Mathematical and numerical study of a corrosion model Claire Chainais-Hillairet

Claire Chainais-Hillairet · Christian Bataillon

Received: date / Accepted: date

Abstract In this paper, we consider a PDE system arising in corrosion modelling.

This system consists in two convection-diffusion equations on the densities of charge carriers and a Poisson equation on the electric potential. Boundary conditions are Robin boundary conditions. We discretize each equation by a finite volume scheme and we prove the convergence of the scheme towards a weak solution to the initial system. Finally, we provide numerical results describing the behaviour of the solutions with respect to an applied voltage.

Keywords corrosion modelling·drift-diffusion system·Scharfetter-Gummel scheme

1 Introduction

1.1 Presentation of the model

In this paper we study the Diffusion Poisson Coupled Model introduced by C. Bataillon et al.in [5]. It is a corrosion model of iron based alloy in nuclear waste repository. The whole system consists in a solution in contact with an oxide layer which covers a metal. It appears that this passive layer protects the metal from its environment. The oxide layer is a mixed electronic and ionic conductor and therefore it is thought of as a semiconductor: charge carriers are convected by the electric field and the electric potential is coupled to the charge densities through a Poisson equation. The oxide layer interacts with the metal (electronic conductor) and the solution (ionic conductor) at its interfaces: the inner interface correspond to the metal/oxide interface and the outer interface corresponds to the oxide/solution interface.

C. Chainais-Hillairet

Laboratoire de Math´ematiques, UMR CNRS 6620, Universit´e Blaise Pascal, 63177 Aubi`ere Cedex, France Tel.: +33-4-63-40-70-76

E-mail: [email protected] C. Bataillon

CEA/DEN/DANS/DPC/SCCME/LECA, 91191 Gif sur Yvette Cedex, France

In most industrial cases, the shape of metal pieces is not flat (container or pipe).

But the thickness of the oxide layer ranges from nanometer to micrometer while the size of the exposed surface is high (square centimeters to square meters). In practice, the available data are averaged over the whole exposed surface. So, the microscopic or even macroscopic heterogeneities of materials are not taken into account. The material is considered as homogeneous. For these reasons, a 1D modelling is sufficient to describe a real system.

We consider here a simple model where only two charge carriers are taken into account: electrons and cationsFe³⁺. The unknowns of the system are the density of electronsN, the density of cationsP and the electric potentialΨ. The current densities of electrons and cations are respectively denoted byJN and JP; they contain a drift part and a diffusion part. Therefore, the equations onNandPare convection-diffusion equations coupled with a Poisson equation onΨ.

Charge carriers are created and consumed at both interfaces: x = 0 is the outer interface (oxide/solution) andx= 1 is the inner interface (oxide/metal). The kinetics of the electrochemical reactions at interfaces, following Butler-Volmer laws (see for instance [15]), leads to the boundary conditions. A Butler-Volmer law is the difference between a forward and a backward probability laws which depend on the voltage drop at the interface. In this first modelling, there is only one probability law at x = 0 because the electrochemical reactions at x= 0 are assumed as irreversible. This is a usual assumption in corrosion field at the oxide/solution interface.

The boundary conditions for the Poisson equation depict that the metal and the solution can be charged because they are respectively electronic and ionic conductors.

Such accumulation of charges induces a field given by the Gauss law. These accumu- lations of charges depend on the voltage drop at the interface by the usual Helmoltz law which rely the charge to the voltage drop through a capacitance. The parameters

∆Ψ₁^pzcand∆Ψ₀^pzcare the voltage drop corresponding to no accumulation of charges respectively in the metal and in the solution.

Finally, the evolutive dimensionless system writes:

– Equation and boundary conditions for the density of cationsP:

∂tP+∂xJ_P = 0, J_P=−∂xP−3P ∂xΨ in (0,1), J_P =−P k⁰₁exp (3a⁰₁Ψ) onx= 0, JP =m^L₁Pexp (−3b^L₁(V −Ψ))−(Pm−P)k₁^Lexp (3a^L₁(V −Ψ)) onx= 1.

– Equation and boundary conditions for the density of electronsN: D1

D2

∂tN+∂xJN = 0, JN =−∂xN+N ∂xΨ in (0,1), J_N =−N k⁰₂exp (−a⁰₂Ψ) onx= 0, J_N =m^L₂Nexp (b^L₂(V −Ψ))−(Nm−N)k^L₂ exp (−a^L₂(V −Ψ)) onx= 1.

– Equation and boundary conditions for the potentialΨ:

−λ²∂²_xxΨ =−N+ 3P+ρ_hl in (0,1), Ψ−α0∂xΨ =∆Ψ₀^pzc onx= 0, Ψ+α1∂xΨ =V −∆Ψ₁^pzc onx= 1.

where

– D1 and D2 are diffusion coefficients. Due to the difference of size and then of mobility of electrons and cations, we haveD1D2.

