Convergence of a finite-volume scheme for the Cahn-Hilliard equation with dynamic boundary conditions

HAL Id: hal-01096996

https://hal.archives-ouvertes.fr/hal-01096996

Submitted on 17 Jun 2015

HAL

is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire

destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Convergence of a finite-volume scheme for the Cahn-Hilliard equation with dynamic boundary

conditions

Flore Nabet

To cite this version:

Flore Nabet. Convergence of a finite-volume scheme for the Cahn-Hilliard equation with dynamic boundary conditions. IMA Journal of Numerical Analysis, Oxford University Press (OUP), 2015,

�10.1093/imanum/drv057�. �hal-01096996�

EQUATION WITH DYNAMIC BOUNDARY CONDITIONS

F. NABET^∗

Abstract. This work is devoted to the numerical study of the Cahn-Hilliard equation with dynamic boundary conditions. A spatial finite-volume discretization is proposed which couples a 2d-method in a smooth connected domain and a 1d-method on its boundary. The convergence of the sequence of approximate solutions is proved and various numerical simulations are given.

Key words.Cahn-Hilliard equation, Dynamic boundary conditions, Finite-volume method, Convergence anal- ysis.

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

1. Introduction.

1.1. The Cahn-Hilliard equation. The Cahn-Hilliard equation describes the process of phase separation when, for example, a binary allow is cooled down sufficiently. This model is a diffuse interface model because the interface thickness between the two phases is small but non-zero. The system is described by a smooth functionccalled the order parameter, which is equal to1in one of the two phases,0in the other and which varies continuously between 0and1in the interfaces.

We consider a connected and bounded domainΩ⊂R²with aC^3,1-continuous boundaryΓ and we choose an orientation ofΓsuch that~nis the unit normal vector outward toΩ.

For a given final timeT >0, the problem is written as follows: To find the concentration of one of the two phasesc: [0, T]×Ω→Rsuch that,

(1.1a) (1.1b)





∂tc=Γb∆µ, µ=−3

2εσb∆c+12

εσbfb^′(c);

whereε >0accounts for the interface thickness (see Fig. 1.1b),fbis the bulk Cahn-Hilliard potential,Γb >0 is a bulk mobility coefficient andσb > 0is a fluid-fluid surface tension coefficient. This is supplemented by an initial condition inΩ,

(1.2) c(0, .) =c0.

In order to solve this equation, we have to add two boundary conditions onΓ =∂Ω. With respect to the chemical potentialµ, we assure that there cannot be any mass exchange through the boundary, thus we consider the homogeneous Neumann boundary condition on(0, T)×Γ,

(1.3) ∂nµ= 0.

Usually, the boundary condition associated with the order parametercis the Neumann bound- ary condition. However, for some physical systems this condition is too restrictive. Indeed, the homogeneous Neumann boundary condition oncimplies that the contact angle between the interface and the wall is equal to ^π₂. But in some physical systems, for example for bi- nary mixture, the dynamic contact angle deviate from the static contact angle ^π₂. In order to

∗flore.nabet@univ-amu.fr, Aix Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France

1

describe this phenomenon, physicists [12, 13, 16] have introduced the following boundary condition on(0, T)×Γ, called the dynamic boundary condition,

(1.4) 1

64 ε³ ΓbΓs

∂tc^pΓ= 3

8ε²σbσs∆Γc^pΓ−6σbfs^′(c^pΓ)−3 2εσb∂nc.

The trace ofconΓis notedc^pΓ,∆Γis the Laplace-Beltrami operator onΓ,∂n is the normal derivative at the boundary,Γs > 0 is a surface kinetic coefficient andσs > 0is a surface capillarity coefficient. The bulk potentialfband the surface potentialfssatisfy the following assumptions.

• Dissipativity

(1.5) lim inf

|c|→∞ fb^′′(c)>0 and lim inf

|c|→∞fs^′′(c)>0.

These conditions imply that there existα1>0andα2≥0such that for allx∈R, fb(c)≥α1c²−α2 and fs(c)≥α1c²−α2.

• Polynomial growth forfb: there existCb>0and a realp≥2such that, (1.6)

fb^(m)(c)≤Cb 1 +|c|^p−m

, m={0,1,2}.

A typical choice for the bulk potential is the polynomial double-well function (see Fig. 1.1a).

0 1

fb(c) =c²(1−c)²

(a) Bulk potential

0.0 0.5 1.0

Interface:ε

(b) Interface thickness

Fig. 1.1: Double-well structure offband definition of the interface thickness

For the sake of simplicity, we note A^∆=3

2εσb, A^fb =12

ε σb, A^∂t= 1 64

ε³ ΓbΓs

, A∆Γ =3

8ε²σbσs and A^fs= 6σb, and we write the Cahn-Hilliard equation with dynamic boundary conditions as follows

(1.7)

















∂tc= Γb∆µ; in(0, T)×Ω;

µ=−A^∆∆c+A^fbf_b^′(c); in(0, T)×Ω;

A^∂t∂tc^pΓ=A∆Γ∆Γc^pΓ− A^fsfs^′(c^pΓ)− A^∆∂nc; on(0, T)×Γ;

∂nµ= 0; on(0, T)×Γ;

c(0, .) =c⁰; inΩ.

