S˙_h = ˙A⁻¹_h .

We also define the L²-orthogonal projector P_h :L²(Ω)→V_h, i.e., (Phq, ϕh) = (q, ϕh), ∀q∈L²(Ω), ∀ϕh ∈Vh.

By Pythagoras’ theorem and assumption (H2), for any q∈L²(Ω), we have kq−P_hqk= inf

rh∈Vh

kq−r_hk →0, ash→0. (3.19) Since V_h ⊂H_per¹ , the operatorP_h has a natural extension toH_per⁻¹ (also denotedP_h), by setting

P_hq∈V_h, (P_hq, ϕ_h) =hq, ϕ_hi_H⁻¹

per,H_per¹ , ∀q∈H_per⁻¹, ∀ϕ_h ∈H_per¹ .

TheH¹-orthogonal projector Π_h:H_per¹ →V_h is defined as follows: for anyq∈H_per¹ , Π_hq∈V_h is uniquely defined by

hΠ_hqi=hqi and (∇Π_hq,∇ϕ_h) = (∇q,∇ϕ_h), ∀q ∈H_per¹ , ∀ϕ_h ∈V_h. (3.20) By Pythagora’s theorem and assumption (H2), for any q∈H_per¹ , we get

|q−Π_hq|₁ = inf

rh∈Vh

|q−r_h|₁ →0, ash→0. (3.21) We point out that the space ˙V_h is invariant by Π_h and by P_h.

By using a L²-orthonormal basis (eⁱ_h)_1≤i≤Nh of V_h as in Section 3.1, with e¹_h ≡1, we see that the matrix ofA_h isA, so that the energyE_h from (3.10) can be rewritten

E_h(u_h) = 1

2kA_hu_hk²− |u˙_h|²₁+ (F(u_h),1);

the norm | · |₋₁ from (3.9) becomes for any element ˙r_h ∈V˙_h

|r˙_h|_−1,h:= ( ˙r_h,A˙⁻¹_h r˙_h) =|A˙⁻¹_h r˙_h|₁. It is an Euclidean norm for the following scalar product on ˙V_h

( ˙r_h,q˙_h)_−1,h:= ( ˙r_h,A˙⁻¹_h q˙_h) = ( ˙A⁻¹_h r˙r,A˙⁻¹_h q˙_h)1. Thus, the discrete pseudo-energy (3.11) reads

E_h(u_h, v_h) :=E_h(u_h) +β

2|v˙_h|²_−1,h, ∀u_h, v_h∈V_h. (3.22) The Cauchy-Schwarz (3.12) inequality gives

|u|˙ ²₁≤ 1

4kA˙huhk²+kuhk², ∀uh ∈Vh, (3.23) and estimate (3.13) becomes

E_h(u_h)≥ 1

4kA_hu_hk²+ku_hk²−c₁, ∀u_h∈V_h. (3.24)

3.4. Convergence as h→0.

Theorem 3.3. Let (u₀, v₀) ∈ H_per² ×H_per⁻¹. Assume that (u⁰_h, v_h⁰)_h>0 is a family of functions in V_h×V_h such that

u⁰_h →u₀ in H_per¹ , A_hu⁰_h → Au₀ in L²(Ω), (3.25) hv⁰_hi → hv₀i in R, A˙⁻¹_h v˙_h⁰ →A˙⁻¹v˙₀ in H˙_per¹ , (3.26) as h→0. Then the solution(u_h, ∂_tu_h) of the space semi-discrete scheme (3.1)-(3.2) tends to the energy solution (u, u_t) of (1.1)-(1.2)in the following sense:

u_h → u weakly ⋆ in L^∞(R₊;H_per¹ ),

u_h → u strongly in C([0, T];L²(Ω)), for all T >0, A_hu_h → Au weakly ⋆ in L^∞(R₊;L²(Ω)),

A˙⁻¹_h ∂_tu˙_h → A˙⁻¹∂_tu˙ weakly ⋆ in L^∞(R₊; ˙H_per¹ ) and weakly in L²(R₊; ˙H_per¹ ).

Proof. The idea is to use a priori estimates on the mass and on the discrete energy, and to pass to the limit in the equation by a compactness argument. We first consider the conservation law for the mass. By (3.7), we get

a_h(t) :=h∂_tu_h(t)i=hv⁰_hie^−t/β, t≥0, and

hu_h(t)i=βhv⁰_hi+hu⁰_hi −βa_h(t), t≥0. (3.27) By assumption, hu⁰_hi → hu₀i and hv_h⁰i → hv₀i in R, so that a_h converges uni- formly on R₊ to the function a(t) := hv0ie^−t/β, and hu_hi converges uniformly on R₊ to the function βhv₀i+hu₀i −βa(t). The estimates below show that (u_h)_h>0 is bounded in L^∞(R₊;L²(Ω)), so that, up to a subsequence, u_h converges weakly ⋆ in L^∞(R₊;L²(Ω)) to some u and so hu_hi → hui weakly ⋆ inL^∞(R₊). By uniqueness of the limit, we find

hui=βhv₀i+hu₀i −βa(t).

