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

Clément Cancès (1) , Claire Chainais-Hillairet (2, 1) , Anita Gerstenmayer (3) , Ansgar Jüngel (3)

1 RAPSODI - Reliable numerical approximations of dissipative systems
2 LPP - Laboratoire Paul Painlevé - UMR 8524
3 Institute for Analysis and Scientific Computing [Wien]

An 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.

Domaines: Analyse numérique [math.NA]

Clément Cancès, Claire Chainais-Hillairet, Anita Gerstenmayer, Ansgar Jüngel. Convergence of a Finite-Volume Scheme for a Degenerate Cross-Diffusion Model for Ion Transport. Numerical Methods for Partial Differential Equations, 2019, 35 (2), pp.545-575. ⟨10.1002/num.22313⟩. ⟨hal-01695129⟩