Finite Volume approximation of a two-phase two fluxes degenerate Cahn-Hilliard model

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Finite Volume approximation of a two-phase two fluxes degenerate Cahn-Hilliard model

Clément Cancès, Flore Nabet

To cite this version:

Clément Cancès, Flore Nabet. Finite Volume approximation of a two-phase two fluxes degenerate

Cahn-Hilliard model. ESAIM: Mathematical Modelling and Numerical Analysis, EDP Sciences, 2021,

55 (3), pp.969–1003. �10.1051/m2an/2021002�. �hal-02561981v3�

DEGENERATE CAHN-HILLIARD MODEL

CL ´EMENT CANC `ES AND FLORE NABET

Abstract. We study a time implicit Finite Volume scheme for degenerate Cahn-Hilliard model proposed in [W. E and P. Palffy-Muhoray. Phys. Rev. E, 55:R3844–R3846, 1997] and studied mathematically by the authors in [C. Canc`es, D. Matthes, and F. Nabet. Arch. Ration. Mech.

Anal., 233(2):837-866, 2019]. The scheme is shown to preserve the key properties of the continuous model, namely mass conservation, positivity of the concentrations, the decay of the energy and the control of the entropy dissipation rate. This allows to establish the existence of a solution to the nonlinear algebraic system corresponding to the scheme. Further, we show thanks to compactness arguments that the approximate solution converges towards a weak solution of the continuous problems as the discretization parameters tend to 0. Numerical results illustrate the behavior of the numerical model.

Keywords. two-phase flow, degenerate Cahn-Hilliard system, finite volumes, convergence AMS subjects classification. 65M12, 65M08, 76T99, 35K52, 35K65

1. The two-phase two fluxes degenerate Cahn-Hilliard model

The goal of this paper is to propose a convergent finite volume discretization for a degenerate Cahn-Hilliard model proposed by E and Palffy-Muhoray in [14] and studied in [8] by the authors.

Before considering the numerical scheme, let us describe and discuss the continuous model.

1.1. The continuous model. We consider a mixture made of two incompressible phases evolving in a bounded and connected polygonal open subset Ω of R² and on a time interval [0, T], where T is an arbitrary finite time horizon. The composition of the fluid is described by the volume fractions c= (c1, c2) of the two phases. Since the whole volume Ω is occupied by the two phases, the following constraint on theci holds

(1) c1+c2= 1 in (0, T)×Ω.

The evolution of the volume fractions is prescribed by the following partial differential equations

(2) ∂tci−∇·

ci

ηi

∇(µi+ Ψi)

=θi∆ci inQT := (0, T)×Ω.

In the above equation,ηi>0 denotes the viscosity of the phasei,µiis its chemical potential (which is one of the unknown of the problem), while Ψi ∈H¹(Ω) is a given external potential acting on phase i that is assumed be independent on time for simplicity. For Ψi, one can typically think about gravity, that is Ψi(x) =−%ig·xwith%ithe density of phaseiandgthe gravitational vector.

The coefficient θi≥0 is a given parameter quantifying the thermal agitation of phasei. The limit case θi = 0 is called the deep-quench limit in the Cahn-Hilliard literature. The difference of the

1

phase chemical potentials is given by the following expression (3) µ1−µ2=−α∆c1+κ(1−2c1) inQT,

where α > 0 and κ > 0 are given coefficients governing the characteristic size of the transition layers between patches of pure phases{c₁= 0}and{c₁= 1}. Typically, αis assumed to be small in comparison toκ. Equation (3) is complemented by homogeneous Neumann boundary conditions

(4) ∇ci·n= 0 on (0, T)×∂Ω,

whereas (2) is complemented by no-flux boundary conditions

(5) ci

η_i∇(µi+ Ψi)·n= 0 on (0, T)×∂Ω.

Up to now, the chemical potentials are defined up to a common constant. This degree of freedom is fixed by imposing a zero mean condition on the mean chemical potential µ, i.e.,

(6)

Z

Ω

µ(t,x)dx= 0, ∀t≥0, whereµ=c1µ1+c2µ2.

Finally to close the system, we impose an initial conditionc⁰= (c⁰₁, c⁰₂) on the volume fractions by setting

(7) c_i|t=0 =c⁰_i in Ω.

The initial profilesc⁰_i ∈H¹(Ω) are assumed to be nonnegative withc⁰₁+c⁰₂= 1 in Ω, and we assume that both phases are present at initial time, i.e.,

(8)

Z

Ω

c⁰_idx>0, i∈ {1,2}.

1.2. Fundamental estimates and weak solutions. As a preliminary to the study of the numeri- cal scheme, we derive formally at the continuous level some a priori estimates. Their transposition at the discrete level will be key in the numerical analysis to be proposed in what follows. Equation (2) can be rewritten under the form

(9) ∂tci+∇·Fi= 0, with Fi=−ci

ηi

∇(µi+ Ψi+ηiθilog(ci)).

In view of the boundary conditions (4)–(5), this ensures that the volume occupied by each phase is preserved along time, namely

Z

Ω

ci(t,x)dx= Z

Ω

c⁰_i(x)dx, for allt≥0.

Moreover, it can be shown by testing (2) by−c⁻_i = min(c_i,0) thatc_i≥0 in (0, T)×Ω. Thanks to the constraint (1), this directly provides that

0≤ci≤1 in (0, T)×Ω.

