Simulation of singularities and instabilities arising in thin film flow
GÜNTHER GRÜN a1andMARTIN RUMPF a2 a1 Institut für Angewandte Mathematik, Universität Bonn, Beringstr. 6, 53115 Bonn, Germany a2 Institut für Angewandte Mathematik, Universität Bonn, Wegelerstr. 6, 53115 Bonn, Germany
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.