Asymptotic behavior of a finite volume scheme for the transient drift-diffusion model

doi: 10.1093/imanum/dri017

Asymptotic behavior of a finite volume scheme for the transient drift-diffusion model

CLAIRECHAINAIS-HILLAIRET

Laboratoire de Math´ematiques Appliqu´ees, Universit´e Blaise Pascal, Clermont-Ferrand

24, Avenue des Landais, 63177 Aubi`ere cedex,

FRANCE and FRANCISFILBET

ICJ, Universit´e Lyon I, Claude Bernard 43, boulevard 11 novembre 1918,

69622, Villeurbanne, cedex FRANCE

[Received on 26 June 2006]

In this paper, we propose a finite volume discretization for multidimensional nonlinear drift-diffusion system. Such a system arises in semiconductors modeling and is composed of two parabolic equations and an elliptic one. We prove that the numerical solution converges to a steady state when time goes to infinity. Several numerical tests show the efficiency of the method.

Keywords: Drift-diffusion system, thermal equilibrium, boundary problem, finite volume scheme.

AMS subject classifications. 65M12, 76X05, 82D37

1. Introduction

In the modeling of semiconductor devices, there exists a hierarchy of models ranging from the kinetic trans- port equations to the drift-diffusion equations, see (23). In semiconductor simulations, the drift-diffusion system is the most widely used because it displays both computational efficiency and physical consistency.

This system consists of two continuity equations for the electron density N :=N(t,x)and the hole density P :=P(t,x)and a Poisson equation for the electrostatic potential V :=V(t,x)for t∈R⁺and x∈R^d.

More precisely, let Ω _⊂^Rd (d>1) be an open and bounded domain such thatΩ is polygonal or polyhedral and we setΓ =∂Ω. For T >0, we denote byΩT = (0,T)×Ω andΓT = (0,T)×Γ. Then,

IMA Journal of Numerical Analysis cInstitute of Mathematics and its Applications 2005; all rights reserved.

setting all physical parameters equal to 1, the drift-diffusion system for a bipolar semiconductor reads



















∂^N

∂^{t −div(∇r(N)−N∇V) =0, (t,x)∈}ΩT,

∂^P

∂^{t −div(∇r(P) +P∇V) =0, (t,x)∈}ΩT,

∆^V =N−P−C, (t,x)∈ΩT,

(1.1)

where C∈L^∞(Ω)is the prescribed doping profile characterizing the device under consideration

|C(x)|6C, x∈Ω. (1.2)

The usual considerations on which the isentropic hydrodynamic model are based suggest a pressure of the form

r(s) =s^α, α >1.

The linear case, whereα =1, corresponds to the isothermal model. In the general case, we will assume that r∈C¹(R), r(0) =r^′(0) =0,with r^′(s)>c₀s^α−1.

Equations (1.1) are supplemented with initial data at time t=0

N(0,x) = N⁰(x), P(0,x) =P⁰(x), x∈Ω, (1.3) such that there exist two constants 06m6M satisfying

m 6 N⁰(x),P⁰(x) 6 M, x∈Ω. (1.4)

Moreover, we will consider Dirichlet-Neumann boundary conditions. Indeed, the physically motivated boundary conditions are either Dirichlet boundary conditions on N, P, V or homogeneous Neumann bound- ary conditions on N, P and V . This means that the boundaryΓ is split into two partsΓ =ΓD∪ΓN and, if we denote byνthe outward normal toΓ, that the boundary conditions read on the boundaryΓD





N(t,x) =N^D(x), (t,x)∈(0,T)×ΓD, P(t,x) = P^D(x), (t,x)∈(0,T)×ΓD, V(t,x) =V^D(x), (t,x)∈(0,T)×ΓD

(1.5) and homogeneous Neumann boundary conditions onΓN:

∇r(N)·ν =∇r(P)·ν =∇V·ν =0 onΓN. (1.6) We assume that the Dirichlet boundary conditions satisfy

m 6 N^D(x),P^D(x) 6 M, x∈ΓD. (1.7)

On the one hand, the existence of solutions to the system (1.1)-(1.6) has been proven under natural assumptions. In some situations, the uniqueness of solutions is also obtained, see (3; 13; 15; 17; 20). On the other hand, a lot of numerical algorithms for solving the drift-diffusion system, in the stationary case as well as in the transient case, have already been proposed. It started with 1-D finite difference methods and the so-called Scharfetter-Gummel scheme (26). In the linear pressure case (r(s) =s), finite element methods (1; 8; 7; 9; 10; 16; 25), mixed exponential fitting finite element methods (4) have also been successfully developed. The extension of the mixed exponential fitting finite element methods to the case of nonlinear

pressures (r(s) =s^α) has been considered in (2; 18) and (21) where numerical results are given in 1-D and 2-D respectively. The convergence of finite volume schemes in the nonlinear case has been established in (6).

