A posteriori error control for a finite volume scheme for a cross-diffusion model of ion transport

TL;DR

This paper develops a reliable a posteriori error estimate for a finite volume scheme modeling ion transport via cross-diffusion, introducing a stability framework and numerical validation for the first such estimate.

Contribution

It presents the first a posteriori error estimate for a cross-diffusion system, using a novel stability framework and solution reconstruction methods.

Findings

Error estimator scales with the true error in numerical experiments.

Stability framework is independent of the numerical scheme.

Provides a pointwise a posteriori error estimate for diffusion equations.

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.