Mini-Symposium Talks MS4: Numerical Method for Deterministic and Stochastic Phase Field Models

In this talk, we prove convergence for fully decoupled numerical schemes for diffuse interface models for two-phase flow of immiscible, incompressible viscous fluids with different mass densities. The model under consideration is consistent with thermodynamics and it allows for a solenoidal velocity field (see Abels,Garcke,Grün, M3AS 2012). It couples a novel momentum equation for the velocity field to a Cahn-Hilliard equation for the evolution of the order parameter. A subtle discretization of the convective coupling between the flux of the phase-field and the momentum equation allows to formulate a numerical scheme which satisfies a discrete counterpart of the energy estimate. By higher regularity results for discrete solutions of convective Cahn-Hilliard equations, we prove its convergence in two and in three space dimensions. In addition, we shall present numerical simulations to underline the full practicality of our approach and to identify physical settings for which the new coupling term suggested in (Abels, Garcke, Grün, M3AS 2012) seems to be indispensable for numerical stability. Finally, we discuss various extensions of the model.