High order exactly divergence-free Hybrid Discontinuous Galerkin Methods for unsteady incompressible flows

For the hyperbolic transport problem a spatial discretization with an Upwind Discontinuous Galerkin method and an explicit treatment in the time integration scheme is rather natural and allows for an efficient implementation. The treatment of the Stokes part involves the solution of linear systems. In this case a discretization with Hybrid Discontinuous Galerkin methods is better suited. We consider such a discretization for the Stokes part with two important features: H(div)-conforming finite elements to guarantee exactly divergence-free velocity solutions and a projection operator which reduces the number of globally coupled unknowns. We present the method, discuss implementational aspects and demonstrate the performance on two and three dimensional benchmark problems.