An error estimate for a finite-volume scheme for the Cahn-Hilliard equation with dynamic boundary conditions

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Preprint submitted on 21 Feb 2018 (v2), last revised 3 Sep 2021 (v3)

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An error estimate for a finite-volume scheme for the Cahn-Hilliard equation with dynamic boundary

conditions

Flore Nabet

To cite this version:

Flore Nabet. An error estimate for a finite-volume scheme for the Cahn-Hilliard equation with dynamic

boundary conditions. 2018. �hal-01273945v2�

CAHN-HILLIARD EQUATION WITH DYNAMIC BOUNDARY CONDITIONS

FLORE NABET ^∗

Abstract. In this paper we consider a finite-volume approximation for the Cahn-Hilliard equation with dynamic boundary conditions. We prove an error estimate for the fully-discrete scheme. We also give numerical simulations which validate the theoretical result.

Key words. Cahn-Hilliard equation, dynamic boundary condition, finite-volume method, error estimate.

AMS subject classifications. 35K55, 65M08, 65M15, 76M12

1. Introduction.

1.1. The Cahn-Hilliard model. We consider the following Cahn-Hilliard equation with a dynamic boundary condition which describes the phase separation process of a binary mix- ture. Find the concentration of one of the two components c : (0, T ) ×Ω → R and the chemical potential µ : (0, T ) × Ω → R such that

∂ _t c = Γ _b ∆µ; in (0, T ) × Ω;

(1.1a)

µ = − 3

2 εσ _b ∆c + 12

ε σ _b f _b ⁰ (c); in (0, T ) × Ω;

(1.1b)

ε ³

64Γ _b Γ _s ∂ t c _p Γ = 3

8 ε ² σ b σ s ∆ Γ c _p Γ − 6σ b f _s ⁰ (c _p Γ ) − 3

2 εσ b ∂ n c; on (0, T ) × Γ;

(1.1c)

∂ n µ = 0; on (0, T ) × Γ;

(1.1d)

c(0, .) = c ⁰ ; in Ω;

(1.1e)

where Ω is a connected and bounded domain of R ² , ∂ _n is the normal derivative operator, c _{p Γ} is the trace of c on the boundary Γ = ∂Ω and ∆ _Γ is the Laplace-Beltrami operator.

The Cahn-Hilliard model is a diffuse interface model that means that the interfaces have a small but non-zero thickness ε > 0 (see Fig. 1a). Several physical parameters which describe the physical properties of the mixture components and the wall appear in the model: a diffusion coefficient called the mobility (supposed to be constant here) Γ b > 0, the binary surface tension coefficient σ b > 0 between the two components (which is the density interfacial energy), a capillarity coefficient σ s > 0 and a relaxation coefficient Γ s > 0. The bulk and surface potentials f b (typically f b (c) = c ² (1−c) ² , see Fig. 1b) and f s respectively satisfy the following dissipativity assumption (useful to prove the bounds on the discrete solutions given in Proposition 4.4),

lim inf

|c|→∞ f _b ⁰⁰ (c) > 0 and lim inf

|c|→∞ f _s ⁰⁰ (c) > 0, and the polynomial growth condition for f b

(1.2) |f _b ⁰ (c)| ≤ C(1 + |c| ^p ), ∀c ∈ R , for some C > 0 and p ∈ [1, +∞[.

∗ CMAP, Ecole polytechnique, CNRS, Université Paris-Saclay, 91128, Palaiseau, France ([email protected]).

1

0.0 0.5 1.0

Interface thickness: ε (a) Interface thickness

0 1

f

b

(c) = c

²

(1 − c)

²

(b) Bulk potential

Fig. 1: Definition of the interface thickness and double-well structure of f b

The total Cahn-Hilliard energy is written as follows (1.3) F (c) =

Z

Ω

3

4 εσ _b |∇c| ² + 12 ε σ _b f _b (c)

| {z }

:=F b (c)

+ Z

Γ

3

16 ε ² σ _b σ _s |∇ _Γ c _{p Γ} | ² + 6σ _b f _s (c _{p Γ} )

| {z }

:=F s (c)

,

and this energy is dissipated over time

(1.4) d

dt F(c(t, .)) = −Γ _b Z

Ω

|∇µ(t, .)| ² − ε ³ 64Γ b Γ s

Z

Γ

|∂ _t c _{p Γ} (t, .)| ² , t ∈ [0, T [.

We can remark that the bulk energy F _b is the energy associated with the Cahn-Hilliard equation with Neumann boundary conditions. Definition (1.3) of the total energy induces us to introduce the function spaces H _Γ ¹ (Ω) = {u ∈ H ¹ (Ω) : u _{p Γ} ∈ H ¹ (Γ)} and H _Γ ² (Ω) = {u ∈ H ² (Ω) : u _{p Γ} ∈ H ² (Γ)} and the corresponding norms, for i ∈ {1, 2}

kuk _H i Γ (Ω) =

kuk ² _H i (Ω) + ku _p Γ k ² _H i (Γ)

¹ ₂

, ∀u ∈ H _Γ ⁱ (Ω).

In the analysis to follow, for the sake of simplicity, all the coefficients in problem (1.1) will be taken equal to one (expect for the numerical results given in Section 5).

1.2. Former results and outline. In the past 30 years, the Cahn-Hilliard equation

associated with the homogeneous Neumann boundary condition on the order parameter c

has been extensively studied. Recently physicists [14, 15, 19] have introduced the dynamic

boundary condition which allows to take into account the interaction between the components

and the wall, especially the contact-line dynamics (see [17]). In the case of Neumann boundary

conditions the numerical analysis with finite-difference and finite-element methods is well-

understood (see [3, 4, 5, 6, 8, 7, 9, 11, 12, 13, 16, 21] and the references therein). However,

to our knowledge, for the problem that we study here there is no error estimate for the fully-

discrete scheme on a curved domain. There exist finite-difference methods but without proof

of stability or convergence (see [14, 15, 19]). A numerical analysis for the semi-discrete scheme

using a spatial finite-element scheme is done in [2] in a slab with periodic boundary conditions

in the longitudinal direction. In [20] the author propose a finite-volume scheme and prove the

convergence of the numerical scheme towards a weak solution of problem (1.1) for a smooth

non-polygonal domain. Finite-volume methods have advantage to easily adapt to the non flat

geometry of the boundary and to naturally couple the equation in the domain and the dynamic boundary condition by the flux term ∂ n c. That is why, in order to prove an error estimate on a smooth domain Ω for this problem, we use the same finite-volume scheme that the one introduced in [20]. Therefore in Section 2 we present the finite-volume framework that is the finite-volume notation on a curved domain, the associated discrete inner products and norms and the functional inequalities used in the paper. Then Section 3 is devoted to the presentation of the numerical scheme. In Section 4 (and in the Appendix) we prove the main result of the paper: an error estimate for the fully-discrete scheme (Theorem 4.2). Finally in Section 5 we present a numerical error estimate for this model in accordance with the error estimate theorem proved in the previous section.

2. The finite-volume framework.

2.1. Mesh and notation. We recall here the main finite-volume notations used in the paper (see Fig. 2). The usual notation for a polygonal domain can be found for example in [10]

and the notation associated to a curved domain in [20].

v = L | L 0

x L

x K

y L

x L i

d _K

, _L

i K d

, _L

d L ,v

x L0

y L0

d L 0

, v

~ n KL i

~ n KL

~ n σ K (x L )

Interior vertex Boundary vertex

Interior center Boundary center

Interior mesh M

K

Boundary mesh ∂M

L ∈ ∂M e L chord associated with L

Fig. 2: Mesh T associated with Ω

An admissible mesh T of Ω is given by an interior mesh M and a boundary mesh ∂M.

