A construction of canonical nonconforming finite element spaces for elliptic equations of any order in any dimension

Jia Li, Shuonan Wu
{"title":"A construction of canonical nonconforming finite element spaces for elliptic equations of any order in any dimension","authors":"Jia Li, Shuonan Wu","doi":"arxiv-2409.06134","DOIUrl":null,"url":null,"abstract":"A unified construction of canonical $H^m$-nonconforming finite elements is\ndeveloped for $n$-dimensional simplices for any $m, n \\geq 1$. Consistency with\nthe Morley-Wang-Xu elements [Math. Comp. 82 (2013), pp. 25-43] is maintained\nwhen $m \\leq n$. In the general case, the degrees of freedom and the shape\nfunction space exhibit well-matched multi-layer structures that ensure their\nalignment. Building on the concept of the nonconforming bubble function, the\nunisolvence is established using an equivalent integral-type representation of\nthe shape function space and by applying induction on $m$. The corresponding\nnonconforming finite element method applies to $2m$-th order elliptic problems,\nwith numerical results for $m=3$ and $m=4$ in 2D supporting the theoretical\nanalysis.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"11 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Numerical Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.06134","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0

Abstract

A unified construction of canonical $H^m$-nonconforming finite elements is developed for $n$-dimensional simplices for any $m, n \geq 1$. Consistency with the Morley-Wang-Xu elements [Math. Comp. 82 (2013), pp. 25-43] is maintained when $m \leq n$. In the general case, the degrees of freedom and the shape function space exhibit well-matched multi-layer structures that ensure their alignment. Building on the concept of the nonconforming bubble function, the unisolvence is established using an equivalent integral-type representation of the shape function space and by applying induction on $m$. The corresponding nonconforming finite element method applies to $2m$-th order elliptic problems, with numerical results for $m=3$ and $m=4$ in 2D supporting the theoretical analysis.
构建任意维度任意阶椭圆方程的典型非符合有限元空间
针对任意$m, n \geq 1$的$n$维单纯形,建立了一个统一的典型$H^m$-不符合有限元的构造。当 $m \leq n$ 时,与 Morley-Wang-Xu 元 [Math. Comp. 82 (2013), pp.在一般情况下,自由度和形状函数空间表现出良好匹配的多层结构,确保了它们的对齐。在不符合气泡函数概念的基础上,使用形状函数空间的等效积分型表示,并通过对 $m$ 的归纳,建立了不符合气泡函数。相应的不符合有限元法适用于 2m$-th阶椭圆问题,二维中$m=3$和$m=4$的数值结果支持理论分析。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
自引率
0.00%
发文量
0
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信