– k⁰₁,k⁰₂,k₁^L,k^L₂,m^L₁,m^L₂ are interface kinetic functions. In what follows, we assume that these functions are constant and strictly positive:

k₁⁰, k₂⁰, k₁^L, k^L₂, m^L₁, m^L₂ >0. (1) – a⁰₁,a⁰₂,a^L₁,b^L₁,a^L₂,b^L₂ are positive transfer coefficients, which satisfy

a^L1 +b^L1 = 1, a^L2 +b^L2 = 1. (2) – Pm is the maximum occupancy for octahedral cations in the host lattice andNm

is the density of state in the conduction band. They satisfy

Pm, Nm>0 (3)

– ρ_hlis the net charge density of the ionic species in the host lattice. In what follows, we assume thatρhlis constant in the whole layer.

– ∆Ψ₀^pzc, ∆Ψ₁^pzc are respectively the outer and the inner pzc voltages (voltages of zero charge),V is the applied voltage. Let us setU0=∆Ψ₀^pzcandU1=V−∆Ψ₁^pzc. – λ², α0, α1 are positive dimensionless parameters linked to the physical parame- ters (see Section 5). The analog of the Debye length in plasma physics is λ, the squareroot ofλ².

In what follows, we focus on the steady state corrosion model, which writes

∂xJP= 0, JP =−∂xP−3P ∂xΨ in (0,1), (4)

∂xJ_N = 0, J_N =−∂xN+N ∂xΨ in (0,1), (5)

−λ²∂_xx² Ψ=−N+ 3P+ρ_hl in (0,1), (6) with the boundary conditions

J_P =−P k₁⁰exp (3a⁰₁Ψ) onx= 0, (7) J_P=m^L₁Pexp (−3b^L₁(V −Ψ))−(Pm−P)k₁^Lexp (3a^L₁(V −Ψ)) onx= 1, (8) J_N =−N k⁰₂exp (−a⁰₂Ψ) onx= 0, (9) JN =m^L₂Nexp (b^L₂(V −Ψ))−(Nm−N)k₂^Lexp (−a^L₂(V −Ψ)) onx= 1,(10) Ψ−α0∂xΨ =U0 onx= 0,(11) Ψ+α1∂xΨ =U1 onx= 1.(12) This system of equations is very close to the drift-diffusion system, well-known in semiconductor modelling (see for instance [11–13]) . The main difference is due to the boundary conditions. While the boundary conditions for the semiconductors are mixed Dirichlet/Neumann boundary conditions (ohmic contacts or insulating bound- ary segments), the boundary conditions for the corrosion model are Robin boundary conditions. They come from the kinetics of the interfacial electrochemical reactions and they induce an additional coupling between all variables (due toΨ(0) in (7), (9) and Ψ(1) in (8), (10)).

1.2 Change the variables

In the drift-diffusion framework, the Slotboom change of variables is frequently used, see [12, 13]. It has been introduced because the convection terms in (4), (5) prohibit the use of the maximum principle for elliptic equations in a simple way. It permits to transform the convection-diffusion equations (4), (5) into nonlinear diffusion equations.

The counterpart is that the Poisson equation (6) becomes a semilinear elliptic equation on the potentialΨ.

The change of variables is the following: the so-called Slotboom variablesuandv are defined by

P =e^−3Ψu, N =e^Ψv.

The current relations then reduce to

JP=−e^−3Ψ∂xu, JN =−e^Ψ∂xv and the following system is obtained from (4)–(6):

∂x(e^−3Ψ∂xu) = 0 in (0,1), (13)

∂x(e^Ψ∂xv) = 0 in (0,1), (14)

−λ²∂_xx² Ψ =−e^Ψv+ 3e^−3Ψu+ρ_hl in (0,1), (15) The boundary conditions onΨ, (11), (12), are unchanged and the boundary conditions onuandvwrite:

∂xu−k₁⁰e^3a01Ψu= 0 onx= 0,(16)

∂xu+“

m^L1e^{−3bL1(V−Ψ)}+k1^Le^3aL1(V−Ψ)”

u=Pmk1^Le^3aL1(V−Ψ)e^3Ψ onx= 1,(17)

∂xv−k⁰₂e^−a02Ψv= 0 onx= 0,(18)

∂xv+“

m^L₂e^bL2(V−Ψ)+k^L₂e^{−aL2(V−Ψ)}”

v=Nmk₂^Le^{−aL2(V−Ψ)}e^−Ψ onx= 1.(19) Let us define in which sense (u, v, Ψ) is solution to (11)–(19).