By differentiating, we recover the conservation law for the mass:

β∂tthui+∂thui= 0, t≥0.

We now turn to the energy estimate. As pointed out in Section 3.1, the (unique) solution (u_h, ∂_tu_h) of (3.1)-(3.2) is in fact a solution (u_h, v_h, z_h, w_h) of (3.1)-(3.2). In particular,v_h =∂_tu_h and z_h=A_hu_h. We have (recall (3.22)):

E(u⁰_h, v_h⁰) = 1

2kA_hu⁰_hk²− |u˙⁰_h|²₁+ (F(u⁰_h),1) + β

2|A˙⁻¹_h v˙_h⁰|²₁.

By using assumptions (3.25)-(3.26) and the Sobolev injection H_per¹ ֒→L^2p+2(Ω), we see that E(u⁰_h, v_h⁰) is uniformly bounded as htends to 0. The energy estimate (3.15) shows that there exists a constant C independent ofh such that

E_h(u_h(t), ∂_tu_h(t)) + Z t

0

|∂_tu˙_h(s)|²_−1,hds≤C,

for all t≥0. By (3.24) and (3.23), we obtain that z_h =A_hu_h and u_h are uniformly bounded in L²(Ω), that ˙u_h and ˙r_h := ˙A⁻¹_h ∂_tu˙_h are uniformly bounded in ˙H_per¹ , and

that Z ∞

0

|A˙⁻¹_h ∂_tu˙_h(t)|²₁dt≤C.

This implies that (u_h)_h>0 is precompact in the space C([0, T];L²(Ω)), for allT >0, as a consequence of the Ascoli-Arzel`a Theorem. Indeed, let T > 0. The family (u_h)_h>0 is uniformly bounded from [0, T] with values inH_per¹ , andH_per¹ is compactly embedded into L²(Ω) by Rellich’s Theorem. Moreover, for all 0 ≤ s ≤ t ≤ T, we have

ku˙_h(t)−u˙_h(s)k² = 2 Z t

s

(∂_tu˙_h(σ),u˙_h(σ)−u˙_h(s))dσ

= 2 Z t

s

(∇r˙_h(σ),∇[ ˙u_h(σ)−u˙_h(s)])dσ

≤ 4kr˙_hk_L^∞₍R₊;H_per¹ )ku˙_hk_L^∞₍R₊;H¹_per)|t−s|. (3.28) Moreover, by (3.27) and the mean value theorem, we find

|hu_h(t)i − hu_h(s)i| ≤ |hv⁰_hi||t−s|.

Thus, (u_h)_h>0 is uniformly equicontinuous from [0, T] with values into L²(Ω), and therefore pre-compact inC([0, T];L²(Ω)), as claimed. Up to a subsequence, we have the following convergence results

u_h → u weakly ⋆ inL^∞(R₊;H_per¹ ),

u_h → u strongly in C([0, T];L²(Ω)), for all T >0, u_h → u a.e. in Ω×R₊,

f(u_h) → f(u) weakly inL^q(0, T;L^q(Ω)), for all T >0, z_h → z weakly ⋆ inL^∞(R₊;L²(Ω)),

˙

r_h → r˙ weakly ⋆ inL^∞(R₊; ˙H_per¹ ),

˙

r_h → r˙ weakly inL²(R₊; ˙H_per¹ ),

where q = (2p+ 2)/(2p+ 1) > 1. Let now ˙ψ ∈ H˙_per¹ and let ˙ψ_h = Π_h( ˙ψ) so that ψ˙_h →ψ˙ strongly in ˙H_per¹ . We have

(∂_tu˙_h,ψ˙_h) = ( ˙A_hr˙_h,ψ˙_h) = (∇r˙_h,∇ψ˙_h)→( ˙r,ψ)˙

weakly ⋆ inL^∞(R₊). On the other hand, ∂_t( ˙u_h,ψ˙_h)→ ∂_t( ˙u,ψ) in˙ D^′(0,∞) (i.e. in the sense of distributions), since ( ˙u_h,ψ˙_h)→( ˙u,ψ) in˙ L^∞(R₊) weakly ⋆. Thus,

∂_t( ˙u,ψ) = (∇˙ r,˙ ∇ψ) =˙ hv,˙ ψi˙ _H⁻¹

per,H_per¹ , (3.29) with ˙v= ˙Ar˙∈L^∞(R₊; ˙H_per⁻¹). This shows that∂_tu˙ = ˙v ∈L^∞(R₊; ˙H_per⁻¹).

