A Stabilized Nonconforming Finite Element Method for the Surface Biharmonic Problem

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Numerical Analysis Pub Date : 2025-08-06 DOI:10.1137/24m1707936

Shuonan Wu, Hao Zhou

引用次数: 0

Abstract

SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1642-1665, August 2025.
Abstract. This paper presents a novel stabilized nonconforming finite element method for solving the surface biharmonic problem. The method extends the New-Zienkiewicz-type (NZT) element to polyhedral (approximated) surfaces by employing the Piola transform to establish the connection of vertex gradients across adjacent elements. Key features of the surface NZT finite element space include its [math]-relative conformity and weak [math] conformity, allowing for stabilization without the use of artificial parameters. Under the assumption that the exact solution and the dual problem possess only [math] regularity, we establish optimal error estimates in the energy norm and provide, for the first time, a comprehensive analysis yielding optimal second-order convergence in the broken [math] norm. Numerical experiments are provided to support the theoretical results and indicate that the stabilization term might be unnecessary.

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表面双调和问题的稳定非协调有限元法

SIAM数值分析杂志，第63卷，第4期，1642-1665页，2025年8月。摘要。本文提出了一种求解曲面双调和问题的稳定非协调有限元新方法。该方法将New-Zienkiewicz-type （NZT）单元扩展到多面体（近似）表面，采用Piola变换建立相邻单元间顶点梯度的连接。地面NZT有限元空间的主要特征包括其相对一致性和弱一致性，允许在不使用人工参数的情况下进行稳定。在精确解和对偶问题只具有数学正则性的假设下，我们在能量范数中建立了最优误差估计，并首次提供了在破坏的数学范数中产生最优二阶收敛的综合分析。数值实验结果支持了理论结果，并表明稳定项可能是不必要的。

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.