Abstract
In this paper, we study a finite element computational model for solving the interaction between a fluid and a poroelastic structure that couples the Stokes equations with the Biot system. Equilibrium and kinematic conditions are imposed on the interface. A mixed Darcy formulation is employed, resulting in continuity of flux condition of essential type. A Lagrange multiplier method is used to impose weakly this condition. With the obtained finite element solutions, the error estimators are performed for the fully discrete formulations.
Highlights
Finite element analysis of an arbitrary Lagrangian–Eulerian method for Stokes/parabolic moving interface problem with jump coefficients has been studied in [18]. e authors in [19] studied a numerical solution of the coupled system of the time-dependent Stokes and fully dynamic Biot equations. ey established stability of the scheme and derived error estimates for the fully discrete coupled scheme
In [20], Jing Wen and Yinnian He considered a strongly conservative discretization for the rearranged Stokes–Biot model based on the interior penalty discontinuous Galerkin method and mixed finite element method. e existence and uniqueness of solution of the numerical scheme have been presented. en, the analysis of stability and priori error estimates has been derived. e numerical examples under uniform meshes which well validate the analysis of convergence and the strong mass conservation are presented
A staggered finite element procedure for the coupled Stokes–Biot system with fluid entry resistance has been studied by Bergkamp et al in [21] while Ambartsumyan et al in [22] studied flow and transport in fractured poroelastic media using Stokes flow in the fractures and the Biot model in the porous media
Summary
E weak formulation is obtained by multiplying the equations in each region by suitable test functions, integrating by parts the second-order terms in space, and utilizing the interface and boundary conditions. For the discretization of the fluid velocity and pressure, we choose finite element spaces Vf,h ⊂ Vf and Wf,h ⊂ Wf, which are assumed to be inf-sup stable. Examples of such spaces include the mini-elements, the Taylor–Hood elements, and the conforming Crouzeix–Raviart elements. We introduce the discrete-in-time norms as follows:.
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