Stabilized Finite Element Methods for the Oberbeck–Boussinesq Model

We consider conforming finite element approximations for the time-dependent Oberbeck–Boussinesq model with inf-sup stable pairs for velocity and pressure and use a stabilization of the incompressibility constraint. In case of dominant convection, a local projection stabilization method in streamline direction is considered both for velocity and temperature. For the arising nonlinear semi-discrete problem, a stability and convergence analysis is given that does not rely on a mesh width restriction. Numerical experiments validate a suitable parameter choice within the bounds of the theoretical results.