Sergio Caucao, Gabriel N. Gatica, Ricardo Oyarzua, Ivana Šebestová:
A fully-mixed finite element method for the Navier-Stokes/Darcy coupled problem with nonlinear viscosity
We propose and analyse an augmented mixed finite element method for the coupling of fluid flow with porous media flow. The flows are governed by a class of nonlinear Navier-Stokes and the linear Darcy equations, respectively, and the transmission conditions are given by mass conservation, balance of normal forces, and the Beavers-Joseph-Saffman law. We apply dual-mixed formulations in both domains, and the nonlinearity involved in the Navier-Stokes region is handled by setting the strain and vorticity tensors as auxiliary unknowns. In turn, since the transmission conditions become essential, they are imposed weakly, which yields the introduction of the traces of the porous media pressure and the fluid velocity as the associated Lagrange multipliers. Furthermore, since the convective term in the fluid forces the velocity to live in a smaller space than usual, we augment the variational formulation with suitable Galerkin type terms arising from the constitutive and equilibrium equations of the Navier-Stokes equations, and the relation defining the strain and vorticity tensors. The resulting augmented scheme is then written equivalently as a fixed point equation, so that the well-known Schauder and Banach theorems, combined with classical results on bijective monotone operators, are applied to prove the unique solvability of the continuous and discrete systems. In particular, given an integer $k ge 0$, piecewise polynomials of degree $le k$, Raviart-Thomas spaces of order $k$, continuous piecewise polynomials of degree $le k+1$, and piecewise polynomials of degree $le k$ are employed in the fluid for approximating the strain tensor, stress, velocity, and vorticity, respectively, whereas Raviart-Thomas spaces of order $k$ and piecewise polynomials of degree $le k$ for the velocity and pressure, together with continuous piecewise polynomials of degree $le k+1$ for the traces, constitute feasible choices in the porous medium. Finally, several numerical results illustrating the good performance of the augmented mixed finite element method and confirming the theoretical rates of convergence are reported.