  We propose and analyze a structure-preserving space-time variational
discretization method for the Cahn-Hilliard-Navier-Stokes system. Uniqueness
and stability for the discrete problem is established in the presence of
concentration dependent mobility and viscosity parameters by means of the
relative energy estimates and order optimal convergence rates are established
for all variables using balanced approximation spaces and relaxed regularity
conditions on the solution. Numerical tests are presented to demonstrate the
proposed method is fully practical and yields the predicted convergence rates.
The discrete stability estimates developed in this paper may also be used to
analyse other discretization schemes, which is briefly outlined in the
discussion.
