  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.
