Two Step Time Discretization of Willmore Flow

Based on a natural approach for the time discretization of gradient flows a new time discretization for discrete Willmore flow of polygonal curves and triangulated surfaces is proposed. The approach is variational and takes into account an approximation of the L-distance between the surface at the current time step and the unknown surface at the new time step as well as a fully implicity approximation of the Willmore functional at the new time step. To evaluate the Willmore energy on the unknown surface of the next time step, we first ask for the solution of a inner, secondary variational problem describing a time step of mean curvature motion. The time discrete velocity deduced from the solution of the latter problem is regarded as an approximation of the mean curvature vector and enters the approximation of the actual Willmore functional. To solve the resulting nested variational problem in each time step numerically relaxation theory from PDE constraint optimization are taken into account. The approach is applied to polygonal curves and triangular surfaces and is independent of the co-dimension. Various numerical examples underline the stability of the new scheme, which enables time steps of the order of the spatial grid size.