A posteriori error analysis of a positivity preserving scheme for the power-law diffusion Keller-Segel model

TL;DR

This paper develops an a posteriori error analysis for a finite volume scheme solving a Keller-Segel model with power-law diffusion, providing bounds and numerical validation for the scheme's accuracy.

Contribution

It introduces conditional a posteriori error bounds for a finite volume scheme applied to a Keller-Segel system with power-law diffusion, including stability-based estimates and numerical experiments.

Findings

Error bounds are linear in mesh width.

Error estimator behavior varies with the diffusion exponent b3.

Numerical experiments confirm theoretical error estimates.

Abstract

We study a finite volume scheme approximating a parabolic-elliptic Keller-Segel system with power law diffusion with exponent $\gamma \in [1,3]$ and periodic boundary conditions. We derive conditional a posteriori bounds for the error measured in the $L^\infty(0,T;H^1(\Omega))$ norm for the chemoattractant and by a quasi-norm-like quantity for the density. These results are based on stability estimates and suitable conforming reconstructions of the numerical solution. We perform numerical experiments showing that our error bounds are linear in mesh width and elucidating the behaviour of the error estimator under changes of $\gamma$.