Resumen de la Tesis:

This work is focused on the analysis of a mixed finite element method for a class of natural convection problems in two dimensions. More precisely, a system based on the coupling of the steady-state equations of momentum (Navier-Stokes), mass and thermal energy conservation by means of the Boussinesq approximation (coined the Boussinesq problem) is considered, where it is also taken into account a temperature dependence of the viscosity of the uid. The construction of this finite element method begins with the introduction of the pseudostress and vorticity tensors, and a mixed formulation for the momentum equations, which is augmented with Galerkin-type terms, in order to deal with the nonlinearity of these equations and the convective term in the energy equation, where a primal formulation is considered. The prescribed temperature on the boundary becomes an essential condition, which is weakly imposed, leading to the definition of the normal heat ux through the boundary as a Lagrange multiplier. It can be seen that this highly coupled problem can be uncoupled and analysed as a fixed-point problem, where Banach and Brouwer theorems will serve to provide sufficient conditions to ensure well-posedness of the problems arising from the continuous and discrete formulations, along with several applications of continuous injections guaranteed by the Rellich-Kondrachov and Sobolev embedding theorems. Finally, some numerical results are shown to illustrate the performance of this finite element method, as well as to prove the associated rates of convergence.

Publicaciones Originadas de la Tesis (ISI)

Javier A. ALMONACID, Gabriel N. GATICA, Ricardo OYARZúA, Ricardo RUIZ-BAIER: A new mixed finite element method for the n-dimensional Boussinesq problem with temperature-dependent viscosity. Networks and Heterogeneous Media, vol. 15, 2, pp. 215-245, (2020).

Javier A. ALMONACID, Gabriel N. GATICA: A fully-mixed finite element method for the n-dimensional Boussinesq problem with temperature-dependent parameters. Computational Methods in Applied Mathematics, vol. 20, 2, pp. 187-213, (2020).

Javier A. ALMONACID, Gabriel N. GATICA, Ricardo RUIZ-BAIER: Ultra-weak symmetry of stress for augmented mixed finite element formulations in continuum mechanics. Calcolo, vol. 57, 1, article:2, (2020).

Javier A. ALMONACID, Gabriel N. GATICA, Ricardo OYARZúA: A posteriori error analysis of a mixed-primal finite element method for the Boussinesq problem with temperature-dependant viscosity. Journal of Scientific Computing, vol. 78, 2, pp. 887-917, (2019).

Javier A. ALMONACID, Gabriel N. GATICA, Ricardo OYARZúA: A mixed-primal finite element method for the Boussinesq problem with temperature-dependent viscosity. Calcolo, vol. 55, 3, article:36, (2018)

Otras Publicaciones (ISI)

Javier A. ALMONACID, Hugo S. DíAZ, Gabriel N. GATICA, Antonio MARQUEZ: A fully-mixed finite element method for the coupling of the Stokes and Darcy-Forchheimer problems. IMA Journal of Numerical Analysis, vol. 40, 2, pp. 1454-1502, (2020).