Section: New Results

Advanced discrete functional analysis results and applications

In [38], Claire Chainais-Hillairet, Benoît Merlet and Alexis Vasseur establish a positive lower bound for the numerical solutions of a stationary convection-diffusion equation on a bounded domain. The proof (which is fully detailed) is based on a celebrated method due to Ennio De Giorgi for showing regularity of the solutions of parabolic and elliptic equations. The robustness of the method allows the authors to adapt it to the discrete solutions obtained by standard finite volume discretizations. Further refinements of this work could lead to improve known error estimates for FV discretizations in $L^p$-norms to $L^{\infty }$-norm.

In [14], Marianne Bessemoulin-Chatard and Claire Chainais-Hillairet study the large–time behavior of a numerical scheme discretizing drift–diffusion systems for semiconductors. The numerical method is finite volume in space, implicit in time, and the numerical fluxes are a generalization of the classical Scharfetter–Gummel scheme which allows to consider both linear or nonlinear pressure laws. They study the convergence of approximate solutions towards an approximation of the thermal equilibrium state as time tends to infinity, and obtain a decay rate by controlling the discrete relative entropy with the entropy production. This result is proved under assumptions of existence and uniform-in-time $L^{\infty }$ estimates for numerical solutions, which are then established in [35].

In [43], Marianne Bessemoulin-Chatard and Claire Chainais-Hillairet propose a new proof of existence of a solution to the scheme already introduced in [14] 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. They also 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 [14].

In [12], Boris Andreianov, Clément Cancès, and Ayman Moussa developed a black box to obtain some compactness on the sequence produced by a finite volume discretization for degenerate parabolic problems. Such problems typically appear in the framework of porous media flows or in semi-conductor devices.