On Basis Constructions in Finite Element Exterior Calculus

TL;DR

This paper provides a systematic approach to constructing bases and degrees of freedom in finite element exterior calculus, including a new basis and detailed methods for isomorphisms and duality pairings.

Contribution

It introduces a previously overlooked basis for finite element spaces and details the construction of isomorphisms and duality pairings, enhancing implementation and theoretical understanding.

Findings

New basis for finite element spaces identified

Methods for constructing isomorphisms and duality pairings detailed

Structural insights facilitate transfer of linear dependencies

Abstract

We give a systematic and self-contained account of the construction of geometrically decomposed bases and degrees of freedom in finite element exterior calculus. In particular, we elaborate upon a previously overlooked basis for one of the families of finite element spaces, which is of interest for implementations. Moreover, we give details for the construction of isomorphisms and duality pairings between finite element spaces. These structural results show, for example, how to transfer linear dependencies between canonical spanning sets, or give a new derivation of the degrees of freedom.