A stable and linear time discretization for a thermodynamically consistent model for two-phase incompressible flow

A new time discretization scheme for the numerical simulation of two-phase ow governed by a thermodynamically consistent di use 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 ow velocity, and of the phase eld. The validity of the energy inequality in the fully discrete setting is numerically investigated. ∗The authors gratefully acknowledge the nancial support by the Deutsche Forschungsgemeinschaft through the priority program SPP1506 entitled Transport processes at uidic interfaces . †Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg. ‡Fachbereich Mathematik, Universität Hamburg, Bundesstraÿe 55, 20146 Hamburg. §Fachbereich Mathematik, Universität Hamburg, Bundesstraÿe 55, 20146 Hamburg.