BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY

BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY
BULLETIN (New Series) OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 47, Number 2, April 2010, Pages 281–354
S 0273-0979(10)01278-4
Article electronically published on January 25, 2010
FINITE ELEMENT EXTERIOR CALCULUS:
FROM HODGE THEORY TO NUMERICAL STABILITY
DOUGLAS N. ARNOLD, RICHARD S. FALK, AND RAGNAR WINTHER
Abstract. This article reports on the confluence of two streams of research,
one emanating from the fields of numerical analysis and scientific computation, the other from topology and geometry. In it we consider the numerical
discretization of partial differential equations that are related to differential
complexes so that de Rham cohomology and Hodge theory are key tools for
exploring the well-posedness of the continuous problem. The discretization
methods we consider are finite element methods, in which a variational or
weak formulation of the PDE problem is approximated by restricting the trial
subspace to an appropriately constructed piecewise polynomial subspace. After a brief introduction to finite element methods, we develop an abstract
Hilbert space framework for analyzing the stability and convergence of such
discretizations. In this framework, the differential complex is represented by
a complex of Hilbert spaces, and stability is obtained by transferring Hodgetheoretic structures that ensure well-posedness of the continuous problem from
the continuous level to the discrete. We show stable discretization is achieved
if the finite element spaces satisfy two hypotheses: they can be arranged into a
subcomplex of this Hilbert complex, and there exists a bounded cochain projection from that complex to the subcomplex. In the next part of the paper,
we consider the most canonical example of the abstract theory, in which the
Hilbert complex is the de Rham complex of a domain in Euclidean space. We
use the Koszul complex to construct two families of finite element differential
forms, show that these can be arranged in subcomplexes of the de Rham complex in numerous ways, and for each construct a bounded cochain projection.
The abstract theory therefore applies to give the stability and convergence
of finite element approximations of the Hodge Laplacian. Other applications
are considered as well, especially the elasticity complex and its application
to the equations of elasticity. Background material is included to make the
presentation self-contained for a variety of readers.
1. Introduction
Numerical algorithms for the solution of partial differential equations are an essential tool of the modern world. They are applied in countless ways every day in
problems as varied as the design of aircraft, prediction of climate, development of
cardiac devices, and modeling of the financial system. Science, engineering, and
Received by the editors June 23, 2009, and, in revised form, August 12, 2009.
2000 Mathematics Subject Classification. Primary: 65N30, 58A14.
Key words and phrases. Finite element exterior calculus, exterior calculus, de Rham cohomology, Hodge theory, Hodge Laplacian, mixed finite elements.
The work of the first author was supported in part by NSF grant DMS-0713568.
The work of the second author was supported in part by NSF grant DMS-0609755.
The work of the third author was supported by the Norwegian Research Council.
c
2010
American Mathematical Society
281
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D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER
technology depend not only on efficient and accurate algorithms for approximating
the solutions of a vast and diverse array of differential equations which arise in
applications, but also on mathematical analysis of the behavior of computed solutions, in order to validate the computations, determine the ranges of applicability,
compare different algorithms, and point the way to improved numerical methods.
Given a partial differential equation (PDE) problem, a numerical algorithm approximates the solution by the solution of a finite-dimensional problem which can be
implemented and solved on a computer. This discretization depends on a parameter (representing, for example, a grid spacing, mesh size, or time step) which can
be adjusted to obtain a more accurate approximate solution, at the cost of a larger
finite-dimensional problem. The mathematical analysis of the algorithm aims to
describe the relationship between the true solution and the numerical solution as
the discretization parameter is varied. For example, at its most basic, the theory
attempts to establish convergence of the discrete solution to the true solution in an
appropriate sense as the discretization parameter tends to zero.
Underlying the analysis of numerical methods for PDEs is the realization that
convergence depends on consistency and stability of the discretization. Consistency,
whose precise definition depends on the particular PDE problem and the type of
numerical method, aims to capture the idea that the operators and data defining the
discrete problem are appropriately close to those of the true problem for small values
of the discretization parameter. The essence of stability is that the discrete problem
is well-posed, uniformly with respect to the discretization parameter. Like wellposedness of the PDE problem itself, stability can be very elusive. One might think
that well-posedness of the PDE problem, which means invertibility of the operator,
together with consistency of the discretization, would imply invertibility of the
discrete operator, since invertible operators between a pair of Banach spaces form an
open set in the norm topology. But this reasoning is bogus. Consistency is not and
cannot be defined to mean norm convergence of the discrete operators to the PDE
operator, since the PDE operator, being an invertible operator between infinitedimensional spaces, is not compact and so is not the norm limit of finite-dimensional
operators. In fact, in the first part of the preceding century, a fundamental, and
initially unexpected, realization was made: that a consistent discretization of a
well-posed problem need not be stable [36, 86, 29]. Only for very special classes
of problems and algorithms does well-posedness at the continuous level transfer to
stability at the discrete level. In other situations, the development and analysis
of stable, consistent algorithms can be a challenging problem, to which a wide
array of mathematical techniques has been applied, just as for establishing the
well-posedness of PDEs.
