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.
Highlights
In this paper we develop a new mixed elasticity formulation for the quasi-static Stokes–Biot problem that models the interaction between a free fluid and flow in deformable porous medium
This coupled physical phenomenon is referred to as fluid–poroelastic structure interaction (FPSI)
The free fluid is modeled by the Stokes equations, while the flow in the deformable porous media is modeled by the Biot system of poroelasticity [15]
Summary
In this paper we develop a new mixed elasticity formulation for the quasi-static Stokes–Biot problem that models the interaction between a free fluid and flow in deformable porous medium. In this paper we develop a mixed finite element discretization of the quasi-static Stokes–Biot system using a mixed elasticity formulation with a weakly symmetric poroelastic stress. We consider a mixed velocity–pressure Darcy formulation, resulting in a five-field Biot formulation, which was proposed in [41] and studied further in [6], where a multipoint stress-flux mixed finite element method is developed. We present a semi-discrete continuous-in-time formulation, which is based on employing stable mixed finite element spaces for the Stokes, Darcy, and elasticity equations on grids that may be non-matching along the interface, as well as suitable choices for the Lagrange multiplier finite element spaces. We employ 0 to denote the null vector or tensor, and use C and c, with or without subscripts, bars, tildes or hats, to denote generic constants independent of the discretization parameters, which may take different values at different places
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