Convergence of a Control Volume Finite Element scheme for a cross-diffusion system modeling ion transport
This work addresses the problem of simulating ion transport in complex geometries for applications like electrochemistry or biology, but it is incremental as it builds on existing numerical methods with specific entropy-based enhancements.
The paper tackles the numerical approximation of a cross-diffusion system for ion transport under an electric field using a control volume finite element scheme, proving convergence to a weak solution and demonstrating numerical results in degenerate diffusion cases.
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.