The large time behavior of the solutions to the nonlinear drift-diffusion model (1.1)-(1.6) has been studied in (19). It is proven that the solution to the transient system converges to a solution to the thermal equilibrium state as t →∞if the boundary conditions (1.5) are in thermal equilibrium. The stationary drift-diffusion system reads













−div(∇r(N)−N∇V) =0, x∈Ω,

−div(∇r(P) +P∇V) = 0, x∈Ω,

∆^V =N−P−C, x∈Ω,

(1.8)

with the boundary conditions (1.5)-(1.6). The thermal equilibrium is a steady-state for which electron and hole currents (∇r(N)−N∇V and∇r(P) +P∇V ) vanish. The existence of a thermal equilibrium has been proven in (24): we introduce the enthalpy function h defined by

h(s) = Z s

1

r^′(τ)

τ ^dτ ^(1.9)

and the generalized inverse g of h, defined by g(s) =

h⁻¹(s) if h(0⁺)<s<∞, 0 if s 6h(0⁺),

where we have implicitly assumed that h(+∞) =∞. If the boundary conditions satisfy N^D,P^D>0 and h(N^D)−V^D =αN and h(P^D) +V^D=αPonΓD,

the thermal equilibrium is defined by

N(x) = g(αN+V(x)), P(x) = g(αP−V(x)), x∈Ω, (1.10) whereas V satisfies the following semilinear elliptic problem

∆^V =g(αN+V)−g(αP−V)−C, inΩ, V(x) =V^D(x)onΓD, ∇V·ν =0 onΓN.

(1.11)

In this paper we are concerned by the theoretical study of the large time behavior of the numerical solution given by a finite volume scheme for the transient drift-diffusion model (1.1)-(1.6) . This work is motivated by a very practical question. Indeed, in numerical analysis the numerical solution is classically proven to converge to the exact solution of the continuous model on a fixed time interval when the mesh size goes to zero. However, in engineering the numerical solution is often computed on a fixed mesh where the final time is increasing and goes to infinity. Thus, in such a situation, it becomes crucial to study the stability and consistency of the numerical solution in the long time asymptotic limit. Moreover in engineering numerical simulations are often performed to find stationary solutions, then the question of consistency of the computed solution with respect to the exact one is usually not known.

This article is the first step of a research program in numerical analysis on the long time asymptotic behavior of discrete solutions (spectral methods for Boltzmann’s equation, finite volume for 2-D Navier- Stokes equations, etc). Here, we focus on a drift-diffusion model for semiconductors when thermal equilib- rium holds at the boundary.

We first study the stationary case and propose a finite volume scheme for the steady state problem. On the one hand, we prove existence and uniqueness of a numerical solution. On the other hand, we establish a priori estimates which will lead to the convergence of the numerical solution to the exact solution of the steady state problem. The second part is devoted to the evolution problem (1.1)-(1.6). We construct a new finite volume scheme and rigorously prove that the numerical solution converges to the solution of the discrete steady state problem given in the first part. The proof is based on the control of the discrete energy dissipation.

2. Numerical scheme and main results

In this section, we present the finite volume schemes for the thermal equilibrium (1.11), with (1.10), and for the time evolution drift-diffusion system (1.1)-(1.6). Then we give the main results of the paper.

We first define the space discretization ofΩ. An admissible mesh ofΩ is given by a family T of control volumes (open and convex polygons in 2-D, polyhedra in 3-D), a familyE of edges in 2-D (faces in 3-D) and a family of points(xK)K∈T which satisfy Definition 5.1 in (12). It implies that the straight line between two neighboring centers of cells(xK,xL)is orthogonal to the edgeσ=K|L. As mentioned in (12) Voronoi meshes are admissible meshes. In 2-D, triangular meshes satisfy the admissibility condition if all angles of the triangles are less thanπ/2, while choosing for the(xK)_K∈T the circumcenters of the triangles.

But this condition on the angles can be relaxed to the assumption that the sum of the two opposite angles to a common edge is less thanπ. In this case, the circumcenters of the triangles are not necessary inside the triangles but the scheme still works.

In the set of edgesE, we distinguish the interior edgesσ ∈E_int and the boundary edges σ ∈E_ext. Because of the Dirichlet-Neumann boundary conditions, we splitE_ext intoE_ext =E^D

ext∪E^N

ext whereE^D

ext is the set of Dirichlet boundary edges andE^N

ext is the set of Neumann boundary edges. For a control volume K∈T, we denote byE_K the set of its edges,E_int,K the set of its interior edges,E^D

ext,K the set of edges of K included inΓDandE^N

ext,K the set of edges of K included inΓN.

