## Preprint 2018-18

## 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

### Abstract:

In this paper, we propose a new mixed-primal finite element method for heat-driven flows with temperature-dependent viscosity modeled by the stationary Boussinesq equations. The motivation for this work is to overcome a drawback found by the authors in a recent work where, in order to derive the mixed formulation for the momentum equation, the reciprocal of the viscosity appears multiplied to a tensor product of velocities, making the analysis more restrictive, as it is necessary to use a continuous injection that is guaranteed only in 2D. Therefore, we show in this work that by adding the strain rate tensor as a new unknown in the problem, we get more flexibility in our reasoning and are able to consider the n-dimensional case, as the viscosity now appears multiplied by this new term only. The rest of the analysis is again based on the introduction of the pseudostress and vorticity tensors, the elimination of the pressure (which can be recovered later on via postprocessing), the incorporation of augmented Galerkin-type terms in the mixed formulation for the momentum equations, and the definition of the normal heat flux as a suitable Lagrange multiplier in the primal formulation employed for the energy equation. The resulting problem is analysed by means of the Banach and Brouwer fixed-point theorems, and several numerical examples illustrating the performance of the new scheme and confirming the theoretical rates of convergence are presented.