Multiplying (2) by µi+ Ψi+ηiθilog(ci), integrating over Ω and summing overiyields X

i∈{1,2}

Z

Ω

∂tci(µi+ Ψi+ηiθilog(ci)) dx+D(c,µ) = 0, where the energy dissipationD(c,µ) is given by

D(c,µ) = X

i∈{1,2}

Z

Ω

ci

η_i|∇(µi+ Ψi+ηiθilog(ci))|²dx≥0.

As a consequence of (1), ∂tc2 = −∂tc1, so that the first term in the previous inequality can be rewritten as

X

i∈{1,2}

Z

Ω

∂_tc_i(µ_i+ Ψ_i+η_iθ_ilog(c_i))

= Z

Ω

∂_tc₁(µ₁−µ₂)dx+ X

i∈{1,2}

Z

Ω

∂_tc_i(Ψ_i+η_iθ_ilog(c_i)) dx.

The second term in the right-hand side can be rewritten as Z

Ω

∂tci(Ψi+ηiθilog(ci)) dx= d dt

Z

Ω

X

i∈{1,2}

[ciΨi+ηiθiH(ci)] dx with

(10) H(c) =clog(c)−c+ 1≥0, c≥0,

while we can make use of (3) to rewrite the first term as Z

Ω

∂tc1(µ1−µ2)dx= d dt

Z

Ω

α

2|∇c1|²+κc1(1−c1) dx.

Therefore, we obtain the energy / energy dissipation relation

(11) d

dtE(c) +D(c,µ) = 0 for allt≥0, where the energy functionalE(c) is defined by

(12) E(c) = Z

Ω



 α

2|∇c1|²+κc1(1−c1) + X

i∈{1,2}

[ciΨi+ηiθiH(ci)]



dx.

A straightforward consequence of (11) is that t7→ E(c(t)) is non-increasing along time, and thus that

(13) E(c(t)) + Z t

0

D(c(τ),µ(τ))dτ=E(c⁰)<∞ for allt≥0.

We deduce from previous inequality that the energy is bounded, hence aL^∞((0, T);H¹(Ω)) estimate onci.

The energy / energy dissipation estimate (11) is not sufficient to carry out our mathematical study since it only provides a weighted estimate on the chemical potentials

(14) X

i∈{1,2}

Z Z

Q_T

ci|∇µi|²dxdt≤C.

In order to bypass this difficulty, one needs to quantify the production of mixing entropy. Let us multiply (2) byηilog(ci), integrate overQT and sum overi∈ {1,2}, which using (1) leads to

X

i∈{1,2}

Z

Ω

ηi(H(ci(T,·))−H(c⁰_i))dx+ X

i∈{1,2}

Z Z

QT

∇ci·∇Ψidxdt

+ X

i∈{1,2}

θiηi

Z Z

QT

ci|∇log(ci)|²dxdt+ Z Z

QT

∇c1·∇(µ1−µ2)dxdt= 0.

The first two terms can be bounded thanks to theL^∞(QT) andL^∞((0, T);H¹(Ω)) estimates onci. For the last term of the left-hand side, one makes use of (3) and (4) to rewrite it as

Z Z

Q_T

∇c₁·∇(µ₁−µ₂)dxdt= Z Z

Q_T

(−∆c₁)(−α∆c₁+κ(1−2c₁))dxdt

≥ α 2

Z Z

Q_T

|∆c₁|²dxdt− κ 2α

Z Z

Q_T

(1−2c₁)²dxdt.

The L^∞(QT) estimate on c1 shows that the last term of the right-hand side is bounded. At the end of the day, sinceci|∇log(ci)|²= 4

∇√ ci

2, one gets

(15) α

2 Z Z

QT

|∆c1|²dxdt+ X

i∈{1,2}

4θiηi

Z Z

QT

|∇√

ci|²dxdt≤C.

Combining this estimate with relation (3), we obtain aL²(QT) estimate on µ1−µ2.

The last step aims at obtaining anL²(Q_T) bound on eachµ_i independently. The definition (6) ofµyields

∇µ= (µ₁−µ₂)∇c1+ X

i∈{1,2}

c_i∇µi.

The first term is in L²((0, T);L¹(Ω)) as the product of an element ofL²(QT) with an element of L^∞((0, T);L²(Ω)), while the second term is in L²(QT) since 0 ≤ci ≤1 and thanks to (14). As a consequence, ∇µis bounded in L²((0, T);L¹(Ω)). Making use of the Poincar´e-Sobolev estimate (recall thatµhas zero mean for all time, cf. (6), and that Ω⊂R²), we obtain thatµis bounded in L²(QT). To get the desiredL²(QT) estimate onµ1, it only remains to check that

µ1= (c1+c2)µ1=µ−c2(µ1−µ2)

belongs toL²(QT) thanks to theL²(QT) estimates on µandµ1−µ2together with 0≤c2≤1.

The interest of the above formal calculations is twofold. First, our scheme has been designed so that all these calculations can be transposed to the discrete setting. The corresponding a priori estimates will be at the basis of the numerical analysis proposed in this paper. Second, these estimates provide enough regularity on the solution to give a proper notion of weak solution to the problem.

