FINITE ELEMENT EXTERIOR CALCULUS,
HOMOLOGICAL TECHNIQUES, AND APPLICATIONS
By
Douglas N. Arnold
Richard S. Falk
and
Ragnar Winther
IMA Preprint Series # 2094
(February 2006 )
INSTITUTE FOR MATHEMATICS AND ITS APPLICATIONS
UNIVERSITY OF MINNESOTA
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Preprint version, February 2006 to appear in Acta Numerica (2006)
Finite element exterior calculus,
homological techniques, and applications
Douglas N. Arnold
Institute for Mathematics and its Applications
and School of Mathematics,
University of Minnesota, Minneapolis, MN 55455, USA
E-mail: arnold@ima.umn.edu
Richard S. Falk
Department of Mathematics,
Rutgers University, Piscataway, NJ 08854, USA
E-mail: falk@math.rutgers.edu
Ragnar Winther
Centre of Mathematics for Applications
and Department of Informatics,
University of Oslo, 0316 Oslo, Norway
E-mail: ragnar.winther@cma.uio.no
Dedicated to Carme, Rena, and Rita.
Finite element exterior calculus is an approach to the design and understand-
ing of finite element discretizations for a wide variety of systems of partial
differential equations. This approach brings to bear tools from differential
geometry, algebraic topology, and homological algebra to develop discretiza-
tions which are compatible with the geometric, topological, and algebraic
structures which underlie well-posedness of the PDE problem being solved.
In the finite element exterior calculus, many finite element spaces are re-
vealed as spaces of piecewise polynomial differential forms. These connect
to each other in discrete subcomplexes of elliptic differential complexes, and
are also related to the continuous elliptic complex through projections which
commute with the complex differential. Applications are made to the finite
element discretization of a variety of problems, including the Hodge Lapla-
cian, Maxwell’s equations, the equations of elasticity, and elliptic eigenvalue
problems, and also to preconditioners.
2 D. N. Arnold, R. S. Falk and R. Winther
CONTENTS
1 Introduction 2
Part 1: Exterior calculus, finite elements, and homology
2 Exterior algebra and exterior calculus 8
3 Polynomial differential forms and the Koszul complex 27
4 Polynomial differential forms on a simplex 41
5 Finite element differential forms and their cohomology 58
6 Differential forms with values in a vector space 74
Part 2: Applications to discretization of differential equations
7 The Hodge Laplacian 78
8 Eigenvalue problems 94
9TheHΛ projection and Maxwell’s equations 101
10 Preconditioning 106
11 The elasticity equations 121
References 149
1. Introduction
The finite element method is one of the greatest advances in numerical
computing of the past century. It has become an indispensable tool for sim-
ulation of a wide variety of phenomena arising in science and engineering. A
tremendous asset of finite elements is that they not only provide a method-
ology to develop numerical algorithms for simulation, but also a theoretical
framework in which to assess the accuracy of the computed solutions. In
this paper we survey and develop the finite element exterior calculus, a new
theoretical approach to the design and understanding of finite element dis-
cretizations for a wide variety of systems of partial differential equations.
This approach brings to bear tools from differential geometry, algebraic
topology, and homological algebra to develop discretizations which are com-
patible with the geometric, topological, and algebraic structures which un-
derlie well-posedness of the PDE problem being solved. Applications treated
here include the finite element discretization of the Hodge Laplacian (which
includes many problems as particular cases), Maxwell’s equations of elec-
tromagnetism, the equations of elasticity, and elliptic eigenvalue problems,
and the construction of preconditioners.
