Low-Order Finite Element Complex with Application to a Fourth-Order Elliptic Singular Perturbation Problem

IF 3.2 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Numerical Analysis Pub Date : 2026-06-08 DOI:10.1137/25m1772320

Xuewei Cui, Xuehai Huang

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引用次数: 0

Abstract

SIAM Journal on Numerical Analysis, Volume 64, Issue 3, Page 828-852, June 2026.
Abstract. A low-order nonconforming finite element discretization of a smooth de Rham complex starting from the [math] space in three dimensions is proposed, involving an [math]-nonconforming finite element space, a new tangentially continuous [math]-nonconforming vector-valued finite element space, the lowest-order Raviart–Thomas space, and piecewise constant functions. While nonconforming for the smooth complex, the discretization conforms to the classical de Rham complex. It is applied to develop a decoupled mixed finite element method for a fourth-order elliptic singular perturbation problem, focusing on the discretization of a generalized singularly perturbed Stokes-type equation. In contrast to Nitsche’s method, which requires additional stabilization to handle boundary layers, the nodal interpolation operator for the lowest-order Nédélec element of the second kind is introduced into the discrete bilinear forms. This modification yields a decoupled mixed method that achieves optimal convergence rates uniformly with respect to the perturbation parameter, even in the presence of strong boundary layers, without requiring any additional stabilization.

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低阶有限元复形在四阶椭圆奇异摄动问题中的应用

SIAM数值分析杂志，64卷，第3期，828-852页，2026年6月。摘要。提出了一种从[math]空间出发的光滑de Rham复合体的三维低阶非协调有限元离散方法，包括[math]-非协调有限元空间、新的切连续[math]-非协调向量值有限元空间、最低阶Raviart-Thomas空间和分段常数函数。对于光滑复形，离散化符合经典的de Rham复形。应用该方法建立了求解四阶椭圆型奇异摄动问题的解耦混合有限元方法，重点研究了广义奇异摄动stokes型方程的离散化问题。Nitsche的方法需要额外的稳定化来处理边界层，与此相反，第二类最低阶nsamdsamlec单元的节点插值算子被引入离散双线性形式。这种修正产生了一种解耦混合方法，即使在存在强边界层的情况下，也可以相对于扰动参数均匀地获得最佳收敛速率，而无需任何额外的稳定化。

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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.