A stabilized hybrid mixed DGFEM naturally coupling Stokes–Darcy flows

Abstract

The coupling of free fluid and porous medium flows, governed by the Stokes and Darcy systems in their mixed velocity–pressure formulations, is addressed in two space dimensions. A stabilized mixed hybrid finite element method is proposed using discontinuous finite element spaces for the velocity and pressure approximations. Lagrange multipliers, associated with the traces of the velocity fields, are introduced to weakly impose the transmission conditions on the edges of the elements in both domains. The Beavers–Joseph–Saffman interface conditions on the interface between Stokes and Darcy domains also are naturally imposed by the same Lagrange multipliers. By condensing the velocity and pressure degrees-of-freedom at the element level the global system is assembled involving only the degrees-of-freedom associated with the multipliers. Stability of the proposed formulation is proved for ample choices of the finite element spaces including equal order of approximations for all fields on triangle and quadrilateral elements. Numerical experiments are performed illustrating flexibility, robustness and optimal rates of convergence.