Definition 1.1. (c,µ) is said to be aweak solutionto the problem (1)–(7)if

• c_i∈L^∞(Q_T)∩L^∞((0, T);H¹(Ω)) withc_i≥0 andc₁+c₂= 1 a.e. inQ_T;

• µ_i∈L²(Q_T) withc_i∇µ_i∈L²(Q_T)andR

Ωµ(t,x)dx= 0for a.e. t∈(0, T);

• For allϕ∈C_c^∞([0, T)×Ω), there holds (16)

Z Z

Q_T

c_i∂_tϕdxdt+ Z

Ω

c⁰_iϕ(0,·)dx− Z Z

Q_T

c_i ηi

∇(µi+ Ψ_i) +θ_i∇ci

·∇ϕdxdt= 0, as well as

(17)

Z Z

QT

(µ₁−µ₂)ϕdxdt= Z Z

QT

[α∇c₁·∇ϕ+κ(1−2c₁)ϕ] dxdt.

The existence of a weak solution has been established in [8] by showing the convergence of a minimizing movement scheme`a laJordan, Kinderlehrer and Otto [24]. Note that in [8], the case of a convex three-dimensional domain Ω is also addressed, but it relies on the fact that the L²(QT) estimate on ∆c1yields a L²((0, T);H²(Ω)) estimate onci for which we dont have an equivalent at

the discrete level. This is why we restrict our attention on the case Ω ⊂R² (but not necessarily convex) in this paper.

1.3. Some words about the model. Before entering the core of the paper, which is devoted to the convergence analysis of a finite volume scheme, let us briefly discuss the model under consideration, and in particular its difference with respect to the usual Cahn-Hilliard system. We refer to [30]

and [8] for a more developed discussions on this purpose. The classical degenerate Cahn-Hilliard equation which is the closest one to our system (1)–(7) writes

(18) ∂tc1−∇·(λ(c1)∇(µg+ Ψ1−Ψ2)) = ∆r(c1) in (0, T)×Ω, where the degenerate mobility is given byλ(c) =_η ^c(1−c)

1+c(η₂−η₁), the function governing the nonlinear diffusionr is such thatr⁰(c) =λ(c)

θ1η1

c +^θ_1−c^2η2

, and the generalized chemical potential (19) µ_g=µ₁−µ₂=−α∆c1+κ(1−2c₁) in (0, T)×Ω.

This system has to be completed with one initial condition onc⁰₁ and boundary conditions

−λ(c₁)∇(µ_g+ Ψ₁−Ψ₂)·n= 0 and ∇c₁·n= 0 on (0, T)×∂Ω.

The existence of a solution to this problem has been addressed in [15] and [26].

Let us come back to our system (1)–(7). Denote the total flux by Ftot = F1 +F2, then summing (9) overi∈ {1,2}yields

(20) ∇·Ftot=−∂t(c1+c2) = 0 in (0, T)×Ω

owing to (1). After some elementary calculations, the conservation (2) for the phase 1 rewrites (21) ∂tc1+∇·(f(c1)Ftot−λ(c1)∇(µg+ Ψ1−Ψ2)) = ∆r(c1) in (0, T)×Ω,

where f(c) = _η ^η2c

1+(η2−η1)c. The equation (21) differs from (18) by the addition of a nonlinear transport term driven by a divergence free vector field. Both systems can be reinterpreted as Wasserstein-type gradient flows [1] of the energyE(c) for different geometries:

• the Wasserstein distance with quadratic cost with the constraintFtot=0for the classical Cahn-Hilliard system, cf [26];

• the Wasserstein distance with quadratic cost with the less stringent constraint∇·Ftot = 0 for the system (1)–(7), cf. [30, 8].

The additional degree of freedom Ftot allows the energyE(c) to decrease faster along the trajec- tories, as highlighted in [8]. Finally, let us point the recent contribution [7] where the convergence of a minimizing movement scheme is addressed for a closely related model where the Cahn-Hilliard energy is replaced by the singular de Gennes-Flory-Higgins energy.

2. Finite Volume approximation and main results

Prior to presenting the scheme and stating our main results, that are the existence of a discrete solution to the scheme and the convergence of the corresponding approximate solutions towards a weak solution to the problem (1)–(7), we introduce some notations and requirements concerning the mesh.

2.1. (Super)-admissible mesh of Ωand time discretization. Let us first give a definition of what we call an admissible mesh.

Definition 2.1. An admissible mesh of Ω is a triplet T,E,(x_K)_K∈T

such that the following conditions are fulfilled.

(i) Each control volume (or cell)K ∈ T is non-empty, open, polygonal and convex. We assume that

K∩L=∅ ifK, L∈ T withK6=L, while [

K∈T

K= Ω.

We denote bym_K the 2-dimensional Lebesgue measure of K.

(ii) Each edgeσ∈ E is closed and is contained in a hyperplane ofR², with positive1-dimensional Hausdorff (or Lebesgue) measure denoted bymσ=H¹(σ)>0. We assume thatH¹(σ∩σ⁰) = 0 forσ, σ⁰ ∈ E unlessσ⁰=σ. For allK∈ T, we assume that there exists a subsetEK ofE such that ∂K =S

σ∈EKσ. Moreover, we suppose that S

K∈T EK =E. Given two distinct control volumes K, L∈ T, the intersection K∩L either reduces to a single edge σ∈ E denoted by K|L, or its1-dimensional Hausdorff measure is0.

(iii) The cell-centers(xK)K∈T are pairwise distinct withxK ∈K, and are such that, ifK, L∈ T share an edgeK|L, then the vector n_KL= _|x^xL−xK

K−xL| is orthogonal toK|L.

