Novel $H(\mathrm{sym} \mathrm{Curl})$-conforming finite elements for the relaxed micromorphic sequence

TL;DR

This paper introduces new $H( ext{sym} ext{Curl})$-conforming finite elements for the relaxed micromorphic sequence, enabling accurate numerical modeling of metamaterials within this advanced mathematical framework.

Contribution

It presents the first construction of $H( ext{sym} ext{Curl})$-conforming finite elements, including proofs and numerical validation, for the relaxed micromorphic sequence.

Findings

Elements respect $H( ext{Curl})$-regularity

Optimal convergence for specific fields

Application to metamaterials modeling

Abstract

In this work we construct novel $H(\mathrm{sym} \mathrm{Curl})$-conforming finite elements for the recently introduced relaxed micromorphic sequence, which can be considered as the completion of the $\mathrm{div} \mathrm{Div}$-sequence with respect to the $H(\mathrm{sym} \mathrm{Curl})$-space. The elements respect $H(\mathrm{Curl})$-regularity and their lowest order versions converge optimally for $[H(\mathrm{sym} \mathrm{Curl}) \setminus H(\mathrm{Curl})]$-fields. This work introduces a detailed construction, proofs of linear independence and conformity of the basis, and numerical examples. Further, we demonstrate an application to the computation of metamaterials with the relaxed micromorphic model.