Block preconditioners for finite element discretization of incompressible flow with thermal convection

We derive block preconditioners for a finite element discretization of incompressible flow coupled to heat transport by the Boussinesq approximation. Our techniques rely on effectively approximating the Schur complement obtained by eliminating the fluid variables to obtain an equation for temperature alone. Additionally, the method utilizes existing block-structured preconditioners and multilevel methods for the Navier---Stokes equations and scalar convection-diffusion. We find that the preconditioner remains robust and scalable even when the subsolves are applied quite inexactly.