A refined mixed finite element method for the Boussinesq equations in polygonal domains

This paper deals with the mixed formulation of the Boussinesq equations in two-dimensional polygonal domains and its numerical approximation. The stea dy solution has a singular behaviour near the corner points so that we show that it belongs to appropriate weighted Sobolev spaces. Since uniform mesh es lead to a slow convergence rate, we derive appropriate refinement rules on the meshes near the corner points in order to restore the quasi-optimal rate of convergence. A numerical test is finally presented which confirms t he theoretical convergence rates.