From a cell model with active motion to a Hele-Shaw-like system. A numerical approach

TL;DR

This paper presents a numerical approach for solving a Hele-Shaw-like system using a cell model with active motion, ensuring convergence and positivity preservation through finite element methods.

Contribution

It introduces a finite element-based numerical scheme with positivity preservation for Hele-Shaw-like systems derived from cell models with active motion.

Findings

Proves convergence of the numerical approximations.

Establishes positivity preservation in the numerical scheme.

Provides uniform-in-time a priori estimates.

Abstract

In this paper we deal with the numerical solution of a Hele--Shaw-like system via a cell model with active motion. Convergence of approximations is established for well-posed initial data. These data are chosen in such a way the time derivate is positive at the initial time. The numerical method is constructed by means of a finite element procedure together with the use of a closed-nodal integration. This gives rise to an algorithm which preserves positivity whenever a right-angled triangulation is considered. As a result, uniform-in-time a priori estimates are proven which allows us to pass to limit towards a solution to the Hele--Shaw problem.