In the sequel, we denote by d the distance inR^d, m the measure inR^d orR^d−1. We assume that the family of mesh considered satisfies the following regularity constraint there existsξ>0 such that

d(xK,σ) > ξd(xK,xL), for K∈T,forσ _∈^Eint,K,σ=K|L. (2.1) The size of the mesh is defined by

δ =max

K∈T(diam(K)). (2.2)

For allσ∈E, we define the transmissibility coefficient:

τσ=











m(σ)

d(xK,x_L), forσ∈E_int,σ=K|L, m(σ)

d(xK,σ), forσ∈E_ext,K. Then, we set

G(x,V) = g(αN+V)−g(αP−V)−C(x).

The scheme corresponding to the equation (1.11) on the potential V reads

σ∈

∑

τσDV_K,σ =m(K)G_K(VK), K∈T, (2.3)

where the(DV_K,σ)_σ∈^E are defined by

DV_K,σ=















V_L−V_K, ifσ∈E_int, σ=K|L, V_σ−V_K ifσ∈E^D

ext,K,

0 ifσ∈E^N

ext,K.

(2.4)

with

V_σ= 1 m(σ)

Z

σV^D(x)dx, σ∈E^D

ext (2.5)

and

G_K(V) = 1 m(K)

Z

KG(x,V)dx, K∈T. (2.6)

Then, we define an approximate solution V_δ associated to the discretizationT (we recall thatδ is the size of the discretization), which is a piecewise constant function :

V_δ(x) = VK x∈K. (2.7)

The scheme leads to a system of nonlinear algebraic equations. In the next section, we will establish existence and uniqueness of a solution to the scheme (2.3)-(2.7) and a priori estimates giving some com- pactness and allowing to pass to the limit on the sequence of approximate solutions(V_δ)_δ>0towards the solution V ∈H¹(Ω)∩L^∞(Ω)of (1.11) coupled with boundary conditions (1.5)-(1.6). The result is the following:

THEOREM2.1 Assume that the boundary conditions satisfy (1.7) with m>0 and the thermal equilibrium onΓD

h(N^D)−V^D =αN, and h(P^D) +V^D=αP, where the enthalpy h is given by (1.9).

The scheme (2.3)-(2.6) admits a unique solution, which satisfies the following L^∞estimate and discrete H¹estimate : there exists a constantC >0, only depending on V^Dand g, such that for all K ∈T

|V_K| 6 C ∀K ∈T

K

∑

∑

σ∈E_K

τσ|DV_K,σ|² 6 C.

We may now define the finite volume approximation of the drift-diffusion system (1.1)-(1.6) in the case of mixed Dirichlet-Neumann boundary conditions. The scheme is almost the same as the one proposed in (5) except that the diffusion is approximated in a different way.

Let(T,E,(xK)K∈T)be an admissible space discretization ofΩ and let us define the time step∆t and MT =E(T/∆t)in order to get a space-time discretization of ΩT. First of all, the initial and boundary

conditions and the doping profile are approximated by their L²projections on control volumes or on edges:

for U ={N⁰,P⁰,C}

U_K = 1 m(K)

Z K

U(x)dx, K∈T (2.8)

and for U ={N,P,V}

U_σ = 1 m(σ)

Z

σU^D(s)ds, σ∈E^D

ext. (2.9)

For n ∈N, we construct the approximate potential Vⁿfrom the density(Nⁿ,Pⁿ)and then we update the density(Nⁿ⁺¹,Pⁿ⁺¹)at iteration n+1. On the one hand, for the potential Vⁿwe use a classical finite volume scheme

σ∈

∑

τσDV_K,σⁿ =m(K) (N_Kⁿ−P_Kⁿ−C_K), K∈T, (2.10) where DV_K,σⁿ are defined analogously to (2.4). On the other hand, for the scheme on Nⁿ⁺¹ and Pⁿ⁺¹, we choose a fully implicit discretization, with a standard upwinding for the convective fluxes and a new nonlinear approximation for the diffusive fluxes. Then the scheme for Nⁿ⁺¹and Pⁿ⁺¹is given for K∈T by