Definition 1 (u, v, Ψ) is solution to the corrosion model (11)–(19) if u∈H¹(0,1), v∈H¹(0,1), Ψ ∈H¹(0,1) and for all test functionϕ∈Cc^∞([0,1])

Z 1 0

e^−3Ψ∂xu ∂xϕ+k1⁰e^3a01Ψ(0)e^−3Ψ(0)u(0)ϕ(0)+

“

m^L₁e^{−3bL1(V−Ψ(1))}+k^L₁e^{3aL1(V−Ψ(1))}”

e^−3Ψ(1)u(1)ϕ(1) = Pmk^L₁e^{3aL1(V−Ψ(1))}ϕ(1)

(20)

Z 1 0

e^Ψ∂xv ∂xϕ+k⁰₂e^−a02Ψ(0)e^Ψ(0)v(0)ϕ(0)+

“

m^L₂e^{bL2(V−Ψ(1))}+k^L₂e^{−aL2(V−Ψ(1))}”

e^Ψ(1)v(1)ϕ(1) = Nmk^L₂e^{−aL2(V−Ψ(1))}ϕ(1)

(21)

Z 1 0

∂xΨ ∂xϕ+ 1 α1

Ψ(1)ϕ(1) + 1 α0

Ψ(0)ϕ(0) = U1

α1

ϕ(1) +U0

α0

ϕ(0)− 1 λ²

Z 1 0

“

e^Ψv−3e^−3Ψu−ρ_hl” ϕ

(22) We note that (20), (21) and (22) are well defined. Indeed, asH¹(0,1) is embedded inC([0,1]), the valuation ofu,vandΨ at the points 0 and 1 has a sense. Furthermore, if (u, v, Ψ) is a solution to (11)–(19) in the sense of Definition 1, then we can define P =e^−3ΨuandN=e^Ψvand (N, P, Ψ) is a solution to the system (4)–(12).

1.3 Outline of the paper. Main results

The aim of this paper is to propose a numerical scheme for the corrosion model, to prove its convergence towards a solution to the corrosion model and then to obtain the existence of a solution to the system (11)–(19) in the sense of Definition 1 as a by-product of the convergence of the scheme.

The scheme we consider is a finite volume scheme. It is first written for the initial system, in the (N, P, Ψ) variables. Indeed, it is well known, see for instance the book by R. Eymard, T. Gallou¨et and R. Herbin [9], that finite volume schemes are well adapted for convection-diffusion equations. Furthermore, convergence of finite volume schemes for the drift-diffusion system with Dirichlet boundary conditions has already been established in [6–8] by C. Chainais-Hillairet, J. G. Liu and Y. J. Peng. Section 2 is devoted to the presentation of the scheme. Starting from the (N, P, Ψ) variables, it is then rewritten in the Slotboom variables (u, v, Ψ).

The scheme we write provides a system of nonlinear equations. In Section 3, we prove a priori estimates on the approximate solutions :L^∞and discreteH¹estimates onu,vandΨ. Thanks to the Brouwer fixed point theorem, these estimates lead to the proof of the existence of a solution to the numerical scheme. Then, Section 4 is devoted to the proof of the convergence of the scheme. In Section 5, we provide numerical experiments in some real-case situation.

The main results of the paper are the existence of a solution to the numerical scheme with the estimates on the approximate solution (Theorem 1) and the convergence of the scheme towards a weak solution of the system (11)–(19) (Theorem 2).

2 Numerical scheme 2.1 Presentation of the scheme

The system (4)–(12) consists in a Poisson equation for the potentielΨ and two convec- tion-diffusion equations for the charge densities N and P, supplemented with Robin boundary conditions. We propose to write a finite volume scheme for this system.

Therefore, let us introduce some notations. We consider a mesh for the domain [0,1], which is not necessarily uniform,i.ea family of given points (xi)_0≤i≤I+1satisfying

x0= 0< x1< x2< . . . < xI < xI+1= 1.

Then, for 1 ≤ i ≤ I −1, we define x_i+1

2 = xi+xi+1

2 and we set x1

2 = x0 = 0, x_I+1

2 =xI+1= 1. Let us set h_i=x_i+1

2 −x_i−1

2,for 1≤i≤I, h_i+1

2 =xi+1−xi, for 0≤i≤I andh= max{h_i, 1≤i≤I}is the size of the mesh.

We first write the scheme for the system in the (N, P, Ψ) variables (4)-(6). It is obtained after integrating formally each equation (4), (5), (6) on the cell ]x_i−1

2, x_i+1 2[.

The discrete unknowns are (Ni, Pi, Ψi)_0≤i≤I+1.The main point remains the discretiza- tion of the fluxes through the interfaces:∂xΨ(x_i+1

2) will be approximated by dΨ_i+1 2, JP(x_i+1

2) byF_i+1

2 andJN(x_i+1

2) byG_i+1

2. We get Scheme forP

F_i+1 2 − F_i−1

2 = 0, 1≤i≤I, (23)

F1

2 =−k⁰1P0e^3a01Ψ0, (24)

F_I+1

2 =P_I+1“

k^L₁e^{3aL1(V−ΨI+1)}+m^L₁e^{−3bL1(V−ΨI+1)}”

−k^L₁Pme^{3aL1(V−ΨI+1)}.(25) Scheme forN

G_i+1 2 − G_i−1

2 = 0, 1≤i≤I, (26)

G1

2 =−k⁰₂N0e^−a02Ψ0, (27)

G_I+1

2 =NI+1

“

m^L2e^{bL2(V−ΨI+1)}+k^L2e^{−aL2(V−ΨI+1)}”

