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In the first part, we study the convergence of discrete solutions to splitting schemes for two-phase flow with different mass densities suggested in [Guillen-Gonzalez, Tierra, J.Comput.Math. (6)2014]. They have been formulated for the diffuse interface model in [Abels, Garcke, Grün, M3AS, 2012, DOI:10.1142/S0218202511500138] which is consistent with thermodynamics. Our technique covers various discretization methods for phase-field energies, ranging from convex-concave splitting to difference quotient approaches for the double-well potential. In the second part of the paper, numerical experiments are presented in two space dimensions to identify discretizations of Cahn-Hilliard energies which are ϕ-stable and which do not reduce the acceleration of falling droplets. Finally, 3d simulations in axial symmetric geometries are shown to underline even more the full practicality of the approach.
We present a finite element scheme for nonlinear fourth-order diffusion equations that arise
for example in lubrication theory for the time evolution of thin films of viscous fluids. The
equations are in general fourth-order degenerate parabolic, but in addition singular terms of
second order may occur which model the effects of intermolecular forces or thermocapillarity.
Discretizing the arising nonlinearities in a subtle way allows us to establish discrete counterparts
of the essential integral estimates found in the continuous setting. As a consequence,
the algorithm is efficient, and results on convergence, nonnegativity or even strict positivity of
discrete solutions follow in a natural way. Applying this scheme to the numerical simulation
of different models shows various interesting qualitative effects, which turn out to be in good
agreement with physical experiments.
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