Next, we set ϕ ∈ H_per² and we let ϕ_h = Π_h(ϕ), so that ϕ_h → ϕ strongly in H_per¹ . Let (eⁱ_h)_1≤i≤Nh be an orthonormal basis of V_h with e¹_h ≡ 1. We let ϕ_h = PNh

i=1ϕ_ieⁱ_h and Φ = (ϕ₁, . . . , ϕ_Nh)^tbe the vector associated to ϕ_h, as in Section 3.1.

On multiplying (3.8) by ˙Φ^tA˙⁻¹ and usingz_h =A_hu_h, we find

β(∂_ttu˙_h,ϕ˙_h)_−1,h+ (∂_tu˙_h,ϕ˙_h)_−1,h+ (∇z_h,∇ϕ_h)−2(∇u_h,∇ϕ_h) + (f(u_h),ϕ˙_h) = 0, (3.30) for all t≥0. We have that

(∂_tu˙_h,ϕ˙_h)_−1,h= ( ˙r_h,ϕ˙_h)→( ˙r,ϕ) = (∂˙ _tu,˙ ϕ)˙ ₋₁,

by (3.29). The convergence above holds inL^∞(R₊) weak ⋆, so that (∂ttu˙_h,ϕ˙_h)_−1,h=∂t(∂tu˙_h,ϕ˙_h)_−1,h→∂t(∂tu,˙ ϕ)˙ −1 inD^′(0,∞).

Moreover,

(∇u_h,∇ϕ_h)→(∇u,∇ϕ) in L^∞(R₊) weak ⋆ . Since ϕ_h= Π_hϕ, we have

(∇z_h,∇ϕ_h) = (∇z_h,∇ϕ) = (z_h,Aϕ)→(z,Aϕ), inL^∞(R₊) weak ⋆. Concerning the last term in (3.30), we have

(f(u_h),ϕ˙_h)→(f(u),ϕ)˙ weakly in L^q(0, T), ∀T >0.

Summing up, we have proved that

β∂_t(∂_tu,˙ ϕ)˙ ₋₁+ (∂_tu,˙ ϕ)˙ ₋₁+ (z,Aϕ)−2(∇u,∇ϕ) + (f(u), ϕ) =hf(u)ihϕi. (3.31) The equality holds in D^′(0,∞), for all ϕ ∈ H_per² . Moreover, ∂_tu˙ ∈ L^∞(R₊;H_per⁻¹), z ∈ L^∞(R₊;L²(Ω)), u ∈ L^∞(R₊;H_per¹ ) and f(u) ∈ L^∞(R₊;H_per⁻¹) so that ∂_ttu belongs to L^∞(R₊;H_per⁻⁴) and

A˙⁻¹(β∂_ttu˙+∂_tu) +˙ Az−2Au+f(u)− hf(u)i= 0

in ˙H_per⁻², a.e. in R₊. Now, recall thatz_h = A_hu_h. Let ζ ∈H_per¹ and ζ_h = Π_hζ, so that ζ_h →ζ strongly in H_per¹ . We have

(z_h, ζ_h) = (A_hu_h, ζ_h) = (∇u_h,∇ζ_h)→(∇u,∇ζ), on one hand, and (z_h, ζ_h)→(z, ζ), on the other hand. Thus, we deduce

(z, ζ) = (∇u,∇ζ),

in L^∞(R₊). The equality holds for every ζ ∈ H_per¹ , soz =Au,u ∈L^∞(R₊;H_per² ) and (u, u_t) is an energy solution of (1.1)-(1.2). By uniqueness of the limit (u, u_t), the whole family (u_h, ∂_tu_h) converges to (u, u_t).

Remark 3.4. Let (u₀, v₀) ∈ H_per² ×H_per⁻¹. If u⁰_h = Π_h(u₀) and v_h⁰ = P_h(v₀), then assumptions (3.25)-(3.26) are satisfied. Indeed, for all ϕ_h∈V_h,

(Au₀, ϕ_h) = (∇u₀,∇ϕ_h) = (∇Π_hu₀,∇ϕ_h) = (A_h(Π_hu₀), ϕ_h).

Then A_h(Π_hu₀) =P_h(Au₀) and by (3.19) we obtain

A_h(Π_hu₀)→ Au₀ inL²(Ω), ash→0.

By definition, observe that

hP_h(v₀)i= (P_h(v₀),1) =hv₀,1i_H⁻¹

per,H_per¹ =hv₀i.