The interior mesh M is a set of disjoint open subsets of Ω, denoted by K and called interior control volumes, which satisfy:

• Ω = ∪ K∈M K ;

• if K , L ∈ M, K 6= L , then K ∩ L = ∅;

• if K , L ∈ M, K 6= L such that the dimension of K ¯ ∩ ¯ L is equal to 1, then ¯ K ∩ L ¯ is the edge of the mesh separating the control volumes K and L ;

• if K ∩ Γ contains a finite number of points, then K is polygonal;

• for any K ∈ M, we associate a point x K (referred to as the center of K ) such that if K , L

are two neighboring interior control volumes, the edge which separates K and L , which

is denoted by σ = K | L , is orthogonal to the straight line going through x K and x L .

Let E be the set of edges of the interior mesh M. We decompose E into two disjoint subsets:

the set of interior (flat) edges E int = {σ ∈ E : σ 6⊂ Γ} and the set of exterior (curved) edges E ext = {σ ∈ E : σ ⊂ Γ}. Similarly we use the notations E _K ^int and E _K ^ext for the edges of a given control volume K ∈ M. For any σ ∈ E, we note m σ its length. For each edge σ ∈ E, we associate a diamond cell D defined as follows:

• D = D _σ the quadrangle whose diagonals are the edge σ and the line segment [x K , x L ] if σ ∈ E int ;

• D = D _σ = {tx + (1 − t)x K , t ∈ [0, 1], x ∈ σ} if σ ∈ E ext ∩ E K .

We note m D the Lebesgue measure of D and D is the set of all diamond cells.

Since the domain Ω is not polygonal, we have to introduce an approximate domain Ω =

∪ K∈M K of Ω where

• K = K if E K ∩ E ext = ∅;

• K is the polygon obtained by joining all the vertex of K if E K ∩ E _ext 6= ∅. We can notice that in this case K may be not convex and that K may be not included in Ω.

We denote by m K (resp. m K ) the Lebesgue measure of K (resp. K ) and by m _M ∈ R ^M (resp.

m _M ∈ R ^M ) the vector (m K ) K∈M (resp. (m K ) K∈M ). The Lebesgue measures of K and K are then related by the following relation.

Proposition 2.1. For any interior control volume K ∈ M, we have m K − m K = O diam( K ) ³

. Therefore, there exists C ₁ , C ₂ > 0 depending only on Γ such that

C 1 m K ≤ m K ≤ C 2 m K .

The boundary mesh ∂M is equal to the set of exterior edges E ext . Thus, the exterior edges are also boundary control volumes. When we consider them as edges, we denote them by σ ∈ E _ext , and, when we consider them as control volumes of the boundary mesh, we denote them by L ∈ ∂M (and its length by m L ). The chord associated with L is then denoted by e L

(and its length by m _{e L} ), and, the quantities m L and m _{e L} are related by the following relation.

Proposition 2.2. For any boundary control volume L ∈ ∂M, we have m e _L − m L = O m ³ _L

.

In particular, there exists C ₃ > 0 independent of size(T ) such that m _{e L} ≤ m L ≤ C ₃ m _{e L} .

Let m ∂M (resp. m ∂M ) be the vector (m L ) L∈∂M (resp. (m e L ) L∈∂M ), then we note m T = (m M , m ∂M ) ∈ R ^T (resp. m T = (m M , m ∂M ) ∈ R ^T ).

For any control volume L ∈ ∂M, we associate a point x _L ∈ L , called the center of the control volume. For any boundary control volume L ∈ ∂M, let K ∈ M the interior control volume such that L = σ is an edge of K . Then, we impose that the straight line going through x K and x L is orthogonal to the chord e L associated with L . Moreover, we define y L as the orthogonal projection of x L on the chord e L .

Let V be the set of vertices of the mesh M which belongs to Γ. We denote by v = L | L 0

the vertex which separates the boundary control volumes L and L 0 . For any v = L | L 0 , let

d L ,v = d(x L , v) be the approximation of the length m γ _Lv of the arc γ L v included in the

boundary control volume L whose ends are x L and v. The measure of the arc γ LL0 (which is

the arc whose ends are x L and x L0 and passing through the vertex v) is then approximate by

the distance d L , L0 = d L ,v + d L0 ,v .

If σ = K | L ∈ E int is an interior edge, we note d K , L the distance between the centers x K and x L ; and ~ n KL the normal vector to σ going from K to L . If σ = L ∈ E ext ∩ E K is an exterior edge, we note d K , L the distance between the center x K and the point y L ; and ~ n KL the normal vector to the chord e L outward to K . In this case, the distance d(x K , x L ) and its approximation d K , L

satisfy the following relation.

Proposition 2.3. Let us consider a boundary control volume L ∈ ∂M such that L = σ ∈ E K where K ∈ M is an interior control volume, then we have

d(x L , y L ) = |d(x K , x L ) − d K , L | = O(m L m γ Lv ).

Let size(T ) be the maximum of the diameters of the interior control volumes K . We introduce a positive number reg(T ) that measures the regularity of a given mesh and is useful to perform the numerical analysis of finite-volume schemes

(2.1) reg(T ) := max max

K∈M

diam( K )

√ m K

, max

K∈M σ ∈EK

diam( K ) d(x K , σ) , max

K∈M σ ∈EK

d K , L

d(x K , σ)

! . The number reg(T ) should be uniformly bounded when size(T ) → 0.

For the needs of the proof of error estimate theorem, we also define a family of quasi- uniform meshes.

Definition 2.4 (Quasi-uniform mesh family of Ω).

For a given mesh T , we define the number reg unif (T ) as follows reg _unif (T ) ^def = sup

reg(T ), sup

K∈M

size(T ) ² m K

. We say that a mesh family T ⁽ⁱ⁾

i∈N is quasi-uniform if reg unif (T ⁽ⁱ⁾ ) is bounded.

2.2. Inner-products and norms. Since the domain Ω is not polygonal, we introduce a L ² -inner product on the domain Ω (and on its boundary Γ = ∂Ω) but also on the approximate polygonal domain Ω (and on its boundary ∂Ω).

For the space discretization, the finite-volume method associates an unknown value u K ∈ R (resp. u L ∈ R ) to each interior (resp. boundary) control volume K ∈ M (resp. L ∈ ∂M). Thus we note

u T = (u M , u ∂M ) = ((u K ) K∈M , (u L ) L∈∂M ) ∈ R ^T = R ^M × R ^∂M .

Definition 2.5 (Discrete L ² -inner products).

• We define the inner product (., .) _{L2 (Ω)} on L ² (Ω) and the inner product (., .) _M on L ² (Ω) as follows: for any u _M , v _M ∈ R ^M , we have

(u _M , v _M ) _{L2 (Ω)} = P

K ∈M

m K u K v K and (u _M , v _M ) _M = P

K ∈M

m K u K v K . We denote by k.k _{L2 (Ω)} and k.k _{0, M} the associated norms.

• We define the inner product (., .) _{L2 (Γ)} on L ² (Γ) and the inner product (., .) _∂M on L ² (∂Ω) as follows: for any u ∂M , v ∂M ∈ R ^∂M , one has

(u _∂M , v _∂M ) _{L2 (Γ)} = P

L ∈∂M

m _L u _L v _L and (u _∂M , v _∂M ) _∂M = P

L ∈∂M

m _{e L} u _L v _L .

We denote by k.k _{L2 (Γ)} and k.k _{0, ∂M} the associated norms.

Owing to Proposition 2.1 (resp. Proposition 2.2), the norms k.k _{L2 (Ω)} and k.k _{0, M} (resp. k.k _{L2 (Γ)} and k.k _{0, ∂M} ) are equivalent (with constants independent on the mesh size).

