author = {Gr{\"{u}}n, G{\"{u}}nther and Rumpf, Martin},
  title = {Simulation of singularities and instabilities arising in thin film
  journal = {European Journal of Applied Mathematics},
  year = {2001},
  volume = {12},
  pages = {293--320},
  abstract = {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.}