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 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.

Hybridizable discontinuous Galerkin methods for the coupled Stokes--Biot problem

“…1 to 3. The results compare well to those obtained by the locking-free method of [30]; the solution does not exhibit locking or oscillations despite Poisson ratio ν = 0.4999995 (for parameter set 2) and despite modeling a very stiff poroelastic medium (parameter set 3). Furthermore, we observe from figs.…”

“…Coupling of surface/subsurface flow. In this final example we consider an example proposed in [30,Section 8.2]. For this we consider the domain Ω = (0, 2) × (−1, 1) with Ω s = (0, 2) × (0, 1) and Ω b = (0, 2) × (−1, 0) and the time interval J = (0, T ) with T = 3.…”

“…Eliminating the Darcy velocity z, they consider stable Stokes elements for (u s , p s ) and (u b , p b ), where p b is the total pressure, and a piecewise continuous and polynomial space for p p . Finally, let us mention that a stress tensor based approach using a weakly symmetric mixed method for poroelasticity [29] and a conforming stable mixed method for Stokes was studied in [1,30], and that partitioned time discretization methods, focusing on efficient time stepping and stability of partitioned schemes, are studied in [16,15,33].…”

A multipoint stress-flux mixed finite element method for the Stokes-Biot model

“…All of the above mentioned works are based on displacement formulations for the elasticity equation. In a recent work [45], the first mathematical and numerical study of a stress-displacement mixed elasticity formulation for the Stokes-Biot model is presented.…”

“…Our five-field dual mixed Biot formulation is based on the model developed in [43] and studied further in [7]. It is also considered in [45] for the Stokes-Biot problem. Our analysis also extends to the strongly symmetric mixed four-field Biot formulation developed in [57].…”