Thesis Summary:

In this Thesis we present, analyze and test a new stabilized finite element method for the coupled Darcy–Stokes problem, which models the flows of a fluid in a free medium into a porous medium. For the first medium, the Stokes equations are considered and for the second one, the Darcy equations, coupled at the interface of both domains by the mass conservation, the balance of normal forces and the Beavers–Joseph–Saffman condition. First, a mixed primal variational formulation of the model problem is proposed together with a result of existence and uniqueness of weak solution. Next, it is shown that the stabilized finite element method is well proposed, in addition a convergence analysis with optimal rates, under higher regularity assumption, for each of the unknowns studied. Moreover, we introduce and analyze a residual based a posteriori error estimator for the stabilized scheme. Finally, the quality of our scheme is verified by means of multiple numerical results in two and three dimensions.