An Energy-Stable Parametric Finite Element Method for the Planar Willmore Flow

IF 2.8 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Numerical Analysis Pub Date : 2025-01-13 DOI:10.1137/24m1633893

Weizhu Bao, Yifei Li

引用次数: 0

Abstract

SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 103-121, February 2025.
Abstract. We propose an energy-stable parametric finite element method (PFEM) for the planar Willmore flow and establish its unconditional energy stability of the full discretization scheme. The key lies in the introduction of two novel geometric identities to describe the planar Willmore flow: the first involves the coupling of the outward unit normal vector [math] and the normal velocity [math], and the second concerns the time derivative of the mean curvature [math]. Based on these, we derive a set of new geometric partial differential equations for the planar Willmore flow, leading to our new fully discretized and unconditionally energy-stable PFEM. Our stability analysis is also based on the two new geometric identities. Extensive numerical experiments are provided to illustrate our PFEM’s efficiency and validate its unconditional energy stability.

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平面Willmore流的能量稳定参数有限元法

SIAM数值分析杂志，第63卷，第1期，第103-121页，2025年2月。摘要。提出了平面Willmore流的能量稳定参数有限元方法，并建立了其完全离散化方案的无条件能量稳定性。关键在于引入两个新的几何恒等式来描述平面Willmore流：第一个涉及向外单位法向量[math]和法向速度[math]的耦合，第二个涉及平均曲率的时间导数[math]。在此基础上，导出了一组新的平面Willmore流的几何偏微分方程，从而得到了一种新的完全离散的无条件能量稳定PFEM。我们的稳定性分析也是基于这两个新的几何恒等式。大量的数值实验证明了该方法的有效性，并验证了其无条件能量稳定性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

SIAM Journal on Numerical Analysis 数学-应用数学

CiteScore

4.80

自引率

6.90%

发文量

110

审稿时长

4-8 weeks

期刊介绍： SIAM Journal on Numerical Analysis (SINUM) contains research articles on the development and analysis of numerical methods. Topics include the rigorous study of convergence of algorithms, their accuracy, their stability, and their computational complexity. Also included are results in mathematical analysis that contribute to algorithm analysis, and computational results that demonstrate algorithm behavior and applicability.