The free energy functional associated with this equation is decomposed into a bulk and a surface contribution

(1.8) F(c) =F^b(c) +F^s(c)

where the bulk contribution is the usual energy functional associated with the Cahn-Hilliard equation with Neumann boundary conditions

F^b(c) = Z

Ω

ÅA^∆

2 |∇c|²+A^fbfb(c) ã

. As regards the surface contribution, we have

F^s(c) = Z

Γ

ÅA∆Γ

2 |∇^ΓcpΓ|²+A^fsfs(cpΓ) ã

and we can note that the dynamic boundary condition (1.4) is obtained by requiring that the system tends to minimize its total free energyF. Indeed, with this definition, the free energy functional is dissipated as follows,

(1.9) d

dtF(c(t, .)) =−Γb

Z

Ω|∇µ(t, .)|²− A^∂t Z

Γ|∂tc^pΓ(t, .)|², t∈[0, T[.

We can see that the natural energy space for this problem is the following function space, (1.10) HΓ¹(Ω) ={u∈H¹(Ω) : Tru=u^pΓ∈H¹(Γ)}.

From a mathematical point of view, the Cahn-Hilliard equation with dynamic boundary con- ditions is now well understood. We refer the reader to [7, 17, 18, 19, 20] and the references therein for details on the existence, uniqueness and regularity of solutions, existence of attrac- tors and convergence to stationary states. Moreover, in [15] the authors prove (in a more gen- eral framework) the existence of solutions to continuous Problem (1.7) in the energy spaces.

From a numerical point of view, there are less results. In [12, 13, 16] the authors consider finite-difference methods and give numerical results without proof of convergence. A spa- tial finite-element semi-discretization is proposed in [6] where the authors prove convergence results and optimal error estimates for the space semi-discrete scheme. All these results are obtained in a slab by imposing periodic conditions in lateral directions. Thus, more complex geometries of the domain are not considered.

In this article, we investigate a finite-volume scheme for solving Problem (1.7) with a smooth domain Ω. This spatial discretization allows to easily couple the dynamics in the domain and those on the boundary by the flux term∂nc. Furthermore, finite-volume schemes account naturally for the non-flat geometry of the boundary and for the associated Laplace- Beltrami operator. Moreover, this kind of scheme preserves the mass.

1.2. Outline. The article is organized as follows. In Section 2, we first give the finite- volume notation associated with the particular geometry of the domain and the discrete un- knowns and inner products. Section 3 is dedicated to introducing the finite-volume scheme and the associated energy estimates. In Section 4, we first prove the existence of a solution to the discrete scheme and then, we state a convergence theorem. The key-point is to obtain strong compactness for the approximate solutions and their traces. To this end, we introduce a new space translation operator and we prove a suitable space translation estimate which gives a limit in L^∞(0, T, H¹(Ω)) whose trace is inL^∞(0, T, H¹(Γ)). Finally, in Section 5, we give numerical error estimates for the Cahn-Hilliard equation with dynamic boundary conditions. We also present qualitative results which are in agreement with the numerical simulations observed in the literature.

2. The Finite-Volume framework.

2.1. Main notation. We recall that the domainΩis not polygonal and that we have to solve an equation on the boundaryΓ. Thus, the notation (see Fig. 2.1) are slightly different from the usual finite-volume notation (introduced for example in [10]).

Admissible mesh. We say thatT is an admissible mesh ofΩifT is constituted of an interior meshMand a boundary mesh∂Msatisfying the following properties.

• The interior meshMis given by a family of disjoint open subsets ofΩcalled control volumes and denoted byKsuch that,

– Ω =∪^K∈^MK;

– ifK,L ∈M,K6=L, then˚K∩˚L=∅;

– ifK,L ∈M,K6=Lsuch that the dimension ofK¯∩¯Lis equal to1, thenK¯∩L¯ is the edge of the mesh separating the control volumesKandL;

– for anyK ∈M, we associate a pointxKsuch that ifK,Lare two neighboring interior control volumes, the edge which separatesKandLis orthogonal to the straight line going throughxKandxL.

• The boundary mesh∂Mis the set of edges of the control volumes inMincluded in Γ. We remark that these edges are not segments but curved sections. We note that the elements of∂Mare both edges of control volumes inMand boundary control volumes. Thus, when we consider them as control volumes belonging to∂Mwe noteL ∈∂Mand when we consider them as edges of an interior control volume we noteσ. This mesh must also satisfies an orthogonality condition: for anyL ∈∂M which is an edge of the interior control volumeK, we definexLas the intersection betweenLand the straight line passing throughxKand orthogonal to the chorde_L associated withL. Then,yL is the intersection between the chorde_L and the line (xKxL).

The set of edges. LetEbe the set of edges of the meshT,E^intis the set of interior (flat) edges andE^extis the set of exterior (curved) edges (we note thatE^ext=∂M). LetE^Kbe the set of edges of a given control volumeK ∈M.

For anyσ∈ E, we note

• mσits length;

• σ=K|^Lifσ∈ E^intis the edge which separates the control volumesKandL;

• ^D=Dσthe quadrangle whose diagonals are the edgeσand the line segment[xK, xL] ifσ∈ E^int;

• ^D=Dσ={tx+ (1−t)xK, t∈[0,1], x∈σ}ifσ∈ E^ext∩ E^K;

• mDthe Lebesgue measure ofD.