We also define semi-inner products in R ^T and R ^∂M .

Definition 2.6 (Discrete H ¹ -semidefinite inner products). We define the H ¹ -semidefinite inner product in R ^T as follows: for any u T , v T ∈ R ^T ,

J u T , v T K ^{1, T} = P

σ = K | L ∈E _int

m σ

d K , L

(u K − u L )(v K − v L ) + P

σ = L ∈E ext

m e L

d K , L

(u K − u L )(v K − v L ).

We also define the H ¹ -semidefinite inner product in R ^∂M as follows:

J u ∂M , v ∂M K 1, ∂M = P

v=L|L0 ∈V

1 d L , L0

(u L − u L0 )(v L − v L0 ), ∀u ∂M , v ∂M ∈ R ^∂M .

We denote by |.| _{1, T} and |.| _{1, ∂M} the associated seminorms. Moreover, we define the H ¹ -norms in R ^T and R ^∂M as follows: for any u T ∈ R ^T , u ∂M ∈ R ^∂M ,

ku T k _{1, T} =

ku T k ² _{0, M} + |u T | ² _{1, T} ¹ ₂

and ku ∂M k _{1, ∂M} =

ku ∂M k ² _{0, ∂M} + |u ∂M | ² _{1, ∂M} ¹ ₂ . 2.3. Functional inequalities. We give here without proofs some functional inequalities available in the literature and that we will use all along the paper. We consider an admissible mesh T of Ω.

Lemma 2.7 (Discrete mean Poincaré inequality, [10, Lemma 3.7]). There exists C 4 > 0 depending only on Ω such that for any u T ∈ R ^T ,

ku M − m Ω (u M )k _{L2 (Ω)} ≤ C 4 |u T | _{1, T} with m Ω (u M ) = 1

|Ω|

P

K ∈M

m K u K , ku _M − m _Ω (u _M )k _{0, M} ≤ C ₄ |u T | _{1, T} with m _Ω (u _M ) = 1

|Ω|

P

K ∈M

m K u K . Thus, we also have

(2.2) ku M k _{L2 (Ω)} ≤ C 4 |u T | _{1, T} + |Ω| ^{1 2} |m Ω (u M )|, and ku M k _{0, M} ≤ C 4 |u T | _{1, T} + |Ω| ^{1 2} |m Ω (u M )|.

Lemma 2.8 (Poincaré-Sobolev inequality, [1, Theorem 3.2]). Let 1 ≤ q < +∞, there exists C ₅ > 0 depending only on q, Ω and reg(T ) such that

(2.3) ku _M k _Lq(Ω) :=

P

K ∈M

m K |u K | ^{q 1} _q

≤ C ₅ ku T k _{1, T} , ∀u T ∈ R ^T .

We can also easily prove the following Sobolev inequality on the one dimensional manifold Γ.

Lemma 2.9. There exists C ₆ > 0 depending only on Γ such that for any u _∂M ∈ R ^∂M , sup

L∈∂M

|u L | ≤ C ₆ ku _∂M k _{1, ∂M} .

The proof of the following lemma for a polygonal domain is done in [10, Lemma 3.10] but it can be adapted in our case.

Lemma 2.10 (Trace inequality). There exists C 7 > 0 depending only on Ω such that for any u T ∈ R ^T we have

ku ∂M k _{L2 (Γ)} ≤ C 7

|u T | _{1, T} + ku M k _{L2 (Ω)}

.

We can also remark that for a quasi-uniform mesh family T ⁽ⁱ⁾

i∈N (see Definition 2.4) and for any q ≥ 1, there exists a uniform constant C 8 > 0 (depending on q and reg unif (T )) such that

(2.4) sup

K∈M

|u K | ≤ C 8

size(T ⁽ⁱ⁾ ) ^{2 / q} ku M k _Lq(Ω) , ∀u M ∈ R ^M .

3. The finite-volume scheme . This section is devoted to the presentation of the nu- merical scheme. We refer the reader to [20] for the proofs of the energy stability, the existence of a discrete solution and the convergence analysis.

For the time discretization, let N ∈ N and ∆t = _N ^T be the time step. For any n ∈ {0, · · · , N} we define t ⁿ = n∆t. Then, at time t ⁿ , the unknowns are denoted by

c ⁿ _T =

(c ⁿ _K ) K∈M

(c ⁿ _L ) L∈∂M

and µ ⁿ _T =

(µ ⁿ _K ) K∈M

(µ ⁿ _L ) L∈∂M

.

Since µ is associated with the homogeneous Neumann boundary condition, for any L ∈ ∂M we have µ ⁿ _L = µ ⁿ _K where K ∈ M is the interior control volume such that L ⊂ ∂ K .

To obtain the finite-volume approximation of problem (1.1) we integrate the continuous equations (1.1a) and (1.1b) on all interior control volumes K ∈ M and we use a consistent two- point flux approximation for the Laplace operators (associated with the Neumann boundary condition for µ). Then we integrate dynamic boundary condition (1.1c) on all boundary control volumes L ∈ ∂M and we use a consistent two-point flux approximation for the Laplace-Beltrami operator.

As regards the discretization of nonlinear terms f _b ⁰ and f _s ⁰ (denoted by d ^{f b} and d ^{f s} re- spectively) we use two different discretizations (see Definition 3.1): the classical implicit dis- cretization and a semi-implicit discretization which enables us to obtain an energy estimate unconditionally stable.

The problem is then written as follows. For a given c ⁿ _T ∈ R ^T , find (c ⁿ⁺¹ _T , µ ⁿ⁺¹ _T ) ∈ R ^T × R ^T such that for any K ∈ M, L ∈ ∂M,

m K

c ⁿ⁺¹ _K − c ⁿ _K

∆t = − P

σ ∈E _K ^int

m _σ d K , L

µ ⁿ⁺¹ _K − µ ⁿ⁺¹ _L (3.1a) ;

m K µ ⁿ⁺¹ _K = P

σ ∈E _K ^int

m σ

d K , L

c ⁿ⁺¹ _K − c ⁿ⁺¹ _L

+ P

σ ∈E ^ext _K

m e L

d K , L

c ⁿ⁺¹ _K − c ⁿ⁺¹ _L (3.1b)

+ m K d ^{f b} (c ⁿ _K , c ⁿ⁺¹ _K );

m e L

c ⁿ⁺¹ _L − c ⁿ _L

∆t = − P

v ∈VL

c ⁿ⁺¹ _L − c ⁿ⁺¹ _L0 d L , L0

− m e L d ^{f s} (c ⁿ _L , c ⁿ⁺¹ _L ) − m e L

d K , L

c ⁿ⁺¹ _L − c ⁿ⁺¹ _K . (3.1c)

Definition 3.1 (Discretization of nonlinear terms). The implicit discretization is defined as follows: for any K ∈ M, L ∈ ∂M,

d ^{f b} (c ⁿ _K , c ⁿ⁺¹ _K ) = f _b ⁰ (c ⁿ⁺¹ _K ) and d ^{f s} (c ⁿ _L , c ⁿ⁺¹ _L ) = f _s ⁰ (c ⁿ⁺¹ _L ).

As regards the semi-implicit discretization, for ∗ ∈ {b, s} we note

d ^{f ∗} (x, y) = f _∗ (x) − f _∗ (y)

x − y , ∀x, y, x 6= y and d ^{f ∗} (x, x) = f _∗ ⁰ (x), ∀x.

We remark that in practice we use a polynomial function for the potential f _∗ and that d ^{f ∗} (x, y)

is a polynomial function in the variables x, y. Thus, from a computational point of view, we

do not have numerical instability when x is too close to y.

