Resumen de la Tesis:

This work is divided in two main parts. In the first part we consider the Navier-Stokes-Brinkman equations, which constitute one of the most common nonlinear models utilized to simulate viscous fluids through porous media, and propose and analyze a Banach spaces-based approach yielding new mixed finite element methods for its numerical solution. In addition to the velocity and pressure, the strain rate tensor, the vorticity, and the stress tensor are introduced as auxiliary unknowns, and then the incompressibility condition is used to eliminate the pressure, which is computed afterwards by a postprocessing formula depending on the stress and the velocity. The resulting continuous formulation becomes a nonlinear perturbation of, in turn, a perturbed saddle point linear system, which is then rewritten as an equivalent fixed-point equation whose operator involved maps the velocity space into itself. The well-posedness of it is then analyzed by applying the classical Banach fixed point theorem, along with a smallness assumption on the data, the Babuška--Brezzi theory in Banach spaces, and a slight variant of a recently obtained solvability result for perturbed saddle point formulations in Banach spaces as well. The resulting Galerkin scheme is momentum-conservative. Its unique solvability is analyzed, under suitable hypotheses on the finite element subspaces, using a similar fixed-point strategy as in the continuous problem. A priori error estimates are rigorously derived, including also that for the pressure. We show that PEERS and AFW elements for the stress, the velocity and the rotation, together with piecewise polynomials of a proper degree for the strain rate tensor, yield stable discrete schemes. Then, the approximation properties of these subspaces and the Céa estimate imply the respective rates of convergence. Finally, we include two and three dimensional numerical experiments that serve to corroborate the theoretical findings, and these tests illustrate the performance of the proposed mixed finite element methods. This part yielded the following work, presently submitted: G.N. Gatica, N. Núñez and R. Ruiz-Baier, New non-augmented mixed finite element methods for the Navier-Stokes-Brinkman equations using Banach spaces. Preprint 2022-14, Centro de Investigación en Ingeniería Matemática (CI²MA), Universidad de Concepción, Chile, (2022). On the other hand, in the second part we consider a steady phase change problem for non-isothermal incompressible viscous flow in porous media with an enthalpy-porosity-viscosity coupling mechanism, and introduce and analyze a Banach spaces-based variational formulation yielding a new mixed-primal finite element method for its numerical solution. The momentum and mass conservation equations are formulated in terms of velocity and the tensors of strain rate, vorticity, and stress; and the incompressibility constraint is used to eliminate the pressure, which is computed afterwards by a postprocessing formula depending on the stress and the velocity. The resulting continuous formulation for the flow becomes a nonlinear perturbation of a perturbed saddle point linear system. The energy conservation equation is written as a nonlinear primal formulation that incorporates the additional unknown of boundary heat flux. The whole mixed-primal formulation is regarded as a fixed-point operator equation, so that its well-posedness hinges on Banach's theorem, along with smallness assumptions on the data. In turn, the solvability analysis of the uncoupled problem in the fluid employs the Babuška-Brezzi theory, a recently obtained result for perturbed saddle-point problems, and the Banach-Nečas--Babuška Theorem, all them in Banach spaces, whereas the one for the uncoupled energy equation applies a nonlinear version of the Babuška-Brezzi theory in Hilbert spaces. An analogue fixed-point strategy is employed for the analysis of the associated Galerkin scheme, using in this case Brouwer's theorem and assuming suitable conditions on the respective discrete subspaces. The error analysis is conducted under appropriate assumptions, and selecting specific finite element families that fit the theory. We finally report on the verification of theoretical convergence rates with the help of numerical examples. This part yielded the following work, presently submitted: G.N. Gatica, N. Núñez and R. Ruiz-Baier, Mixed-primal methods for natural convection driven phase change with Navier-Stokes-Brinkman equations. Preprint 2022-30, Centro de Investigación en Ingeniería Matemática (CI²MA), Universidad de Concepción, Chile, (2022).