The interior mesh. We note that ifKis a control volume with one edge, at least, on the boundary, thenKis not polygonal and may be not convex.

For anyK ∈M, we note:

• ^Kthe polygon shaped by the vertices ofK; we remark thatK=KifE^K∩ E^ext =∅ and thatKcan be not included inΩifE^K∩ E^ext6=∅;

• mK(respectivelymK) the Lebesgue measure ofK(respectivelyK).

The boundary mesh. Lete_Lbe the chord associated withL. We notemL(respectively me_L) the length ofL(respectivelye_L).

LetV be the set of vertices included in the boundaryΓandV^L is the set of vertices of the boundary control volumeL.

Distances and normals. For an edgeσ∈ E, we note:

• ~n_KLthe unit normal vector toσgoing fromKtoLanddK,Lthe distance between xKandxLifσ∈ E^int;

• ~nKL the unit normal vector toe_L outward toKanddK,L the distance betweenxK

andyLifσ∈ E^ext.

• ~nKσ(x)the unit normal vector toσat the pointx∈σoutward toK(we remark that ifσ∈ E^int,~nKσ(x) =~nKLfor allx∈σ).

For a vertexv=L|^L′which separates the boundary control volumesL,L^′ ∈∂M, we note

• dL,vthe distance between the vertexvand the centeryL;

• dL,L′ the sum ofdL,v anddL′,v: it is the approximation of the lengthmγ_LL′ of the arcγLL′⊂Γpassing through the vertexv=L|^L′and whose ends arexLandxL′;

• nvLgives an orientation of the curveγLL′in function of the orientation of the curve Γ : nvL = 1 if the orientation ofΓ going fromL toL^′. ThusnvL = ±1 and nvL=−nvL′.

v=L|^L′ xL

xK

yL

xLi

d_K

,L i Kd

,_L

d^L,v

xL′

yL′

dL ′,v

~ n_KLi

~n_KL

~n_K

σ(xL)

Interior vertex Boundary vertex

Interior center Boundary center

Interior meshM

K

Boundary mesh∂M

L ∈∂M e_Lchord associated withL

Fig. 2.1: MeshT associated withΩ

As regards the boundary mesh, we can remark that in the proposed scheme (Section 3.1) we only use the coordinates of the vertices of the mesh. All the other quantities are approximations and we do not use the equation ofΓ. However, to pass to the limit in the convergence theorem we reason with the exact quantities. In the sequel (in particular in the proof of the convergence theorem) the following relations will be very useful and frequently use even if it is not expressly mentioned. For anyL ∈∂M,

m^eL−mL=O m³L

, mγ_LL′−dL,L′ =O mγ_LL′(mL+mL′) and for anyK ∈Msuch thatσ=L∈ E^K∩ E^ext,

mK−mK=O diam(K)³

, |d(xK, xL)−dK,L|=d(xL, yL) =O m²L

.

We can also note that for allx∈^L⊂Γ,~n(x)−~n_KL =O(mL). The proof of these results can be obtained by using the Taylor formulas and a parametrization ofΓ.

The mesh size is defined bysize(T) = sup{diam(K),K ∈M}. We introduce a positive numberreg(T)which measures the regularity of a given mesh,

(2.1) reg(T) := max Ñ

N,max

K∈M σ∈EK

diam(K) diam(Dσ),max

D

diam(D)

√mD

,max

K∈M

diam(K)

√mK

é

whereN is the maximum of edges incident to any vertex.

All the constants in the results below depends on this quantity which is very useful to perform the convergence analysis of finite-volume schemes. The numberreg(T)must be uniformly bounded when the mesh size tends to0for the convergence results hold.

2.2. Discrete unknowns. Regarding the space discretization, we define the piecewise constant functionsuM∈R^Mandu∂M∈R^∂Mas follows

u^M= P

K∈M

uK1K∈L^∞(Ω) and u∂M= P

L∈∂M

uL1L∈L^∞(Γ),

where1K(respectively1L) is the indicator function of the control volumeK(respectivelyL).

We also define the discrete functionuT ∈ R^T which we associate with the coupleuT = (uM, u∂M).

As regards the time discretization, we setN ∈N^∗and then∆t= _N^T. For alln∈ {1, . . . , N}, we definetⁿ = n∆t. Then, we define u^∆tM (respectivelyu^∆t∂M) as the piecewise constant function in(0, T)×Ω(respectively(0, T)×Γ) such that for anyt∈[tⁿ, tⁿ⁺¹[,

u^∆tM(t, x) =uⁿ⁺¹K ifx∈^K and u^∆t∂M(t, x) =uⁿ⁺¹L ifx∈^L.

For a given time steptⁿ, the finite-volume scheme associates with each interior control volumeK ∈Man unknown valuecⁿ_Kand with each boundary control volumeL ∈∂Man unknown valuecⁿ_Lfor the concentration. Regarding the chemical potential, the same notation are used with an interior unknownµⁿK for allK ∈ Mand a boundary unknownµⁿL for all L ∈∂M. However the homogeneous Neumann boundary condition is associated withµ, so we impose the value of the boundary unknownsµⁿ∂M∈R^∂Mas follows

µⁿL =µⁿK, ∀L ∈∂Msuch thatL=σ∈ E^K∩ E^ext.

