full
search.png

Search

close.png

Cancel

arrow
Volume 38, Issue 2
Energy-Stable Parametric Finite Element Methods for the Generalized Willmore Flow with Axisymmetric Geometry: Closed Surfaces

Meng Li & Yifei Li

DOI: 10.4208/cicp.OA-2024-0268

Commun. Comput. Phys., 38 (2025), pp. 575-602.

Published online: 2025-08

Preview Purchase PDF 73 653

Cited by

google scholar semantic scholar
Export citation
  • Abstract

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.

  • Keywords

Generalized Willmore flow, parametric finite element method, energy-stable, geometric identity.

  • AMS Subject Headings

65M60, 65M12, 53C44, 35K55

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@Article{CiCP-38-575, author = {Li , Meng and Li , Yifei}, title = {Energy-Stable Parametric Finite Element Methods for the Generalized Willmore Flow with Axisymmetric Geometry: Closed Surfaces}, journal = {Communications in Computational Physics}, year = {2025}, volume = {38}, number = {2}, pages = {575--602}, abstract = {

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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2024-0268}, url = {http://global-sci.org/intro/article_detail/cicp/24308.html} }
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 -

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.

Li , Meng and Li , Yifei. (2025). Energy-Stable Parametric Finite Element Methods for the Generalized Willmore Flow with Axisymmetric Geometry: Closed Surfaces. Communications in Computational Physics. 38 (2). 575-602. doi:10.4208/cicp.OA-2024-0268
Copy to clipboard
BibteX RIS TXT
The citation has been copied to your clipboard