A twofold saddle point approach for the coupling of fluid flow with nonlinear porous media flow

In this paper we develop the a priori and a posteriori error analyses of a mixed finite element method for the coupling of fluid flow with nonlinear porous media flow. Flows are governed by the Stokes and nonlinear 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 dual-mixed formulations in both domains, and in order to handle the nonlinearity involved, we introduce the pressure gradient in the Darcy region as an auxiliary unknown. 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. As a consequence, the resulting variational formulation can be written, conveniently, as a twofold saddle point operator equation. Thus, a well-known generalization of the classical Babuska-Brezzi theory is applied to show the well-posedness of the continuous and discrete formulations and to derive the corresponding a priori error estimate. In particular, the set of feasible finite element subspaces includes Raviart-Thomas elements of lowest order and piecewise constants for the velocities and pressure, respectively, in both domains, together with piecewise constant vectors for the Darcy pressure gradient and continuous piecewise linear elements for the traces. Then, we employ classical approaches and use known estimates to derive a reliable and efficient residual-based a posteriori error estimator for the coupled problem. Finally, several numerical results confirming the good performance of the method and the theoretical properties of the a posteriori error estimator, and illustrating the capability of the corresponding adaptive algorithm to localize the singularities of the solution, are reported.