m(K)N_Kⁿ⁺¹−N_Kⁿ

∆^{t (2.11)}

−

∑

σ∈EK, σ=K|L

τσ

min(N_Kⁿ⁺¹,N_Lⁿ⁺¹)Dh(Nⁿ⁺¹)_K,σ −(DV_K,σⁿ )⁺N_Kⁿ⁺¹−(DV_K,σⁿ )⁻N_Lⁿ⁺¹

−

∑

σ∈E^D

ext,K

τσ

min(N_Kⁿ⁺¹,N_σ)Dh(Nⁿ⁺¹)_K,σ −(DV_K,σⁿ )⁺N_Kⁿ⁺¹−(DV_K,σⁿ )⁻N_σ

=0,

m(K)P_Kⁿ⁺¹−P_Kⁿ

∆^{t (2.12)}

−

∑

σ∈E K, σ=K|L

τ_σmin(P_Kⁿ⁺¹,P_Lⁿ⁺¹)Dh(Pⁿ⁺¹)_K,σ + (DV_K,σⁿ )⁺P_Lⁿ⁺¹+ (DV_K,σⁿ )⁻P_Kⁿ⁺¹

−

∑

σ∈ED ext,K

τσ

min(P_Kⁿ⁺¹,P_σ)Dh(Pⁿ⁺¹)_K,σ + (DV_K,σⁿ )⁺P_σ+ (DV_K,σⁿ )⁻P_Kⁿ⁺¹

=0,

where Dh(P)_K,σis defined by

Dh(P)_K,σ =















h(PL)−h(PK), ifσ∈E_int, σ=K|L, h(P_σ)−h(PK) ifσ∈E^D

ext,K,

0 ifσ∈E^N

ext,K.

(2.13)

and u⁺=max{u,0}and u⁻=min{u,0}.

Then, the approximate solution(N_δ,P_δ,V_δ)to the problem (1.1)-(1.6) associated to the discretization Dis defined as piecewise constant function by

N_δ(t,x) =N_Kⁿ⁺¹, P_δ(t,x) =P_Kⁿ⁺¹, V_δ(t,x) =V_Kⁿ⁺¹ (t,x)∈[tⁿ,tⁿ⁺¹)×K,

where{(N_Kⁿ,P_Kⁿ,V_Kⁿ), K∈T, 06n6M_T+1}is the solution to the scheme (2.10)-(2.12).

Before presenting our main result, let us comment on the derivation of such a scheme. It is based on a classical time implicit scheme and an up-wind finite volume scheme in space, whereas the diffusion is written in such a way that the scheme preserves the steady state and that the numerical solution satisfies an energy estimate. The present scheme can be compared with the well-known Scharfetter-Gummel scheme (26) which is second order, preserves the nonnegativity of the solution(N,P)and also the steady states. In the following, we will prove that (even if our algorithm is formally first order); the numerical solution is much more robust : the density(N,P)satisfies a maximum principle, the energy is decreasing, the energy dissipation is well controlled and steady states are exactly preserved.

We may now state our main result.

THEOREM2.2 We assume that there is no doping profile (C=0), that the initial and boundary conditions satisfy (1.4) and (1.7) with 0<m6M and that the following condition on the time step is fulfilled

∆tβ < 1, whereβ:=M²

m . (2.14)

Then, the solution(N_δ,P_δ,V_δ)given by the finite volume scheme (2.8)-(2.12) satisfies for each K∈T (N_Kⁿ,P_Kⁿ) → (NK,PK) when n→∞,

V_Kⁿ → V_K when n→∞,

where(NK,P_K,VK)is an approximation to the solution of the steady state equation (1.10)-(1.11) given by (2.3)-(2.4).

The proof is based as in the continuous case on the energy estimate and the control of its dissipation. Let us mention that this result holds on a restrictive assumption of C=0 (vanishing doping profile). However, this restriction also holds in the continuous case for the same reason. Indeed, the energy estimate and the control of its dissipation are still true, but those properties are not enough to prove (rigorously) convergence to equilibrium. The case C=0 allows us to prove uniform lower bound on the density(N,P)avoiding regions where the density vanishes when time goes to infinity.

In the last section, we perform numerical simulations and give numerical evidence of convergence to a steady state even when this condition is not satisfied.

3. Drift diffusion system at thermal equilibrium

In this section, we study the numerical solution corresponding to the steady state (1.8) with boundary conditions (1.5), (1.6) in the thermal equilibrium case where the steady state rewrites (1.10)-(1.11).

3.1 A semilinear elliptic problem

The aim of this section is to prove the convergence of a finite volume scheme for a semilinear elliptic problem like (1.11). More precisely, we are interested in problems of the form:

( ∆^V =G(x,V), x∈Ω,

V =V^DonΓD, ∇V·ν =0 onΓN. (3.1)

The assumptions are the following:

G(x,V)is monotonically increasing with respect to V for all x∈Ω. (3.2)

There exist functions G₁(V)and G₂(V)monotically increasing such that

G₁(V)6G(x,V)6G₂(V)for all x∈Ω. (3.3) Moreover,

there exist V₁and V₂satisfying G₁(V1) =0 and G₂(V2) =0. (3.4) Finally, the function V^Dcan be extended in the whole domainΩ and satisfies

V^D∈H¹(Ω). (3.5)

Under such assumptions, the problem (3.1) admits a unique solution V∈H¹(Ω)∩L^∞(Ω). The proof of this result can be found in (22). For the thermal equilibrium (1.11), the assumptions (3.2), (3.3) are clearly satisfied. Indeed,

G(x,V) = g(αN+V)−g(αP−V)−C(x)

is monotonically increasing with respect to V . The functions G₁and G₂are the following G₁(V) =g(αN+V)−g(αP−V)−C, G₂(V) =g(αN+V)−g(αP−V)−C,

where C=inf_x∈ΩC(x), C=sup_x∈ΩC(x)and since limV→−∞g(V) =0 and limV→+∞g(V) = +∞, we have

Vlim→±∞G₁(V) =±∞, lim

V→±∞G₂(V) =±∞

therefore from the continuity of G we show that there exist V₁and V₂such that G₁(V1) =G₂(V2) = 0 and (3.4) is satisfied.

