This thesis considers conforming finite element discretizations for the time-dependent Oberbeck-Boussinesq model with a pressure-correction projection scheme of second order in time. Discrete inf-sup stability of the ansatz spaces for velocity and pressure is assumed. For handling poor mass conservation, a stabilization of the incompressibility constraint, the so called grad-div stabilization, is considered. Furthermore, a local projection stabilization method in streamline direction (LPS SU) is applied for velocity and temperature for dealing with dominating convection. Numerical analysis is performed both with respect to the semi-discretization in space and for the fully discretized model: Stability and convergence results are given and a suitable design of stabilization parameters is proposed. Here, grad-div stabilization proves to be essential for robustness of this approach.
These findings are validated by various numerical experiments. Analytical examples are considered to verify convergence rates in space and time. In addition, more realistic isothermal and non-isothermal flow examples are investigated and a suitable parameter choice within the bounds of the theoretical results is obtained experimentally.
Keywords: Oberbeck-Boussinesq Model; Navier-Stokes Equations; Local Projection Stabilization; Non-Isothermal Flow; Stabilized Finite Element Methods; Numerical Analysis; Pressure-Correction Projection Method
