Dual-mixed finite element methods for the stationary Boussinesq problem

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.