Finite element spaces of double forms

TL;DR

This paper develops finite element spaces for double forms, extending existing methods and introducing new spaces for tensors with specific symmetries, crucial for solving PDEs involving complex tensor fields.

Contribution

It constructs finite element spaces for all subspaces of double forms, including new spaces for Riemann curvature tensor symmetries, enhancing PDE discretization techniques.

Findings

Recovered known finite element spaces for symmetric and trace-free matrices.

Introduced new finite element spaces for tensors with Riemann curvature symmetries.

Excluded one subspace that cannot be discretized with piecewise constants.

Abstract

The tensor product of two differential forms of degree $p$ and $q$ is a multilinear form that is alternating in its first $p$ arguments and alternating in its last $q$ arguments. These forms, which are known as double forms or $(p,q)$-forms, play a central role in certain differential complexes that arise when studying partial differential equations. We construct piecewise polynomial finite element spaces for all of the natural subspaces of the space of $(p,q)$-forms, excluding one subspace which fails to admit a piecewise constant discretization. As special cases, our construction recovers known finite element spaces for symmetric matrices with tangential-tangential continuity (the Regge finite elements), symmetric matrices with normal-normal continuity, and trace-free matrices with normal-tangential continuity. It also gives rise to new spaces, like a finite element space for tensors possessing the symmetries of the Riemann curvature tensor.