Non conforming vector finite elements for H(curl) intersected with H(div)

TL;DR

This paper introduces a new family of nonconforming vector finite elements for 2D problems in the intersection of curl and H(div) spaces, confirming a conjecture in 2D but disproving its 3D extension.

Contribution

It presents a novel family of nonconforming finite elements for 2D curl and H(div) spaces, validating a conjecture in 2D and disproving its 3D counterpart.

Findings

Validated the conjecture for 2D nonconforming elements.

Disproved the extension of the conjecture to 3D domains.

Provided a construction of finite elements for specific function spaces.

Abstract

We present a family of nonconforming vector finite elements of arbitrary order for problems posed on the space (curl) intersected with H(div) on a bidimensional domain. This result was first stated as a conjecture by Brenner and Sung. In contrast an extension of the same conjecture to three dimensional domains is disproved.