(iv) Given two cellsK, L∈ T sharing an edgeσ=K|L, we assume that the straight line joining xK andxL crosses the edgeσ in its midpointxσ.

Let us introduce some additional notations, some of them being depicted on Fig. 1. The size of the meshT (which is intended to tend to 0 in the convergence proof) is defined byh_T = max_K∈T hK, with hK = diam(K). Given two neighboring cellsK, L∈ T sharing an edge σ=K|L, we denote bydσ =|xK−xL| whereasdKσ =|xK−xσ| ≤dσ. The transmissivities τσ andτKσ of the edge σ are respectively defined byτσ= ^m_d^σ

σ andτKσ= _d^mσ

Kσ. The diamondDσ and half diamond DKσ

cells are defined as the convex hulls of {xK,xL, σ} and {xK, σ} respectively. Denoting by mD_σ

(resp. mD_Kσ) the 2-dimensional Lebesgue measure ofDσ (resp. DKσ), we will use many time the following elementary geometric properties: dσmσ = 2mD_σ anddKσmσ= 2mD_Kσ. We also denote by EK,int the subset of EK made of the internal edges σ such that there exists L ∈ T such that σ=K|L, and byEint=S

K∈T EK,int.

Even though this is absolutely not necessary, we choose to restrict our attention to the case of uniform time discretizations in the mathematical proofs in order to reduce the amount of notations.

In what follows, we set ∆t=T /N andtⁿ =n∆t forn∈ {0, . . . , N}. The integer N is intended to be large and even to tend to +∞in the convergence proof.

Remark 2.2. Condition (iv)above enforces an additional restriction with respect to the classical definition of finite volumes meshes with orthogonality condition (iii). Meshes satisfying this condi- tion in addition to the more classical assumptions (i)–(iii) is called super-admissible following the terminology introduced in [18]. It is for instance satisfied by cartesian grids or by acute triangu- lations. However, (iv) is in general not satisfied by Vorono¨ı meshes. This condition appears for technical reasons related to the construction of a strongly convergent SUSHI discrete gradient [18], see Proposition 4.2 later on. On the other hand, this condition was recently pushed forward in [21]

to show the consistency of the discrete optimal transportation geometry[27]hidden behind our work with the continuous optimal transportation geometry[32]in which our system(1)–(7)has a gradient flow structure [8].

•

•

•

x_K

xσ

xL

σ= K|L dKσ

dLσ

dσ

=

= •

x_K•

xL

σ= K|L

DKσ

DLσ

m

σ

Figure 1. Illustration of an admissible mesh in the sense of Definition 2.1. Each pointxK belongs to the cellKforK∈ T. For anyσ=K|L, the segment [xK,xL] intersects σ at its midpoint xσ in an orthogonal way. This properties hold for meshes made of triangles with acute angles if xK is chosen as the center of the circumcircle of the triangleK. On the right figure, the dashed area is the diamond cell Dσ corresponding to the edge σ = K|L. It is made ofDK,σ =Dσ∩K (in green),DLσ =Dσ∩L(in red), and of the edge σ=K|L(in blue), the length of which is equal tomσ.

2.2. A two-point flux approximation finite volume scheme. The scheme we propose is a cell- centered scheme based on two-point flux approximation (TPFA) finite volumes. At each time step n∈ {1, . . . , N}, then unknowns are located at the centers xK of the cells K ∈ T. Given discrete volume fractions cⁿ⁻¹ =

cⁿ⁻¹_1,K, cⁿ⁻¹_2,K

K∈T at time tⁿ⁻¹, we look for updated volume fractions cⁿ= cⁿ_1,K, cⁿ_2,K

K∈T and chemical potentialsµⁿ= µⁿ_1,K, µⁿ_2,K

K∈T at time tⁿ that are expected to approximate the mean values on K of their continuous counterparts c(tⁿ) and µ(tⁿ). At time t= 0, we initialize the procedure by setting

(22) c⁰_i,K = 1

mK

Z

K

c⁰_idx, ∀K∈ T, i∈ {1,2}.

As highlighted in the formal calculations presented in Section 1.2, the analysis requires the use of the logarithm of the volume fractions. To this end, the volume fractions cⁿ_i,K have to be strictly positive forn≥1. To ensure this property, some thermal diffusion is needed, see Lemma 3.2. In the case whereθi= 0, then one needs to introduce a small amount of numerical diffusion by setting (23) θ_i,T = max(θi, ρh_T)>0, i∈ {1,2},

whereρ >0 is a parameter that can be fixed by the user. Equation (2) is then discretized into (24) m_Kcⁿ_i,K−cⁿ⁻¹_i,K

∆t + X

σ∈EK,int

σ=K|L

τ_σ cⁿ_i,σ

ηi

µⁿ_i,K+ Ψ_i,K−µⁿ_i,L−Ψ_i,L

+θ_i,T(cⁿ_i,K−cⁿ_i,L)

= 0

for all K ∈ T and i ∈ {1,2}. In the above relation, we used the following discretization of the external potential:

Ψi,K = 1 mK

Z

K

Ψi(x)dx, ∀K∈ T, i∈ {1,2}.