2.3. Inner products and norms. Here, we define the inner products and the norms used in the paper. We define the discreteL²-inner products and the discreteH¹-semi-definite inner products onR^T andR^∂M.

DEFINITION2.1 (DiscreteL²-inner products).

For anyu^M, v^M∈R^Mand for anyu∂M, v∂M∈R^∂M, we define (u^M, v^M)M= P

K∈M

mKuKvK and (u∂M, v∂M)_∂M= P

L∈∂M

m^eLuLvL. The associatedL²-norms are notedk.k0,^Mandk.k0,∂M.

For anyu^M∈R^Mand for anyu∂M∈R^∂M, we define the discreteL^p-norms onR^Mand R^∂Mas follows

ku^Mk^p0,p,^M= P

K∈M

mK|uK|^p and ku^Mk^p0,p,∂M= P

L∈∂M

m^eL|uL|^p.

We can remark that the normk.k0,p,^M(respectivelyk.k0,p,∂M) is equivalent to the usualL^p normk.k^pLp(Ω) =P

K∈MmK|uK|^pinΩ(respectivelyk.k^pLp(Γ)=P

L∈∂MmL|uL|^pinΓ) with constants independent of the mesh size.

DEFINITION2.2 (DiscreteH¹-semi-definite inner products).

• For anyuT, vT ∈R^T, we define theH¹-semi-definite inner product inR^T as follows

JuT, vTK1,T = P

σ=K|L∈Eint

mσ

dK,L

(uK−uL) (vK−vL) + P

σ=L∈Eext

m^eL

dK,L

(uK−uL) (vK−vL).

• For anyu∂M, v∂M∈R^∂M, we define theH¹-semi-definite inner product inR^∂M Ju∂M, v∂MK1,∂M= P

v=L|L′ ∈V

dL,L′

ÅuL−uL′

dL,L′

ã ÅvL−vL′

dL,L′

ã . The associatedH¹-seminorms are noted|.|1,T and|.|1,∂M.

From this, for anyuT ∈R^T,u∂M∈R^∂Mwe can define theH¹-norms as follows kuTk²1,T =kuMk²0,^M+|uT|²1,T and ku∂Mk²1,∂M=ku∂Mk²0,∂M+|u∂M|²1,∂M. Moreover, we have to define norms in time and space.

DEFINITION2.3 (Discrete norms in time and space).

LetN be a discrete norm on a spaceB, then we define

• The discreteL^p(0, T;N(B))norm by

ku^∆tk^Lp(0,T;N(B))=

NX−1 n=0

∆t(N(uⁿ⁺¹))^p

!¹p

.

• The discreteL^∞(0, T;N(B))norm by

ku^∆tk^L∞(0,T;N(B)) = sup

n≤NN(uⁿ).

3. The Finite-Volume scheme and the discrete energy.

3.1. The numerical scheme.

DEFINITION3.1 (Discrete mean projection).

Letube an integrable function onΩwhich admits a traceu^pΓintegrable onΓ, we setP^m_Tu= (P^m_Mu,P^m_∂Mu^pΓ)with

P^m_Mu= Å 1

mK

Z

K

u(x)dx ã

K∈M

and P^m_∂Mu^pΓ= Å 1

mL

Z

L

u^pΓ(x)dσ(x) ã

L∈∂M

. With this definition, choosingc⁰ ∈ HΓ¹(Ω) (where HΓ¹(Ω) is defined by (1.10)), we can define the discrete initial concentration as follows,

(3.1) c⁰T =P^m_Tc⁰.

In order to obtain the finite-volume scheme associated with the Cahn-Hilliard model (1.7), we have to integrate the continuous equations. We integrate equations (1.1) fort∈[tⁿ, tⁿ⁺¹] and forK ∈ M. For Laplace operators, we use a consistent two-point flux approximation together with the homogeneous Neumann boundary condition (1.3) for the equation (1.1a).

The dynamic boundary condition (1.4) is integrated fort ∈[tⁿ, tⁿ⁺¹]and forL ∈∂M. We use a consistent two-point flux approximation for the Laplace-Beltrami operator onΓ.

With respect to the non-linear terms, in the paper we use a semi-implicit discretization de- scribed in Subsection 3.3. We denote byd^fb(respectivelyd^fs) the discretization of the po- tentialfb(respectivelyfs).

The scheme we propose reads: For anyn≥0, find(cⁿ⁺¹_T , µⁿ⁺¹_T )∈R^T ×R^T such that for anyuT, vT ∈R^T,

(3.2a)

(3.2b)

(3.2c)





















































Åcⁿ⁺¹M −cⁿM

∆t , v^M ã

M

=−ΓbJµⁿ⁺¹T , vTK1,T, µⁿ⁺¹M , u^M

M=A^∆ P

σ=K|^L∈Eint

mσ

dK,L

cⁿ⁺¹K −cⁿ⁺¹L

(uK−uL)

+A^∆ P

σ=L∈Eext

me_L

dK,L

cⁿ⁺¹_K −cⁿ⁺¹_L uK

+A^fb

P

K∈M

mKd^fb(cⁿK, cⁿ⁺¹K )uK, A^∂t

Åcⁿ⁺¹∂M −cⁿ∂M

∆t , u∂M

ã

∂M

=− A∆ΓJcⁿ⁺¹∂M , u∂MK1,∂M

− A^fs

P

L∈∂M

me_Ld^fs(cⁿ_L, cⁿ⁺¹_L )uL

− A^∆ P

σ=L∈Eext

m^eL

dK,L

cⁿ⁺¹L −cⁿ⁺¹K

uL.

