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IMA Journal of Numerical Analysis Advance Access published online on February 16, 2008
IMA Journal of Numerical Analysis, doi:10.1093/imanum/drm049
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© The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

A conforming mixed finite-element method for the coupling of fluid flow with porous media flow

Gabriel N. Gatica{dagger}

CI2MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile

Salim Meddahi{ddagger}

Departamento de Matemáticas, Universidad de Oviedo, Calvo Sotelo s/n, 33007 Oviedo, Spain

Ricardo Oyarzúa§

Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile

{dagger} Corresponding author. Email: ggatica{at}ing-mat.udec.cl

{ddagger} Email: salim{at}uniovi.es

§ Email: royarzua{at}ing-mat.udec.cl

Received on 25 June 2006. Revised on 10 November 2007.


  Abstract

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 show 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.

Key Words: mixed finite elements; Stokes equation; Darcy equation; conforming method


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