In this paper we will consider PDEs that are related to differential complexes,
for which de Rham cohomology and Hodge theory are key tools for exploring the
well-posedness of the continuous problem. These are linear elliptic PDEs, but they
are a fundamental component of problems arising in many mathematical models,
including parabolic, hyperbolic, and nonlinear problems. The finite element exterior
calculus, which we develop here, is a theory which was developed to capture the
key structures of de Rham cohomology and Hodge theory at the discrete level and
to relate the discrete and continuous structures, in order to obtain stable finite
element discretizations.
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FINITE ELEMENT EXTERIOR CALCULUS
283
1.1. The finite element method. The finite element method, whose development as an approach to the computer solution of PDEs began over 50 years ago
and is still flourishing today, has proven to be one of the most important technologies in numerical PDEs. Finite elements not only provide a methodology to
develop numerical algorithms for many different problems, but also a mathematical
framework in which to explore their behavior. They are based on a weak or variational form of the PDE problem, and they fall into the class of Galerkin methods,
in which the trial space for the weak formulation is replaced by a finite-dimensional
subspace to obtain the discrete problem. For a finite element method, this subspace
is a space of piecewise polynomials defined by specifying three things: a simplicial
decomposition of the domain, and, for each simplex, a space of polynomials called
the shape functions, and a set of degrees of freedom for the shape functions, i.e., a
basis for their dual space, with each degree of freedom associated to a face of some
dimension of the simplex. This allows for efficient implementation of the global
piecewise polynomial subspace, with the degrees of freedom determining the degree
of interelement continuity.
For readers unfamilar with the finite element method, we introduce some basic
ideas by considering the approximation of the simplest two-point boundary value
problem
(1)
−u (x) = f (x),
−1 < x < 1,
u(−1) = u(1) = 0.
Weak solutions to this problem are sought in the Sobolev space H 1 (−1, 1) consisting
of functions in L2 (−1, 1) whose first derivatives also belong to L2 (−1, 1). Indeed,
the solution u can be characterized as the minimizer of the energy functional
1
1 1 |u (x)|2 dx −
f (x)u(x) dx
(2)
J(u) :=
2 −1
−1
over the space H̊ 1 (−1, 1) (which consists of H 1 (−1, 1) functions vanishing at ±1),
or, equivalently, as the solution of the weak problem: Find u ∈ H̊ 1 (−1, 1) such that
1
1
u (x)v (x) dx =
f (x)v(x) dx, v ∈ H̊ 1 (−1, 1).
(3)
−1
−1
It is easily seen by integrating by parts that a smooth solution of the boundary value
problem satisfies the weak formulation and that a solution of the weak formulation
which possesses appropriate smoothness will be a solution of the boundary value
problem.
Letting Vh denote a finite-dimensional subspace of H̊ 1 (−1, 1), called the trial
space, we may define an approximate solution uh ∈ Vh as the minimizer of the
functional J over the trial space (the classical Ritz method), or, equivalently by
Galerkin’s method, in which uh ∈ Vh is determined by requiring that the variation
given in the weak problem hold only for functions in Vh , i.e., by the equations
1
1
uh (x) v (x) dx =
f (x) v(x) dx, v ∈ Vh .
−1
−1
By choosing a basis for the trial space Vh , the Galerkin method reduces to a linear
system of algebraic equations for the coefficients of the expansion of uh in terms
M
of the basis functions. More specifically, if we write uh = j=1 cj φj , where the
functions φj form a basis for the trial space, then the Galerkin equations hold if
and only if Ac = b, where the coefficient matrix of the linear system is given by
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284
D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER
1
1
Aij = −1 φj φi dx and bi = −1 f φi dx. Since this is a square system of linear
equations, it is nonsingular if and only if the only solution for f = 0 is uh = 0. This
follows immediately by choosing v = uh .
The simplest finite element method is obtained by applying Galerkin’s method
with the trial space Vh consisting of all elements of H̊ 1 (−1, 1) that are linear on
each subinterval of some chosen mesh −1 = x0 < x1 < · · · < xN = 1 of the
domain (−1, 1). Figure 1.1 compares the exact solution u = cos(πx/2) and the
finite element solution uh in the case of a uniform mesh with N = 14 subintervals.
The derivatives are compared as well. For this simple problem, uh is simply the
orthogonal projection of u into Vh with respect to the inner product defined by the
left-hand side of (3), and the finite element method gives a good approximation
even with a fairly coarse mesh. Higher accuracy can easily be obtained by using a
finer mesh or piecewise polynomials of higher degree.