−k^L2Nme^{−aL2(V−ΨI+1)}.(28) Scheme forΨ

−λ²(dΨ_i+1

2−dΨ_i−1

2) =hi(3Pi−Ni+ρhl), 1≤i≤I, (29) dΨ_i+1

2 = Ψi+1−Ψi

h_i+1 2

, 0≤i≤I, (30)

Ψ0−α0dΨ1

2 =U0, (31)

Ψ_I+1+α₁dΨ_I+1

2 =U₁. (32)

There are many possibilities for the definition of the numerical fluxesF_i+1 2,G_i+1

2. We choose here the Scharfetter-Gummel fluxes [16] defined by

F_i+1 2 =

B(3h_i+1 2dΨ_i+1

2)P_i−B(−3h_i+1 2dΨ_i+1

2)P_i+1 h_i+1

2

0≤i≤I, (33)

G_i+1 2 =

B(−h_i+1 2dΨ_i+1

2)N_i−B(h_i+1 2dΨ_i+1

2)N_i+1 h_i+1

2

0≤i≤I, (34) whereB is the Bernoulli function defined onRby

B(x) = x

e^x−1 forx6= 0, B(0) = 1. (35) In what follows, we denote by (S) the scheme (23)–(35).

Remark 1 The classical upwind fluxes can be written under the form (33), (34) but with the following functionB

B(x) = 1 + (−x)⁺= 1 +x⁻, wherex⁺= max(x,0) andx⁻= max(−x,0). (36)

2.2 The scheme in the Slotboom variables

The scheme (S) is a system of nonlinear equations. Before proving existence of a solution to this system and later convergence of the scheme, we rewrite it in the (u, v, Ψ) variables.

As in the continuous case, we define

u_i=e^3ΨiP_i, v_i=e^−ΨiN_i, ∀0≤i≤I+ 1. (37) Using the definition of the Bernoulli function (35) and (30), we get that

B(3h_i+1 2dΨ_i+1

2) = 8

><

>:

3(Ψ_i+1−Ψ_i)

e^3Ψi+1−e^3Ψie^3Ψi ifΨi6=Ψi+1, 1 ifΨ_i=Ψ_i+1,

B(−3h_i+1 2dΨ_i+1

2) = 8

><

>:

3(Ψi+1−Ψi)

e^3Ψi+1−e^3Ψie^3Ψi+1 ifΨi6=Ψi+1, 1 ifΨ_i=Ψ_i+1. Therefore, the numerical fluxesF_i+1

2 defined by (33) can be rewritten F_i+1

2 = 1

f(Ψi, Ψi+1)

ui−ui+1

h_i+1 2

,∀0≤i≤I, (38)

withf :R²→Rdefined by

f(x, y) = 8

><

>:

e^3x−e^3y

3(x−y) ifx6=y, e^3x ifx=y.

In the new variables, the boundary conditions (24), (25) become F1

2 =−k⁰₁u0e^−3Ψ0e^3a01Ψ0, (39)

F_I+1

2 =u_I+1e^−3ΨI+1“

k₁^Le^{3aL1(V−ΨI+1)}+m^L₁e^{−3bL1(V−ΨI+1)}”

−k₁^LPme^{3aL1(V−ΨI+1)}. (40) Therefore, the scheme foruis given by (23), (38)–(40). Similarly, the numerical fluxes G_i+1

2 defined by (34) rewrite G_i+1

2 = 1

g(Ψ_i, Ψ_i+1)

v_i−v_i+1 h_i+1

2

,∀0≤i≤I, (41)

withg:R²→Rdefined by

g(x, y) = 8

><

>:

−e^−x−e^−y

x−y ifx6=y, e^−x ifx=y.

The boundary conditions (27), (28) become G1

2 =−k⁰₂e^Ψ0v0e^−a02Ψ0, (42)

G_I+1

2 =vI+1e^ΨI+1“

m^L2e^{bL2(V−ΨI+1)}+k2^Le^{−aL2(V−ΨI+1)}”

−k2^LNme^{−aL2(V−ΨI+1)}. (43) Therefore, the scheme forvis given by (26), (41)–(43). Finally, the scheme forΨ (29) rewrites in the new variables

−λ²(dΨ_i+1

2 −dΨ_i−1

2) =hi(3uie^−3Ψi−vie^Ψi+ρ_hl), 1≤i≤I, (44) with the numerical fluxes (30) and the boundary conditions (31)–(32).

2.3 Algebraic expression ofuandv

Proposition 1 We assume thatΨ= (Ψi)_0≤i≤I+1 is a given vector. Let set A⁰_u=k⁰₁e^3a01Ψ0, A¹_u=k^L₁e^{3aL1(V−ΨI+1)}+m^L₁e^{−3bL1(V−ΨI+1)},

A⁰v=k₂⁰e^−a02Ψ0, A¹v=k^L₂e^{−aL2(V−ΨI+1)}+m^L₂e^{bL2(V−ΨI+1)}, Bu¹=k^L₁Pme^{3aL1(V−ΨI+1)}, B¹v=k^L₂Nme^{−aL2(V−ΨI+1)}.

