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For a hypersurface in ${\mathbb R}^3$, Willmore flow is defined as the$L^2$--gradient flow of the classical Willmore energy: the integral of thesquared mean curvature. This geometric evolution law is of interest indifferential geometry, image reconstruction and mathematical biology. In thispaper, we propose novel numerical approximations for the Willmore flow ofaxisymmetric hypersurfaces. For the semidiscrete continuous-in-time variants weprove a stability result. We consider both closed surfaces, and surfaces with aboundary. In the latter case, we carefully derive weak formulations of suitableboundary conditions. Furthermore, we consider many generalizations of theclassical Willmore energy, particularly those that play a role in the study ofbiomembranes. In the generalized models we include spontaneous curvature andarea difference elasticity (ADE) effects, Gaussian curvature and line energycontributions. Several numerical experiments demonstrate the efficiency androbustness of our developed numerical methods.

Stable approximations for axisymmetric Willmore flow for closed and open surfaces