Thesis Summary:

This thesis aims at the formulation, analysis and implementation of new mixed finite element methods for a set of partial differential equations arising in the context of fluid mechanics. More precisely, we extend the study of a Banach spaces–based mixed formulation recently introduced for the Navier–Stokes problem allowing conservation of momentum, and first develop an a posteriori error analysis for the corresponding Galerkin scheme. By extending standard techniques commonly used on Hilbert spaces to the case of Banach spaces, such us local estimates, and suitable Helmholtz decompositions, we prove reliability of the estimator, whereas inverse inequalities, the localization technique based on bubble functions, among other tools, are employed to prove efficiency. Next, we present a mixed finite element method for a class of steady-state natural convection models describing the behavior of non-isothermal incompressible fluids subject to a heat source. Our approach is based on the introduction of a modified pseudostress tensor depending on the pressure, and the diffusive and convective terms of the Navier-Stokes equations for the fluid and a vector unknown involving the temperature, its gradient and the velocity. The introduction of these further unknowns lead to a mixed formulation where the aforementioned pseudostress tensor and vector unknown, together with the velocity and the temperature, are the main unknowns of the system. For both, the continuous and discrete problems, we make use of the Banach–Necas–Babuška and Banach’s fixed point theorems to prove unique solvability. Using the techniques developed for the a posteriori error analysis of the momentum conservative formulation for the Navier–Stokes problem, we complement the study of the aforementioned mixed finite element scheme for the natural convection model and derive a reliable and efficient residual-based a posteriori error estimator for the corresponding Galerkin scheme. Finally, we study a mixed formulation for the unsteady Brinkman–Forchheimer equations. Our approach is based on the introduction of the velocity gradient and the aforementioned pseudostress tensor, as further unknowns. As a consequence, we obtain a mixed formulation where the velocity together with its gradient and the pseudostress tensor, are the main unknowns of the system. We establish existence and uniqueness of a solution to the weak formulation in a Banach space setting, employing classical results on nonlinear monotone operators. We then present the well-posedness and error analysis for a semidiscrete continuous-in-time scheme and a fully discrete finite element approximation. For all the problems described above, several numerical experiments are provided illustrating the good performance of the proposed methods and confirming the theoretical results.

ISI Publications from the Thesis

Sergio CAUCAO, Ricardo OYARZúA, Segundo VILLA-FUENTES: A posteriori error analysis of a momentum and thermal energy conservative mixed-FEM for the Boussinesq equations. Calcolo, vol. 59, 4, article: 45, (2022).

Sergio CAUCAO, Ricardo OYARZúA, Segundo VILLA-FUENTES, Ivan YOTOV: A three-field Banach spaces-based mixed formulation for the unsteady Brinkman-Forchheimer equations. Computer Methods in Applied Mechanics and Engineering, vol. 394, Paper No. 114895, (2022).

Jessika CAMAñO, Sergio CAUCAO, Ricardo OYARZúA, Segundo VILLA-FUENTES: A posteriori error analysis of a momentum conservative Banach-spaces based mixed-FEM for the Navier-Stokes problem. Applied Numerical Mathematics, vol. 176, pp. 134-158, (2022).

Sergio CAUCAO, Ricardo OYARZúA, Segundo VILLA-FUENTES: A new mixed-FEM for steady-state natural convection models allowing conservation of momentum and thermal energy. Calcolo, vol. 57, 4, article:36, (2020).