Energy-Stable Parametric Finite Element Methods for the Generalized Willmore Flow with Axisymmetric Geometry: Closed Surfaces

In this paper, we extend the work [2] to establish an energy-stable parametric finite element approximation for the generalized axisymmetric Willmore flow with closed surfaces. A crucial aspect of our approach is the introduction of two novel geometric identities involving the weighted normal velocity, $\vec{x} \cdot \vec{e_1}(\vec{x_t} \cdot \vec{v})\vec{v},$ and the curvature variable, $G_S =\varkappa_S−\overline{\varkappa},$ where $\varkappa_S$ represents the mean curvature and $\overline{\varkappa}$ denotes the spontaneous curvature. We theoretically prove that the numerical method preserves the stability of the original energy. Several numerical tests are provided to demonstrate the energy stability, as well as the accuracy and efficiency of our developed numerical scheme.