Mesh-Preserving and Energy-Stable Parametric FEM for Geometric Flows of Surfaces

IF 2.8 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Numerical Analysis Pub Date : 2025-03-27 DOI:10.1137/24m1671542

Beiping Duan

引用次数: 0

Abstract

SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 619-640, April 2025.
Abstract. Mesh quality is crucial in the simulation of surface evolution equations using parametric finite element methods (FEMs). Energy-diminishing schemes may fail even when the surface remains smooth due to poor mesh distribution. In this paper, we aim to develop mesh-preserving and energy-stable parametric finite element schemes for the mean curvature flow and surface diffusion of two-dimensional surfaces. These new schemes are based on a reformulation of general surface evolution equations, achieved by coupling the original equation with a modified harmonic map heat flow. We demonstrate that our Euler schemes are energy-diminishing, and the proposed BDF2 schemes are energy-stable under a mild assumption on the mesh distortion. Numerical tests demonstrate that the proposed schemes perform exceptionally well in maintaining mesh quality.

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曲面几何流动的保网格和能量稳定参数有限元

SIAM数值分析杂志，第63卷，第2期，619-640页，2025年4月。摘要。在参数化有限元法模拟曲面演化方程时，网格质量至关重要。由于网格分布不好，即使表面保持光滑，能量递减方案也可能失败。在本文中，我们的目标是为二维表面的平均曲率流动和表面扩散建立保网格和能量稳定的参数有限元格式。这些新格式是基于一般表面演化方程的重新表述，通过将原始方程与修正的谐波映射热流耦合来实现。我们证明了我们的欧拉格式是能量递减的，并且所提出的BDF2格式在网格畸变的温和假设下是能量稳定的。数值试验表明，所提方案在保持网格质量方面表现优异。

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.