Volume 54, Number 4, July-August 2020
|Page(s)||1339 - 1372|
|Published online||18 June 2020|
Chemotaxis on networks: Analysis and numerical approximation
Department of Mathematics, TU Darmstadt, Darmstadt, Germany
* Corresponding author: firstname.lastname@example.org
Accepted: 16 September 2019
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.
Mathematics Subject Classification: 35K15 / 35R02 / 65M60 / 92C17
Key words: Chemotaxis / partial differential equations on networks / global solutions / finite elements / mass lumping / upwind discretization
© EDP Sciences, SMAI 2020
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