 
Summary: © 2007 WILEYVCH Verlag GmbH & Co. KGaA, Weinheim
PAMM · Proc. Appl. Math. Mech. 7, 10219011021902 (2007) / DOI 10.1002/pamm.200700549
Finite element differential forms
Douglas N. Arnold1,
, Richard S. Falk2
, and Ragnar Winther3
Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN, USA.
2
Department of Mathematics, Rutgers University, Piscataway, NJ, USA.
3
Centre of Mathematics for Applications, University of Oslo, Oslo, Norway.
A differential form is a field which assigns to each point of a domain an alternating multilinear form on its tangent space. The
exterior derivative operation, which maps differential forms to differential forms of the next higher order, unifies the basic first
order differential operators of calculus, and is a building block for a great variety of differential equations. When discretizing
such differential equations by finite element methods, stable discretization depends on the development of spaces of finite
element differential forms. As revealed recently through the finite element exterior calculus, for each order of differential
form, there are two natural families of finite element subspaces associated to a simplicial triangulation. In the case of forms of
order zero, which are simply functions, these two families reduce to one, which is simply the wellknown family of Lagrange
finite element subspaces of the first order Sobolev space. For forms of degree 1 and of degree n  1 (where n is the space
dimension), we obtain two natural families of finite element subspaces, unifying many of the known mixed finite element