3.2 Existence and uniqueness

First we prove that if(VK,V_σ)solution to the scheme (2.3)-(2.6) exists, it satisfies an L^∞-estimate. The following lemma is the discrete version of the L^∞-estimate in the continous case proved in (22).

LEMMA3.1 We assume that (3.2), (3.3) , (3.4) and (3.5) are satisfied. Let us set e

V=max{V₁,sup

ΓD

V^D}, V

e =min{V₂,inf

ΓD

V^D}. (3.6)

If the scheme (2.3)-(2.6) admits a solution, then it satisfies the following L^∞estimate : V

e

6 VK 6 V,e ∀K ∈T. (3.7)

Proof. The definition (3.6), combined with the monotonicity of G₁and G₂and with (3.3) lead to G₁(eV)>G₁(V1) =0 and G₂(V

e)6G₂(V2) = 0.

Then, we defineVe_K=V for Ke ∈T andVe_σ=V fore σ∈E^D

ext, andW bye e

W= eW_K =V_K−Ve_K, for K∈T, We_σ =V_σ−Ve_σ, forσ∈E^D

ext. From the definitions of G₁, (3.3) andV, it follows that for Ke ∈T

σ∈

∑

τσdVe_K,σ −m(K)G_K(eV_K) 6 0−m(K)G₁(eV_K) 6 −m(K)G₁(V1) = 0

and using that V is a solution to (2.3)-(2.6), it yields for all K ∈T

σ∈

∑

τσdWe_K,σ > m(K)

G_K(VK)−G_K(eV_K)

. (3.8)

On the one hand, using the definition ofV (3.6), we know thate We_σ60 for allσ_∈^E_ext^D. On the other hand, we denote byWe_K0 =max

K∈TWe_K and assume that e

W_K0 =V_K0−Ve_K0 >0.

Then, writing (3.8) for K=K₀and using that G_K(V)is nondecreasing with respect to V , the right hand side is positive whereas the left hand side is negative. Therefore, we have shown that for all K ∈T,We_K 60, hence the upper bound

V_K 6V,e ∀K ∈T.

The lower bound is obtained by the same way.

The result of existence and uniqueness of a solution to the numerical scheme (2.3)-(2.6) is a consequence of the L^∞-estimate (3.7) and comes from an application of Leray-Schauder fixed point theorem.

PROPOSITION3.1 We assume that (3.2), (3.3), (3.4) and (3.5) are satisfied. Then, the numerical scheme (2.3)-(2.6) admits a unique solution V= (VK)K∈T which satisfies the L^∞-estimate (3.7).

Proof. We start by uniqueness and consider two solutions U¹ and U²to (2.3)-(2.6). Multiplying by U_K¹−U_K²and summing over K∈T, it follows

K

∑

∑

σ∈E_K

τσ

D(U¹−U²)_K,σ2

+

∑

K∈T

m(K)

G_K(U_K¹)−G_K(U_K²) U_K¹−U_K²

=0.

Since G(x,V)is increasing with respect to V

G_K(U_K¹)−G_K(U_K²) U_K¹−U_K²

>0, ∀K ∈T; we conclude that

K

∑

∑

σ∈E_K

τσ

D(U¹−U²)_K,σ2

60 and since(U¹−U²)_σ=0, forσ ∈E^D

ext, then U¹=U².

For the existence proof, we introduce the application T : (V,λ)→W where W is the solution to the linear system

σ∈

∑

τσDW_K,σ = λm(K)G_K(VK), ∀K∈T, with

W_σ= 1 m(σ)

Z

σλ^VD(x)dγ.

The operator T is a linear mapping fromR^θ×[0,1]→R^θ , whereθ is the number of control volumes, continuous and compact. Furthermore, it satisfies :

• T(V,0) =0,

• for all(V,λ)∈R^θ×[0,1]such that T(V,λ) =V , we have V

e

6V_K6V.e

Thanks to the Leray-Schauder fixed point theorem, it follows that T₁: V 7→T(V,1)admits a unique fixed

point, which concludes the proof of Proposition 3.1.

From the L^∞ bound, we can now establish a discrete H¹ estimate giving strong compactness on the approximation. Assume that(u_σ)_σ∈ED

ext is given on the boundaryΓ^{D. For u}= (uK)K∈T, we define the L²- norm and the H¹-seminorm as follows:

kuk²0,Ω =

∑

K∈T

m(K)|uK|²

|u|²_1,Ω =

∑

σ∈Eint σ=K|L

τσ|u_K−u_L|²+

∑

K∈T

∑

σ∈E^D

ext,K

τσ|u_K−u_σ|².

