Analysis of fully-mixed finite element methods for the Stokes-Darcy coupled problem

Abstract

In this paper we analyze fully-mixed finite element methods for the coupling of fluid flow with porous media flow in 2D. 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. The fully-mixed concept employed here refers to the fact that we consider dual-mixed formulations in both media, which means that the main unknowns are given by the pseudostress and the velocity in the fluid, together with the velocity and the pressure in the porous medium. In addition, the transmission conditions become essential, which leads to the introduction of the traces of the porous media pressure and the fluid velocity as the associated Lagrange multipliers. We apply the Fredholm and Babuška-Brezzi theories to derive sufficient conditions for the unique solvability of the resulting continuous and discrete formulations. In particular, we show that the existence of uniformly bounded discrete liftings of the normal traces simplifies the derivation of the corresponding stability estimates. A feasible choice of subspaces is given by Raviart-Thomas elements of lowest order and piecewise constants for the velocities and pressures, respectively, in both domains, together with continuous piecewise linear elements for the Lagrange multipliers. Finally, several numerical results illustrating the good performance of the method with these discrete spaces, and confirming the theoretical rate of convergence, are provided.