European Journal of Applied Mathematics



Simulation of singularities and instabilities arising in thin film flow


GÜNTHER GRÜN a1 and MARTIN 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

Abstract

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.

(Received September 18 2000)