We recall the discrete Poincar´e inequality:

LEMMA3.2 LetΩ be an open convex bounded polygonal or polyhedral subset ofR^d(d=2 or 3). Then, there exists C_Ω ∈R₊only depending onΩsuch that, for all admissible mesh ofΩ satisfying the regularity assumption (2.1), for all(uK)K∈T and(u_σ)_σ∈E_ext^D satisfying u_σ=0 for allσ∈E_ext^D, we have

kuk0,Ω 6 C_Ω√ p d

ξ ^{|u|1,Ω (3.9)}

Proof. We perform a similar proof as in (14). LetT be an admissible mesh and denote by X(T)the set of functions fromΩ toRwhich are constant over each control volume K∈T and which are zero on the set of edgesσ _⊂ΓD. We consider v∈X(T)and since the function v is piecewise constant and has a finite number of jumps (which corresponds to the number of edges), we get that v∈BV(Ω). Moreover in dimension d, the space of BV functions which are zero on the boundaryΓDis continuously embedded in L^d−1d (Ω)(11, Theorem 3.5). Then, there exists a constant C_Ω >0, depending only onΩ, such that

Z

Ω|v(x)|^d−1d dx 6 C_Ω [BV_Ω(v)]^d−1d , where

BV_Ω(v) = sup Z

Ωv(x)divϕ(x)dx, ϕ_∈C_o^∞(Ω), |ϕ(x)|61, ∀x∈Ω. Applying this latter result to our function v∈X(T), we get

K

∑

m(K)|vK|^dd−1

!^d−1_d

6C_ΩBV_Ω(v)

and since v is piecewise constant, for allϕ∈C_o^∞(Ω) Z

Ωv(x)divϕ(x)dx =

∑

K∈T

v_K Z

K

divϕ(x)dx.

Thus, applying the Green formula to the smooth and compactly supported functionϕ Z

Ωv(x)divϕ(x)dx =

∑

K∈T

v_K

∑

σ∈E_int,K Z

σϕ(γ)·νK,σdγ,

whereνK,σ is the unit normal to the edgeσ, oriented outwards K. Next, we perform a discrete integration by part

Z

Ωv(x)divϕ(x)dx =

∑

σ∈Eint, σ=K|L

(vK−v_L) Z

σϕ(γ)·νK,σdγ,

6

∑

σ∈E int, σ=K|L

m(σ)|vK−vL|||ϕ_||∞,

6

∑

σ∈Eint, σ=K|L

m(σ)|v_K−v_L|.

Hence, we get

K

∑

m(K)|v_K|^d−1d

!^d−1_d

6 C_Ω

∑

σ∈Eint, σ=K|L

m(σ)|v_K−v_L|.

Now, we take v=|u|^2(dd−1) and use that

|u_K|^2(dd−1)− |u_L|^2(d−1)d 6 2(d−1) d

|u_K|^d−2d +|u_L|^d−2d

|u_K−u_L|. Integrating by parts and applying the Cauchy-Schwarz inequality, it yields, thanks to (2.1),

K

∑

m(K)|u_K|²

!^d−_d¹

6 C_Ω

∑

K∈T

∑

σ∈Eint,K σ=K|L

m(σ)|u_K|^d−2d |u_K−u_L|

6 C_Ω

pξ ^|u|1,Ω

∑

K∈T

∑

σ∈E_K

m(σ)d(xK,σ)|u_K|^2(dd−2)

!1/2

.

Since∑_σ∈E_Km(σ)d(xK,σ) =d m(K), this gives

K

∑

m(K)|u_K|²

!^d−_d¹

6 C_Ω√ p d

ξ ^|u|1,Ω

∑

K∈T

m(K)|u_K|^2(d−2)d

!1/2

.

Finally using the H¨older inequality, we get

K

∑

m(K)|uK|²

!^d−_d¹

6 C_Ω√ p d

ξ ^|u|1,Ω

∑

K∈T

m(K)|uK|²

!^d_2d⁻²

and then (3.9).

The next lemma provides an L²estimate and an H¹estimate on the numerical solution to the scheme (2.3)-(2.6). We let its proof to the reader while it is close to the proof of Lemma 4.1 in (5).