Edge values cⁿ_i,σ of the discrete volume fractions also appear in (24). Rather than using upstream values of the volume fractions as in our previous work [9], we make use of a logarithmic mean, i.e.,

(25) cⁿ_i,σ=















cⁿ_i,K ifcⁿ_i,K=cⁿ_i,L≥0, 0 if min(cⁿ_i,K, cⁿ_i,L)≤0,

cⁿ_i,K−cⁿ_i,L

log(cⁿ_i,K)−log(cⁿ_i,L) otherwise,

for allσ=K|L∈ E_int andi∈ {1,2}. This particular choice of the edge value is closely related to the one introduced in the early work [23] for the approximation of the thin film equation, and fits with the one suggested in [27, 29] and used in a closely related context to ours in [28]. Equation (3) is discretized into

(26) µⁿ_1,K−µⁿ_2,K= α m_K

X

σ∈EK,int

σ=K|L

τσ cⁿ_1,K−cⁿ_1,L

+κ(1−2cⁿ⁻¹_1,K), ∀K∈ T.

Note that the repulsive term (second in the right-hand side) is discretized in an explicit way for stability issues that will appear clearly later on. The constraint (1) is discretized in a straightforward way by imposing

(27) cⁿ_1,K+cⁿ_2,K= 1, ∀K∈ T.

The last equation to be transposed in the discrete setting is (6), which is translated into

(28) X

K∈T

m_Kµⁿ_K = 0, whereµⁿ_K =cⁿ_1,Kµⁿ_1,K+cⁿ_2,Kµⁿ_2,K.

2.3. Main results. Before addressing the convergence of the scheme, we focus first on the case of a fixed mesh and time discretization. The scheme (24)–(28) yields a nonlinear system on (cⁿ,µⁿ).

The existence of a solution to this nonlinear system is far from being obvious. The existence of such a solution and some important properties of the discrete solution mimicking the properties highlighted in Section 1.2 are gathered in the first theorem of this paper.

Theorem 2.3. Assume that the inverse CFL condition (53)is fulfilled, then there exists (at least) one solution (cⁿ,µⁿ)_n≥1 to the scheme (24)–(28). Moreover, this solution satisfies the following properties:

(i) mass conservation:

X

K∈T

m_Kcⁿ_i,K= Z

Ω

c⁰_idx, n≥0, i∈ {1,2};

(ii) positivity:

0< cⁿ_i,K <1, K∈ T, i∈ {1,2}, n≥1;

(iii) energy decay:

ET(cⁿ)−ET(cⁿ⁻¹)

∆t +D(cⁿ,µⁿ)≤0, ∀n≥1, where the discrete energyE(cⁿ) is defined by

E_T(cⁿ) = α 2

X

σ∈Eint

σ=K|L

τσ cⁿ_1,K−cⁿ_1,L2

(29)

+ X

K∈T

m_K







κcⁿ_1,Kcⁿ_2,K+ X

i={1,2}

cⁿ_i,KΨ_i,K+θ_i,Tη_iH(cⁿ_i,K)





 ,

whereH is given by (10), and the discrete dissipation is defined by (30)

D_T(cⁿ,µⁿ) = X

i∈{1,2}

X

σ∈Eint

σ=K|L

τ_σcⁿ_i,σ ηi

µⁿ_i,K+ Ψ_i,K−µⁿ_i,L−Ψ_i,L+θ_i,Tη_i log(cⁿ_i,K)−log(cⁿ_i,L)

!² .

The existence of a solution to the scheme for each time step allows to reconstruct an approximate solution (c_T,∆t,µ_T,∆t) withc_T,∆t= (c_1,T,∆t, c_2,T,∆t) andµ_T,∆t= (µ_1,T,∆t, µ_2,T,∆t) by setting (31) c_i,T,∆t(t,x) =cⁿ_i,K and µ_i,T,∆t(t,x) =µⁿ_i,K if (t,x)∈(tⁿ⁻¹, tⁿ]×K, i∈ {1,2}.

The approximate solutions are expected to approximate the continuous solution to (1)–(7). Our second theorem gives a mathematical foundation to this statement. It requires the introduction of a suitable sequence of discretizations of QT. In what follows, we denote by Tm,Em,(xK)_K∈T

m

m≥1

a sequence of admissible meshes of Ω is the sense of Definition 2.1. We assume thath_Tm tends to 0 asm→ ∞as well as the following regularity requirements:

• shape regularity of the cells: there exists a finiteζ >1 such that (32) dσ≤ζdKσ, ∀m≥1, ∀K∈ Tm, ∀σ∈ EK,int,m,

and such that

(33) m_K ≥ 1

ζ(h_K)², ∀m≥1, ∀K∈ T_m;

• boundedness of the number of edges per element: there exists`^?≥3 such that

(34) #EK ≤`^?, ∀m≥1, ∀K∈ Tm;

• control on the transmissivities: there existτ^?, τ?≥0 such that (35) τ^?≥τ_σ ≥τ_?>0, ∀m≥1, ∀σ∈ E_int,m. The combination of a sequence Tm,Em,(xK)_K∈T

m

m≥1 fulfilling (32)–(35) together with a time step ∆t_m = T /N_m and N_m → +∞ as m → ∞ is said to be a regular discretization of Q_T if it moreover satisfies the inverse CFL condition (53).

Theorem 2.4. Let Tm,Em,(xK)_K∈T

m,∆tm

m≥1 be a sequence of regular discretizations of QT, and let c_Tm,∆t_m,µ_Tm,∆tm

m≥1 be a corresponding sequence of approximate solutions. Then there exists a weak solution(c,µ)to (1)–(7)in the sense of Definition 1.1 such that, up to a subsequence,

c_i,Tm,∆tm −→

m→∞c_i a.e. inQ_T and µ_i,Tm,∆tm −→

m→∞µ_i weakly inL²(Q_T).

