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

Authors:       Douglas N. Arnold, Richard S. Falk, and Ragnar Winther

Status:        Computer Methods in Applied Mechanics and Engineering
               198 (2009), pp. 1660-1672.               

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.

Keywords:      finite element exterior calculus, finite element bases,
               Bernstein bases

Subj. class.:  65N30

URL:           http://www.math.rutgers.edu/~falk/papers/decomp.pdf