Computational Parametric Willmore Flow with Spontaneous Curvature and Area Difference Elasticity Effects

Abstract

A new stable continuous-in-time semidiscrete parametric finite element method for Willmore flow is introduced. The approach allows for spontaneous curvature and area difference elasticity effects, which are important for many applications, in particular, in the context of membranes. The method extends ideas from Dziuk and the present authors to obtain an approximation that allows for a tangential redistribution of mesh points, which typically leads to better mesh properties. Moreover, we consider volume and surface area preserving variants of these schemes and, in particular, we obtain stable approximations of Helfrich flow. We also discuss fully discrete variants and present several numerical computations.