On the coupling of the Stokes and the porous media equations: G-convergence and domain decomposition methods

To model the transport of substances (back and forth) between surface and ground water, we study the coupling of the Stokes and the Darcy equations. We formulate the problem as a substructuring or interface problem and we solve it by proposing a) an iterative method similar to the Dirichlet-Neumann; b) a preconditioner for elliptic problems with rapidly varying coefficients. We prove the convergence of the iterative method that we introduce with a classical Banach fixed point method argument. Regarding the numerical analysis, we use the P 1 (cross-grid) - P 0 finite elements for the Stokes problem, while in the porous region we use the classical P 1 finite elements (in other words, introducing a suitable method, we do not need to use the mixed formulation). We can use this formulation, based on classical variational principles, since by using a preconditioner based on homogenized (or effective) coefficients, we replace the problem with oscillating coefficients by another one with constant coefficients. Numerical results for some test cases are also provided.