  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.
