Modeling and simulation of faceting effects on surfaces are topics of growing importance in modern nanotechnology. Such effects pose various theoretical and computational challenges, since they are caused by non-convex surface energies, which lead to ill-posed evolution equations for the surfaces. In order to overcome the illposedness, regularization of the energy by a curvature-dependent term has become a standard approach, which seems to be related to the actual physics, too. The use of curvature-dependent energies yields higher order partial differential equations for surface variables, whose numerical solution is a very challenging task.
In this paper we investigate the numerical simulation of anisotropic growth with curvature-dependent energy by level set methods, which yield flexible and robust surface representations. We consider the two dominating growth modes, namely attachment-detachment kinetics and surface diffusion. The level set formulations are given in terms of metric gradient flows, which are discretized by finite element methods in space and in a semi-implicit way as local variational problems in time. Finally, the constructed level set methods are applied to the simulation of faceting of embedded surfaces and thin films.

@Article{BHSV07,
  author       = {Burger, M. and Hausser, F. and Stöcker, C. and Voigt, A.},
  title        = {A level set approach to anisotropic flows with curvature regularization},
  journal      = {J. Comp. Phys.},
  volume       = {225},
  pages        = {183-205},
  year         = {2007},
  url          = \{/2007/BHSV07},
}