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 the squared
mean curvature. This geometric evolution law is of interest in differential
geometry, image reconstruction and mathematical biology. In this paper, we
propose novel numerical approximations for the Willmore flow of axisymmetric
hypersurfaces. For the semidiscrete continuous-in-time variants we prove a
stability result.
We consider both closed surfaces, and surfaces with a boundary. In the latter
case, we carefully derive
weak formulations of suitable boundary conditions. Furthermore, we
consider many generalizations of the classical Willmore energy, particularly
those that play a role in the study of biomembranes. In the generalized
models we include
spontaneous curvature and area difference elasticity (ADE) effects,
Gaussian curvature and line energy contributions.
Several numerical experiments demonstrate the efficiency and robustness of our
developed numerical methods.