[1] G. Grün and M. Rumpf. Simulation of singularities and instabilities arising in thin film flow. European Journal of Applied Mathematics, 12:293-320, 2001.
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We present a finite element scheme for nonlinear fourth order diffusion equations that arise e.g. in lubrication theory for the time evolution of thin films of viscous fluids. The equations being in general fourth order degenerate parabolic, in addition singular terms of second order may occur which are to model effects of intermolecular forces or thermocapillarity. Discretizing the arising nonlinearities in a subtle way allows 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 fine accordance with physical experiments.