An exactly divergence-free finite element method for a generalized Boussinesq problem

We propose and analyse a mixed finite element method with exactly divergence-free velocities for the numerical simulation of a generalized Boussinesq problem, describing the motion of a nonisothermal incompressible fluid subject to a heat source. The method is based on using divergence-conforming elements of order k for the velocities, discontinuous elements of order k−1 for the pressure, and standard continuous elements of order k for the discretization of the temperature. The H1 conformity of the velocities is enforced by a discontinuous Galerkin approach. The resulting numerical scheme yields exactly divergence-free velocity approximations; thus, it is provably energy stable without the need to modify the underlying differential equations. We prove the existence and stability of discrete solutions, and derive optimal error estimates in the mesh size for small and smooth solutions.