A mixed elasticity formulation for fluid-poroelastic structure interaction

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 structure velocity and Darcy pressure as Lagrange multipliers. Existence and uniqueness of a solution are established for the continuous weak formulation. Stability and error analysis is performed for the semi-discrete continuous-in-time mixed finite element scheme. Numerical experiments are presented in confirmation of the theoretical results.

Parameter-robust methods for the Biot-Stokes interfacial coupling without Lagrange multipliers

“…We use the conditions proposed in [1] (although, other forms and dedicated phenomena could be incorporated without much effort, such as fluid entry resistance [2,3]). The recent literature contains various numerical methods for (Navier-)Stokes/Biot interface formulations including mixed, double mixed, monolithic, segregated, conforming, non-conforming, and DG discretizations [4,5,6,7,8,9,10,11,12,13,14,15].…”