 
Summary: Published in Computer Methods in Applied Mechanics and Engineering 198 (2009) 16601672
GEOMETRIC DECOMPOSITIONS AND LOCAL BASES FOR SPACES OF FINITE
ELEMENT DIFFERENTIAL FORMS
DOUGLAS N. ARNOLD, RICHARD S. FALK, AND RAGNAR WINTHER
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 RaviartThomas, BrezziDouglasMarini, and
NŽedŽelec 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.
1. Introduction
The study of finite element exterior calculus has given increased insight into the construction of stable
and accurate finite element methods for problems appearing in various applications, ranging from electro
magnetics to elasticity. Instead of considering the design of discrete methods for each particular problem
separately, it has proved beneficial to simultaneously study approximations of a family of problems, tied
together by a common differential complex.
