Comparison of a finite-element and finite-volume scheme for a degenerate cross-diffusion system for ion transport
For researchers in numerical methods for ion transport, this provides a rigorous analysis and comparison of two schemes, but the results are incremental.
The paper analyzes a structure-preserving finite-element scheme for a degenerate cross-diffusion system for ion transport, proving existence, convergence, and properties like entropy dissipation. Numerical simulations compare it with a finite-volume scheme, discussing advantages and drawbacks.
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.