New developments of the devising of DG methods for incompressible fluid flow

We discuss two new techniques for devising discontinuous Galerkin (DG) methods for incompressible fluid flow.
Their introduction is motivated by the fact that DG methods for the incompressible Navier-Stokes equations are locally conservative and energy-stable only if the approximate velocity is exactly divergence free. How to compute exactly divergence free velocities by using low order polynomial spaces or in three-space dimension has remained an open problem from a long time.
The first technique allows us to work with completely discontinuous velocities; it is based on the use of a simple, local postprocessing to obtain a divergence-free velocity. The second technique allows us to work with exactly divergence-free velocities without having to actually construct the corresponding spaces.