A mixed finite element method for nearly incompressible multiple-network poroelasticity

TL;DR

This paper introduces a new mixed finite element method for nearly incompressible multi-network poroelasticity, providing robust solutions and error estimates, with applications to biological fluid-tissue interactions.

Contribution

A novel mixed finite element formulation with an additional total pressure variable that remains stable in nearly incompressible regimes and complex network interactions.

Findings

The method is robust in incompressible limits.

Energy and error estimates are established.

Numerical experiments confirm theoretical results.

Abstract

In this paper, we present and analyze a new mixed finite element formulation of a general family of quasi-static multiple-network poroelasticity (MPET) equations. The MPET equations describe flow and deformation in an elastic porous medium that is permeated by multiple fluid networks of differing characteristics. As such, the MPET equations represent a generalization of Biot's equations, and numerical discretizations of the MPET equations face similar challenges. Here, we focus on the nearly incompressible case for which standard mixed finite element discretizations of the MPET equations perform poorly. Instead, we propose a new mixed finite element formulation based on introducing an additional total pressure variable. By presenting energy estimates for the continuous solutions and a priori error estimates for a family of compatible semi-discretizations, we show that this formulation is robust in the limits of incompressibility, vanishing storage coefficients, and vanishing transfer between networks. These theoretical results are corroborated by numerical experiments. Our primary interest in the MPET equations stems from the use of these equations in modelling interactions between biological fluids and tissues in physiological settings. So, we additionally present physiologically realistic numerical results for blood and tissue fluid flow interactions in the human brain.