## Preprint 2016-07

## Eligio Colmenares, Michael Neilan:

### Dual-mixed finite element methods for the stationary Boussinesq problem

### Abstract:

We propose and analyze two mixed approaches for numerically solving the stationary Boussinesq model describing heat driven flows. For the fluid equations, the velocity gradient and a Bernoulli stress tensor are introduced as auxiliary unknowns. For the heat equation, we consider primal and mixed-primal formulations; the latter, incorporating additionally the normal component of the temperature gradient on the Dirichlet boundary. Both dual-mixed formulations exhibit the same classical structure of the Navier--Stokes equations. We derive a priori estimates and the existence of continuous and discrete solutions for the formulations. In addition, we prove the uniqueness of solutions and optimal-order error estimates provided the data is sufficiently small. Numerical experiments are given which back up the theoretical results and illustrate the robustness and accuracy of both methods for a classic benchmark problem.

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

**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).