Coupling Darcy and Stokes Equations for Porous Media with Cracks

In order to handle the flow of a viscous incompressible fluid in a porous medium with cracks, the thickness of which cannot be neglected, we consider a model which couples the Darcy equations in the medium with the Stokes equations in the cracks by a new boundary condition at the interface, namely the continuity of the pressure. We prove that this model admits a unique solution and propose a mixed formulation of it. Relying on this formulation, we describe a finite element discretization and derive a priori and a posteriori error estimates. We present some numerical experiments that are in good agreement with the analysis. Mathematics Subject Classification. 65N30, 65N50, 76D07, 76S05. Received: December 2, 2002. Revised: March 2, 2003.