The convergence properties stated in Theorem 2.4 are weaker than what is practically proved in the paper. The statement of optimal convergence properties would require the introduction of additional material that we postpone to the proof in order to optimize the readability of the paper.

The remaining of the paper is organized as follows. In Section 3, we work at fixed mesh and time step. We derive some a priori estimates and show the existence of (at least) one solution to the scheme thanks to a topological degree argument. Next in Section 4, we show thanks to compactness arguments that the approximate solution converge towards a weak solution to the scheme. Finally, we present in Section 5 some numerical simulations.

3. A priori estimates and existence of a discrete solution

In Section 3.1, we first derive some a priori estimates on the solutions to the scheme (24)–(28).

These estimates will be at the basis of the existence proof for a discrete solution to the scheme in Section 3.2, but also of the convergence proof carried out in Section 4.

3.1. A priori estimates. This section is devoted to the derivation to all the a priori estimates needed in the numerical analysis of the scheme. The first of them is the global conservation of mass, which is a consequence of the local conservativity of the scheme.

Lemma 3.1. For all n≥0andi∈ {1,2}, there holds

(36) X

K∈T

mKcⁿ_i,K= Z

Ω

c⁰_idx>0.

Proof. Summing (24) overK∈ T and using the conservativity of the scheme leads to X

K∈T

mKcⁿ_i,K = X

K∈T

mKcⁿ⁻¹_i,K , i∈ {1,2}, n∈ {1, . . . , N}.

A straightforward induction and the definition (22) ofc⁰_i,K then provides (36).

Our second lemma shows that the volume fractions are positive.

Lemma 3.2. Let n≥1, and let(cⁿ,µⁿ) be a solution to the scheme (24)–(28), then (37) 0< cⁿ_i,K <1, ∀K∈ T, ∀n≥1, ∀i∈ {1,2}.

Proof. Assume by induction that 0≤cⁿ⁻¹_i,K andP

Km_Kcⁿ⁻¹_i,K >0 (the later having been established for alln≥1 in Lemma 3.1), and suppose for contradiction that

cⁿ_i,K= min

L∈Tcⁿ_i,L≤0,

so that cⁿ_i,σ = 0 for all σ ∈ EK,int. Therefore, on this specific control volume K, the scheme (24) reduces to

m_Kcⁿ_i,K−cⁿ⁻¹_i,K

∆t +θ_i,T X

σ∈EK,int

σ=K|L

τ_σ(cⁿ_i,K−cⁿ_i,L) = 0.

The left-hand side is nonpositive, and even negative unlesscⁿ_i,L=cⁿ_i,K≤0 for all the neighbouring cellsLofK. We can thus iterate the argument and show thatcⁿ_i,L≤0 for allL∈ T, which provides a contradiction with the property P

KmKcⁿ_i,K >0 established in Lemma 3.1.

As a consequence of Lemma 3.2, the quantities log(cⁿ_i,K) have a sense. They will be used many times along the paper. Our next lemma consists in discrete counterparts of the energy / energy dissipation relations (11)–(13).

Lemma 3.3. Let(cⁿ,µⁿ)be a solution to the scheme (24)–(28), then the following discrete energy dissipation relation holds

E_T(cⁿ)−E_T(cⁿ⁻¹)

∆t +D_T(cⁿ,µⁿ)≤0, ∀n≥1,

where the discrete energy ET and the discrete dissipation DT are defined by (29)and (30)respec- tively.

Proof. Multiplying (24) byµⁿ_i,K+ Ψi,K+θ_i,Tηilog(cⁿ_i,K) and summing overi∈ {1,2}and K∈ T yields

(38) A₁+A₂+A₃+D_T(cⁿ,µⁿ) = 0, where

A1= X

K∈T

mK

∆t X

i∈{1,2}

(cⁿ_i,K−cⁿ⁻¹_i,K )µⁿ_i,K, A2= X

K∈T

mK

∆t X

i∈{1,2}

(cⁿ_i,K−cⁿ⁻¹_i,K )θ_i,Tηilog(cⁿ_i,K), and

(39) A3= X

K∈T

mK

∆t X

i∈{1,2}

(cⁿ_i,K−cⁿ⁻¹_i,K )Ψi,K. It follows from a convexity inequality that

(40) A2≥ X

K∈T

m_K

∆t X

i∈{1,2}

θi,Tηi

H(cⁿ_i,K)−H(cⁿ⁻¹_i,K) .

Using (27) and (26), the termA₁ rewrites

(41) A1= X

K∈T

mK

∆t

cⁿ_1,K−cⁿ⁻¹_1,K

µⁿ_1,K−µⁿ_2,K

=A11+A12, with

A11=α

∆t X

σ∈Eint

σ=K|L

τσ cⁿ_1,K−cⁿ_1,L

cⁿ_1,K−cⁿ_1,L−cⁿ⁻¹_1,K +cⁿ⁻¹_1,L

A12=κ X

K∈T

mK

∆t(1−2cⁿ⁻¹_1,K)

cⁿ_1,K−cⁿ⁻¹_1,K .

