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The purpose of this article is to study the convergence of a low order finiteelement approximation for a natural convection problem. We prove that thediscretization based on P1 polynomials for every variable (velocity, pressureand temperature) is well-posed if used with a penalty term in the divergenceequation, to compensate the loss of an inf-sup condition. With mild assumptionson the pressure regularity, we recover convergence for theNavier-Stokes-Boussinesq system, provided the penalty term is chosen inaccordance with the mesh size. We express conditions to obtain optimal order ofconvergence. We illustrate theoretical convergence results with extensiveexamples. The computational cost that can be saved by this approach is alsoassessed.

On the convergence of a low order Lagrange finite element approach for natural convection problems