From a cell model with active motion to a Hele-Shaw-like system. A numerical approach
Provides a numerical approach and convergence proof for a specific Hele-Shaw-type system, but the problem is domain-specific and the contribution is incremental.
The paper develops a numerical method for solving a Hele-Shaw-like system derived from a cell model with active motion, proving convergence for well-posed initial data with positive time derivative at the start.
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.