Finite elements for symmetric and traceless tensors in three dimensions

TL;DR

This paper develops finite element sub-complexes for symmetric, traceless tensors in 3D, ensuring exactness and stability, with applications in continuum mechanics and general relativity.

Contribution

It introduces a new family of finite element sub-complexes that include symmetric, traceless tensors and proves their exactness and stability properties.

Findings

Constructed finite element sub-complexes on tetrahedral meshes.

Proved the exactness of the complexes on contractible domains.

Established inf-sup stability of the finite element spaces.

Abstract

We construct a family of finite element sub-complexes of the conformal complex on tetrahedral meshes and show their exactness on contractible domains. This complex includes vector fields and symmetric and traceless tensor fields, connected through the conformal Killing operator, the linearized Cotton-York operator, and the divergence operator, respectively. This leads to discrete versions of transverse traceless (TT) tensors, i.e., symmetric, traceless and divergence-free matrix fields, in continuum mechanics and general relativity. We also show the inf-sup stability of the $H(\operatorname{div})$-conforming finite element symmetric and traceless tensors paired with discontinuous vectors.