Javier A. Almonacid, Hugo S. Díaz, Gabriel N. Gatica, Antonio Marquez:
A fully-mixed finite element method for the coupling of the Stokes and Darcy-Forchheimer problems
In this paper we introduce and analyze a fully-mixed formulation for the nonlinear problem given by the coupling of the Stokes and Darcy-Forchheimer equations with the Beavers-Joseph-Saffman condition on the interface. This new approach yields non-Hilbert normed spaces and a twofold saddle point structure for the corresponding operator equation, whose continuous and discrete solvabilities are analyzed by means of a suitable abstract theory developed for this purpose. In particular, feasible choices of finite element subspaces include PEERS of lowest order for the stress of the fluid, Raviart-Thomas of lowest order for the Darcy velocity, piecewise constants for the pressures, and continuous piecewise linear elements for the vorticity. A priori error estimates and associated rates of convergence are derived, and several numerical results illustrating the good performance of the method are reported.