构建任意维度任意阶椭圆方程的典型非符合有限元空间

Jia Li, Shuonan Wu
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引用次数: 0

摘要

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