 
Summary: A COMPUTATONAL METHOD FOR APPROXIMATING
A DARCYSTOKES SYSTEM
GOVERNING A VUGGY POROUS MEDIUM
TODD ARBOGAST AND DANA S. BRUNSON
Abstract. We develop and analyze a mixed finite element method for the solution of an elliptic
system modeling a porous medium with large cavities, called vugs. It consists of a second order
elliptic (i.e., Darcy) equation on part of the domain coupled to a Stokes equation on the rest of the
domain, and a slip boundary condition (due to BeaversJosephSaffman) on the interface between
them. The tangential velocity is not continuous on the interface. We consider a 2D vuggy porous
medium with many small cavities throughout its extent, so the interface is not isolated. We use a
certain conforming Stokes element on rectangles, slightly modified near the interface to account for
the tangential discontinuity. This gives a mixed finite element method for the entire DarcyStokes
system with a regular sparsity pattern that is easy to implement, independent of the vug geometry,
as long as it aligns with the grid. We prove optimal global first order L2 convergence of the velocity
and pressure, as well as the velocity gradient in the Stokes domain. Numerical results verify these
rates of convergence, and even suggest somewhat better convergence in certain situations. Finally,
we present a lower dimensional space that uses RaviartThomas elements in the Darcy domain and
uses our new modified elements near the interface in transition to the Stokes elements.
Key words. mixed finite elements, DarcyStokes system, vuggy porous media, BeaversJoseph
boundary condition, error estimates
