Analysis of a Discontinuous Finite Element Method for the Coupled Stokes and Darcy Problems

Modeling the interaction between surface and subsurface flow is a challenging environmental problem. One such example is the simulation of transport of contaminants through rivers into the aquifers. Mathematically, this complex problem can be modeled by the coupled system of Stokes and Darcy equations. The aim of this paper is to formulate and analyze a discontinuous finite element method for the coupled Stokes and Darcy problems. The physical domain is decomposed into two regions: the region filled with an incompressible fluid modeled by the Stokes equations and the porous medium region modeled by Darcy’s law. The interface conditions consist of the Beavers–Joseph–Saffman condition, the continuity of flux and the balance of forces. The matching condition of meshes at the interface can be relaxed. The unknowns, namely the fluid velocity and pressure in the fluid region, and the fluid pressure in the porous medium, are approximated by totally discontinuous polynomials of different order. The discontinuous Galerkin (DG) methods considered here, are based on the