| Issue |
ESAIM: M2AN
Volume 60, Number 2, March-April 2026
|
|
|---|---|---|
| Page(s) | 909 - 944 | |
| DOI | https://doi.org/10.1051/m2an/2026014 | |
| Published online | 17 April 2026 | |
A posteriori error control for a finite volume scheme for a cross-diffusion model of ion transport
Department of Mathematics, Technical University of Darmstadt, 64293 Darmstadt, Germany
* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.
Received:
21
July
2025
Accepted:
28
January
2026
Abstract
We derive a reliable a posteriori error estimate for a cell-centered finite volume scheme approximating a cross-diffusion system modeling ion transport through nanopores. To this end, we derive a stability framework that is independent of the numerical scheme and introduce a suitable (conforming) reconstruction of the numerical solution. The stability framework relies on some simplifying assumptions that coincide with those made in weak uniqueness results for this system. Additionally, when electrical forces are present, we assume that the solvent concentration is uniformly bounded from below. This is the first a posteriori error estimate for a cross-diffusion system. Along the way, we derive a pointwise a posteriori error estimate for a finite volume scheme that approximates the diffusion equation. We conduct numerical experiments showing that the error estimator scales with the same order as the true error.
Mathematics Subject Classification: 35K65 / 35K51 / 65M08 / 35K40 / 65M15
Key words: Cross-diffusion / ion transport / finite-volume approximation / a posteriori error estimates / diffusion equation
© The authors. Published by EDP Sciences, SMAI 2026
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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