A space–time discontinuous Galerkin method for the incompressible Navier–Stokes equations

We introduce a space–time discontinuous Galerkin (DG) finite element method for the incompressible Navier–Stokes equations. Our formulation can be made arbitrarily high-order accurate in both space and time and can be directly applied to deforming domains. Different stabilizing approaches are discussed which ensure stability of the method. A numerical study is performed to compare the effect of the stabilizing approaches, to show the method’s robustness on deforming domains and to investigate the behavior of the convergence rates of the solution. Recently we introduced a space–time hybridizable DG (HDG) method for incompressible flows [S. Rhebergen, B. Cockburn, A space–time hybridizable discontinuous Galerkin method for incompressible flows on deforming domains, J. Comput. Phys. 231 (2012) 4185–4204]. We will compare numerical results of the space–time DG and space–time HDG methods. This constitutes the first comparison between DG and HDG methods.