On higher-order mixed FEM for low Mach number flows: application to a natural convection benchmark problem

We consider higher-order mixed finite elements with continuous pressures for the computation of stationary compressible flows at low Mach number. The proposed approach is based on a fully coupled treatment of the governing equations and therefore, for steady-state calculations, does not rely on time-stepping techniques. The non-linear problem is solved by means of a quasiNewton iteration. The strongly coupled system resulting from higher-order discretization of the linearized equations requires adequate solvers. We propose a new scheme based on multigrid methods with varying FEM ansatz orders on the grid hierarchy as well as multiplicative smoothers based on blocking techniques. Computational results are described for a benchmark configuration including a flow with heat transfer in the low Mach number regime. Furthermore, the issue of anisotropic grids is addressed in that context.