  A new time discretization scheme for the numerical simulation of two-phase
flow governed by a thermodynamically consistent diffuse interface model is
presented. The scheme is consistent in the sense that it allows for a discrete
in time energy inequality. An adaptive spatial discretization is proposed that
conserves the energy inequality in the fully discrete setting by applying a
suitable post processing step to the adaptive cycle. For the fully discrete
scheme a quasi-reliable error estimator is derived which estimates the error
both of the flow velocity, and of the phase field. The validity of the energy
inequality in the fully discrete setting is numerically investigated.