We notice that we write the scheme in a "variational" formulation which is equivalent to the classical finite-volume formulation and which will be more useful in the analysis.

Let us remark that in equation (3.2a) the finite-volume approximation of the term∆µ only uses the interior edges of the meshMwhile in equation (3.2b) the approximation of the term∆cuses all the edges of the mesh (interior and exterior). The reason for this is thatµ satisfies the homogeneous Neumann boundary condition (so the exterior edges do not step in) whilecsatisfies the dynamic boundary condition (1.4) where the exterior edges intervene (this term is essential to allow the coupling with equation (3.2c) on the boundary mesh∂M).

SettingvT ≡1in equation (3.2a), we note that we have the conservation of the volume at the discrete level,

(3.3) P

K∈M

mKcⁿK= P

K∈M

mKc⁰K, ∀n∈ {1, . . . , N}.

3.2. Energy estimates. We define the discrete free energy associated with the continu- ous free energy (1.8). As in the continuous case, the discrete free energy is decomposed into a bulk contributionF^b,T and a surface contributionF^s,∂M: For anycT ∈R^T, we set

F^T(cT) = A^∆

2 |cT|²1,T +A^fb P

K∈M

mKfb(cK)

| {z }

:=Fb,T(cT)

+A∆Γ

2 |c∂M|²1,∂M+A^fs P

L∈∂M

me_Lfs(cL)

| {z }

:=Fs,∂M(c∂M)

.

Regardless of the choice of the discretization of non-linear terms, we have a general energy estimate.

PROPOSITION3.2 (General energy equality).

Letcⁿ_T ∈R^T. We assume that there exists a solution(cⁿ⁺¹_T , µⁿ⁺¹_T )to discrete Problem(3.2).

Then, the following equality holds

(3.4)

F^T(cⁿ⁺¹T )− F^T(cⁿT) + ∆tΓb

µⁿ⁺¹T

²

1,T +A^∂t

∆t

cⁿ⁺¹∂M −cⁿ∂M

²

0,∂M

+A^∆ 2

cⁿ⁺¹T −cⁿT

²

1,T +A∆Γ

2

cⁿ⁺¹∂M −cⁿ∂M

²

1,∂M

=A^fb

P

K∈M

mK fb(cⁿ⁺¹_K )−fb(cⁿ_K)−d^fb(cⁿ_K, cⁿ⁺¹_K )(cⁿ⁺¹_K −cⁿ_K) +A^fs

P

L∈∂M

m^eL fs(cⁿ⁺¹L )−fs(cⁿL)−d^fs(cⁿL, cⁿ⁺¹L )(cⁿ⁺¹L −cⁿL) .

Proof. We consider the scheme (3.2) withuT = cⁿ⁺¹_T −cⁿ_T andvT =−∆tµⁿ⁺¹_T as test

functions and we add the three equations.

The definition of the discrete energy cannot give a discrete counterpart to the equality (1.9) which gives the dissipation of the continuous energy. However, with a good choice for the discretization of non-linear terms, we can obtain the dissipation of the discrete energy.

3.3. Discretization of non-linear terms. In order to obtain an energy estimate without any condition on the time step∆t, we choose a time discretization for the non-linear terms such that the right hand side of (3.4) is equal to0. Namely, we set

d^fb(x, y) = fb(y)−fb(x)

y−x and d^fs(x, y) = fs(y)−fs(x)

y−x , ∀x, y, x6=y.

In practice, we mostly used polynomial functions for the potentialsfbandfs. Then, the terms d^fb(x, y)andd^fs(x, y)can be written as polynomial functions in the variablesx, y. Thus, we do not have numerical instability whenxis too close fromy.

In effect, we have

d^fb(x, y) =fb^′

x+y 2

+ (x−y)²P(x, y);

whereP is a polynomial function in the variablesx, y.

Thus, we remark thatd^fb(x, x) =f_b^′(x)andd^fssatisfies the same properties.

PROPOSITION3.3 (Discrete free energy equality).

Letcⁿ_T ∈R^T. We assume that there exists a solution(cⁿ⁺¹_T , µⁿ⁺¹_T )to discrete Problem(3.2).

Then, the following equality holds

(3.5) F^T(cⁿ⁺¹T )− F^T(cⁿT) + ∆tΓb

µⁿ⁺¹T

²

1,T +A^∂t

∆t

cⁿ⁺¹∂M −cⁿ∂M

²

0,∂M

+A^∆ 2

cⁿ⁺¹T −cⁿT

²

1,T +A∆Γ

2

cⁿ⁺¹∂M −cⁿ∂M

²

1,∂M= 0.

