@INCOLLECTION{PePoRu13,
author = {Perl, Ricardo and Pozzi, Paola and Rumpf, Martin},
title = {A Nested Variational Time Discretization for Parametric Anisotropic
Willmore Flow},
booktitle = {Singular Phenomena and Scaling in Mathematical Models},
publisher = {Springer, Cham},
year = {2014},
editor = {Michael Griebel},
pages = {221--241},
abstract = {A variational time discretization of anisotropic Willmore flow combined
with a spatial discretization via piecewise affine finite elements
is presented. Here, both the energy and the metric underlying the
gradient flow are anisotropic, which in particular ensures that Wulff
shapes are invariant up to scaling under the gradient flow. In each
time step of the gradient flow a nested optimization problem has
to be solved. Thereby, an outer variational problem reflects the
time discretization of the actual Willmore flow and involves an approximate
anisotropic $L^2$-distance between two consecutive time steps and
a fully implicit approximation of the anisotropic Willmore energy.
The anisotropic mean curvature needed to evaluate the energy integrand
is replaced by the time discrete, approximate speed from an inner,
fully implicit variational scheme for anisotropic mean curvature
motion. To solve the nested optimization problem a Newton method
for the associated Lagrangian is applied. Computational results for
the evolution of curves underline the robustness of the new scheme,
in particular with respect to large time steps.},
doi = {10.1007/978-3-319-00786-1_10},
}