A posteriori error control for a Discontinuous Galerkin approximation of a Keller-Segel model

TL;DR

This paper develops a posteriori error estimates for a discontinuous Galerkin method applied to the Keller-Segel model, enabling adaptive mesh refinement and proof of solution existence based on numerical data.

Contribution

It introduces a conditional, optimal error estimator for the Keller-Segel system that supports adaptive refinement and solution existence proofs.

Findings

Error estimator decays with mesh refinement

Estimator can verify existence of weak solutions

Supports adaptive numerical methods for Keller-Segel model

Abstract

We provide a posteriori error estimates for a discontinuous Galerkin scheme for the parabolic-elliptic Keller-Segel system in 2 or 3 space dimensions. The estimates are conditional, in the sense that an a posteriori computable quantity needs to be small enough - which can be ensured by mesh refinement - and optimal in the sense that the error estimator decays with the same order as the error under mesh refinement. A specific feature of our error estimator is that it can be used to prove existence of a weak solution up to a certain time based on numerical results.