Bound-preserving finite element approximations of the Keller-Segel equations

TL;DR

This paper develops two finite element algorithms for the Keller-Segel equations that preserve key physical bounds and energy laws at the discrete level, validated through numerical experiments on blowup phenomena.

Contribution

It introduces two stabilized finite element methods with nonlinear artificial diffusion and shock detection that ensure bound-preserving and energy law adherence for Keller-Segel models.

Findings

Both algorithms maintain positivity and non-negativity of densities.

The second algorithm achieves a discrete energy law on acute meshes.

Numerical experiments reveal a locking phenomenon limiting blowup growth.

Abstract

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 nonnegative variable. We propose two algorithms, which combine a stabilized finite element method and a semi-implicit time integration. The stabilization consists of a nonlinear artificial diffusion that employs a graph-Laplacian operator and a shock detector that localizes local extrema. As a result, both algorithms turn out to be nonlinear.Both algorithms can generate cell and chemoattractant numerical densities fulfilling lower bounds. However, the first 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. We report some numerical experiments to validate the theoretical results on blowup and non-blowup phenomena. In the blowup setting, we identify a \textit{locking} phenomenon that relates the $L^\infty(\Omega)$-norm to the $L^1(\Omega)$-norm limiting the growth of the singularity when supported on a macroelement.