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Two Families of $n$-Rectangle Nonconforming Finite Elements for Sixth-Order Elliptic Equations

Two Families of $n$-Rectangle Nonconforming Finite Elements for Sixth-Order Elliptic Equations

Year:    2025

Author:    Xianlin Jin, Shuonan Wu

Journal of Computational Mathematics, Vol. 43 (2025), Iss. 1 : pp. 121–142

Abstract

In this paper, we propose two families of nonconforming finite elements on $n$-rectangle meshes of any dimension to solve the sixth-order elliptic equations. The unisolvent property and the approximation ability of the new finite element spaces are established. A new mechanism, called the exchange of sub-rectangles, for investigating the weak continuities of the proposed elements is discovered. With the help of some conforming relatives for the $H^3$ problems, we establish the quasi-optimal error estimate for the triharmonic equation in the broken $H^3$ norm of any dimension. The theoretical results are validated further by the numerical tests in both 2D and 3D situations.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/jcm.2309-m2023-0052

Journal of Computational Mathematics, Vol. 43 (2025), Iss. 1 : pp. 121–142

Published online:    2025-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    22

Keywords:    Nonconforming finite element method $n$-Rectangle element Sixth-order elliptic equation Exchange of sub-rectangles.

Author Details

Xianlin Jin

Shuonan Wu