Figure 1.1. Approximation of −u = f by the simplest finite
element method. The left plot shows u and the right plot shows
−u , with the exact solution in blue and the finite element solution
in green.
The weak formulation (3) associated to minimization of the functional (2) is not
the only variational formulation that can be used for discretization, and in more
complicated situations other formulations may bring important advantages. In this
simple situation, we may, for example, start by writing the differential equation
−u = f as the first-order system
σ = −u ,
σ = f.
The pair (σ, u) can then be characterized variationally as the unique critical point
of the functional
1
1
1
( σ 2 − uσ ) dx +
f u dx
I(σ, u) =
−1 2
−1
over H 1 (−1, 1) × L2 (−1, 1). Equivalently, the pair is the solution of the weak
formulation: Find σ ∈ H 1 (−1, 1), u ∈ L2 (−1, 1) satisfying
1
1
στ dx −
uτ dx = 0, τ ∈ H 1 (−1, 1),
−1
−1
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FINITE ELEMENT EXTERIOR CALCULUS
1
−1
σ v dx =
285
1
f v dx,
−1
v ∈ L2 (−1, 1).
This is called a mixed formulation of the boundary value problem. Note that for the
mixed formulation, the Dirichlet boundary condition is implied by the formulation,
as can be seen by integration by parts. Note also that in this case the solution is a
saddle point of the functional I, not an extremum: I(σ, v) ≤ I(σ, u) ≤ I(τ, u) for
τ ∈ H 1 (−1, 1), v ∈ L2 (−1, 1).
Although the mixed formulation is perfectly well-posed, it may easily lead to a
discretization which is not. If we apply Galerkin’s method with a simple choice of
trial subspaces Σh ⊂ H 1 (−1, 1) and Vh ⊂ L2 (−1, 1), we obtain a finite-dimensional
linear system, which, however, may be singular, or may become increasingly unstable as the mesh is refined. This concept will be formalized and explored in the
next section, but the result of such instability is clearly visible in simple computations. For example, the choice of continuous piecewise linear functions for both
Σh and Vh leads to a singular linear system. The choice of continuous piecewise
linear functions for Σh and piecewise constants for Vh leads to stable discretization
and good accuracy. However choosing piecewise quadratics for Σh and piecewise
constants for Vh gives a nonsingular system but unstable approximation (see [25]
for further discussion of this example). The dramatic difference between the stable
and unstable methods can be seen in Figure 1.2.
Figure 1.2. Approximation of the mixed formulation for −u = f
in one dimension with two choices of elements, piecewise constants
for u and piecewise linears for σ (a stable method, shown in green),
or piecewise constants for u and piecewise quadratics for σ (unstable, shown in red). The left plot shows u and the right plot shows
σ, with the exact solution in blue. (In the right plot, the blue curve
essentially coincides with the green curve and hence is not visible.)
In one dimension, finding stable pairs of finite-dimensional subspaces for the
mixed formulation of the two-point boundary value problem is easy. For any integer
r ≥ 1, the combination of continuous piecewise polynomials of degree at most r for
σ and arbitrary piecewise polynomials of degree at most r−1 for u is stable as can be
verified via elementary means (and which can be viewed as a very simple application
of the theory presented in this paper). In higher dimensions, the problem of finding
stable combinations of elements is considerably more complicated. This is discussed
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286
D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER
in Section 2.3.1 below. In particular, we shall see that the choice of continuous
piecewise linear functions for σ and piecewise constant functions for u is not stable
in more than one dimension. However, stable element choices are known for this
problem and again may be viewed as a simple application of the finite element
exterior calculus developed in this paper.
1.2. The contents of this paper. The brief introduction to the finite element
method just given will be continued in Section 2. In particular, there we formalize
the notions of consistency and stability and establish their relation to convergence.
We shall also give several more computational examples. While seemingly simple,
some of these examples may be surprising even to specialists, and they illustrate
the difficulty in obtaining good methods and the need for a theoretical framework
in which to understand such behaviors.
Like the theory of weak solutions of PDEs, the theory of finite element methods is based on functional analysis and takes its most natural form in a Hilbert
space setting. In Section 3 of this paper, we develop an abstract Hilbert space
framework, which captures key elements of Hodge theory and which can be used
to explore the stability of finite element methods. The most basic object in this
framework is a cochain complex of Hilbert spaces, referred to as a Hilbert complex.
Function spaces of such complexes will occur in the weak formulations of the PDE
problems we consider, and the differentials will be differential operators entering
into the PDE problem. The most canonical example of a Hilbert complex is the L2
de Rham complex of a Riemannian manifold, but it is a far more general object with
other important realizations. For example, it allows for the definition of spaces of
harmonic forms and the proof that they are isomorphic to the cohomology groups.
