Preprint 2025-08
Isaac Bermudez, Gabriel N. Gatica, Juan P. Silva:
A new Banach spaces-based mixed finite element method for the coupled Navier-Stokes and Darcy equations
Abstract:
In this paper we propose and analyze a new fully-mixed finite element method for the coupled model arising from the Navier-Stokes equations, with variable viscosity, in an incompressible fluid, and the Darcy equations in an adjacent porous medium, so that suitable transmission conditions are considered on the corresponding interface. The approach is based on the introduction of the further unknowns in the fluid given by the velocity gradient and the pseudostress tensor, where the latter includes the respective diffusive and convective terms. The above allows the elimination from the system of the fluid pressure, which can be calculated later on via a postprocessing formula. In addition, the traces of the fluid velocity and the Darcy pressure become the Lagrange multipliers enforcing weakly the interface conditions. In this way, the resulting variational formulation is given by a nonlinear perturbation of a threefold saddle point operator equation, where the saddle-point in the middle of them is, in turn, perturbed. A fixed-point strategy along with the generalized Babuv ska-Brezzi theory, a related abstract result for perturbed saddle-point problems, the Banach-Nev cas-Babuv ska theorem, and the Banach fixed-point theorem, are employed to prove the well-posedness of the continuous and Galerkin schemes. In particular, Raviart-Thomas and piecewise polynomial subspaces of the lowest degree for the domain unknowns, as well as continuous piecewise linear polynomials for the Lagrange multipliers on the interface, constitute a feasible choice of the finite element subspaces. Optimal error estimates and associated rates of convergence are then established. Finally, several numerical results illustrating the good performance of the method in 2D and confirming the theoretical findings, are reported.