LEMMA 3.3 We assume that (3.2), (3.3) , (3.4) and (3.5) are satisfied. Then, there existsC >0 such that the solution(VK)K∈T,(V_σ)_σ∈E_ext^D to the scheme (2.3)-(2.6) satisfies

K

∑

m(K)|V_K|² +

∑

σ∈Eint σ=K|L

τσ|V_K−V_L|²+

∑

K∈T

∑

σ∈E^D

ext,K

τσ|V_K−V_σ|² 6 C. (3.10)

4. Asymptotic behavior of the time dependent approximate solution 4.1 Classical a priori estimates

We do not detail here the proof of the convergence of the scheme (2.8)-(2.12) when space and time steps go to 0. Indeed, this scheme is very close to the scheme studied in (5): the only difference is the discretization of the diffusive fluxes. Therefore the proof of the convergence of the scheme towards a weak solution of the problem (1.1)-(1.6) is similar to the proof done in (5). Let us recall the required hypotheses:

(H1) N⁰, P⁰ ∈L^∞(Ω), N^D, P^D ∈L²(ΩT)∩H¹(ΩT)and V^D ∈L^∞(R⁺; H¹(Ω));

(H2) there exist two constants m and M such that

0<m<N⁰,P⁰<M, inΩ, and m<N^D,P^D<M, inΩT; (H3) r∈C²(R)is strictly increasing on(0,+∞);

(H4) C∈L^∞(ΩT)with C=kCk∞.

The result is the following. We insist on the a priori estimates which will be used in the proof of Theorem 2.2.

THEOREM 4.1 Let(H1)−(H4)hold andT be an admissible mesh ofΩ. Assume that the following stability condition is fulfilled

∆tβT < 1, whereβT:=M exp(C T) +C. (4.1) Then, there exists a unique approximate solution(N_δ,P_δ,V_δ)to the scheme (2.8)-(2.12), which satisfies for all K∈T and all n=0,1, ...,M_T,

m exp(−C T) 6 N_Kⁿ,P_Kⁿ 6 M exp(C T).

In particular, if C=0, the maximum principle holds for N_δ and P_δ, i.e.;

m 6 N_Kⁿ,P_Kⁿ 6M, ∀(n,K)∈N×T. (4.2)

and

kVⁿk²1,Ω = kVⁿk²0,Ω +|Vⁿ|²1,Ω 6 4 m(Ω)²M², ∀n∈N. (4.3) Moreover, the approximate solution (N_δ,P_δ,V_δ)converges to(N,P,V)as space and time steps go to 0, where(N,P,V)is a weak solution to (1.1)-(1.6).

4.2 Preliminary results

As in the continuous case, see (19), the study of the large time behavior of the scheme (2.8)-(2.12) is based on an energy estimate with the control of the energy dissipation.

First, let us recall some notations. We denote by(NK,P_K,VK)the solution to the discrete thermal equi- librium. This means that(VK)is the solution to (2.3)-(2.7) and

N_K =g(αN+VK), and P_K=g(αP−V_K), which is equivalent to

h(NK)−VK = αN, h(PK) +VK = αP.

The solution to the time-dependent scheme (2.8)-(2.12) is denoted(N_Kⁿ,P_Kⁿ,V_Kⁿ).

For the sequel, we need to define

H(s) = Z s

1 h(τ)dτ, 0 6 s

(with the convention h(0) =h(0⁺)). Then we can introduce the discrete version of the deviation of the total energy (sum of the internal energies for the electron and hole densities and the energy due to the electrostatic potential) from the thermal equilibrium, see (19): for n>0,

Eⁿ :=

∑

K∈T

m(K) [H(N_Kⁿ)−H(NK)−h(NK) (N_Kⁿ−N_K)]

+

∑

K∈T

m(K) [H(P_Kⁿ)−H(PK)−h(PK) (P_Kⁿ−PK)]

+1

2|Vⁿ−V|²_1,Ω.

As H is a convex function, we haveEⁿ>0 for n>0. We also introduce the energy dissipationI(Nⁿ⁺¹,Pⁿ⁺¹,Vⁿ):

I(Nⁿ⁺¹,Pⁿ⁺¹,Vⁿ) :=

∑

σ∈E σ=K|Lint

τ_σmin(N_Kⁿ⁺¹,N_Lⁿ⁺¹)h

D h(Nⁿ⁺¹)−Vⁿ

K,σ

i2

+

∑

K∈T

∑

σ∈E_ext,K

τ_σ min(N_Kⁿ⁺¹,N_σ)h

D h(Nⁿ⁺¹)−Vⁿ

K,σ

i2

+

∑

σ∈Eint σ=K|L

τσmin(P_Kⁿ⁺¹,P_Lⁿ⁺¹)h

D h(Pⁿ⁺¹) +Vⁿ

K,σ

i2

+

∑

K∈T

∑

σ∈E_ext,K

τσ min(P_Kⁿ⁺¹,P_σ)h

D h(Pⁿ⁺¹) +Vⁿ

K,σ

i2

The proof of Theorem 2.2 relies on the control of energy and energy dissipation given by the following Proposition.