Proposition 3.2. For both discretizations the discrete energy is dissipated as follows: if c ⁿ _T is given and (c ⁿ⁺¹ _T , µ ⁿ⁺¹ _T ) is solution to problem (3.1), then there exists C 9 > 0 independent of ∆t and T such that

F T (c ⁿ⁺¹ _T ) − F T (c ⁿ _T ) + C ₉

∆t µ ⁿ⁺¹ _T

2 1, T + 1

∆t

c ⁿ⁺¹ _∂M − c ⁿ _∂M

2 0, ∂M

+ 1 2

c ⁿ⁺¹ _T − c ⁿ _T

2 1, T + 1

2

c ⁿ⁺¹ _∂M − c ⁿ _∂M

2 1, ∂M

≤ 0,

with a condition ∆t ≤ ∆t ₀ (with ∆t ₀ depending on the parameters of the equation) for the implicit discretization.

We note that summing equation (3.1a) for all K ∈ M we have the volume conservation at the discrete level

(3.2) |Ω|m Ω (c ⁿ _M ) := P

K ∈M

m K c ⁿ _K = P

K ∈M

m K c ⁰ _K := |Ω|m Ω c ⁰ _M

, ∀n ∈ {1, . . . , N }.

4. Error estimate for the fully-discrete scheme. We can now enter in the heart of the matter. The most delicate point in the proof of the error estimate (Theorem 4.2) comes from the nonlinear term f _b ⁰ (c) for which we have to pay special attention. For this, we are inspired by methods described in [8, 18] for the Neumann boundary condition and a finite- element approximation. However some supplementary difficulties arise in our case. First, the finite-volume framework complicates the study of this term. Indeed, when we use a conform finite-element method, we work (for the space discretization at least) on H ¹ -conformal spaces.

That is not the case in the framework of finite-volume method where we resort to discrete spaces. Moreover, we use here two different discretizations for the nonlinear term f _b ⁰ (see Definition 3.1). The second one, that is the semi-implicit discretization, is more difficult to study, which complicates again the proof of Theorem 4.2.

4.1. Main result. In this section we present the error estimate (Theorem 4.2) between the center-value projection (see Definition 4.1) of the exact solution and the discrete solution obtained by solving problem (3.1).

Definition 4.1 (Center-value projection). The center-value projection P ^c T : C ⁰ (Ω) → R ^T is defined as follows. For any u ∈ C ⁰ (Ω), we set P ^c T u = ( P ^c M u, P ^c ∂M u) with

P ^c M u = ( P ^c K u) _K∈M = (u(x K )) _K∈M and P ^c ∂M u = ( P ^c L u) _L∈∂M = (u _p Γ (x L )) _L∈∂M .

Theorem 4.2 (Error estimate). Let (c, µ) ∈ C ³ ([0, T ] × Ω) × C ² ([0, T ] × Ω) be a solution to the Cahn-Hilliard equation (1.1) associated with the initial data c ⁰ ∈ C ² (Ω). Let M > 0 be such that kck _L ∞ (0,T;L ^∞ (Ω)) ≤ M and M ⁰ > M . Then setting c ⁰ _T = P ^c T c ⁰ , for any solution (c ⁿ _T , µ ⁿ _T ) to problem (3.1) satisfying

(4.1) sup

sup

K∈M

|c ⁿ _K |, sup

L∈∂M

|c ⁿ _L |

≤ M ⁰ , ∀n ∈ J 0, N K ,

there exists C ₁₀ > 0 (depending on M and M ⁰ ) such that the following estimate holds (with ∆t small enough),

(4.2) sup

n∈ J 0,N K

k P ^c T c(t ⁿ ) − c ⁿ _T k _{1, T} + sup

n∈ J 0,N K

k P ^c ∂M c(t ⁿ ) − c ⁿ _∂M k _{1, ∂M} ≤ C ₁₀ (∆t + size(T )).

In order to prove Theorem 4.2 we need some Lipschitz type regularity properties on the poten-

tials. So we introduce truncated Lipschitz continuous functions in the following way. The exact

solution c of continuous problem (1.1) is supposed to belong to C ³ ([0, T ] × Ω). Thus, there exists M > 0 such that kck _L ∞ (0,T ;L ^∞ (Ω)) ≤ M . Let M ⁰ > M , we choose truncated potentials f e _b and f e _s of the initial potentials f _b and f _s satisfying f e _b = f _b and f e _s = f _s on [−M ⁰ , M ⁰ ], and which are constant at the infinity. These truncated functions, and all their derivatives, are Lipschitz continuous. Moreover, the definition of the semi-implicit discretization of nonlinear terms implies

d ^{f b} (x, y) = Z 1

0

f _b ⁰ (x + s(y − x))ds and d ^{f s} (x, y) = Z 1

0

f _s ⁰ (x + s(y − x))ds.

Thus the semi-implicit discretization of f e _b ⁰ (resp. f e _s ⁰ ) coincides with d ^{f b} (resp. d ^{f s} ) on [−M ⁰ , M ⁰ ] ² and is Lipschitz continuous.

Therefore, we can derive the proof of Theorem 4.2 with the truncated potentials instead of the initial potentials (and so with c ⁿ _T a solution to discrete problem (3.1) but with the truncated potentials). We obtain estimate (4.2) for this problem with constant C 10 depending on M ⁰ . Thus for a mesh belonging to a quasi-uniform mesh family, gathering estimate (2.4), Lemma 2.8 and Theorem 4.2 we deduce

sup

K∈M

| P ^c K c(t ⁿ ) − c ⁿ _K | ≤ C 5 C 10 C 8

∆t

size(T ) ^{2 / q} + size(T ) ^{1− 2 / q}

. Moreover, thanks to Lemma 2.9 and Theorem 4.2 we also have

sup

L∈∂M

| P ^c L c(t ⁿ ) − c ⁿ _L | ≤ C 6 C 10 (∆t + size(T )).

Thus, if ∆t and size(T ) tend to 0 with ∆t ≤ Csize(T ) ^α for some α > 0, we have that bound (4.1) is satisfied for any approximate solution. Moreover, since the functions f b (resp.

f s ) and f e b (resp. f e s ) coincide on [−M ⁰ , M ⁰ ], if c ⁿ _T is solution to discrete problem (3.1) with the truncated potentials, it also holds with the initial potentials (and reciprocally).

In conclusion, for a quasi-uniform mesh family, if ∆t and size(T ) are so that ∆t ≤ size(T ) ^α (for an arbitrary value α > 0), then assumption (4.1) is still satisfied for ∆t and size(T ) small enough. Therefore there exists at least one solution c ⁿ _T to discrete problem (3.1) which satisfies (4.1) (although there may exist solutions of problem (3.1) for which (4.1) does not hold). Thus we can carry out the proof of Theorem 4.2 with the truncated functions which satisfy all the necessary regularity assumptions. For the sake of simplicity we omit the tilde sign in the sequel.

Remark 4.3. From a computational point of view, we can check that assumption (4.1) holds when we use the potentials f b and f s .

In order to prove the error estimate theorem we have to use a priori bounds on the discrete solution (obtained thanks to the discrete energy estimate given in Proposition 3.2).

