A nested variational time discretization for parametric Willmore flow

Abstract

A novel variational time discretization of isotropic and anisotropic Willmore flow combined with a spatial parametric finite element discretization is applied to the evolution of polygonal curves and triangulated surfaces. In the underlying natural approach for the discretization of gradient flows a nested optimization problem has to be solved at each time step. Thereby, an outer variational problem reflects the time discretization of the actual Willmore flow and involves an approximate $L^2$-distance between two consecutive time steps and a fully implicit approximation of the Willmore energy. The mean curvature needed to evaluate the integrant of the latter energy is replaced by the time discrete, approximate speed from an inner, fully implicit variational scheme for mean curvature motion. To solve the resulting PDE constrained optimization problem at every time step duality techniques from PDE optimization are applied. Computational results underline the robustness of the new scheme, in particular with respect to large time steps, and show applications to surface restoration and blending.