Using again elementary convexity inequalities, one gets that

(42) A₁₁≥ α

2∆t X

σ∈Eint

σ=K|L

τ_σ

(cⁿ_1,K−cⁿ_1,L)²−(cⁿ⁻¹_1,K −cⁿ⁻¹_1,L)² ,

and

A12≥κX

K∈T

mK

∆t n

cⁿ_1,K(1−cⁿ_1,K)−cⁿ⁻¹_1,K(1−cⁿ⁻¹_1,K)o .

The relation (27) allows to rewrite the right-hand side of the above inequality, so that

(43) A12≥κ X

K∈T

mK

∆t n

cⁿ_1,Kcⁿ_2,K−cⁿ⁻¹_1,Kcⁿ⁻¹_2,Ko .

The combination of (39)–(43) in (38) concludes the proof of Lemma 3.3.

The boundedness of the discrete energy E(cⁿ) provides a discrete L^∞((0, T);H¹(Ω)) estimate on the volume fractions, as established in the next corollary.

Corollary 3.4. There existsC1 depending only onΩ,α,κ,Ψ,θi,c⁰_i, andζsuch that

(44) X

σ∈Eint

σ=K|L

τσ(cⁿ_1,K−cⁿ_1,L)²≤C1.

Proof. As a straightforward consequence of Lemma 3.3, the energy is decaying along the time steps, so that

X

σ∈Eint

σ=K|L

τ_σ(cⁿ_1,K−cⁿ_1,L)²+ 2 α

X

i∈{1,2}

X

K∈T

m_Kcⁿ_i,KΨ_i,K≤ 2

αE_T(cⁿ)≤ 2 αE_T(c⁰)

≤ X

σ∈Eint

σ=K|L

τσ(c⁰_1,K−c⁰_1,L)²+ 2 α

X

K∈T

mK







κc⁰_1,Kc⁰_2,K+ X

i∈{1,2}

Ψi,Kc⁰_i,K+θi,TηiH(c⁰_i,K)





 .

Sincec⁰_1,K+c⁰_2,K = 1, there holds X

K∈T

mKκc⁰_1,Kc⁰_2,K ≤κ 4|Ω|.

Owing to [17, Lemma 9.4], there existsC2 depending only onζ such that X

σ∈E_int σ=K|L

τσ(c⁰_1,K−c⁰_1,L)²≤C2

Z

Ω

|∇c⁰₁|²dx,

whereas Jensen’s inequality ensures that X

K∈T

mKθ_i,TηiH(c⁰_i,K)≤θ_i,Tηi

Z

Ω

H(c⁰_i)dx≤θ_i,Tηi|Ω|.

Finally, since 0≤cⁿ_i,K ≤1 forn≥0, we have

X

σ∈E_int σ=K|L

τσ(cⁿ_1,K−cⁿ_1,L)²≤C2

Z

Ω

|∇c⁰₁|²dx+|Ω|

α



 κ

2 + 2 X

i∈{1,2}

θ_i,Tηi+ 2

|Ω|kΨik_L1(Ω)



.

Let us now focus on the quantification of the production of mixing entropy at the discrete level.

Our next lemma provides a discrete counterpart to Estimate (15).

Lemma 3.5. There existsC3 depending only onΩ,α,κ,T,Ψi,ηi,θi,ζ andc⁰₁ such that

(45)

N

X

n=1

∆t X

K∈T

mK







 1 m_K

X

σ∈E_K,int σ=K|L

τσ(cⁿ_1,K−cⁿ_1,L)









2

+

N

X

n=1

∆t X

i∈{1,2}

ηiθi,T

X

σ∈Eint

σ=K|L

τσ(cⁿ_i,K−cⁿ_i,L)(log(cⁿ_i,K)−log(cⁿ_i,L))≤C3.

As a consequence, there existsC4 depending only onΩ,α,κ,T,Ψi,ηi,θi,ζ andc⁰₁ such that (46)

N

X

n=1

∆t X

K∈T

mK µⁿ_1,K−µⁿ_2,K2

≤C4.

Proof. Multiplying (24) by ∆tηilog(cⁿ_i,K) and summing overi∈ {1,2},n∈ {1, . . . , N}andK∈ T yields

(47) A₁+A₂+A₃+A₄= 0,

where we have set A₁= X

i∈{1,2}

η_i

N

X

n=1

X

K∈T

m_K(cⁿ_i,K−cⁿ⁻¹_i,K) log(cⁿ_i,K),

A₂= X

i∈{1,2}

N

X

n=1

∆t X

σ∈Eint

σ=K|L

τ_σcⁿ_i,σ µⁿ_i,K−µⁿ_i,L

log(cⁿ_i,K)−log(cⁿ_i,L) ,

A₃= X

i∈{1,2}

N

X

n=1

∆t X

σ∈Eint

σ=K|L

τ_σcⁿ_i,σ(Ψ_i,K−Ψ_i,L) log(cⁿ_i,K)−log(cⁿ_i,L) ,

A₄=

N

X

n=1

∆t X

i∈{1,2}

η_iθ_i,T X

σ∈Eint

σ=K|L

τ_σ(cⁿ_i,K−cⁿ_i,L)(log(cⁿ_i,K)−log(cⁿ_i,L)).

(48)

It follows from the convexity ofH that

(49)

A1≥ X

i∈{1,2}

ηi N

X

n=1

X

K∈T

mK

H(cⁿ_i,K)−H(cⁿ⁻¹_i,K )

= X

i∈{1,2}

ηi

X

K∈T

mK H(c^N_i,K)−H(c⁰_i,K)

≥ − X

i∈{1,2}

η_i X

K∈T

m_K ≥ −C.