If the kinetic functions satisfy the positivity assumption (1), the scheme for u (23), (38)–(40) and the scheme forv(26), (41)–(43) can be solved explicitly. We obtain

ui=−F A⁰u

e^3Ψ0− F

i−1

X

j=0

f(Ψj, Ψj+1)h_j+1 2, withF=−B¹u

A¹u

e^3ΨI+1 1

e^3ΨI+1 A¹_u +e^3Ψ0

A⁰_u +

I

X

i=0

f(Ψ_i, Ψ_i+1)h_i+1 2

, (45)

v_i=− G

A⁰_ve^−Ψ0− G

i−1

X

j=0

g(Ψ_j, Ψ_j+1)h_j+1 2, withG=−Bv¹

A¹v

e^−ΨI+1 1

e^−ΨI+1 A¹v

+e^−Ψ0 A⁰v

+

I

X

i=0

g(Ψi, Ψi+1)h_i+1 2

. (46)

Proof

We prove (45), the proof of (46) is similar. The scheme (23) leads toF_i+1

2 =Ffor 0≤i≤I. Thanks to (1), we haveA⁰u, A¹u>0 and from the boundary conditions (39), (40), we deduce

u0=−F A⁰u

e^3Ψ0, u_I+1= F A¹u

e^3ΨI+1+B_u¹ A¹u

e^3ΨI+1. But, thanks to (38), we have

F A¹u

e^3ΨI+1+B_u¹ A¹u

e^3ΨI+1+ F A⁰u

e^3Ψ0=

I

X

i=0

(ui+1−ui) =−F

I

X

i=0

f(Ψi, Ψi+1)h_i+1 2. We get

F=−B¹_u A¹u

e^3ΨI+1 1

e^3ΨI+1 A¹u

+e^3Ψ0 A⁰u

+

I

X

i=0

h_i+1

2f(Ψ_i, Ψ_i+1) .

Then, writingui=u0+

i−1

X

j=0

(uj+1−uj) =u0− F

i−1

X

j=0

f(Ψj, Ψj+1)h_j+1

2 completes the proof of (45).

2 Corollary 1 The scheme for the corrosion model(S)written in the(N, P, Ψ)variables is equivalent, through the change of variables (37), to the coupled system of nonlinear equations (45), (46), (44), (30)–(32) written in the (u, v, Ψ) variables. This system, equivalent to(S)will be denoted by(S⁰) in all the sequel.

3 Existence of a solution to the scheme, estimates

Theorem 1 Let us assume (1), (2) and (3). The scheme(S⁰)admits a solution u= (u_i)_0≤i≤I+1=,v= (v_i)_0≤i≤I+1,Ψ= (Ψ_i)_0≤i≤I+1 which satisfies the followingL^∞ and discreteH¹-estimates:

0< ui≤ k^L₁

m^L₁e^3VPm, 0< vi≤ k^L₂

m^L₂e^−VNm

and there existsM >0, depending only onλ², α₀, α₁,∆Ψ₀^pzc, ∆Ψ₁^pzc,V, Pm, Nm, ρ_hl and the interface kinetic functions(k^0,L_i )_i=1,2,(m^0,L_i )_i=1,2, such that

|Ψi| ≤M and

I

X

i=0

(Ψi+1−Ψi)² h_i+1

2

+Ψ0²+Ψ_I+1² ≤M and

I

X

i=0

(ui−ui+1)² h_i+1

2

≤M,

I

X

i=0

(vi−vi+1)² h_i+1

2

≤M.

The outline of the proof of Theorem 1 is the following. We first proveL^∞estimates on uandvwhenΨis given. Then, foruandvgiven, we prove existence and uniqueness of Ψsolution to (44), (31), (32) with estimates on Ψ. The use of the Brouwer fixed point theorem will conclude the proof.

3.1 Estimates onuandv

Lemma 1 Assume (1), (2), (3) and thatΨ is given. Then u andv defined by (45) and (46) verify

0< ui ≤ k₁^L

m^L₁e^3VPm, (47)

0< v_i ≤ k₂^L

m^L₂e^−VNm. (48)

Proof

Thanks to (1) and (3), we have A⁰_u, A¹_u, B_u¹ >0. From (45), we get that F <0 and that (ui)_0≤i≤I+1 is an increasing sequence (see also (38)). Therefore,

0< u₀≤u_i≤u_I+1≤B¹u

A¹_ue^3ΨI+1, ∀0≤i≤I+ 1.

However, Bu¹

A¹u

e^3ΨI+1 = k^L₁Pme^{3aL1(V−ΨI+1)}

k^L₁e^{3aL1(V−ΨI+1)}+m^L₁e^{−3bL1(V−ΨI+1)}e^3ΨI+1,

= Pm

1 +m^L₁

k^L₁ e^{−3(aL1+bL1)(V−ΨI+1)} e^3ΨI+1.

