Abstract. We introduce a finite volume scheme for multi-dimensional drift-diffusion equations. Such equations arise from the theory of semiconductors and are composed of two continuity equations coupled with a Poisson equation. In the case that the continuity equations are non degenerate, we prove the convergence of the scheme and then the existence of solutions to the problem. The key point of the proof relies on the construction of an approximate gradient of the electric potential which allows us to deal with coupled terms in the continuity equations. Finally, a numerical example is given to show the efficiency of the scheme.Mathematics Subject Classification. 65M60, 76X05.

Abstract. In this paper, we study some finite volume schemes for the nonlinear hyperbolic equation ut(x, t) + divF (x, t, u(x, t)) = 0 with the initial condition u0 ∈ L ∞ (R N ). Passing to the limit in these schemes, we prove the existence of an entropy solution u ∈ L ∞ (R N × R+). Proving also uniqueness, we obtain the convergence of the finite volume approximation to the entropy solution into an "h 1 4 " error estimate between the approximate and the entropy solutions (where h defines the size of the mesh).Résumé. Dans cet article, onétudie des schémas volumes finis pour l'équation hyperbolique non linéaire ut(x, t) + divF (x, t, u(x, t)) = 0, avec comme condition initiale u0 ∈ L ∞ (R N ). En passantà la limite dans ces schémas numériques, on obtient l'existence d'une solution entropique u ∈ L ∞ (R N ×R+), puis son unicité. On montre aussi la convergence dans L

We prove several discrete Gagliardo-Nirenberg-Sobolev and Sobolev-Poincaré inequalities for some approximations with arbitrary boundary values on finite volume admissible meshes. The keypoint of our approach is to use the continuous embedding of the space BV (Ω) into L N/(N−1) (Ω) for a Lipschitz domain Ω ⊂ R N , with N ≥ 2. Finally, we give several applications to discrete duality finite volume (DDFV) schemes which are used for the approximation of nonlinear and non isotropic elliptic and parabolic problems.

We consider a convective-diffusive elliptic problem with Neumann boundary conditions: the presence of the convective term entails the non-coercivity of the continuous equation and, because of the boundary conditions, the equation has a kernel. We discretize this equation with finite volume techniques and in a general framework which allows to consider several treatments of the convective term: either via a centered scheme, an upwind scheme (widely used in fluid mechanics problems) or a Scharfetter-Gummel scheme (common to semiconductor literature). We prove that these schemes satisfy the same properties as the continuous problem (one-dimensional kernel spanned by a positive function for instance) and that their kernel and solution converge to the kernel and solution of the PDE. We also present several numerical implementations, studying the effects of the choice of one scheme or the other in the approximation of the solution or the kernel.

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.

We study a finite volume discretization of a strongly coupled elliptic-parabolic PDE system describing miscible displacement in a porous medium. We discretize each equation by a finite volume scheme which allows a wide variety of unstructured grids (in any space dimension) and gives strong enough convergence for handling the nonlinear coupling of the equations. We prove the convergence of the scheme as the time and space steps go to 0. Finally, we provide numerical results to demonstrate the efficiency of the proposed numerical scheme.

No abstract

We propose a nonlinear Discrete Duality Finite Volume scheme to approximate the solutions of drift diffusion equations. The scheme is built to preserve at the discrete level even on severely distorted meshes the energy / energy dissipation relation. This relation is of paramount importance to capture the long-time behavior of the problem in an accurate way. To enforce it, the linear convection diffusion equation is rewritten in a nonlinear form before being discretized. We establish the existence of positive solutions to the scheme. Based on compactness arguments, the convergence of the approximate solution towards a weak solution is established. Finally, we provide numerical evidences of the good behavior of the scheme when the discretization parameters tend to 0 and when time goes to infinity.