 
Summary: Preprint: DOI: 10.1007/s105960089121y
To appear in Computational Geosciences, 2009.
The original publication is available at www.springerlink.com
A DISCRETIZATION AND MULTIGRID SOLVER FOR A
DARCYSTOKES SYSTEM OF THREE DIMENSIONAL
VUGGY POROUS MEDIA
TODD ARBOGAST AND MARIO SAN MARTIN GOMEZ
Abstract. We develop a finite element discretization and multigrid solver for a DarcyStokes
system of three dimensional vuggy porous media, i.e., porous media with cavities. The finite ele
ment method uses low order mixed finite elements in the Darcy and Stokes domains, and special
transition elements near the DarcyStokes interface to allow for tangential discontinuities implied by
the BeaversJoseph boundary condition. We design a multigrid method to solve the resulting sad
dle point linear system. The intertwining of the Darcy and Stokes subdomains makes the resulting
matrix highly illconditioned. The velocity field is very irregular, and its discontinuous tangential
component at the DarcyStokes interface makes it difficult to define intergrid transfer operators. Our
definition is based on mass conservation and the analysis of the orders of magnitude of the solution.
The coarser grid equations are defined using the Galerkin method. A new smoother of Uzawa type is
developed based on taking an optimal step in a good search direction. Our algorithm has a measured
convergence factor independent of the size of the system, at least when there are no disconnected
vugs. We study the macroscopic effective permeability of a vuggy medium, showing that the influ