Proposition 4.4 (Bounds of the discrete solutions, [20, Proposition 4.4]). For any c ⁰ ∈ C ² (Ω), let c ⁰ _T = P ^c T c ⁰ and (c ⁿ _T , µ ⁿ _T ) ∈ R ^T × R ^T be a solution to problem (3.1). Then, there exist positive constants M ₁ , M ₂ , M ₃ , M ₄ and M ₅ independent of ∆t and size(T ) such that

sup

n≤N

kc ⁿ _T k _{1, T} ≤ M 1 , sup

n≤N

kc ⁿ _∂M k _{1, ∂M} ≤ M 2 ,

N −1

X

n=0

∆t µ ⁿ⁺¹ _T

2

1, T ≤ M 3 ,

N−1

X

n=0

∆t

c ⁿ⁺¹ _T − c ⁿ _T

∆t

2

1, T

≤ M 4

∆t ,

and

N −1

X

n=0

∆t

c ⁿ⁺¹ _∂M − c ⁿ _∂M

∆t

2

0, ∂M

+ ∆t ²

N−1

X

n=0

c ⁿ⁺¹ _∂M − c ⁿ _∂M

∆t

2

1, ∂M

≤ M ₅ .

We can remark that in [20] the proposition is proved by choosing the mean-value projec- tion on all control volumes as discrete initial data for the initial concentration c ⁰ . In fact, when the initial data is not enough regular, we have to choose this projection to obtain the a priori bounds. However, when the initial data belongs to C ² (Ω) the center-value projection is sufficient. Indeed, thanks to the mean-value theorem, we obtain a bound on the H ¹ -seminorm of c ⁰ _T which allows us to prove the proposition as in [20].

4.2. Discrete projections. To prove Theorem 4.2 we have to define another projection:

the elliptic projection. These projection is in fact the solution of a suitable Laplace problem and thus depends on the boundary condition that we want to impose. Therefore, the definition of the elliptic projection for the chemical potential (see Definition 4.5) and for the order parameter (see Definition 4.6) are different.

Definition 4.5 (Elliptic projection with Neumann boundary conditions).

We define the space H _N ² (Ω) = {u ∈ H ² (Ω) : ∇u · ~ n = 0 on Γ}, then the elliptic projection P ^ell,N T : H _N ² (Ω) → R ^T is defined as follows. For any u ∈ H _N ² (Ω), P ^ell,N T u is the solution to the following discrete Laplace problem.

Find v T ∈ R ^T such that P

K ∈M

m K v K = Z

Ω

u and

(4.3) P

σ ∈E _K ^int

m σ

v K − v L

d K , L

= − Z

K

∆u(x)dx, ∀ K ∈ M.

Definition 4.6 (Elliptic projection with Ventcell boundary conditions).

The elliptic projection P ^ell,D T : H _Γ ² (Ω) → R ^T is defined as follows. For any u ∈ H _Γ ² (Ω), P ^ell,D T u is the solution to the following discrete Laplace problem.

Find v T ∈ R ^T such that P

K ∈M

m K v K = Z

Ω

u and for any K ∈ M, L ∈ ∂M,

(4.4)



 

 

 

  P

σ ∈E _K ^int

m σ

v K − v L

d K , L

+ P

σ ∈E _K ^ext

m e L

v K − v L

d K , L

= − Z

K

∆u(x)dx;

P

v ∈VL

v L − v L0

d L , L0

+ m e L

v L − v K

d K , L

= − Z

L

∆ Γ u _p Γ (x)dσ(x) + Z

L

∇u(x) · ~ n(x)dσ(x).

Remark 4.7. The elliptic projection is the solution to the finite-volume two-point flux approximation of the continuous problem

(4.5)

( −∆v = f in Ω,

−α∆ Γ v + ∂ n v = g on Γ, with f = −∆u and

• α = 0, g = 0 for the elliptic projection P ^{ell,N T} ;

• α = 1, g = −∆ _Γ u + ∂ _n u for the elliptic projection P ^{ell,D T} .

When u is a time-dependent function, for a fixed time t ∈ R , we denote P ^{ell,N T} (u(t)) and P ^ell,D T (u(t)) the elliptic projections of the function v = u(t, .).

In order to prove Theorem 4.2 we have to relate the different discrete projections to the

solution of the continuous problem (1.1). With this in mind we give below several properties

which will be used all along the proof in Section 4.3.

Thanks to the Taylor’s formulas and the Jensen inequality we can easily prove the following estimates between an arbitrary function and its center-value projection.

Lemma 4.8. Let u ∈ H ² (Ω), there exists C 11 > 0 independent of size(T ) such that ku − P ^c M uk _{L2 (Ω)} ≤ C 11 size(T ) k∇uk _{H1 (Ω)} .

Moreover, there exists C 12 > 0 independent of size(T ) such that for any u _p Γ ∈ H ¹ (Γ), ku _p Γ − P ^c ∂M u _p Γ k _{L2 (Γ)} ≤ C 12 size(T ) k∇ Γ u _p Γ k _{L2 (Γ)} .

Since the introduction of the elliptic projection is essential to prove the error estimate theorem, we also need to control this projection.

Proposition 4.9. Let u ∈ H _N ² (Ω), there exists C 13 > 0 depending only on Ω such that

P ^ell,N M u − u

L2 (Ω) +

P ^ell,N T u − P ^c T u _1,

T ≤ C ₁₃ size(T ) kuk _H 2 (Ω) .

Proof. Thanks to Definition 4.5 of the elliptic projection, P ^{ell,N T} u is solution to discrete problem (4.3). Thus the difference P ^{ell,N T} u − P ^c T u is the error associated to problem (4.5) with α = g = 0. Thanks to the error estimate for the Laplace problem (4.5) for the two-point flux approximation scheme (see for example [10, Section 3.2.3]) we obtain the expected estimates.

As regards the analogous proposition in the case of Ventcell boundary condition, the proof does not seem available in the literature and so we propose a complete proof in the Appendix.

Proposition 4.10. Let u ∈ H _Γ ² (Ω), there exists C 14 > 0 depending only on Ω and reg(T ) such that

P ^ell,D T u − u

_{L2 (Ω)} +

P ^ell,D ∂M u − u

_{L2 (Γ)} ≤ C 14 size(T ) kuk _H 2 Γ (Ω)

and

P ^ell,D T u − P ^c T u _1,

T +

P ^ell,D ∂M u − P ^c ∂M u _1,

∂M ≤ C ₁₄ size(T ) kuk _H 2 Γ (Ω) .

4.3. Proof of Theorem 4.2. This section is devoted to the proof of Theorem 4.2. We decompose the proof in three steps. First, we use the scheme to decompose the different components of the error. Then, we control all the terms and prove that they tend to 0 when the mesh size and the time step tend to 0. Finally we use the discrete Gronwall lemma to conclude the proof.

In the proof different components of the total error appear. Thus we decompose the error as follows.

Definition 4.11 (Error). Let u : (0, T )×Ω → R and u ⁿ _T be a finite-volume approximation of u at time t ⁿ . We denote by e ^u,n _T ∈ R ^T the error associated with u at time t = t ⁿ defined as follows

e ^u,n _T = ¨ e ^u,n _T + ˙ e ^u,n _T with ¨ e ^u,n _T = P ^c T u(t ⁿ ) − P ^ell,∗ T u(t ⁿ ) and e ˙ ^u,n _T = P ^ell,∗ T u(t ⁿ ) − u ⁿ _T , with ∗ = {N, D} depending on the boundary condition associated with u.

We also define e ¯ ^u,n T = u(t ⁿ , ·) − ^m _m ^T

T P ^ell,∗ T u(t ⁿ ).

4.3.1. Different contributions of the error. First, subtracting the scheme and the continuous problem we identify the different components of the error (Proposition 4.12). Then, we separate the error into two parts: the error e ˙ T between the elliptic projection of the exact solution and the approximate solution in the left-hand side, and all the other contributions in the right-hand side (Proposition 4.13).

