arXiv subjects

Claire Chainais-Hillairet

Publications and source records attributed to Claire Chainais-Hillairet.

25 records · Page 2Linked to original sources

arXiv2017

Numerical analysis of a nonlinear free-energy diminishing Discrete Duality Finite Volume scheme for convection diffusion equations

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.

math.AP↗

arXiv2017

Uniform L $\infty$ estimates for approximate solutions of the bipolar drift-diffusion system

We establish uniform L $\infty$ bounds for approximate solutions of the drift-diffusion system for electrons and holes in semiconductor devices, computed with the Schar-fetter-Gummel finite-volume scheme. The proof is based on a Moser iteration technique adapted to the discrete case.

math.NA↗

arXiv2015

Convergence of a Finite Volume Scheme for a Corrosion Model

In this paper, we study the numerical approximation of a system of partial dif-ferential equations describing the corrosion of an iron based alloy in a nuclear waste repository. In particular, we are interested in the convergence of a numerical scheme consisting in an implicit Euler scheme in time and a Scharfetter-Gummel finite volume scheme in space.

math.AP↗

arXiv2015

A Finite-Volume Scheme for a Spinorial Matrix Drift-Diffusion Model for Semiconductors

An implicit Euler finite-volume scheme for a spinorial matrix drift-diffusion model for semiconductors is analyzed. The model consists of strongly coupled parabolic equations for the electron density matrix or, alternatively, of weakly coupled equations for the charge and spin-vector densities, coupled to the Poisson equation for the elec-tric potential. The equations are solved in a bounded domain with mixed Dirichlet-Neumann boundary conditions. The charge and spin-vector fluxes are approximated by a Scharfetter-Gummel discretization. The main features of the numerical scheme are the preservation of positivity and L $\infty$ bounds and the dissipation of the discrete free energy. The existence of a bounded discrete solution and the monotonicity of the discrete free energy are proved. For undoped semiconductor materials, the numerical scheme is uncon-ditionally stable. The fundamental ideas are reformulations using spin-up and spin-down densities and certain projections of the spin-vector density, free energy estimates, and a discrete Moser iteration. Furthermore, numerical simulations of a simple ferromagnetic-layer field-effect transistor in two space dimensions are presented.

math.NA↗

arXiv2014

On discrete functional inequalities for some finite volume schemes

We prove several discrete Gagliardo-Nirenberg-Sobolev and Poincaré-Sobolev inequalities for some approximations with arbitrary boundary values on finite volume 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\subset \mathbb{R}^{N}$, with $N \geq 2$. 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.

math.NA↗

arXiv2013

Entropy-dissipative discretization of nonlinear diffusion equations and discrete Beckner inequalities

The time decay of fully discrete finite-volume approximations of porous-medium and fast-diffusion equations with Neumann or periodic boundary conditions is proved in the entropy sense. The algebraic or exponential decay rates are computed explicitly. In particular, the numerical scheme dissipates all zeroth-order entropies which are dissipated by the continuous equation. The proofs are based on novel continuous and discrete generalized Beckner inequalities. Furthermore, the exponential decay of some first-order entropies is proved in the continuous and discrete case using systematic integration by parts. Numerical experiments in one and two space dimensions illustrate the theoretical results and indicate that some restrictions on the parameters seem to be only technical.

math.NA↗

arXiv2012

On the existence of solutions for a drift-diffusion system arising in corrosion modelling

In this paper, we consider a drift-diffusion system describing the corrosion of an iron based alloy in a nuclear waste repository. In comparison with the classical drift-diffusion system arising in the modeling of semiconductor devices, the originality of the corrosion model lies in the boundary conditions which are of Robin type and induce an additional coupling between the equations. We prove the existence of a weak solution by passing to the limit on a sequence of approximate solutions given by a semi-discretization in time.

math.AP↗