A stabilized total pressure-formulation of the Biot's poroelasticity equations in frequency domain: numerical analysis and applications

TL;DR

This paper introduces a stabilized finite element method for solving Biot's poroelastic equations in the frequency domain, ensuring stability and accuracy across various permeabilities and applications like brain elastography.

Contribution

It develops a new stabilized equal order finite element scheme with pressure stabilization for the total pressure formulation of Biot's equations, extending stability analysis to the discrete case.

Findings

Method is stable and accurate for a wide permeability range.

Numerical experiments validate optimal convergence.

Application demonstrated on realistic brain geometry.

Abstract

This work focuses on the numerical solution of the dynamics of a poroelastic material in the frequency domain. We provide a detailed stability analysis based on the application of the Fredholm alternative in the continuous case, considering a total pressure formulation of the Biot's equations. In the discrete setting, we propose a stabilized equal order finite element method complemented by an additional pressure stabilization to enhance the robustness of the numerical scheme with respect to the fluid permeability. Utilizing the Fredholm alternative, we extend the well-posedness results to the discrete setting, obtaining theoretical optimal convergence for the case of linear finite elements. We present different numerical experiments to validate the proposed method. First, we consider model problems with known analytic solutions in two and three dimensions. As next, we show that the method is robust for a wide range of permeabilities, including the case of discontinuous coefficients. Lastly, we show the application for the simulation of brain elastography on a realistic brain geometry obtained from medical imaging.