  Grad-div stabilization is a classical remedy in conforming mixed finite
element methods for incompressible flow problems, for mitigating velocity
errors that are sometimes called poor mass conservation. Such errors arise due
to the relaxation of the divergence constraint in classical mixed methods, and
are excited whenever the spatial discretization has to deal with comparably
large and complicated pressures. In this contribution, an analogue of grad-div
stabilization for Discontinuous Galerkin methods is studied. Here, the key is
the penalization of the jumps of the normal velocities over facets of the
triangulation, which controls the measure-valued part of the distributional
divergence of the discrete velocity solution. Our contribution is twofold:
first, we characterize the limit for arbitrarily large penalization parameters,
which shows that the stabilized nonconforming Discontinuous Galerkin methods
remain robust and accurate in this limit; second, we extend these ideas to the
case of non-simplicial meshes; here, broken grad-div stabilization must be used
in addition to the normal velocity jump penalization, in order to get the
desired pressure robustness effect. The analysis is performed for the Stokes
equations, and more complex flows and Crouzeix-Raviart elements are considered
in numerical examples that also show the relevance of the theory in practical
settings.
