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IMA Journal of Numerical Analysis Advance Access published online on December 2, 2008
IMA Journal of Numerical Analysis, doi:10.1093/imanum/drn054
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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 new finite-element discretization of the Stokes problem coupled with the Darcy equations

Christine Bernardi{dagger} and Frédéric Hecht{ddagger}

Laboratoire Jacques-Louis Lions, CNRS & Université Pierre et Marie Curie, Boîte courrier 187, 4 place Jussieu, 75252 Paris Cedex 05, France

Fatma Zohra Nouri§

Département de Mathématiques, Faculté des Sciences, Université Badji-Mokhtar, BP 12, 23000 Annaba, Algérie

{dagger} Corresponding author. Email: bernardi{at}ann.jussieu.fr

{ddagger} Email: hecht{at}ann.jussieu.fr

§ Email: fz_nouri{at}yahoo.fr

Received on 30 October 2007. Accepted for publication 18 April 2008.


  Abstract

The flow in a rigid porous medium with a crack is usually modelled by the Darcy equations coupled with the Stokes problem. We first propose a new variational formulation of the Stokes system, where the unknowns are the vorticity, the velocity and the pressure, and describe the corresponding finite-element discretization. We extend this discretization to the case where the Darcy and the Stokes equations are coupled and prove optimal a priori and a posteriori error estimates. We conclude with some numerical experiments.

Key Words: stokes problem; Darey equations; finite elements


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