Proposition 4.12. Let us consider a couple (c, µ) solution to the continuous Cahn-Hilliard equation (1.1) and a couple (c ⁿ⁺¹ _T , µ ⁿ⁺¹ _T ) solution to the finite-volume scheme (3.1). Then, the following equality holds

(4.6)

∆t e ˙ ^µ,n+1 _T

2 1, T + 1

2

e ˙ ^c,n+1 _T

2

1, T − | e ˙ ^c,n _T | ² _{1, T} +

e ˙ ^c,n+1 _T − e ˙ ^c,n _T

2 1, T

+ 1

∆t

e ˙ ^c,n+1 _∂M − e ˙ ^c,n _∂M

2 0, ∂M + 1

2

e ˙ ^c,n+1 _∂M

2

1, ∂M − | e ˙ ^c,n _∂M | ² _{1, ∂M} +

e ˙ ^c,n+1 _∂M − e ˙ ^c,n _∂M

2 1, ∂M

=∆t R ⁿ⁺¹ _c , e ˙ ^µ,n+1 _M

L2 (Ω) − ¯ e ^c,n+1 _M − e ¯ ^c,n _M , e ˙ ^µ,n+1 _M

L2 (Ω) + ¯ e ^µ,n+1 _M , e ˙ ^c,n+1 _M − e ˙ ^c,n _M

L2 (Ω)

+ R ⁿ⁺¹ _c

pΓ , e ˙ ^c,n+1 _∂M − e ˙ ^c,n _∂M

L2 (Γ) − ¯ e ^c,n+1 _∂M − e ¯ ^c,n _∂M

∆t , e ˙ ^c,n+1 _∂M − e ˙ ^c,n _∂M

!

L2 (Γ)

−

f _b ⁰ (c(t ⁿ⁺¹ , ·)) − m _M m M

d ^{f b} (c ⁿ _M , c ⁿ⁺¹ _M ), e ˙ ^c,n+1 _M − e ˙ ^c,n _M

L2 (Ω)

−

f _s ⁰ (c _p Γ (t ⁿ⁺¹ , ·)) − m ∂M

m ∂M

d ^{f s} (c ⁿ _∂M , c ⁿ⁺¹ _∂M ), e ˙ ^c,n+1 _∂M − e ˙ ^c,n _∂M

L2 (Γ)

; where the terms R ⁿ⁺¹ _c and R ⁿ⁺¹ _{c pΓ} are defined as follows

(4.7) R ⁿ⁺¹ _c

(pΓ) = c _{( p Γ )} (t ⁿ⁺¹ , x) − c _{( p Γ )} (t ⁿ , x)

∆t − ∂ t c _{( p Γ )} (t ⁿ⁺¹ , x).

Proof. Let (c, µ) be a solution to the Cahn-Hilliard equation (1.1). Applying Definition 4.5 of the elliptic projection with Neumann boundary condition to µ, for any K ∈ M we have (4.8)

Z

K

∂ t c(t ⁿ⁺¹ , x)dx + P

σ ∈E ^int _K

m σ

P ^ell,N K µ(t ⁿ⁺¹ ) − P ^ell,N L µ(t ⁿ⁺¹ ) d K , L

= 0.

In the same way, applying Definition 4.6 of the elliptic projection with Ventcell boundary condition to c, for any K ∈ M we have

(4.9) P

σ ∈E _K ^int

m _σ P ^{ell,D K} c(t ⁿ⁺¹ ) − P ^{ell,D L} c(t ⁿ⁺¹ ) d K , L

+ P

σ ∈E _K ^ext

m _{e L} P ^{ell,D K} c(t ⁿ⁺¹ ) − P ^{ell,D L} c(t ⁿ⁺¹ ) d K , L

+ Z

K

f _b ⁰ (c(t ⁿ⁺¹ , x))dx − Z

K

µ(t ⁿ⁺¹ , x)dx = 0, and for any L ∈ ∂M,

(4.10) P

v ∈VL

P ^ell,D L c(t ⁿ⁺¹ ) − P ^ell,D L0 c(t ⁿ⁺¹ ) d L , L0

+ m _{e L} P ^ell,D L c(t ⁿ⁺¹ ) − P ^ell,D K c(t ⁿ⁺¹ ) d K , L

+ Z

L

f _s ⁰ (c _{p Γ} (t ⁿ⁺¹ , x))dσ(x) + Z

L

∂ _t c _{p Γ} (t ⁿ⁺¹ , x)dσ(x) = 0,

where K ∈ M is the interior control volume such that L ∈ ∂ K . Subtracting equation (3.1a) of discrete problem and equation (4.8) and using definition (4.7) of R _c ⁿ⁺¹ imply

(4.11) P

σ ∈E _K ^int

m _σ e ˙ ^µ,n+1 K − e ˙ ^µ,n+1 L

d K , L

+ m K

˙

e ^c,n+1 K − e ˙ ^c,n K

∆t =

Z

K

R _c ⁿ⁺¹ (x)dx

− Z

K

¯

e ^c,n+1 K (x) − ¯ e ^c,n K (x)

∆t dx.

Now, we subtract equation (3.1b) of the discrete problem and equation (4.9), then we obtain

(4.12) P

σ ∈E _K ^int

m σ

˙

e ^c,n+1 _K − e ˙ ^c,n+1 _L d K , L

+ P

σ ∈E _K ^ext

m e L

˙

e ^c,n+1 _K − e ˙ ^c,n+1 _L d K , L

− m K e ˙ ^µ,n+1 _K

= Z

K

¯

e ^µ,n+1 _K dx − Z

K

f _b ⁰ (c(t ⁿ⁺¹ , x)) − m K

m K

d ^{f b} (c ⁿ _K , c ⁿ⁺¹ _K )

dx.

Subtracting equation (3.1c) of the discrete problem and equation (4.10) we have

(4.13) P

v ∈VL

˙

e ^c,n+1 L − e ˙ ^c,n+1 _L0 d L , L0

+ m _{e L} e ˙ ^c,n+1 L − e ˙ ^c,n+1 K

d K , L

+ m _{e L} e ˙ ^c,n+1 L − e ˙ ^c,n L

∆t =

Z

L

R ⁿ⁺¹ _c

pΓ (x)dσ(x)

− Z

L

¯

e ^c,n+1 L (x) − ¯ e ^c,n L (x)

∆t dσ(x) −

Z

L

f _s ⁰ (c _{p Γ} (t ⁿ⁺¹ , x)) − m e _L

m L

d ^{f s} (c ⁿ _L , c ⁿ⁺¹ _L )

dσ(x).

We multiply equation (4.11) by v K and we sum up over all interior control volumes K ∈ M, we obtain

(4.14) J e ˙ ^µ,n+1 _T , v T K 1, T + e ˙ ^c,n+1 _M − e ˙ ^c,n _M

∆t , v _M

!

M

= R ⁿ⁺¹ _c , v _M

L2 (Ω) − e ¯ ^c,n+1 _M − ¯ e ^c,n _M

∆t , v _M

!

L2 (Ω)

.

Now, we first multiply equation (4.12) by u K and we sum up over all interior control volumes

K ∈ M. Then, we multiply equation (4.13) by u L and we sum up over all boundary control volumes L ∈ ∂M. Summing the resulting equalities, we have

(4.15)

J e ˙ ^c,n+1 _T , u T K 1, T − e ˙ ^µ,n+1 _M , u _M

M + J e ˙ ^c,n+1 _∂M , u _∂M K 1, ∂M + e ˙ ^c,n+1 _∂M − e ˙ ^c,n _∂M

∆t , u _∂M

!

∂M

= ¯ e ^µ,n+1 _M , u M

L2 (Ω) +

R ⁿ⁺¹ _{c pΓ} , u ∂M

L2 (Γ) − e ¯ ^c,n+1 ∂M − e ¯ ^c,n ∂M

∆t , u ∂M

!

