Convergence of a family of Galerkin discretizations for the Stokes-Darcy coupled problem

Abstract

In this paper we analyze the well-posedness (unique solvability, stability, and Cea’s estimate) of a family of Galerkin schemes for the coupling of fluid flow with porous media flow. Flows are governed by the Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers-Joseph-Saffman law. We consider the usual primal formulation in the Stokes domain and the dual-mixed one in the Darcy region, which yields a compact perturbation of an invertible mapping as the resulting operator equation. We then apply a classical result on projection methods for Fredholm operators of index zero to show that use of any pair of stable Stokes and Darcy elements implies the well-posedness of the corresponding Stokes-Darcy Galerkin scheme. This extends previous results showing well-posedness only for BernardiRaugel and Raviart-Thomas elements. In addition, we show that under somewhat more demanding hypotheses, an alternative approach that makes no use of compacteness arguments, can also be applied. Finally, we provide several numerical results illustrating the good performance of the Galerkin method for different geometries of the problem using the MINI element and the Raviart-Thomas subspace of lowest order.