Using the assumption (2) on the transfer coefficients, we get B¹_u

A¹u

e^3ΨI+1≤ Pme^3ΨI+1 1 +m^L₁

k^L₁ e^−3Ve^3ΨI+1

≤ k^L₁

m^L₁e^3VPm.

It concludes the proof of (47) and the proof of (48) is similar.

2

3.2 Estimates onΨ

Lemma 2 We consider that u and v are given and satisfy the L^∞-estimates (47), (48). Then the scheme on Ψ written as the nonlinear system of equations (44), (30)–

(32) admits a unique solutionΨ= (Ψi)_0≤i≤I+1. Moreover, there existsM >0depend- ing only on the data (λ²,α0, α1, ∆Ψ₀^pzc,∆Ψ₁^pzc, V, Pm, Nm, ρ_hl and the interface kinetic functions) such that :

|Ψi| ≤M, (49)

I

X

i=0

(Ψi+1−Ψi)² h_i+1

2

+Ψ0²+Ψ_I+1² ≤M. (50)

Proof

We first prove the uniqueness of the solution. Let assume that there exist two solutionsΨ¹andΨ²to the system (44), (30)–(32) and setΦ=Ψ¹−Ψ².Φsatisfies

−λ²(Φ_i+1−Φ_i h_i+1

2

−Φ_i−Φ_i−1 h_i−1

2

) =h_i“

3u_i(e^−3Ψi1−e^−3Ψi2)−v_i(e^Ψi1−^Ψi2)” , (51)

Φ0−α0

Φ1−Φ0

h1 2

!

= 0, ΦI+1+α1

ΦI+1−ΦI

h_I+1 2

!

= 0.

Multiplying (51) byΦiand summing overi, we obtain, because the functionsx→e^−3x andx→ −e^x are decreasing andui, viare positive for all 1≤i≤I,

I

X

i=0

h_i+1 2

Φ_i+1−Φ_i h_i+1

2

!2

+Φ²₀

α₀ +Φ²_I+1 α₁ ≤0.

It followsΦ= 0 andΨ¹=Ψ².

Let us now prove the existence of Ψ= (Ψi)_0≤i≤I+1, solution to (44), (30)–(32).

We introduce the functionI:R^I+2→Rdefined by I(Ψ) = 1

2

I

X

i=0

h_i+1 2

Ψ_i+1−Ψ_i h_i+1

2

!2

+ 1

2α₁Ψ_I+1² + 1

2α₀Ψ₀²−U1

α₁Ψ_I+1−U0

α₀Ψ0

+1 λ²

I

X

i=1

hi

“−ρhlΨi+e^Ψivi+e^−3Ψiui

” . (52)

It is clear thatIis a continuous, convex function onR^I+2. Let us give a lower bound onI(Ψ). Asuandvsatisfy (47) and (48), we have

I

X

i=1

hi

“

e^Ψivi+e^−3Ψiui

”

≥0.

Then, using Young inequality, we get I(Ψ)≥min

„1 2, 1

4α²₀, 1 4α²₁

« 0

@

I

X

i=0

h_i+1 2

Ψ_i+1−Ψ_i h_i+1

2

!2

+Ψ₀²+Ψ_I+1² 1 A

−ρhl

λ²

I

X

i=1

h_iΨ_i−α0U₀²−α1U₁².

But, thanks to Cauchy-Schwarz inequality,

˛

˛

˛

˛

˛

I

X

i=1

hiΨi

˛

˛

˛

˛

˛

≤

I

X

i=1

hiΨ_i²

!

1 2

.

Rewriting

Ψi=

i−1

X

j=0

h_j+1 2

Ψ_j+1−Ψ_j h_j+1

2

! +Ψ0,

we get

Ψi² ≤ 2 0

@

i−1

X

j=0

h_j+1 2

Ψj+1−Ψj

h_j+1 2

!1 A

2

+2Ψ0² ≤ 2

I

X

j=0

h_j+1 2

Ψj+1−Ψj

h_j+1 2

!2

+2Ψ0² (53) and

I

X

i=1

hiΨi²≤2

I

X

j=0

h_j+1 2

Ψj+1−Ψj

h_j+1 2

!2

+ 2Ψ0². (54)

Inequality (54) is a discrete version of the Poincar´e inequality. Using (54), we get that there existµ >0 andν >0, only depending onα0,α1,U0,U1,λ²andρ_hl such that for allΨ∈R^I+2

I(Ψ)≥µ 0

@

I

X

i=0

h_i+1 2

Ψ_i+1−Ψ_i h_i+1

2

!2

+Ψ₀²+Ψ_I+1² 1

A−ν (55)

≥ µ 2

I

X

i=1

hiΨi²

!

