A five-field augmented fully-mixed finite element method for the Navier-Stokes/Darcy coupled problem

Abstract In this work we introduce and analyze a new augmented fully-mixed formulation for the stationary Navier–Stokes/Darcy coupled problem. Our approach employs, on the free-fluid region, a technique previously applied to the stationary Navier–Stokes equations, which consists of the introduction of a modified pseudostress tensor involving the diffusive and convective terms, together with the pressure. In addition, by using the incompressibility condition, the pressure is eliminated, and since the convective term forces the free-fluid velocity to live in a smaller space than usual, we augment the resulting formulation with suitable Galerkin type terms arising from the constitutive and equilibrium equations. On the other hand, in the Darcy region we apply the usual dual-mixed formulation, which yields the introduction of the trace of the porous media pressure as an associated Lagrange multiplier. The latter is connected with the fact that one of the transmission conditions involving mass conservation becomes essential and must be imposed weakly. In this way, we obtain a five-field formulation where the pseudostress and the velocity in the fluid, together with the velocity and the pressure in the porous medium, and the aforementioned Lagrange multiplier, are the corresponding unknowns. The well-posedness analysis is carried out by combining the classical Babuska–Brezzi theory and the Banach fixed-point theorem. A proper adaptation of the arguments exploited in the continuous analysis allows us to state suitable hypotheses on the finite element subspaces ensuring that the associated Galerkin scheme is well-posed and convergent. In particular, Raviart–Thomas elements of lowest order for the pseudostress and the Darcy velocity, continuous piecewise linear polynomials for the free-fluid velocity, piecewise constants for the Darcy pressure, together with continuous piecewise linear elements for the Lagrange multiplier, constitute feasible choices. Finally, we provide several numerical results illustrating the performance of the Galerkin method and confirming the theoretical rates of convergence.