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

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.