−ν

This inequality implies that I(Ψ) tends to +∞when Ψ tends to +∞. Therefore it admits a minimum and this minimum is a solution to∇I(Ψ) = 0, which corresponds to the numerical scheme (44), (30)–(32). It concludes the proof of the existence of a solutionΨto (44), (30)–(32). Furthermore, letΨ be the minimum of the functionI, i.e.the solution to the numerical scheme. It satisfies I(Ψ)≤ I(Ψ) for alle Ψe ∈R^I+2 anda fortioriforΨe = 0. But

I(0) = 1 λ²

I

X

i=1

hi(vi+ui)≤ 1

λ²(kuk∞+kvk∞). (56) Then, from (55) and (56), we obtain the discrete H¹-estimate (50) and theL^∞- estimate (49) is a straightforward consequence of (53).

2 As a consequence of Lemma 2, we getH¹-estimates onuandv.

Lemma 3 IfΨsatisfies the L^∞-estimate (49) and the H¹-estimate (50), then there exists M >0 only depending on the data (λ², α0, α1, ∆Ψ₀^pzc, ∆Ψ₁^pzc, V, Pm, Nm, ρhl and the interface kinetic functions) such thatuandv satisfy :

I

X

i=0

(u_i−u_i+1)² h_i+1

2

≤M and

I

X

i=0

(v_i−v_i+1)² h_i+1

2

≤M. (57) Proof

We focus on the estimate onu. Thanks to (38), we have

I

X

i=0

(ui−ui+1)² h_i+1

2

=F²

I

X

i=0

h_i+1

2(f(Ψi, Ψi+1))².

But,f(Ψi, Ψi+1) = e^3δi with δi ∈[min(Ψi, Ψi+1),max(Ψi, Ψi+1)]. Therefore, the L^∞- estimate onΨimplies that|f(Ψi, Ψi+1)|is bounded and we get (57). 2

3.3 Existence result

We now conclude the proof of Theorem 1 with the proof of existence of a solution (u,v,Ψ) to the system of nonlinear equations (S⁰). Let us defineT :R^I+2→R^I+2by T(Ψ) =Φ, whereΦ= (Φi)_0≤i≤I+1 is solution to

−λ²(Φi+1−Φi

h_i+1 2

−Φi−Φ_i−1 h_i−1

2

) = hi(3uie^−3Ψi−vie^Ψi+ρhl), Φ₀−α₀Φ1−Φ0

h1 2

=U₀, ΦI+1+α1

Φ_I+1−Φ_I h_I+1

2

=U1,

with u and v given by (45) and (46). T is clearly a continuous map from R^I+2 to R^I+2. Furthermore, thanks to Lemma 2, T maps a closed ball of R^I+2 into itself.

Applying Brouwer fixed point theorem, we get the existence of a solution to the system of nonlinear equations (45), (46), (44), (31), (32) which is equivalent to (S).

Remark 2 With the Brouwer fixed point theorem, we do not get the uniqueness of a solution to the numerical scheme. Indeed, it is not clear that this solution might be unique for any applied voltage V. But it is also true for the continuous model: the question of the uniqueness of the solution is still open.

The problem of uniqueness of the stationary state or existence of multiple stationary states is well known in the semiconductor framework. There exist some uniqueness or non-uniqueness results for the drift-diffusion system for semiconductor devices (with Dirichlet-Neumann boundary conditions). We refer for instance to the works by Mock [14] and Markowich [12] (uniqueness results close to the thermal equilibrium, for small spplied voltages), by Gajewski [10] (uniqueness for large Debye lengths). F. Alabau established uniqueness results for large applied voltages but with specific conditions on the doping profile ([1–3]). Existence of multiple steady-state solutions was also proved in [4].

4 Convergence of the scheme

4.1 Approximate solutions and theorem of convergence

We first construct the sequences of approximate solutions. In general finite volume methods lead to piecewise constant approximate solutions.

Definition 2 For a given mesh (xi)_0≤i≤I+1 of size h, we associate to each vector w= (wi)_0≤i≤I+1a functionw^hdefined by:

w^h(x) =w_i, ∀x∈]x_i−1 2, x_i+1

2[.

It is also possible to use a continuous piecewise affine reconstruction for the approximate solutions.

Definition 3 For a given mesh (xi)_0≤i≤I+1 of size h, we associate to each vector w= (wi)_0≤i≤I+1a functionw^hdefined by:

– w^h∈ C([0,1]),

– w^his an affine function in [x_i, x_i+1], – w^h(xi) =wi.

The difference between the two different approximate solutions is of order h(size of the mesh). We have

|w^h(x)−w^h(x)| ≤h wi+1−wi

h_i+1 2

, ∀x∈(xi, xi+1), ∀0≤i≤I. (58) Theorem 2 Assume (1), (2), (3). Let us consider a sequence of meshes indexed by h, the size of the mesh. For each mesh,(u,v,Ψ)is a solution to the system(S⁰) and (u^h,v^h,Ψ^h)is the associated approximate solution in the sense of Definition 3. Then, up to a subsequence, ash→0,

u^h→u, v^h→v, Ψ^h→Ψ ash→0inC([0,1]).

and(u, v, Ψ)is a solution to the corrosion model (11)–(19) in the sense of Definition 1.

The outline of the proof of Theorem 2 is the following. First, we establish the compactness of the sequence of approximate solutions. It yields the convergence of an extracted subsequence. Then, we prove that the limit of the extracted subsequence is a weak solution to the corrosion model.

