Bound-preserving finite element approximations of the Keller-Segel equations
This paper aims to develop numerical approximations of the Keller{Segel equations that mimic at the discrete level the lower bounds and the energy law of the continuous problem. We solve these equations for two unknowns: the organism (or cell) density, which is a positive variable, and the chemoattr...
| Autores: | , , |
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| Tipo de documento: | artigo |
| Estado: | Versión aceptada para publicación |
| Data de publicação: | 2023 |
| País: | España |
| Recursos: | Universidad de Sevilla (US) |
| Repositório: | idUS. Depósito de Investigación de la Universidad de Sevilla |
| OAI Identifier: | oai:idus.us.es:11441/168309 |
| Acesso em linha: | https://hdl.handle.net/11441/168309 https://doi.org/10.1142/S0218202523500148 |
| Access Level: | Acceso aberto |
| Palavra-chave: | Keller-Segel equations Nonlinear parabolic equations Shock detector Lower bounds Energy law |
| Resumo: | This paper aims to develop numerical approximations of the Keller{Segel equations that mimic at the discrete level the lower bounds and the energy law of the continuous problem. We solve these equations for two unknowns: the organism (or cell) density, which is a positive variable, and the chemoattractant density, which is a non-negative variable. We propose two algorithms, which combine a stabilized nite element method and a semi-implicit time integration. The stabilization consists of a nonlinear arti cial di usion that employs a graph-Laplacian operator and a shock detector that localizes local extrema. As a result, both algorithms turn out to be nonlinear and can generate cell and chemoattractant numerical densities ful lling lower bounds. However, the rst algorithm requires a suitable constraint between the space and time discrete parameters, whereas the second one does not. We design the latter to attain a discrete energy law on acute meshes. |
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