A conforming mixed finite element method for the coupling of fluid flow with porous media flow. (English) Zbl 1157.76025

Gatica, Gabriel N.; Meddahi, Salim; Oyarzúa, Ricardo

IMA J. Numer. Anal. 29, No. 1, 86-108 (2009).

Summary: We consider a porous medium entirely enclosed within a fluid region and present a well-posed conforming mixed finite element method for the corresponding coupled problem. The interface conditions refer to mass conservation, balance of normal forces and the Beavers-Joseph-Saffman law, which yields the introduction of the trace of the porous medium pressure as a suitable Lagrange multiplier. The finite element subspaces defining the discrete formulation employ Bernardi-Raugel and Raviart-Thomas elements for the velocities, piecewise constants for the pressures and continuous piecewise-linear elements for the Lagrange multiplier. We prove stability, convergence and a priori error estimates for the associated Galerkin scheme. Finally, we provide several numerical results illustrating the good performance of the method and confirming the theoretical rates of convergence.

MSC:

76M10	Finite element methods applied to problems in fluid mechanics
76S05	Flows in porous media; filtration; seepage
76D07	Stokes and related (Oseen, etc.) flows

Keywords:

Stokes equation; Darcy equation; Beavers-Joseph-Saffman law; convergence; a priori error estimates