Chemotaxis on networks: Analysis and numerical approximation

TL;DR

This paper extends the analysis of the Keller-Segel chemotaxis model to one-dimensional networks, establishing existence, bounds, and convergence of numerical schemes with positivity preservation and optimal accuracy.

Contribution

It introduces a variational framework for weak solutions on networks and develops a finite element discretization with positivity and convergence guarantees.

Findings

Global existence of weak solutions on networks.

Numerical schemes preserve positivity and provide uniform bounds.

Order optimal convergence rates of the numerical approximations.

Abstract

We consider the Keller-Segel model of chemotaxis on one-dimensional networks. Using a variational characterization of solutions, positivity preservation, conservation of mass, and energy estimates, we establish global existence of weak solutions and uniform bounds. This extends related results of Osaki and Yagi to the network context. We then analyze the discretization of the system by finite elements and an implicit time-stepping scheme. Mass lumping and upwinding are used to guarantee the positivity of the solutions on the discrete level. This allows us to deduce uniform bounds for the numerical approximations and to establish order optimal convergence of the discrete approximations to the continuous solution without artificial smoothness requirements. In addition, we prove convergence rates under reasonable assumptions. Some numerical tests are presented to illustrate the theoretical results.