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

TL;DR

This paper introduces a unified approach to constructing canonical nonconforming finite element spaces for elliptic equations of any order in any dimension, ensuring consistency and unisolvence.

Contribution

It develops a general construction method for $H^m$-nonconforming finite elements applicable in any dimension and order, extending previous specific cases.

Findings

Constructed finite element spaces for any order and dimension.

Proved unisolvence using integral representations and induction.

Numerical results validate theoretical properties for 2D cases.

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.