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International audience; 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.

Der Transport von Ionen durch biologische Kan��le kann auf makroskopischer Ebene durch partielle Differentialgleichungen f��r die Evolution der Ionenkonzentrationen beschrieben werden. Bezieht man bei der Modellierung F��llungseffekte mit ein, die durch den beschr��nkten Raum in den engen Ionenkan��len hervorgerufen werden, so f��hrt dies zu Kreuzdiffusionstermen in den Gleichungen. Das resultierende Modell f��r die Ionenkonzentrationen besteht aus einem stark gekoppelten System nichtlinearer parabolischer Diffusionsgleichungen, die zus��tzlich einen Driftterm beinhalten. Dieser Term verh��lt sich proportional zum Gradienten des elektrischen Potentials, welches ��ber eine Poisson-Gleichung mit den Ionenkonzentrationen gekoppelt ist. Das Ziel dieser Arbeit ist die mathematische Analyse des vorliegenden Modells, sowie die Entwicklung zuverl��ssiger numerischer Verfahren f��r die approximative L��sung der Gleichungen., The transport of ions through biological channels, on a macroscopic level, can be described with a system of partial differential equations (PDEs) for the evolution of the ion concentrations. If one considers additional size exclusion effects, occurring in narrow pores, cross-diffusion terms appear in the system. The ion concentrations then solve strongly coupled diffusion equations with a drift term involving the electric potential which is coupled to the concentrations through a Poisson equation. The aim of this thesis is to close gaps in the analysis of this cross-diffusion model and provide reliable numerical approximations that capture the behavior of its solutions.