Convergence of a Finite-Volume Scheme for a Degenerate Cross-Diffusion Model for Ion Transport

International audienceAn implicit Euler finite-volume scheme for a degenerate cross-diffusion system describing the ion transport through biological membranes is analyzed. The strongly coupled equations for the ion concentrations include drift terms involving the electric potential , which is coupled to the concentrations through the Poisson equation. The cross-diffusion system possesses a formal gradient-flow structure revealing nonstandard degen-eracies, which lead to considerable mathematical difficulties. The finite-volume scheme is based on two-point flux approximations with " double " upwind mobilities. It preserves the structure of the continuous model like nonnegativity, upper bounds, and entropy dis-sipation. The degeneracy is overcome by proving a new discrete Aubin-Lions lemma of " degenerate " type. Under suitable assumptions, the existence and uniqueness of bounded discrete solutions, a discrete entropy inequality, and the convergence of the scheme is proved. Numerical simulations of a calcium-selective ion channel in two space dimensions indicate that the numerical scheme is of first order