Resumen de la Tesis:

This work consists of two main parts. In the first part we propose and analyze a mixed variational formulation for the Navier-Stokes equations with variable viscosity that depends nonlinearly on the velocity gradient. Differently from previous works in which augmented terms are added to the formulation, here we employ a technique that had been previously applied to the stationary Boussinesq problem and the Navier-Stokes equations with constant viscosity. Firstly, a modified pseudostress tensor is introduced involving the diffusive and convective terms, and the pressure. Secondly, by using the incompressibility condition, the pressure is eliminated, and the gradient of velocity is incorporated as an auxiliary unknown to handle the aforementioned nonlinearity. As a consequence, a Banach spaces-based formulation is obtained, which can be written as a perturbed twofold saddle point operator equation. We address the continuous and discrete solvability of this problem by linearizing the perturbation and employing a fixed-point approach along with a particular case of a known abstract theory. Given an integer ℓ ⩾ 0, feasible choices of finite element subspaces include discontinuous piecewise polynomials of degree ⩽ ℓ for each entry of the velocity gradient, Raviart-Thomas spaces of order ℓ for the pseudostress, and discontinuous piecewise polynomials of degree ⩽ ℓ for the velocity as well. Finally, optimal a priori error estimates are derived, and several numerical results confirming in general the theoretical rates of convergence, and illustrating the good performance of the scheme, are reported. This part yielded the following work already published: I. Bermúdez, C.I. Correa, G.N. Gatica and J.P. Silva, A perturbed twofold saddle point-based mixed finite element method for the Navier-Stokes equations with variable viscosity. Appl. Numer. Math. 201 (2024), 465–487. On the other hand, in the second part 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 ity 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 Babuška-Brezzi theory, a related abstract result for perturbed saddle-point problems, the Banach-Nečas-Babuška 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. This part yielded the following work, presently submitted: I. Bermúdez, G.N. Gatica and J.P. Silva, A new Banach spaces-based mixed finite element method for the coupled Navier-Stokes and Darcy equations. Preprint 2025-08, Centro de Investigación en Ingeniería Matemática (CI2MA), Universidad de Concepción, Chile, (2025).