Uniform-in-time bounds for approximate solutions of the drift–diffusion system

摘要： In this paper, we consider a numerical approximation of the Van Roosbroeck’s drift–diffusion system given by a backward Euler in time and finite volume in space discretization, with Scharfetter–Gummel fluxes. We first propose a proof of existence of a solution to the scheme which does not require any assumption on the time step. The result relies on the application of a topological degree argument which is based on the positivity and on uniform-in-time upper bounds of the approximate densities. Secondly, we establish uniform-in-time lower bounds satisfied by the approximate densities. These uniform-in-time upper and lower bounds ensure the exponential decay of the scheme towards the thermal equilibrium as shown in Bessemoulin-Chatard (Numer Math 25(3):147–168, 2016).