The stationary Boussinesq problem under singular forcing

Abstract

In Lipschitz two and three dimensional domains, we study the existence for the so--called Boussinesq model of thermally driven convection under singular forcing. By singular we mean that the heat source is allowed to belong to $H^{-1}(\varpi ,\Omega )$, where $\varpi$ is a weight in the Muckenhoupt class $A_2$ that is regular near the boundary. We propose a finite element scheme and, under the assumption that the domain is convex and ${\varpi }^{-1}\in A_1$, show its convergence. In the case that the thermal diffusion and viscosity are constants, we propose an a posteriori error estimator and show its reliability and local efficiency.