## Preprint 2019-37

## Eligio Colmenares, Gabriel N. Gatica, Sebastian Moraga, Ricardo Ruiz-Baier:

### A fully-mixed finite element method for the steady state Oberbeck-Boussinesq system

### Abstract:

We propose a new fully-mixed formulation for the stationary Oberbeck-Boussinesq problem when viscosity depends on both temperature and concentration. Following similar ideas applied previously to the Boussinesq and Navier-Stokes equations, we incorporate the velocity gradient and the Bernoulli stress tensor as auxiliary unknowns of the fluid equations. In turn, the gradients of temperature and of concentration, in addition to a Bernoulli vector, are introduced as further variables of the heat and mass transfer equations. Consequently, a dual-mixed approach with Dirichlet data is defined in each sub-system, and the well-known Banach and Brouwer theorems are combined with the Babuska-Brezzi theory in each independent set of equations, yielding the solvability of the continuous and discrete schemes. Next, we describe specific finite element subspaces satisfying appropriate stability requirements, and derive optimal a priori error estimates. Finally, several numerical examples illustrating the performance of the fully-mixed scheme and confirming the theoretical rates of convergence are presented.

This preprint gave rise to the following definitive publication(s):

**Eligio COLMENARES, Gabriel N. GATICA, Sebastian MORAGA, Ricardo RUIZ-BAIER: ***A fully-mixed finite element method for the steady state Oberbeck-Boussinesq system*. SMAI Journal of Computational Mathematics, vol. 6, pp. 125-157, (2020).