Pre-Publicación 2025-15
Sergio Caucao, Ricardo Oyarzúa, Segundo Villa-Fuentes:
A priori and a posteriori error analyses of a fully-mixed finite element method for the coupled Navier–Stokes/Darcy problem
Abstract:
We propose and analyze a fully mixed finite element formulation for coupling free fluid flow with porous media flow, governed respectively by the Navier–Stokes and Darcy equations. The coupling is enforced through continuity of the normal velocity (mass conservation), balance of normal forces, and the Beavers–Joseph–Saffman law. For the Navier–Stokes region, we adopt a pseudostress– velocity–vorticity formulation in a Banach space setting, where the pseudostress tensor depends on the pressure as well as on the diffusive and convective terms of the equations, and the trace of the velocity on the interface is also included as an independent unknown. For the Darcy region, we employ the standard dual-mixed formulation, with velocity, pressure, and the trace of the latter on the interface as primary unknowns. The resulting scheme can be written as a nonlinear perturbation of a two-fold saddle-point problem. Well-posedness of both the continuous and discrete formulations is established under smallness assumptions on the data, by means of a fixed-point strategy combined with the Banach–Nečas–Babuška theorem and Banach’s fixed-point theorem. These results hold for arbitrary finite element subspaces satisfying suitable stability conditions. Specific choices of finite element spaces are identified that fulfill these requirements, and we derive optimal-order a priori error estimates. In addition, we develop a reliable and efficient residual-based a posteriori error estimator for the proposed method. The proofs of reliability and efficiency rely on the global inf-sup condition, Helmholtz decompositions, inverse inequalities, and well-known properties of bubble functions. Several two-dimensional numerical experiments, with and without manufactured solutions, are presented to confirm the theoretical convergence rates and to illustrate the accuracy and flexibility of the method.
