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A. Gerstenmayer, A. Jüngel:
"Comparison of a finite-element and finite-volume scheme for a degenerate cross-diffusion system for ion transport";
in: "ASC Report 32/2018", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2018, ISBN: 978-3-902627-11-7, 1 - 25.

A structure-preserving implicit Euler finite-element scheme for a degenerate cross-diffusion system for ion transport is analyzed. The scheme preserves the nonnegativ-ity 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.

Ion transport, finite-element method, entropy method, existence of discrete solutions, convergence of the scheme, calcium-selective ion channel, bipolar ion channel.

http://www.asc.tuwien.ac.at/preprint/2018/asc32x2018.pdf