Robust finite element methods and solvers for the Biot--Brinkman equations in vorticity form

TL;DR

This paper introduces a new finite element formulation for steady viscous flow in deformable porous media, offering enhanced robustness and accuracy, especially under challenging parameter regimes, validated through numerical experiments.

Contribution

A novel divergence-conforming finite element method using parameter-weighted spaces for coupled flow in deformable porous media, with proven robustness and optimal error estimates.

Findings

Method achieves optimal error estimates in relevant norms.

Robustness demonstrated for large Lamé parameters and small permeability.

Numerical examples confirm convergence and effectiveness of preconditioners.

Abstract

In this paper, we propose a new formulation and a suitable finite element method for the steady coupling of viscous flow in deformable porous media using divergence-conforming filtration fluxes. The proposed method is based on the use of parameter-weighted spaces, which allows for a more accurate and robust analysis of the continuous and discrete problems. Furthermore, we conduct a solvability analysis of the proposed method and derive optimal error estimates in appropriate norms. These error estimates are shown to be robust in the case of large Lam\'e parameters and small permeability and storativity coefficients. To illustrate the effectiveness of the proposed method, we provide a few representative numerical examples, including convergence verification, poroelastic channel flow simulation, and test the robustness of block-diagonal preconditioners with respect to model parameters.