Mathematical and computational modeling of coupled Stokes and Darcy flows

Ivan Yotov

University of Pittsburgh

Coupled Stokes and Darcy flows occur in a number of science and engineering applications, including ground water - surface water interactions, flows through fractured porous media, flows through vuggy rocks, and flows through industrial filters. We discuss a mathematical model and several numerical models for coupling the Stokes and the Darcy equations through the Beavers-Joseph-Saffman interface conditions. We prove existence and uniqueness a weak solution. Optimal order error estimates are established for a finite element discretization based on conforming or discontinuous Stokes elements in the Stokes domain and mixed finite elements in the Darcy domain. The formulation utilizes a Lagrange multiplier to impose the interface conditions. A non-overlapping domain decomposition algorithm is developed which reduces the coupled algebraic system to an interface problem for the normal stress. Each interface iteration requires solving Stokes and Darcy subdomain problems. It is shown that the interface problem is symmetric and positive definite and that its condition number is $O(1/h)$, where $h$ is the discretization parameter. Extensions to coupling with transport are also discussed. Numerical results are presented.