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We present various variational approximations of Willmore flow in $\mathbb{R}^d$, $d=2,3$. As well as the classic Willmore flow, we also consider variants that are (a) volume preserving and (b) volume and area preserving. The latter evolution law is the so-called Helfrich flow. In addition, we consider motion by Gauß curvature. The presented fully discrete schemes are easy to solve as they are linear at each time level, and they have good properties with respect to the distribution of mesh points. Finally, we present numerous numerical experiments, including simulations for energies appearing in the modeling of biological cell membranes.

Parametric Approximation of Willmore Flow and Related Geometric Evolution Equations