A Nested Variational Time Discretization for Parametric Anisotropic Willmore Flow

  • Ricardo PerlEmail author
  • Paola Pozzi
  • Martin Rumpf


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.


Euler Lagrange Equation Curvature Vector Gradient Flow Finsler Geometry Curvature Motion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



Ricardo Perl was supported by the DFG project Ru 567/14-1 and Martin Rumpf acknowledges support by the SFB 611.


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© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Institut für Numerische SimulationRheinische Friedrich-Wilhelms-Universität BonnBonnGermany
  2. 2.Fakultät für MathematikUniversität DuisburgDuisburgGermany

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