Error analysis of a backward Euler positive preserving stabilized scheme for a Chemotaxis system

TL;DR

This paper analyzes a stabilized backward Euler finite element scheme for a chemotaxis model, proving error bounds and demonstrating the scheme's effectiveness through numerical experiments.

Contribution

It introduces a positivity-preserving, stabilized finite element scheme for the Keller-Segel chemotaxis model with rigorous error analysis.

Findings

Error bounds in L2 for cell density

Error bounds in H1 for chemical concentration

Numerical experiments confirming theoretical results

Abstract

For a Keller-Segel model for chemotaxis in two spatial dimensions we consider a modification of a positivity preserving fully discrete scheme using a local extremum diminishing flux limiter. We discretize space using piecewise linear finite elements on an quasiuniform triangulation of acute type and time by the backward Euler method. We assume that initial data are sufficiently small in order not to have a blow-up of the solution. Under appropriate assumptions on the regularity of the exact solution and the time step parameter we show existence of the fully discrete approximation and derive error bounds in $L^{2}$ for the cell density and $H^{1}$ for the chemical concentration. We also present numerical experiments to illustrate the theoretical results.