A mixed elasticity formulation for fluid-poroelastic structure interaction

TL;DR

This paper introduces a new mixed finite element method for simulating fluid flow interacting with deformable porous media, ensuring stability, accuracy, and robustness across various physical parameters.

Contribution

It develops a fully mixed formulation combining elasticity, Darcy flow, and Stokes equations with interface conditions, and provides theoretical analysis and numerical validation.

Findings

Proves existence and uniqueness of the solution.

Derives stability and error estimates for the method.

Numerical experiments confirm robustness and accuracy.

Abstract

We develop a mixed finite element method for the coupled problem arising in the interaction between a free fluid governed by the Stokes equations and flow in deformable porous medium modeled by the Biot system of poroelasticity. Mass conservation, balance of stress, and the Beavers--Joseph--Saffman condition are imposed on the interface. We consider a fully mixed Biot formulation based on a weakly symmetric stress-displacement-rotation elasticity system and Darcy velocity-pressure flow formulation. A velocity-pressure formulation is used for the Stokes equations. The interface conditions are incorporated through the introduction of the traces of the structure velocity and the Darcy pressure as Lagrange multipliers. Existence and uniqueness of a solution are established for the continuous weak formulation. Stability and error estimates are derived for the semi-discrete continuous-in-time mixed finite element approximation. Numerical experiments are presented to verify the theoretical results and illustrate the robustness of the method with respect to the physical parameters.