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 discrete level lower bounds and energy law continuous problem. We solve these for two unknowns: organism (or cell) density, which is a positive variable, chemoattractant non-negative variable. propose algorithms, combine stabilized finite element method semi-implicit time integration. The stabilization consists nonlinear artificial diffusion employs graph-Laplacian operator shock detector localizes local extrema. As result, both algorithms turn out be can generate cell densities fulfilling bounds. However, first algorithm requires suitable constraint between space parameters, whereas second one does not. design latter attain on acute meshes. report some experiments validate theoretical results blowup nonblowup phenomena. In setting, we identify locking phenomenon relates [Formula: see text]-norm limiting growth singularity when supported macroelement.