Remark:We can also use a fully implicit discretization for non-linear terms, namely d^fb(cⁿ_K, cⁿ⁺¹_K ) =f_b^′(cⁿ⁺¹_K ), ∀K ∈M and d^fs(cⁿ_L, cⁿ⁺¹_L ) =f_s^′(cⁿ⁺¹_L ), ∀L ∈∂M. In this case, we obtain the dissipation of the discrete energy for any∆t ≤ ∆t0where∆t0

only depends on the parameters of the equation. All the results given in the paper are true for this discretization if we assume that the time step satisfies∆t≤∆t0.

4. Existence and convergence theorems. The existence and convergence results proved below are true for other choices of discretization for the non-linear terms. Thus, we give here general assumptions (which are satisfied by the semi-implicit discretization) of the discretiza- tion of the non-linear potentiald^fb to obtain these results: d^fb is ofC¹class and there exist Cb≥0and a realpsuch that2≤p <+∞,

(4.1)

d^fb(a, b)≤Cb 1 +|a|^p−1+|b|^p−1 ,

D d^fb(a, .)

(b)≤Cb 1 +|a|^p−2+|b|^p−2 .

4.1. Preliminary results. In this subsection, we consider a bounded connected Lips- chitz domainΩ⊂R²and we denote byΓ =∂Ωits boundary. LetT be an admissible mesh associated withΩas described in the subsection 2.1.

First, we recall the discrete Poincaré inequality and the discrete Poincaré-Sobolev in- equality which will be very useful in the sequel. The proofs can be found in [2] in the case whereΩis polygonal and with Neumann boundary conditions but these results can be easily adapted to our case.

LEMMA4.1 (Poincaré inequality,[2, Theorem 5]).

There existsC1>0depending only onΩandreg(T)such that

(4.2) ku^M−m^M(u^M)k0,^M≤C1|uT|1,T, ∀uT ∈R^T; wherem^M(u^M) =_M¹_ΩP

K∈MmKuKandMΩ=P

K∈MmK. This Lemma gives for anyuT ∈R^T,

(4.3) kuTk²1,T ≤ 2C₁²+ 1

|uT|²1,T + 2MΩ(mM(uM))². LEMMA4.2 (Poincaré-Sobolev inequality,[2, Theorem 3]).

Let1≤q <+∞, then there existsC2>0depending only onq,Ωandreg(T)such that (4.4) ku^Mk0,q,^M≤C2kuTk1,T , ∀uT ∈R^T.

Now, we give a Sobolev inequality for the one dimension manifoldΓ.

LEMMA4.3.There existsC3>0depending only onΓandreg(T)such that ku∂MkL^∞(Γ)≤C3ku∂Mk1,∂M.

Proof. LetL1∈∂M, then uL1=

Å

uL1− 1 MΓ

P

L∈∂M

me_LuL

ã + 1

MΓ

P

L∈∂M

me_LuL with MΓ = P

L∈∂M

me_L. The Cauchy-Schwarz inequality implies

|uL1|²≤ 2 MΓ

Å P

L∈∂M

m^eL|uL1−uL|²+ku∂Mk²0,∂M

ã . IfuL1, uL2 ∈∂M, the triangle inequality and the Cauchy-Schwarz inequality give

|uL1−uL2| ≤ Ç P

v=L|L′ ∈V

dL,L′

å¹₂

|u∂M|1,∂M.

Thus, there existsCΓ>0depending only onΓsuch that

|uL1|²≤2 Å

CΓ|u∂M|²1,∂M+ 1

MΓ ku∂Mk²0,∂M

ã ,

and the proof is complete.

4.2. Existence theorem. This subsection is devoted to state general existence theorem.

THEOREM4.4 (Existence of a discrete solution).

Letcⁿ_T ∈R^T. We assume that:

• the potentialsfbandfbsatisfy dissipativity assumption(1.5)and the bulk potential satisfies the growth condition(1.6);

• the discretization of non-linear terms satisfies growth condition(4.1)forfb. Then, there exists at least one solution(cⁿ⁺¹_T , µⁿ⁺¹_T )∈R^T ×R^T to Problem(3.2).

Proof.

The proof is very similar to the one given in [4, Theorem 2.9], thus we do not give the details here. The key-point is the use of the topological degree theory [8].

As regards thea prioriestimates, we have three key-points.

• We consider the Problem (3.2) withδd^fb(respectivelyδd^fs) instead ofd^fb(respec- tivelyd^fs) withδ∈[0,1].

• Using the energy estimate (3.5), the quantitiescⁿ⁺¹T

1,T,cⁿ⁺¹∂M

1,∂Mandµⁿ⁺¹T

1,T

are bounded independently ofδ.

• ChoosinguT ≡ 1 as test function in the equations (3.2b) and (3.2c), we obtain a bound (independent ofδ) on the mean-value of µⁿ⁺¹T and thanks to the Poincaré estimate (4.3), we have the expected bound onµⁿ⁺¹T

1,T. The proof of the well-posedness of the scheme whenδ= 0is classical.

4.3. The convergence theorem. In order to give the convergence theorem, we have to recall the definition of a solution to the continuous equation (1.7) in a weak sense.

DEFINITION4.5 (Weak formulation).