L2 (Ω)

−

f _b ⁰ (c(t ⁿ⁺¹ , ·)) − m M

m M

d ^{f b} (c ⁿ _M , c ⁿ⁺¹ _M ), u M

L2 (Ω)

−

f _s ⁰ (c _{p Γ} (t ⁿ⁺¹ , ·) − m _∂M

m _∂M d ^{f s} (c ⁿ _∂M , c ⁿ⁺¹ _∂M ), u _∂M

L2 (Γ)

.

By choosing v T = ∆t e ˙ ^µ,n+1 T in equation (4.14) and u T = ˙ e ^c,n+1 T − e ˙ ^c,n T in equation (4.15) and

adding the two resulting equalities the claim follows.

Proposition 4.13. Let (c, µ) be solution to the Cahn-Hilliard equation (1.1) and (c ⁿ⁺¹ _T , µ ⁿ⁺¹ _T ) be solution to the discrete scheme (3.1). Then, for any n 0 ∈ J 0, N K we have

(4.16) 1 2

n ₀

P

n=0

∆t e ˙ ^µ,n+1 _T

2 1, T + 1

2 1

2 e ˙ ^c,n _T ^{0 +1}

2 1, T −

e ˙ ^c,0 _T

2 1, T +

n ₀

P

n=0

e ˙ ^c,n+1 _T − e ˙ ^c,n _T

2 1, T

+ 1 2

n ₀

P

n=0

∆t

˙

e ^c,n+1 _∂M − e ˙ ^c,n _∂M

∆t

2

0, ∂M

+ 1 2

e ˙ ^c,n _∂M ^{0 +1}

2 1, ∂M −

e ˙ ^c,0 _∂M

2 1, ∂M +

n ₀

P

n=0

e ˙ ^c,n+1 _∂M − e ˙ ^c,n _∂M

2 1, ∂M

≤ T _c 0 +

n 0

P

n=1

∆t | e ˙ ^c,n _T | ² _{1, T} + T _R n+1

c + T e ¯ _T + T g _T + T f _b + T f _s + T T , where the different error terms are defined as follows

T c ⁰ = e ˙ ^c,0 _M

2

L2 (Ω) + 5|Ω|

8C 1 C ₄ ² m Ω e ˙ ^c,0 _M ² , T _R n+1

c = 7 2 C ₄ ²

n 0

P

n=0

∆t R _c ⁿ⁺¹

2

L2 (Ω) + 2C 3 n 0

P

n=0

∆t R ⁿ⁺¹ _{c pΓ}

2

L2 (Γ) , T e ¯ T = 7

2 C ₄ ²

n ₀

P

n=0

∆t

¯

e ^c,n+1 M − e ¯ ^c,n M

∆t

2

L2 (Ω)

+ 2C 3 n ₀

P

n=0

∆t

¯

e ^c,n+1 ∂M − ¯ e ^c,n ∂M

∆t

2

L2 (Γ)

+ 1 2

¯ e ^µ,1 _M

2

L2 (Ω) + 2C ₄ ²

¯ e ^µ,n _M ^{0 +1}

2

L2 (Ω) + C ₄ ² 2

n ₀

P

n=1

∆t

¯

e ^µ,n _M − ¯ e ^µ,n+1 _M

∆t

2

L2 (Ω)

, T _{g T} = 1

2

g _M (t ¹ , ·)

2

L2 (Ω) + 4C ₂ C ₄ ²

g _M (t ^{n 0 +1} , ·)

2

L2 (Ω)

+ C 2 C ₄ ²

n 0

P

n=1

∆t

g M (t ⁿ , ·) − g M (t ⁿ⁺¹ , ·)

∆t

2

L2 (Ω)

+ 2C 3 n 0

P

n=0

∆t

g ∂M (t ⁿ⁺¹ , ·)

2

L2 (Γ) , T f _b =4

n 0

P

n=0

∆t

f _b ⁰ ( P ^c T (c(t ⁿ⁺¹ ))) − f _b ⁰ (c ⁿ⁺¹ _T )

2 1, T +

d ^{f b} (c ⁿ⁺¹ _T , c ⁿ⁺¹ _T ) − d ^{f b} (c ⁿ _T , c ⁿ⁺¹ _T )

2 1, T

, T f _s =4C 3

n ₀

P

n=0

∆t

f _s ⁰ P ^c ∂M (c(t ⁿ⁺¹ ))

− d ^{f s} (c ⁿ _∂M , c ⁿ⁺¹ _∂M )

2

L2 (Γ) , T T = 4

n 0

P

n=0

∆t

m T − m T

m T

f _b ⁰ ( P ^c T (c(t ⁿ⁺¹ )))

2

1, T

+ 4C 3 n 0

P

n=0

∆t

m ∂M − m ∂M

m _∂M d ^{f s} (c ⁿ _∂M , c ⁿ⁺¹ _∂M )

2

L2 (Γ)

. The term g M (resp. g ∂M ) is such that for any K ∈ M (resp. L ∈ ∂M),

(4.17) g M (t, x) = f _b ⁰ (c(t, x)) − f _b ⁰ (c(t, x K )) , ∀x ∈ K , t ∈ R ; g _∂M (t, x) = f _s ⁰ (c _{p Γ} (t, x)) − f _s ⁰ (c _{p Γ} (t, x L )) , ∀x ∈ L , t ∈ R .

Proof. To begin we sum identity (4.6) for n going from 0 to n 0 and we use definition (4.17) of the functions g M and g ∂M for the terms where the nonlinear potentials appear. To obtain estimate (4.16), we have to apply the Young inequality to all the terms in the right-hand side of the resulting equality.

• Let us begin by the terms where the L ² -inner product with e ˙ ^µ,n+1 _M appears. Noting that if v _M ∈ R ^M has a zero mean-value, for any u _M ∈ R ^M , we have

(u _M , v _M ) _{L2 (Ω)} = (u _M − m _Ω (u _M ), v _M ) _{L2 (Ω)} ≤ ku M − m _Ω (u _M )k _{L2 (Ω)} kv M k _{L2 (Ω)} ,

then thanks to the discrete mean Poincaré inequality given in Lemma 2.7, we obtain (4.18) (u M , v M ) _{L2 (Ω)} ≤ C 4 |u T | _{1, T} kv M k _{L2 (Ω)} .

The couple (c, µ) is solution to the Cahn-Hilliard equation (1.1) with the homogeneous Neumann boundary condition for µ so for any n ∈ J 0, n 0 K , m Ω R ⁿ⁺¹ _c

= 0. In the same way, thanks to Definition 4.6 of the elliptic projection, for any n ∈ J 0, n 0 K we have m _Ω (¯ e ^c,n _M ) = 0 and so m _Ω

¯

e ^c,n+1 _M − ¯ e ^c,n _M

= 0. Thus, owing to estimate (4.18) and the Young inequality, for any n ∈ J 0, n 0 K one has

(4.19) ∆t R ⁿ⁺¹ _c , e ˙ ^µ,n+1 _M

L2 (Ω) ≤ 1 8 ∆t

e ˙ ^µ,n+1 _T

2

1, T + 2C ₄ ² ∆t R ⁿ⁺¹ _c

2

L2 (Ω) , and

(4.20) e ¯ ^c,n+1 _M − ¯ e ^c,n _M , e ˙ ^µ,n+1 _M

L2 (Ω) ≤ 1 8 ∆t

e ˙ ^µ,n+1 _T

2

1, T + 2C ₄ ² ∆t

¯

e ^c,n+1 _M − e ¯ ^c,n _M

∆t

2

L2 (Ω)

.

