Conforming Finite Elements for $H(\text{sym}\,\text{Curl})$ and $H(\text{dev}\,\text{sym}\,\text{Curl})$

TL;DR

This paper develops conforming finite element methods for specialized matrix-valued function spaces relevant in advanced solid mechanics, ensuring conformity, unisolvence, and optimal approximation error bounds.

Contribution

It introduces the first conforming finite elements for the spaces $H( ext{sym} ext{Curl})$ and $H( ext{dev} ext{sym} ext{Curl})$, with rigorous proofs and error analysis.

Findings

Constructed conforming finite elements for $H( ext{sym} ext{Curl})$ and $H( ext{dev} ext{sym} ext{Curl})$

Proved conformity and unisolvence of the elements

Established optimal approximation error bounds

Abstract

We construct conforming finite elements for the spaces $H(\text{sym}\,\text{Curl})$ and $H(\text{dev}\,\text{sym}\,\text{Curl})$. Those are spaces of matrix-valued functions with symmetric or deviatoric-symmetric $\text{Curl}$ in a Lebesgue space, and they appear in various models of nonstandard solid mechanics. The finite elements are not $H(\text{Curl})$-conforming. We show the construction, prove conformity and unisolvence, and point out optimal approximation error bounds.