Moreover, for all ϕ_h ∈V_h, we have (P_h( ˙v0),ϕ˙_h) = hv˙0,ϕ˙_hi_H⁻¹

per,H_per¹ = (∇A˙⁻¹v˙0,∇ϕ˙_h)

= (∇Π_h( ˙A⁻¹v˙₀),∇ϕ˙_h) = ( ˙A_h(Π_h( ˙A⁻¹v˙₀)),ϕ˙_h), so that

P_h( ˙v0) = ˙A_h(Π_h( ˙A⁻¹v˙0)).

Thus, thanks to (3.21), we deduce

A˙⁻¹_h (P_h( ˙v0)) = Π_h( ˙A⁻¹v˙0)→A˙⁻¹v˙0 inH_per¹ .

4. The fully discrete problem

4.1. The fully discrete scheme. For the time discretization, we use the following decomposition:

(H3) F =F₊+F₋, whereF₊andF₋are polynomials such thatF₊^(iv)≥0,F₋^(iv)≤ 0, anddeg(F₋)< deg(F) (here,deg denotes the degree of the polynomial).

As a consequence, deg(F₊) = deg(F) and F₊, F have the same leading coefficient.

We denotef =F^′ =f₊+f₋, wheref₊ =F₊^′ andf₋=F₋^′ . For the energy estimate, we will use the fact that there exist two constants c₅ >0 and c₆ ≥0 which depend only on f and on the decomposition f =f₊+f₋ such that

1

2(|f(r)|+|f(s)|) + 1

12(s−r)²(|f₊^′′(r)|+|f₋^′′(s)|)≤c₅(F(r) +F(s)) +c₆, (4.1) for all r, s∈R.

Remark 4.1. A decomposition (H3) is always possible for a polynomial potential such as (2.2). Indeed, for a quartic polynomial (for instance (2.6)), we can always choose F+ = F and F− = 0. For a polynomial with higher degree, we notice that F^(iv), being a polynomial of even degree with strictly positive leading coefficient, is bounded from below, i.e.

F^(iv)(s)≥ −c₄ ∀s∈R,

for some constant c₄ ≥ 0. A possible (but not unique !) choice is then F₊^(iv) = F^(iv)+c4 and F₋^(iv)=−c4, i.e. F+(s) =F(s) +c4s⁴/24 and F−(s) =−c4s⁴/24.

We use the same notation as in Section 3. In particular, V_h is a family of finite- dimensional subspaces of H_per¹ which satisfies assumptions (H1)-(H2).

Let τ > 0 denote the time step, and (u⁰_h, v_h⁰) in V_h×V_h the initial datum. The fully discrete scheme reads: for n ≥ 0, find (uⁿ⁺¹, vⁿ⁺¹, zⁿ⁺¹, wⁿ⁺¹) ∈ (V_h)⁴ such that





















((uⁿ⁺¹_h −uⁿ_h)/τ, ϕ_h) = (v_h^n+1/2, ϕ_h)

β((v_hⁿ⁺¹−v_hⁿ)/τ, ψ_h) =−(v_h^n+1/2, ψ_h)−(∇w_hⁿ⁺¹,∇ψ_h) (z_hⁿ⁺¹, ζ_h) = (∇u^n+1/2_h ,∇ζ_h)

(wⁿ⁺¹_h , ξ_h) = (∇z_hⁿ⁺¹,∇ξ_h)−2(∇u^n+1/2_h ,∇ξ_h) + ((f(uⁿ_h) +f(uⁿ⁺¹_h ))/2, ξ_h)

−1

12 (uⁿ⁺¹_h −uⁿ_h)² f₊^′′(uⁿ_h) +f₋^′′(uⁿ⁺¹_h ) , ξ_h

,

(4.2) for all ϕ_h,ψ_h,ζ_h,ξ_h inV_h. Here, we have denoted

u^n+1/2_h = (uⁿ⁺¹_h +uⁿ_h)/2 and v_h^n+1/2 = (v_hⁿ⁺¹+vⁿ_h)/2.

Notice that z_h⁰ and w⁰_h do not need to be defined. In fact, z_hⁿ⁺¹ (resp. (w_hⁿ⁺¹)) is a second-order (in time) approximation of z_h(t_n+1/2) (resp. w_h(t_n+1/2)), where t_n+1/2= (n+ 1/2)τ.

Let (eⁱ_h)_1≤i≤Nh be an L²-orthonormal basis of V_h, with e¹_h ≡1, so that we have the identification V_h∋u_h ≃U ∈R^Nh. InR^Nh, the scheme reads: letU⁰, V⁰ inR^Nh