Finite element approximation and very weak solution existence in a two-dimensional, degenerate Keller-Segel model
Juan Vicente Gutiérrez-Santacreu
TL;DR
The paper addresses a degenerate, cross-diffusion Keller–Segel system with local sensing in two dimensions and develops a first-order finite element method in space with implicit Euler time stepping that preserves discrete positivity, mass conservation, and a discrete maximum principle. It proves that, as the mesh size $h$ and time step $k$ go to zero, a subsequence of discrete solutions converges to a global very weak solution $(u,v)$ of the continuous problem on $\Omega_T$, with $z=\Phi(v)u$ identified in the limit. The main contributions are the rigorous compactness-based convergence analysis for a degenerate diffusion and the construction of a robust numerical framework that yields very weak solvability for 2D Keller–Segel systems on polygonal domains.
Abstract
This paper is devoted to the design and analysis of a numerical algorithm for approximating solutions of a degenerate cross-diffusion system, which models particular instances of taxis-type migration processes under local sensing mechanisms. The degeneracy leads to solutions that are very weak due to the low regularity themselves. Specifically, the solutions satisfy pointwise bounds (such as positivity and the maximum principle), integrability (such as mass conservation), and dual a priori estimates. The proposed numerical scheme combines a finite element spatial discretization with Euler time stepping. The discrete solutions preserve the above-mentioned properties at the discrete level, enabling the derivation of compactness arguments and the convergence (up to a subsequence) of the numerical solutions to a very weak solution of the continuous problem on two-dimensional polygonal domains.