.

The particular choice (25) forcⁿ_i,σ was fixed so that cⁿ_i,σ log(cⁿ_i,K)−log(cⁿ_i,L)

=cⁿ_i,K−cⁿ_i,L, n≥1, σ=K|L.

Therefore, using (27) and Cauchy-Schwarz inequality, we deduce that

A3=

N

X

n=1

∆t X

σ∈Eint

σ=K|L

τσ(cⁿ_1,K−cⁿ_1,L)(Ψ1,K−Ψ1,L−Ψ2,K+ Ψ2,L) (50)

≥ −

N

X

n=1

∆t







 X

σ∈Eint

σ=K|L

τ_σ(cⁿ_1,K−cⁿ_1,L)²









1/2

X

i∈{1,2}







 X

σ∈Eint

σ=K|L

τ_σ(Ψ_i,K−Ψ_i,L)²









1/2

≥ −C,

where the last inequality is a consequence of Corollary 3.4 and of estimate

(51) X

σ∈Eint

σ=K|L

τσ(Ψi,K−Ψi,L)²≤C,

which itself is a consequence of [17, Lemma 9.4] and of theH¹(Ω) regularity of the external potentials Ψ_i. Similarly, one can rewrite

A₂=

N

X

n=1

∆t X

σ∈Eint

σ=K|L

τ_σ(cⁿ_1,K−cⁿ_1,L)(µⁿ_1,K−µⁿ_2,K−µⁿ_1,L+µⁿ_2,L),

=

N

X

n=1

∆t X

K∈T

mK







 1 mK

X

σ∈EK,int

σ=K|L

τσ(cⁿ_1,K−cⁿ_1,L)









µⁿ_1,K−µⁿ_2,K .

Thanks to the relation (26), it turns to

A2=α

N

X

n=1

∆t X

K∈T

mK







 1 mK

X

σ∈EK,int

σ=K|L

τσ(cⁿ_1,K−cⁿ_1,L)









2

+κ

N

X

n=1

∆t X

K∈T

mK







 1 mK

X

σ∈EK,int

σ=K|L

τσ(cⁿ_1,K−cⁿ_1,L)









(1−2cⁿ⁻¹_1,K).

Using the fact that 0≤cⁿ⁻¹_1,K ≤1 and the inequalityab≥ −_2κ^αa²−_2α^κb², we obtain

(52) A₂≥α

2

N

X

n=1

∆t X

K∈T

m_K







 1 mK

X

σ∈EK,int

σ=K|L

τ_σ(cⁿ_1,K−cⁿ_1,L)









2

−C.

The combination of (48)–(52) in (47) provides (45). Let us now focus on estimate (46). Equality (26) gives

N

X

n=1

∆t X

K∈T

mK(µⁿ_1,K−µⁿ_2,K)²≤2α²

N

X

n=1

∆t X

K∈T

mK







 1 mK

X

σ∈EK,int

σ=K|L

τσ(cⁿ_1,K−cⁿ_1,L)









2

+ 2κ²

N

X

n=1

∆t X

K∈T

m_K(1−2cⁿ⁻¹_1,K)².

Since 0≤cⁿ⁻¹_1,K ≤1 and the logarithmic function is increasing, estimate (45) concludes the proof.

The following lemma is a transposition to the discrete setting of the weighted estimate (14) on the chemical potentials.

Lemma 3.6. There existsC5 depending only onα,κ,c⁰_i,Ψi,T,Ω,ηi andζ such that

N

X

n=1

∆t X

i∈{1,2}

X

σ∈Eint

σ=K|L

τσcⁿ_i,σ µⁿ_i,K−µⁿ_i,L²

≤C5.

Proof. Definition (30) ofD_T(cⁿ,µⁿ) together with inequality (a+b+c)²≤3(a²+b²+c²) yield 1

maxiηi N

X

n=1

∆t X

i∈{1,2}

X

σ∈Eint

σ=K|L

τσcⁿ_i,σ µⁿ_i,K−µⁿ_i,L2

≤3

N

X

n=1

∆tDT(cⁿ,µⁿ)

+ 3

N

X

n=1

∆t X

i∈{1,2}

X

σ∈Eint

σ=K|L

τσ

cⁿ_i,σ ηi

(Ψi,K−Ψi,L)²+ (θ_i,Tηi)² log(cⁿ_i,K)−log(cⁿ_i,L)2 .

Owing to Lemma 3.3, the first term of the right-hand side is bounded by

N

X

n=1

∆tDT(cⁿ,µⁿ)≤ET(c⁰)−ET(c^N)≤2ET(c⁰),

which is bounded as already seen in the proof of Corollary 3.4. On the other hand, since 0≤cⁿ_i,σ≤1, one has

N

X

n=1

∆t X

i∈{1,2}

X

σ∈Eint

σ=K|L

τ_σcⁿ_i,σ

η_i (Ψ_i,K−Ψ_i,L)²≤T C₂ X

i∈{1,2}

1

η_ikΨ_ik_H1(Ω). Finally,

N

X

n=1

∆t X

i∈{1,2}

ηi

X

σ∈Eint

σ=K|L

τσcⁿ_i,σ(θi,T)² log(cⁿ_i,K)−log(cⁿ_i,L)2

≤C3max

i θi,T

thanks to Lemma 3.5.