Robust approximation of generalized Biot-Brinkman problems

TL;DR

This paper develops and analyzes a robust finite element method for solving the complex generalized Biot-Brinkman equations, which model interactions of elastic and fluid networks in porous media, ensuring stability across various parameters.

Contribution

It introduces a new three-field finite element formulation with proven robustness and preconditioning strategies for the generalized Biot-Brinkman equations.

Findings

Finite element discretization is robust across parameter regimes.

Preconditioning strategy enhances computational efficiency.

Numerical examples confirm theoretical stability.

Abstract

The generalized Biot-Brinkman equations describe the displacement, pressures and fluxes in an elastic medium permeated by multiple viscous fluid networks and can be used to study complex poromechanical interactions in geophysics, biophysics and other engineering sciences. These equations extend on the Biot and multiple-network poroelasticity equations on the one hand and Brinkman flow models on the other hand, and as such embody a range of singular perturbation problems in realistic parameter regimes. In this paper, we introduce, theoretically analyze and numerically investigate a class of three-field finite element formulations of the generalized Biot-Brinkman equations. By introducing appropriate norms, we demonstrate that the proposed finite element discretization, as well as an associated preconditioning strategy, is robust with respect to the relevant parameter regimes. The theoretical analysis is complemented by numerical examples.