Thesis Summary:

Thesis Summary: The goal of this thesis is to develop, analyze, and implement new mixed finite element methods for coupled and decoupled problems that arise in the context of fluid mechanics. In particular, we focus on models describing the behavior of a fluid through porous media. Firstly, an a priori error analysis of a fully-mixed finite element method based on Banach spaces for a nonlinear coupled problem arising from the interaction between the concentration and temperature of a solute immersed in a fluid moving through a porous medium is developed. The model consists of the coupling of the stationary Brinkman-Forchheimer equations with a double diffusion phenomenon. For the mathematical analysis, a nonlinear mixed formulation for the Brinkman-Forchheimer equation is proposed, where in addition to the velocity, the velocity gradient and the pseudo-stress tensor are introduced as new unknowns. In turn, a dual-mixed formulation for the double diffusion equations is adopted using temperature/concentration gradients and Bernoulli-type vectors as additional unknowns. The solvability of this formulation is established by combining fixed-point arguments, classical results on nonlinear monotone operators, Babuška-Brezzi's theory in Banach spaces, assumptions of sufficiently small data, and Banach's fixed-point theorem. In particular, Raviart-Thomas spaces of order k ≥ 0 are used to approximate the pseudo-stress tensor and Bernoulli vectors, and piecewise discontinuous polynomials of degree k for the velocity, temperature, concentration fields, and their corresponding gradients. Now, an a posteriori error and computational adaptivity analysis is performed for the fully-mixed variational formulation developed for the coupling of Brinkman-Forchheimer and double-diffusion equations. Here, a reliable and efficient residual-based a posteriori error estimator is derived. The reliability analysis of the proposed estimator is mainly based on the strong monotonicity and inf-sup conditions of the operators involved, along with an appropriate assumption on the data, a stable Helmholtz decomposition in non-standard Banach spaces, and local approximation properties of the Raviart-Thomas and Clément interpolants. In turn, the efficiency estimation is a consequence of standard arguments like inverse inequalities, bubble function-based localization technique, and other results available in the literature. Finally, a mixed finite element method for the nonlinear problem given by the stationary convective Brinkman-Forchheimer equations with variable porosity is studied. Here, the pseudostress and the gradient of the porosity times the velocity are incorporated as additional unknowns. As a consequence, a three-field mixed variational formulation based on Banach spaces is obtained, where the aforementioned variables are the main unknowns of the system along with the velocity. The resulting mixed scheme is then equivalently written as a fixed-point equation, so that Banach's well-known theorem, combined with classical results on nonlinear monotone operators and a hypothesis of sufficiently small data, is applied to demonstrate the unique solvability of the continuous and discrete systems. For all the problems described above, several numerical experiments are provided that illustrate the good performance of the proposed methods, and that confirm the theoretical results of convergence as well as the reliability and efficiency of the respective a posteriori error estimators.

ISI Publications from the Thesis

Sergio CAUCAO, Gabriel N. GATICA, Juan P. ORTEGA: A three-field mixed finite element method for the convective Brinkman-Forchheimer problem with varying porosity. Journal of Computational and Applied Mathematics, vol. 451, Paper No. 116090, (2024).

Sergio CAUCAO, Gabriel N. GATICA, Juan P. ORTEGA: A posteriori error analysis of a Banach spaces-based fully mixed FEM for double-diffusive convection in a fluid-saturated porous medium. Computational Geosciences, vol. 27, 2, pp. 289-316, (2023).

Sergio CAUCAO, Gabriel N. GATICA, Juan P. ORTEGA: A fully-mixed formulation in Banach spaces for the coupling of the steady Brinkman-Forchheimer and double-diffusion equations. ESAIM: Mathematical Modelling and Numerical Analysis, vol. 55, 6, pp. 2725-2758, (2021).