To design a finite element method to solve a problem in partial differen-
tial equations, the problem is first given a weak or variational formulation
which characterizes the solution among all elements of a given space of func-
tions on the domain of interest. A finite element method for this problem
proceeds with the construction of a finite-dimensional subspace of the given
Finite element exterior calculus 3
function space where the solution is sought, and then the specification of
a unique element of this subspace as the solution of an appropriate set of
equations on this finite-dimensional space. In finite element methods, the
subspace is constructed from a triangulation or simplicial decomposition of
the given domain, using spaces of polynomials on each simplex, pieced to-
gether by a certain assembly process. Because the finite element space so
constructed is a subspace of the space where the exact solution is sought,
one can consider the difference between the exact and finite element solution
and measure it via appropriate norms, seminorms, or functionals. Generally
speaking, error bounds can be obtained in terms of three quantities: the ap-
proximation error, which measures the error in the best approximation of
the exact solution possible within the finite element space, the consistency
error, which measures the extent to which the equations used to select the
finite element solution from the finite element space reflect the continu-
ous problem, and the stability constant, which measures the well-posedness
of the finite-dimensional problem. The approximation properties of finite
element spaces are well understood, and the consistency of finite element
methods is usually easy to control (in fact, for all the methods considered in
this paper there is no consistency error in the sense that the exact solution
will satisfy the natural extension of the finite element equations to solution
space). In marked contrast, the stability of finite element procedures can
be very subtle. For many important problems, the development of stable
finite element methods remains extremely challenging or even out of reach,
and in other cases it is difficult to assess the stability of methods of interest.
Lack of stable methods not only puts some important problems beyond the
reach of simulation, but has also led to spectacular and costly failures of
numerical methods.
It should not be surprising that stability is a subtle matter. Establishing
stability means proving the well-posedness of the discrete equations, uni-
formly in the discretization parameters. Proving the well-posedness of PDE
problems is, of course, the central problem of the theory of PDEs. While
there are PDE problems for which this is a simple matter, for many im-
portant problems it lies deep, and a great deal of mathematics, including
analysis, geometry, topology, and algebra, has been developed to estab-
lish the well-posedness of various PDE problems. So it is to be expected
that a great deal of mathematics bears as well on the stability of numer-
ical methods for PDE. An important but insufficiently appreciated point
is that approximability and consistency together with well-posedness of the
continuous problem do not imply stability. For example, one may consider
a PDE problem whose solution is characterized by a variational principle,
i.e., as the unique critical point of some functional on some function space,
and define a finite element method by seeking a critical point of the same
functional (so there is no consistency error) on a highly accurate finite el-
4 D. N. Arnold, R. S. Falk and R. Winther
ement space (based on small elements and/or high-order polynomials, so
there is arbitrary low approximability error), and yet such a method will
very often be unstable and therefore not convergent. Analogously, in the
finite difference methodology, one may start with a PDE problem stated in
strong form and replace the derivatives in the equation by consistent finite
differences, and yet obtain a finite difference method which is unstable.
As mentioned, the well-posedness of many PDE problems reflects geo-
metrical, algebraic, topological, and homological structures underlying the
problem, formalized by exterior calculus, Hodge theory, and homological
algebra. In recent years there has been a growing realization that stability
of numerical methods can be obtained by designing methods which are com-
patible with these structures in the sense that they reproduce or mimic them
(not just approximate them) on the discrete level. See, for example, Arnold
(2002), and the volume edited by Arnold, Bochev, Lehoucq, Nicolaides and
Shashkov (2006a). In the present paper, the compatibility is mostly related
to elliptic complexes which are associated with the PDEs under consider-
ation, mostly the de Rham complex and its variants and another complex
associated with the equations of linear elasticity. Our finite element spaces
will arise as the spaces in finite-dimensional subcomplexes which inherit the
cohomology and other features of the exact complexes. The inheritance will
generally be established by cochain projections: projection operators from
the infinite-dimensional spaces in the original elliptic complex which map
onto the finite element subspaces and commute with the differential opera-
tors of the complex. Thus the main theme of the paper is the development
of finite element subcomplexes of certain elliptic differential complexes and
cochain projections onto them, and their implications and applications in
numerical PDEs. We refer to this theme and the mathematical framework
we construct to carry it out, as finite element exterior calculus.
We mention some of the computational challenges which motivated the
development of finite element exterior calculus and which it has helped to
address successfully. These are challenges both of understanding the poor
behaviour of seemingly reasonable numerical methods, and of developing
effective methods. In each case, the finite element exterior calculus provides
an explanation for the difficulties experienced with naive methods, and also
points to a practical finite element solution.
• The system σ =gradu,divu = f (the Poisson equation written in
first-order form) is among the simplest and most basic PDEs. But
even in one dimension, the stability of finite element discretizations
for it are hard to predict. For example, the use of classical continuous
piecewise linear elements for both σ and u is unstable. See Arnold,
Falk and Winther (2006b) for a discussion and numerical examples.
This is one of many problems handled in a unified manner by the finite
