A Stabilized Finite Element Method for Morpho-Visco-Poroelastic Model

TL;DR

This paper introduces a stabilized finite element method for a complex morpho-visco-poroelastic model that combines elastic, viscous, and porous effects with microstructural growth or shrinkage, relevant in biological tissue modeling.

Contribution

It develops a new stabilized finite element approach for the model, ensuring numerical stability and avoiding spurious oscillations, with validation through computer simulations.

Findings

Stability of equilibria assessed for continuous and semi-discrete models

A numerical condition for solution monotonicity established

Stabilization effectively prevents spurious oscillations in simulations

Abstract

We propose a mathematical model that combines elastic, viscous and porous effects with growth or shrinkage due to microstructural changes. This phenomenon is important in tissue or tumor growth, as well as in dermal contraction. Although existence results of the solution to the problem are not given, the current study assesses stability of the equilibria for both the continuous and semi-discrete versions of the model. Furthermore, a numerical condition for monotonicity of the numerical solution is described, as well as a way to stabilize the numerical solution so that spurious oscillations are avoided. The derived stabilization result is confirmed by computer simulations. In order to have a more quantitative picture, the total variation has been evaluated as a function of the stabilization parameter.