4.2 Proof of Theorem 2

Step 1: Compactness of approximate solutions

For a given mesh of sizeh, we define an approximate solution (u^h,v^h,Ψ^h) associated to a solution (u,v,Ψ) to the scheme (S⁰). It provides a sequence of approximate solutions with respect toh: (u^h,v^h,Ψ^h)_h>0. The discrete estimates established in Section 3 lead toH¹-estimates on (u^h,v^h,Ψ^h)_h>0.

Proposition 2 Forh∈(0,1), an approximate solution(u^h,v^h,Ψ^h)verifies:

– u^h∈H¹(0,1),v^h∈H¹(0,1),Ψ^h∈H¹(0,1) – there exists C >0, not depending onh, such that

ku^hk_H1≤C, kv^hk_H1≤C, kΨ^hk_H1≤C.

Proof

This is a consequence of Lemmas 1, 2 and 3. We can choose the following norm on H¹(0,1) (equivalent to the usual norm thanks to a Poincar´e inequality):

kwk²_H1=k∂xwk²_L2+w(0)². Therefore forw^h=u^horv^h orΨ^h, we have

kw^hk²_H1 =

I

X

i=0

(w_i+1−w_i)² h_i+1

2

+w₀².

2 But,H¹(0,1) is compactly embedded inC([0,1]). Therefore a consequence of Propo- sition 2 is the convergence of an extracted subsequence of approximate solutions in C([0,1]).

Corollary 2 Let(u^h,v^h,Ψ^h)h>0be a sequence of approximate solutions. There exists u,v,Ψ ∈ C([0,1])such that, up to a subsequence,

u^h→u, v^h→v, Ψ^h→Ψ ash→0inC([0,1]).

Moreoveru,v,Ψ ∈H¹(0,1)and, up to a subsequence,

∂xu^h* ∂xu, ∂xv^h* ∂xv, ∂xΨ^h* ∂xΨ ash→0inL²(0,1) weak.

Step 2: Passing to the limit in the scheme

Let us now prove that the limit of the extracted subsequence (u, v, Ψ) obtained in Corollary 2 is a solution of the corrosion model (11)–(19) in the sense of Definition 1.

It is a consequence of the following Proposition.

Proposition 3 Let(u,v,Ψ)be a solution to the scheme(S)and(u^h,v^h,Ψ^h) be the corresponding approximate solution. For allϕ∈C_c^∞([0,1]), it satisfies

˛

˛

˛

˛

˛ Z1

0

e^−3Ψh∂xu^h∂xϕ+k⁰1e^3a01Ψh(0)e^−3Ψh(0)u^h(0)ϕ(0)−Pmk^L1e^{3aL1(V−Ψh(1))}ϕ(1)

+“

m^L₁e^{−3bL1(V−Ψh(1))}+k^L₁e^{3aL1(V−Ψh(1))}”

e^−3Ψh(1)u^h(1)ϕ(1)

˛

˛

˛

˛

˛

≤R^u_ϕ(h), (59)

˛

˛

˛

˛

˛ Z1

0

e^Ψh∂xv^h∂xϕ+k2⁰e^−a02Ψh(0)e^Ψh(0)v^h(0)ϕ(0)−Nmk2^Le^{−aL2(V−Ψh(1))}ϕ(1)

+“

m^L₂e^{bL2(V−Ψh(1))}+k^L₂e^{−aL2(V−Ψh(1))}”

e^Ψh(1)v^h(1)ϕ(1)

˛

˛

˛

˛

˛

≤R^v_ϕ(h), (60)

˛

˛

˛

˛

˛ Z 1

0

∂xΨ^h∂xϕ+ 1 α1

Ψ^h(1)ϕ(1) + 1 α0

Ψ^h(0)ϕ(0)

+ 1 λ²

Z1 0

“

e^Ψhv^h−3e^−3Ψhu^h−ρ_hl” ϕ−U1

α₁ϕ(1) +U0

α₀ϕ(0)

˛

˛

˛

˛

˛

≤R^Ψ_ϕ(h), (61) withR_ϕ^u,R^v_ϕ,R^Ψ_ϕ tending to 0 as htends to 0 (for a givenϕ).

Proof

We focus here on the proof of (59). The proof of (60) and (61) is similar. Let ϕ∈C_c^∞([0,1]). We setϕ_i =ϕ(x_i). Multipliying the scheme (23) byϕ_i and summing over 1≤i≤I, we get

I−1

X

i=1

1 f(Ψi, Ψi+1)

ui+1−ui

h_i+1 2

(ϕi+1−ϕi) +k₁⁰u0e^−3Ψ0e^3a01Ψ0ϕ1−Pmk^L₁e^{3aL1(V−ΨI+1)}ϕ_I +“

m^L₁e^{−3bL1(V−ΨI+1)}+k₁^Le^{3aL1(V−ΨI+1)}”

e^−3ΨI+1u_I+1ϕ_I = 0.