Thesis Summary:

This dissertation aims to develop, to mathematically analyze and to computationally implement diverse mixed finite element methods for the numerical simulation of natural convection, or thermally driven flow problems, in the Boussinesq approximation framework; a system given by the NavierStokes and advection-diffusion equations, nonlinearly coupled via buoyancy forces and convective heat transfer. We firstly present two augmented mixed schemes based on the incorporation of parameterized redundant Galerkin terms and the introduction of a modified pseudostress tensor in the fluid equations. As for the heat equation, mixed–primal and mixed formulations are separately considered by defining the normal component of the temperature gradient as an additional unknown on the boundary, and introducing a vectorial variable defined in the domain depending on the fluid velocity, the temperature and its gradient, respectively. In both cases, equivalent fixed–point settings are derived and analyzed to state the well–posedness of the continuous problem by using the classical Banach Theorem combined with the Lax-Milgram Theorem and the Babuška-Brezzi theory, under small data constraint and suitable stabilization parameters. The solvability and convergence of the associated Galerkin schemes are also shown for arbitrary finite element subspaces and, in the mixed–primal case, assuming that those used for approximating the temperature and the boundary unknown are inf–sup compatible. A posteriori error analyses and adaptive computations in two and three dimensions are further carried out for the aforementioned augmented mixed methods. In each case, duality techniques and stable Helmholtz decompositions are the main underlying tools used in our methodology to derive a global error indicator and to show its reliability. A global efficiency property with respect to the natural norms is further proved via usual localization techniques of bubble functions and/or well-known results from previous a posteriori error analyses of related mixed schemes. We finally propose and analyze two new dual–mixed methods that exhibit the same classical structure of the Navier–Stokes equations. Here, we incorporate the velocity gradient and a Bernoulli stress tensor as auxiliary unknowns in the fluid equations, whereas both primal and mixed–primal approaches are considered for the heat equation. Without any constraint on data, we derive a priori estimates and the existence of continuous and discrete solutions for the formulations by the Leray–Schauder principle. Uniqueness is further proven provided the data is sufficiently small. We show that all the techniques described above are quasi–optimally convergent for specific choices of finite element subspaces, and allow high–order approximation not only of the main unknowns but also several physically relevant variables that can be obtained by a simple post-processing, such as the pressure, the vorticity fluid, the shear–stress tensor, and the velocity and the temperature gradients. Numerical experiments are given to confirm the theoretical findings, and to illustrate the robustness and accuracy of each method, including classic benchmark problems.

ISI Publications from the Thesis

Eligio COLMENARES, Gabriel N. GATICA, Ricardo OYARZúA: A posteriori error analysis of an augmented fully-mixed formulation for the stationary Boussinesq model. Computers & Mathematics with Applications, vol. 77, 3, pp. 693-714, (2019).

Eligio COLMENARES, Gabriel N. GATICA, Ricardo OYARZúA: A posteriori error analysis of an augmented mixed-primal formulation for the stationary Boussinesq model. Calcolo, vol. 54, 3, pp. 1055-1095, (2017).

Eligio COLMENARES, Gabriel N. GATICA, Ricardo OYARZúA: An augmented fully-mixed finite element method for the stationary Boussinesq problem. Calcolo, vol. 54, 1, pp. 167-205, (2017).

Eligio COLMENARES, Michael NEILAN: Dual-mixed finite element methods for the stationary Boussinesq problem. Computers & Mathematics with Applications, vol. 72, 7, pp. 1828-1850, (2016).

Eligio COLMENARES, Gabriel N. GATICA, Ricardo OYARZúA: Analysis of an augmented mixed-primal formulation for the stationary Boussinesq problem. Numerical Methods for Partial Differential Equations, vol. 32, 2, pp. 445-478, (2016).

Eligio COLMENARES, Gabriel N. GATICA, Ricardo OYARZúA: Fixed point strategies for mixed variational formulations of the stationary Boussinesq problem. Comptes Rendus Mathematique, vol. 354, 1, pp. 57-62, (2016).