  We design consistent discontinuous Galerkin finite element schemes for the
approximation of a quasi-incompressible two phase flow model of
Allen-Cahn/Cahn-Hilliard/Navier-Stokes-Korteweg type which allows for phase
transitions. We show that the scheme is mass conservative and monotonically
energy dissipative. In this case the dissipation is isolated to discrete
equivalents of those effects already causing dissipation on the continuous
level, that is, there is no artificial numerical dissipation added into the
scheme. In this sense the methods are consistent with the energy dissipation of
the continuous PDE system.
