Convergence of a Control Volume Finite Element scheme for a cross-diffusion system modeling ion transport

TL;DR

This paper introduces a control volume finite element scheme for a cross-diffusion system modeling ion transport, ensuring entropy stability and convergence, with numerical validation in degenerate diffusion scenarios.

Contribution

It develops a novel CVFE scheme that guarantees entropy inequalities and convergence for ion transport models on general meshes.

Findings

Scheme satisfies entropy inequalities.

Proves convergence to weak solutions.

Numerical tests confirm convergence in degenerate cases.

Abstract

An approximation of a system coupling the cross-diffusion of chemical species within a solvent, subjected to an electric field, is obtained through a control volume finite element (CVFE) scheme on general simplicial meshes in two or three space dimensions. The discrete unknowns of the numerical scheme are derived from the chemical potential of the species. The scheme is designed in order to fulfill entropy inequalities, yielding compactness properties for the discrete solutions and convergence to a weak solution of the continuous problem. Numerical illustrations of the convergence properties are provided in situations where diffusion of ionic species degenerates.