maximize


@article{BeGrLeRu02,
  author = {Becker, J{\"{u}}rgen and Gr{\"{u}}n, G{\"{u}}nther and Lenz, Martin
	and Rumpf, Martin},
  title = {Numerical Methods for Fourth Order Nonlinear Degenerate Diffusion
	Problems},
  journal = {Applications of Mathematics},
  year = {2002},
  volume = {47},
  pages = {517--543},
  number = {6},
  abstract = {Numerical schemes are presented for a class of fourth order diffusion
	problems. These problems arise in lubrication theory for thin films
	of viscous fluids on surfaces. The equations being in general fourth
	order degenerate parabolic, additional singular terms of second order
	may occur to model effects of gravity, molecular interactions or
	thermocapillarity. Furthermore, we incorporate nonlinear surface
	tension terms. Finally, in the case of a thin film flow driven by
	a surface active agent (surfactant), the coupling of the thin film
	equation with an evolution equation for the surfactant density has
	to be considered. Discretizing the arising nonlinearities in a subtle
	way enables us to establish discrete counterparts of the essential
	integral estimates found in the continuous setting. As a consequence,
	the resulting algorithms are efficient, and results on convergence
	and nonnegativity or even strict positivity of discrete solutions
	follow in a natural way. The paper presents a finite element and
	a finite volume scheme and compares both approaches. Furthermore,
	an overview over qualitative properties of solutions is given, and
	various applications show the potential of the proposed approach.},
  doi = {10.1023/B:APOM.0000034537.55985.44},
  url = {http://dx.doi.org/10.1023/B:APOM.0000034537.55985.44}
}