PROPOSITION4.2 Let(H1)−(H4)hold andT be an admissible mesh ofΩ. Then, for n>1, Eⁿ⁺¹ +

1−M²

m ∆^t∆^{t I}(Nⁿ⁺¹,Pⁿ⁺¹,Vⁿ) 6 Eⁿ. (4.4) The proof of Proposition 4.2 will be given later. First, we give a result to estimate the energy due to the elecrostatic potential.

LEMMA4.1 Let(H1)−(H4)hold andT be an admissible mesh ofΩ. Then, for n>0, 1

2|Vⁿ⁺¹−V|²_1,Ω − 1

2|Vⁿ−V|²_1,Ω 6 −

∑

K∈T

m(K) N_Kⁿ⁺¹−N_Kⁿ−P_Kⁿ⁺¹+P_Kⁿ

[V_Kⁿ−V_K]

+M²

m ∆^t2I(Nⁿ⁺¹,Pⁿ⁺¹,Vⁿ). (4.5) and

1

2|Vⁿ⁺¹−Vⁿ|1,Ω 6 M²

m ∆^t2I(Nⁿ⁺¹,Pⁿ⁺¹,Vⁿ). (4.6)

Proof. Substituting the discrete Poisson equation (2.10) at time tⁿ⁺¹and tⁿ, we easily obtain for K∈T

σ∈

∑

τσ

h

DV_K,σⁿ⁺¹−DV_K,σⁿ i

=m(K) N_Kⁿ⁺¹−N_Kⁿ−P_Kⁿ⁺¹+P_Kⁿ

. (4.7)

Next, we multiply the latter equality by−[V_Kⁿ−V_K]and sum over K∈T. Performing a discrete integration by part, we classically have

σ

∑

τσ [V_Lⁿ⁺¹−V_Kⁿ⁺¹]−[V_Lⁿ−V_Kⁿ]

[D(Vⁿ−V)_K,σ]

+

∑

K∈T

∑

σ∈E^D

ext,K

τσ [V_σ−V_Kⁿ⁺¹]−[V_σ−V_Kⁿ]

[D(Vⁿ−V)_K,σ]

6 −

∑

K∈T

m(K) N_Kⁿ⁺¹−N_Kⁿ−P_Kⁿ⁺¹+P_Kⁿ

[V_Kⁿ−V_K].

Thus, using the following equality

[a−b]b=a² 2 −b²

2 −1

2[a−b]²,

we take a=D(Vⁿ⁺¹−V)_K,σ, b=D(Vⁿ−V)_K,σand set W=Vⁿ⁺¹−Vⁿ, which give the following inequality 1

2 |Vⁿ⁺¹−V|²_1,Ω− |Vⁿ−V|²_1,Ω

−1 2|W|²_1,Ω

6 −

∑

K∈T

m(K) N_Kⁿ⁺¹−N_Kⁿ−P_Kⁿ⁺¹+P_Kⁿ

[V_Kⁿ−VK]. (4.8)

Now, the main step consists in the control of the residual term|W|²_1,Ω. To this aim, we start again from (4.7), multiply it by−W_Kand sum over K ∈T. We get

|W|²_1,Ω =−

∑

K∈T

m(K) N_Kⁿ⁺¹−N_Kⁿ−P_Kⁿ⁺¹+P_Kⁿ

W_K 6∆^t[I1+I2+I₃+I₄],

where I_α,α∈ {1, ..,4}are obtained using the finite volume scheme (2.11), (2.12) for Nⁿ⁺¹and Pⁿ⁺¹. More precisely,

I₁=

∑

σ∈Eint σ=K|L

τσmin(N_Kⁿ⁺¹,N_Lⁿ⁺¹)Dh(Nⁿ⁺¹)_K,σ−(DV_K,σⁿ )⁺N_Kⁿ⁺¹−(DV_K,σⁿ )⁻N_Lⁿ⁺¹|DW_K,σ|

I₂ =

∑

K∈T

∑

σ∈E^D

ext,K

τσmin(N_Kⁿ⁺¹,N_σ)Dh(Nⁿ⁺¹)_K,σ−(DV_K,σⁿ )⁺N_Kⁿ⁺¹−(DV_K,σⁿ )⁻N_σ|DW_K,σ|

I₃=

∑

σ∈Eint σ=K|L

τσmin(P_Kⁿ⁺¹,P_Lⁿ⁺¹)Dh(Pⁿ⁺¹)_K,σ + (DV_K,σⁿ )⁺P_Lⁿ⁺¹+ (DV_K,σⁿ )⁻P_Kⁿ⁺¹|DW_K,σ|

I₄ =

∑

K∈T

∑

σ∈E^D

ext,K

τσmin(P_Kⁿ⁺¹,P_σ)Dh(Pⁿ⁺¹)_K,σ+ (DV_K,σⁿ )⁺P_σ+ (DV_K,σⁿ )⁻P_Kⁿ⁺¹|DW_K,σ|.