Simulation of singularities and instabilities arising in thin film flow

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.