ON CONVERGENT SCHEMES FOR DIFFUSE INTERFACE MODELS FOR TWO-PHASE FLOW OF INCOMPRESSIBLE FLUIDS WITH GENERAL MASS DENSITIES

We are concerned with convergence results for fully discrete finite-element schemes suggested in [Grun and Klingbeil, J. Comput. Phys., in press, DOI: 10.1016/j.jcp.2013.10.028]. They were developed for the diffuse interface model in [H. Abels, H. Garcke, and G. Grun, Math. Models Methods Appl. Sci., 22 (2012), 1150013], which describes two-phase flow of immiscible, incompressible viscous fluids. We formulate general conditions on discretization spaces and projection operators which allow us to prove compactness of discrete solutions with respect to both time and space and which hence permit us to establish convergence of the scheme to a generalized solution. We identify a simple quantitative and physical criterion to decide whether this generalized solution is in fact a weak solution. In this case, our analysis provides another pathway to establish existence of weak solutions to the aforementioned model in two and in three space dimensions. Our argument is particularly based on higher regularity results for discrete solutions to convective Cahn-Hilliard equations and on discrete versions of Sobolev's embedding theorem.