Abstract: We present and analyze discontinuous Galerkin finite element methods for the discretization of incompressible fluid flow problems. The main advantages of these methods in comparison with standard conforming finite element approaches lie in their robustness in transport-dominated regimes, their local conservation properties, their flexibility in the mesh-design, and their exact satisfaction of the incompressibility condition.

We first discuss discontinuous Galerkin for the incompressible Navier-Stokes equations. The underlying stability mechanisms are presented, and optimal a-priori error estimates are derived. We then develop the a-posteriori error estimation of hp-adaptive discretizations. Finally, we discuss applictions of these results to the numerical approximation of incompressible magneto-hydrodynamics problems that describe electrically conducting incompressible fluids. All our theoretical results are illustrated and verified in numerical experiments.