# Comparison of a finite-element and finite-volume scheme for a degenerate cross-diffusion system for ion transport

## TL;DR

This paper compares finite-element and finite-volume numerical schemes for simulating ion transport in a degenerate cross-diffusion system, emphasizing structure preservation, convergence, and practical performance.

## Contribution

It introduces a structure-preserving finite-element scheme and compares it with a finite-volume scheme for ion transport modeling.

## Key findings

- The finite-element scheme preserves nonnegativity, bounds, and entropy dissipation.
- Existence and convergence of discrete solutions are established.
- Numerical simulations highlight advantages and drawbacks of both schemes.

## Abstract

A structure-preserving implicit Euler finite-element scheme for a degenerate cross-diffusion system for ion transport is analyzed. The scheme preserves the nonnegativity and upper bounds of the ion concentrations, the total relative mass, and it dissipates the entropy (or free energy). The existence of discrete solutions to the scheme and their convergence towards a solution to the continuous system is proved. Numerical simulations of two-dimensional ion channels using the finite-element scheme with linear elements and an alternative finite-volume scheme are presented. The advantages and drawbacks of both schemes are discussed in detail.