Geometric decompositions and local bases for spaces of finite element differential forms

TL;DR

This paper develops geometric decompositions and explicit local bases for finite element differential form spaces on simplicial meshes, generalizing classical finite element bases across arbitrary dimensions and polynomial degrees.

Contribution

It introduces a unified framework for geometric decompositions and local bases for finite element differential forms, extending classical bases to higher dimensions and polynomial degrees.

Findings

Explicit local bases derived for finite element differential forms

Applicable to arbitrary dimensions and polynomial degrees

Framework based on extension operators for geometric decompositions

Abstract

We study the two primary families of spaces of finite element differential forms with respect to a simplicial mesh in any number of space dimensions. These spaces are generalizations of the classical finite element spaces for vector fields, frequently referred to as Raviart-Thomas, Brezzi-Douglas-Marini, and Nedelec spaces. In the present paper, we derive geometric decompositions of these spaces which lead directly to explicit local bases for them, generalizing the Bernstein basis for ordinary Lagrange finite elements. The approach applies to both families of finite element spaces, for arbitrary polynomial degree, arbitrary order of the differential forms, and an arbitrary simplicial triangulation in any number of space dimensions. A prominent role in the construction is played by the notion of a consistent family of extension operators, which expresses in an abstract framework a sufficient condition for deriving a geometric decomposition of a finite element space leading to a local basis.