The stationary Boussinesq problem under singular forcing

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 $H^{-1}(\varpi ,\mathrm{\Omega })$, 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.