• Now we focus on the terms whose term e ˙ ^c,n+1 _M − e ˙ ^c,n _M appears in the inner product on L ² (Ω). For the two first we perform a discrete time integration by parts.

Let us begin by the term where the function ¯ e ^µ,n+1 _M intervenes.

n ₀

P

n=0

¯

e ^µ,n+1 _M , e ˙ ^c,n+1 _M − e ˙ ^c,n _M

L2 (Ω) = − ¯ e ^µ,1 _M , e ˙ ^c,0 _M

L2 (Ω) + ¯ e ^µ,n _M ^{0 +1} , e ˙ ^c,n _M ^{0 +1}

L2 (Ω)

+

n ₀

P

n=1

¯

e ^µ,n _M − ¯ e ^µ,n+1 _M , e ˙ ^c,n _M

L2 (Ω) .

Noting that by definition of the elliptic projection (see Definition 4.5) m Ω (¯ e ^µ,n M ) = 0, the Cauchy-Schwartz inequality, inequality (4.18) and the Young inequality give

(4.21)

n 0

P

n=0

¯

e ^µ,n+1 _M , e ˙ ^c,n+1 _M − e ˙ ^c,n _M

L2 (Ω)

≤ 1 2

¯ e ^µ,1 _M

2

L2 (Ω) + 1 2

e ˙ ^c,0 _M

2

L2 (Ω)

+ C ₄ ² 2

n ₀

P

n=1

∆t

¯

e ^µ,n _M − ¯ e ^µ,n+1 _M

∆t

2

L2 (Ω)

+ 1 2

n ₀

P

n=1

∆t | e ˙ ^c,n _T | ² _{1, T} + 1

8 e ˙ ^c,n _T ^{0 +1}

2

1, T + 2C ₄ ²

¯ e ^µ,n _M ^{0 +1}

2

L2 (Ω) .

Considering now the inner product with the term g M (t ⁿ⁺¹ , ·). Since the mean-value of the function g M (t, ·) is not equal to 0, we cannot apply exactly the same reasoning.

Definition 4.6 of the elliptic projection and the discrete volume conservation (3.2) imply that for any n ∈ J 0, N K , m _Ω ( ˙ e ^c,n _M ) = m _Ω

˙ e ^c,0 _M

. Then the Cauchy-Schwarz inequality, Proposition 2.1, Poincaré inequality (2.2) and the Young inequality get

(4.22)

n ₀

P

n=0

g _M (t ⁿ⁺¹ , ·), e ˙ ^c,n+1 _M − e ˙ ^c,n _M

L2 (Ω)

≤ 1 2

g _M (t ¹ , ·)

2

L2 (Ω) + 1 2

e ˙ ^c,0 _M

2

L2 (Ω)

+ 4C 2 C ₄ ²

g M (t ^{n 0 +1} , ·)

2

L2 (Ω) + 1 8 e ˙ ^c,n _T ^{0 +1}

2

1, T + 5|Ω|

8C ₁ C ₄ ² m Ω e ˙ ^c,0 _M ² + 1

2

n ₀

P

n=1

∆t | e ˙ ^c,n _T | ² _{1, T} + C ₂ C ₄ ²

n ₀

P

n=1

∆t

g _M (t ⁿ , ·) − g _M (t ⁿ⁺¹ , ·)

∆t

2

L2 (Ω)

.

As regards the last term where the inner product with e ˙ ^c,n+1 _M − e ˙ ^c,n _M appears we have to use the scheme. Noting that

f _b ⁰ P ^c M (c(t ⁿ⁺¹ ))

− m M

m M

d ^{f b} (c ⁿ _M , c ⁿ⁺¹ _M ), e ˙ ^c,n+1 _M − e ˙ ^c,n _M

L2 (Ω)

= m M

m _M f _b ⁰ P ^c M (c(t ⁿ⁺¹ ))

− d ^{f b} (c ⁿ _M , c ⁿ⁺¹ _M ), e ˙ ^c,n+1 _M − e ˙ ^c,n _M

M

,

and choosing v T = ∆t

m M

m _M f _b ⁰ ( P ^c T (c(t ⁿ⁺¹ ))) − d ^{f b} (c ⁿ _T , c ⁿ⁺¹ _T )

in identity (4.14) we ob- tain

f _b ⁰ P ^c M (c(t ⁿ⁺¹ ))

− m _M

m _M d ^{f b} (c ⁿ _M , c ⁿ⁺¹ _M ), e ˙ ^c,n+1 _M − e ˙ ^c,n _M

L2 (Ω)

= −∆t J e ˙ ^µ,n+1 _T , m T

m T

f _b ⁰ ( P ^c T (c(t ⁿ⁺¹ ))) − d ^{f b} (c ⁿ _T , c ⁿ⁺¹ _T ) K 1, T

+∆t

R ⁿ⁺¹ _c , m _M m M

f _b ⁰ ( P ^c M (c(t ⁿ⁺¹ ))) − d ^{f b} (c ⁿ _M , c ⁿ⁺¹ _M )

L2 (Ω)

−

¯

e ^c,n+1 _M − ¯ e ^c,n _M , m M

m _M f _b ⁰ ( P ^c M (c(t ⁿ⁺¹ ))) − d ^{f b} (c ⁿ _M , c ⁿ⁺¹ _M )

L2 (Ω)

.

Since m Ω R ⁿ⁺¹ _c

= m Ω

¯

e ^c,n+1 M − e ¯ ^c,n M

= 0, the Cauchy-Schwarz inequality and in- equality (4.18) imply

f _b ⁰ P ^c M (c(t ⁿ⁺¹ ))

− m _M m M

d ^{f b} (c ⁿ _M , c ⁿ⁺¹ _M ), e ˙ ^c,n+1 _M − e ˙ ^c,n _M

L2 (Ω)

≤ ∆t



 e ˙ ^µ,n+1 _T

_1,

T + C 4

R _c ⁿ⁺¹

_{L2 (Ω)} + C 4

¯

e ^c,n+1 _M − ¯ e ^c,n _M

∆t _{L2 (Ω)}





×

m T

m T

f _b ⁰ ( P ^c T (c(t ⁿ⁺¹ ))) − d ^{f b} (c ⁿ _T , c ⁿ⁺¹ _T ) _1,

T

. Recalling that for any x ∈ R , d ^{f b} (x, x) = f _b ⁰ (x) (see Definition 3.1) and applying the Young inequality we deduce

(4.23)

f _b ⁰ P ^c M (c(t ⁿ⁺¹ ))

− m _M

m _M d ^{f b} (c ⁿ _M , c ⁿ⁺¹ _M ), e ˙ ^c,n+1 _M − e ˙ ^c,n _M

L2 (Ω)

≤ 1 4 ∆t

e ˙ ^µ,n+1 _T

2 1, T + 3

2 C ₄ ² ∆t





R ⁿ⁺¹ _c

2

L2 (Ω) +

¯

e ^c,n+1 M − e ¯ ^c,n M

∆t

2

L2 (Ω)



 + 4∆t

m T − m T

m T

f _b ⁰ ( P ^c T (c(t ⁿ⁺¹ )))

2

1, T

+ 4∆t

f _b ⁰ ( P ^c T (c(t ⁿ⁺¹ ))) − f _b ⁰ (c ⁿ⁺¹ _T )

2 1, T

+ 4∆t

d ^{f b} (c ⁿ⁺¹ _T , c ⁿ⁺¹ _T ) − d ^{f b} (c ⁿ _T , c ⁿ⁺¹ _T )

2 1, T .

• Finally we focus on the terms due to the dynamic boundary condition, that is the

terms where the inner product in L ² (Γ) appears.