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

A second-order fully-balanced structure-preserving variational discretization scheme for the Cahn-Hilliard Navier-Stokes system