We say that a couple(c, µ)∈L^∞(0, T;H¹(Ω))×L²(0, T;H¹(Ω))such thatc^pΓ∈L^∞(0, T;H¹(Γ)) is solution to continuous Problem(1.7)in the weak sense if for allφ ∈ Cc² R×R²

such thatφ(T, .) = 0, the following identities hold

Z T 0

Z

Ω

(−∂tφc+ Γb∇µ· ∇φ) dxdt= Z

Ω

c⁰φ(0, .)dx, (4.5)

Z T 0

Z

Ω

(−µφ+A^∆∇c· ∇φ+A^fbfb^′(c)φ) dxdt +

Z T 0

Z

Γ −A^∂t∂tφc^pΓ+A∆Γ∇^Γc^pΓ· ∇^Γφ+A^fsfs^′(c^pΓ)φ

dσ(x)dt

=A^∂t Z

Γ

Tr (c⁰)φ(0, .)dσ(x).

(4.6)

THEOREM4.6 (Convergence theorem).

Letc⁰ ∈ HΓ¹(Ω)(see definition(1.10)) andÄÄ

(c^∆tT )^(m)ä ,Ä

(µ^∆tT )^(m)ää

m∈Na sequence of solutions to Problem(3.2)associated with a sequence of discretizations such that the space and time steps,size(T^(m))and∆t^(m)respectively, tend to0. Then, assuming thatreg(T^(m)) is bounded when m → +∞, there exists a weak solution (c, µ)to Problem (1.7)(in the sense of Definition 4.5) for the initial datac⁰such that, up to a subsequence, the following convergence properties hold, for allq≥1

(c^∆tT )^(m)→cinL²(0, T;L^q(Ω)), (c^∆t∂M)^(m)→c^pΓinL²(0, T;L^q(Γ)), and (µ^∆tT )^(m)⇀ µinL²(0, T;L^q(Ω))weakly.

Because of non-linearities in the equation both inΩand onΓ, to prove this theorem we need strong compactness both inL²((0, T)×Ω)and inL²((0, T)×Γ). Thus, we have to apply the Kolmogorov theorem to obtain the existence of the limit and the strong convergences.

Then, we can pass to the limit in the scheme and, especially in the non-linear terms. To apply the Kolmogorov theorem we have to apply three key elements: the bounds on the discrete

solutions (see Proposition 4.11), an estimate of space translates (see Theorem 4.20) and an estimate of time translates (see Theorem 4.26).

4.3.1. Properties of the mean-value projection. Definition (3.1) of the discrete initial concentration induce us to give the following properties on the discrete mean-value projection which will be useful in the sequel.

PROPOSITION4.7 ([1, Proposition 3.5]).

For anyp≥1, there existsC4>0independent ofsize(T)such that kP^m_Muk0,p,^M≤C4kukLp(Ω), ∀u∈L^p(Ω).

Assumption A:LetP ⊂R²be a pseudo-triangle with one curved sideσ⊂R². There existν1, ν2 > 0such that for any sub-arcσ˜ ⊂ σ, the corresponding sub-triangle P^σ˜ (see Fig. 4.1) satisfies,

ν1≤ mP˜σ

mσ˜ ≤ν2.

σ

P ˜σ

P^σ˜

Fig. 4.1: The pseudo-triangleP and one of its sub-triangles

LEMMA4.8 ([10, Lemma 3.4] and [3]).

For anyp≥1, there existsC5>0depending only onreg(T), ν1, ν2andpsuch that,

• for any segmentσ⊂R²and for any bounded setP ⊂R²with positive measure,

• for any pseudo-triangleP ⊂ R²with one curved sideσ ⊂R²which satisfies the Assumption A,

and for anyu∈H¹(R²), then

|uP −uσ|^p≤C5(mσ+ diam(Q))^p mP

Z

Q|∇u(z)|^pdz, whereuP denotes the mean-value ofuonP,uσthe mean-value ofuonσand

• Q=”P^σis the convex hull ofP ∪σifσis a segment;

• Q=Pifσ⊂∂P is the curved edge of the pseudo-triangleP.

In this study, we want to apply this lemma (specifically the second point) to the control volumesK ∈Mwith one edge belongs toΓ(namely it is a curved edge). Let us remark that we can prove that for a small enough mesh size the assumption A is satisfied for these control volumes (see [3]).

Remark:This lemma is crucial to prove the proposition below which will be used in the proof of Proposition 4.11 which yields the bounds on the discrete solutions. These bounds are one of the key point of the proof of the convergence Theorem 4.6.

Indeed, to obtain the bounds on the discrete solutions we have to project the initial data such that the discreteH¹-norms onΩandΓare both controlled by theH_Γ¹(Ω)-norm of the initial

data. Ifσis a segment (the first point of the lemma) the proof of this result is classical and can be found, for instance in [10, Lemma 3.4]. Thus, we can think that we can avoid to use the case whereσis a curved edge, whose proof is more complicated, by choosing the mean on the chords for the initial data. However, with this choice, we lose theH¹-estimate on the boundary. In fact, in this case we are not able to prove the Proposition 4.10 which allows to obtain the H¹-estimate on the boundary. For this reason, we choose the discrete initial concentration equals to the mean projection on the curved edges (see definition (3.1) ofc⁰T).

Thus, in this case we need to use the second point on the Lemma 4.8 which is a technical result whose proof is given in [3].

