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Volume 43, Issue 1
Two Families of $n$-Rectangle Nonconforming Finite Elements for Sixth-Order Elliptic Equations

Xianlin Jin & Shuonan Wu

DOI: 10.4208/jcm.2309-m2023-0052

J. Comp. Math., 43 (2025), pp. 121-142.

Published online: 2024-11

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  • 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.

  • Keywords

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

  • AMS Subject Headings

65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-43-121, author = {Jin , Xianlin and Wu , Shuonan}, title = {Two Families of $n$-Rectangle Nonconforming Finite Elements for Sixth-Order Elliptic Equations}, journal = {Journal of Computational Mathematics}, year = {2024}, volume = {43}, number = {1}, pages = {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.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2309-m2023-0052}, url = {http://global-sci.org/intro/article_detail/jcm/23532.html} }
TY - JOUR T1 - Two Families of $n$-Rectangle Nonconforming Finite Elements for Sixth-Order Elliptic Equations AU - Jin , Xianlin AU - Wu , Shuonan JO - Journal of Computational Mathematics VL - 1 SP - 121 EP - 142 PY - 2024 DA - 2024/11 SN - 43 DO - http://doi.org/10.4208/jcm.2309-m2023-0052 UR - https://global-sci.org/intro/article_detail/jcm/23532.html KW - Nonconforming finite element method, $n$-Rectangle element, Sixth-order elliptic equation, Exchange of sub-rectangles. AB -

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.

Jin , Xianlin and Wu , Shuonan. (2024). Two Families of $n$-Rectangle Nonconforming Finite Elements for Sixth-Order Elliptic Equations. Journal of Computational Mathematics. 43 (1). 121-142. doi:10.4208/jcm.2309-m2023-0052
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