On high order finite element spaces of differential forms

TL;DR

This paper integrates high order finite element spaces of differential forms into the finite element exterior calculus framework, introduces new low order degrees of freedom, and proposes canonical resolutions using scalar polynomials and Whitney forms.

Contribution

It extends the finite element exterior calculus to high order differential form spaces, introduces new low order degrees of freedom, and offers canonical resolutions as alternatives to existing bases.

Findings

Unified framework for high order finite element differential forms

New low order degrees of freedom based on Bossavit's observations

Canonical resolutions using scalar polynomials and Whitney forms

Abstract

We show how the high order finite element spaces of differential forms due to Raviart-Thomas-N\'edelec-Hiptmair fit into the framework of finite element systems, in an elaboration of the finite element exterior calculus of Arnold-Falk-Winther. Based on observations by Bossavit, we provide new low order degrees of freedom. As an alternative to existing choices of bases, we provide canonical resolutions in terms of scalar polynomials and Whitney forms.