PROPOSITION4.9. There existsC6>0independent ofsize(T)such that for any func- tionu∈H¹(Ω),

|P^m_T u|1,T ≤C6k∇ukL2 (Ω).

Proof. Lemma 4.8 gives,

|P^m_T u|²1,T ≤2C5 P

σ=K|L∈Eint

mσ

dK,L

Å4diam(b^K)² mK

Z b

K|∇u(z)|²dz+4diam(b^L)² mL

Z b

L|∇u(z)|²dz ã

+C5 P

σ=L∈Eext

m^eL

dK,L

(mσ+ diam(K))² mK

Z

K|∇u(z)|²dz.

Then, thanks to definition (2.1) ofreg(T)we conclude the proof.

PROPOSITION 4.10. Letu∈H¹(Γ), there existsC7 >0independent ofsize(T)such that,

|P^m

∂Mu|1,∂M≤C7k∇^ΓukL2 (Γ).

Proof. We consider two neighboring boundary control volumesL,L^′ ∈∂M, then

|P^m_Lu−P^m_L′u|²≤ 1 mLmL′

Z

L

Z

L^′

Z

Ù

xy∇^Γu(z)·~τ(z)dσ(z)

2

dσ(x)dσ(y), where~τ is the unit tangent vector to the curveΓ.

Thanks to the Cauchy-Schwarz inequality, we have

|P^m_Lu−P^m_L′u|²≤(mL+mL′) Z

L∪L^′|∇^Γu(z)|²dσ(z),

and the mesh regularity completes the proof.

4.3.2. Bounds of the solutions. The following proposition is one of the key points of the proof of convergence.

PROPOSITION4.11 (Bounds of the discrete solutions).

Assuming that the assumptions of Theorem 4.6 are satisfied. Then, there exist positive con- stantsM1,M2,M3,M4andM5independent of∆tandsize(T)such that,

(4.7)

sup

n≤NkcⁿTk1,T ≤M1, sup

n≤Nkcⁿ∂Mk1,∂M≤M2,

NX−1 n=0

∆tµⁿ⁺¹M

²

1,T ≤M3,

N−1X

n=0

∆t

cⁿ⁺¹T −cⁿT

∆t

2

1,T

≤M4

∆t and

N−1X

n=0

∆t

cⁿ⁺¹∂M −cⁿ∂M

∆t

2

0,∂M

+ ∆t²

NX−1 n=0

cⁿ⁺¹∂M −cⁿ∂M

∆t

2

1,∂M

≤M5.

Proof.

• The discrete energy estimate (3.5) gives a uniform bound on the discrete free energy,

(4.8) ∀n∈J0, NK, F^T(cⁿT)≤ F^T(c⁰T).

Then, thanks to the polynomial growth assumption (1.6), we have F^T(c⁰T)≤A^∆

2 c⁰T

²

1,T +A^fbCb

ÄMΩ+c⁰T

^p

0,p,^M

ä +A∆Γ

2 c⁰∂M

²

1,∂M+A^fsCΓ max

B 0,kc⁰kL^∞(Γ)

|fs|.

Thus, definition (3.1) ofc⁰T and Propositions 4.7, 4.9 and 4.10 imply that there exists K0>0such that,

(4.9) F^T(c⁰_T)≤K0.

Thanks to dissipativity assumption (1.5),

(4.10) F^T(cⁿ_T)≥A^∆

2 |cⁿ_T|²1,T +A^fbα1kcⁿMk²0,^M− A^fbα2MΩ

+A∆Γ

2 |cⁿ_∂M|²1,∂M+A^fsα1kcⁿ_∂Mk²0,∂M− A^fsα2MΓ. Using (4.8), (4.9) and (4.10) and settingK1:=K0+α2(A^fbMΩ+A^fsMΓ)>0, there exist positive constantsK2,K3,K4andK5such that for alln∈J0, NK,

kcⁿMk²0,^M≤ K1

A^fbα1 :=K2, |cⁿT|²1,T ≤ 2K1

A^∆ :=K3, kcⁿ_∂Mk²0,∂M≤ K1

A^fsα1

:=K4, |cⁿ_T|²1,∂M≤ 2K1

A∆Γ

:=K5. These estimates are established for alln∈J0, NK, thus

(4.11)

sup

n≤NkcⁿTk1,T ≤p

K2+K3:=M1, sup

n≤NkcⁿTk1,∂M≤p

K4+K5:=M2.

• Adding energy estimates (3.5) fornfrom0toN−1, then

(4.12)

F^T(c^N_T) +

NX−1 n=0

Å

∆tΓb

µⁿ⁺¹_T ²_1,

T +A^∂t

∆t

cⁿ⁺¹_∂M −cⁿ_∂M

²_0,

∂M

+A^∆ 2

cⁿ⁺¹T −cⁿT

²

1,T +A∆Γ

2

cⁿ⁺¹∂M −cⁿ∂M

²

1,∂M

ã

=F^T(c⁰T).

Thanks to estimates (4.9) and (4.10), we have (4.13)

NX−1 n=0

∆tµⁿ⁺¹T

²

1,T ≤ K1

Γb

and

NX−1 n=0

1

∆t

cⁿ⁺¹∂M −cⁿ∂M

²

0,∂M≤ K1

A^∂t.