TY - JOUR
T1 - Energy-Stable Parametric Finite Element Methods for the Generalized Willmore Flow with Axisymmetric Geometry: Closed Surfaces
AU - Li , Meng
AU - Li , Yifei
JO - Communications in Computational Physics
VL - 2
SP - 575
EP - 602
PY - 2025
DA - 2025/08
SN - 38
DO - http://doi.org/10.4208/cicp.OA-2024-0268
UR - https://global-sci.org/intro/article_detail/cicp/24308.html
KW - Generalized Willmore flow, parametric finite element method, energy-stable, geometric identity.
AB - <p style="text-align: justify;">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.</p>