Stable Backward Differentiation Formula Time Discretization of BGN-Based Parametric Finite Element Methods for Geometric Flows

IF 4.3 3区材料科学 Q1 ENGINEERING, ELECTRICAL & ELECTRONIC ACS Applied Electronic Materials Pub Date : 2024-09-06 DOI:10.1137/23m1625597

Wei Jiang, Chunmei Su, Ganghui Zhang

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Abstract

SIAM Journal on Scientific Computing, Volume 46, Issue 5, Page A2874-A2898, October 2024.
Abstract. We propose a novel class of temporal high-order parametric finite element methods for solving a wide range of geometric flows of curves and surfaces. By incorporating the backward differentiation formula (BDF) for time discretization into the BGN formulation, originally proposed by Barrett, Garcke, and Nürnberg (J. Comput. Phys., 222 (2007), pp. 441–467), we successfully develop high-order BGN/BDF[math] schemes. The proposed BGN/BDF[math] schemes not only retain almost all the advantages of the classical first-order BGN scheme such as computational efficiency and good mesh quality, but also exhibit the desired [math]th-order temporal accuracy in terms of shape metrics, ranging from second-order to fourth-order accuracy. Furthermore, we validate the performance of our proposed BGN/BDF[math] schemes through extensive numerical examples, demonstrating their high-order temporal accuracy for various types of geometric flows while maintaining good mesh quality throughout the evolution.

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基于 BGN 的几何流参数有限元方法的稳定后向微分公式时间离散化

SIAM 科学计算期刊》，第 46 卷第 5 期，第 A2874-A2898 页，2024 年 10 月。摘要我们提出了一类新的时间高阶参数有限元方法，用于求解各种曲线和曲面的几何流。通过将用于时间离散化的后向微分公式（BDF）纳入 BGN 公式，该公式最初由 Barrett、Garcke 和 Nürnberg 提出（J. Comput.物理》，222 (2007)，第 441-467 页），我们成功开发了高阶 BGN/BDF[math] 方案。所提出的 BGN/BDF[math]方案不仅保留了经典一阶 BGN 方案的几乎所有优点，如计算效率和良好的网格质量，而且在形状度量方面表现出了理想的[math]三阶时间精度，从二阶精度到四阶精度不等。此外，我们还通过大量数值示例验证了我们提出的 BGN/BDF[math] 方案的性能，证明了它们对各种类型的几何流具有高阶时间精度，同时在整个演化过程中保持了良好的网格质量。

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