Error estimates and blow-up analysis of a finite-element approximation for the parabolic-elliptic Keller-Segel system

TL;DR

This paper analyzes a finite-element scheme for the Keller-Segel system, providing error estimates and demonstrating finite-time blowup of solutions under certain initial conditions.

Contribution

It establishes error bounds for a new numerical scheme and proves finite-time blowup for nonradial solutions based on initial data properties.

Findings

Error estimates for the finite-element scheme

Finite-time blowup of nonradial solutions under specific conditions

Validation of scheme's properties like mass conservation and positivity

Abstract

The Keller-Segel equations are widely used for describing chemotaxis in biology. Recently, a new fully discrete scheme for this model was proposed in [46], mass conservation, positivity and energy decay were proved for the proposed scheme, which are important properties of the original system. In this paper, we establish the error estimates of this scheme. Then, based on the error estimates, we derive the finite-time blowup of nonradial numerical solutions under some conditions on the mass and the moment of the initial data.