A Hilbert complex includes enough structure to define an abstract Hodge Laplacian, defined from a variational problem with a saddle point structure. However,
for these problems to be well-posed, we need the additional property of a closed
Hilbert complex, i.e., that the range of the differentials are closed.
In this framework, the finite element spaces used to compute approximate solutions are represented by finite-dimensional subspaces of the spaces in the closed
Hilbert complex. We identify two key properties of these subspaces: first, they
should combine to form a subcomplex of the Hilbert complex, and, second, there
should exist a bounded cochain projection from the Hilbert complex to this subcomplex. Under these hypotheses and a minor consistency condition, it is easy to
show that the subcomplex inherits the cohomology of the true complex, i.e., that
the cochain projections induce an isomorphism from the space of harmonic forms to
the space of discrete harmonic forms, and to get an error estimate on the difference
between a harmonic form and its discrete counterpart. In the applications, this will
be crucial for stable approximation of the PDEs. In fact, a main theme of finite
element exterior calculus is that the same two assumptions, the subcomplex property and the existence of a bounded cochain projection, are the natural hypotheses
to establish the stability of the corresponding discrete Hodge Laplacian, defined by
the Galerkin method.
In Section 4 we look in more depth at the canonical example of the de Rham
complex for a bounded domain in Euclidean space, beginning with a brief summary
of exterior calculus. We interpret the de Rham complex as a Hilbert complex and
discuss the PDEs most closely associated with it. This brings us to the topic of
Section 5, the construction of finite element de Rham subcomplexes, which is the
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FINITE ELEMENT EXTERIOR CALCULUS
287
heart of finite element exterior calculus and the reason for its name. In this section,
we construct finite element spaces of differential forms, i.e., piecewise polynomial
spaces defined via a simplicial decomposition and specification of shape functions
and degrees of freedom, which combine to form a subcomplex of the L2 de Rham
complex admitting a bounded cochain projection. First we construct the spaces
of polynomial differential forms used for shape functions, relying heavily on the
Koszul complex and its properties, and then we construct the degrees of freedom.
We next show that the resulting finite element spaces can be efficiently implemented, have good approximation properties, and can be combined into de Rham
subcomplexes. Finally, we construct bounded cochain projections, and, having verified the hypotheses of the abstract theory, draw conclusions for the finite element
approximation of the Hodge Laplacian.
In the final two sections of the paper, we make other applications of the abstract
framework. In the last section, we study a differential complex we call the elasticity
complex, which is quite different from the de Rham complex. In particular, one of its
differentials is a partial differential operator of second order. Via the finite element
exterior calculus of the elasticity complex, we have obtained the first stable mixed
finite elements using polynomial shape functions for the equations of elasticity, with
important applications in solid mechanics.
1.3. Antecedents and related approaches. We now discuss some of the antecedents of finite element exterior calculus and some related approaches. While
the first comprehensive view of finite element exterior calculus, and the first use
of that phrase, was in the authors’ 2006 paper [8], this was certainly not the first
intersection of finite element theory and Hodge theory. In 1957, Whitney [88] published his complex of Whitney forms, which is, in our terminology, a finite element
de Rham subcomplex. Whitney’s goals were geometric. For example, he used these
forms in a proof of de Rham’s theorem identifying the cohomology of a manifold
defined via differential forms (de Rham cohomology) with that defined via a triangulation and cochains (simplicial cohomology). With the benefit of hindsight,
we may view this, at least in principle, as an early application of finite elements
to reduce the computation of a quantity of interest defined via infinite-dimensional
function spaces and operators, to a finite-dimensional computation using piecewise
polynomials on a triangulation. The computed quantities are the Betti numbers of
the manifold, i.e., the dimensions of the de Rham cohomology spaces. For these
integer quantities, issues of approximation and convergence do not play much of a
role. The situation is different in the 1976 work of Dodziuk [39] and Dodziuk and
Patodi [40], who considered the approximation of the Hodge Laplacian on a Riemannian manifold by a combinatorial Hodge Laplacian, a sort of finite difference
approximation defined on cochains with respect to a triangulation. The combinatorial Hodge Laplacian was defined in [39] using the Whitney forms, thus realizing
the finite difference operator as a sort of finite element approximation. A key result
in [39] was a proof of some convergence properties of the Whitney forms. In [40]
the authors applied them to show that the eigenvalues of the combinatorial Hodge
Laplacian converge to those of the true Hodge Laplacian. This is indeed a finite
element convergence result, as the authors remark. In 1978, Müller [71] further
developed this work and used it to prove the Ray–Singer conjecture. This conjecture equates a topological invariant defined in terms of the Riemannian structure
with one defined in terms of a triangulation and was the original goal of [39, 40].
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288
D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER
(Cheeger [30] gave a different, independent proof of the Ray–Singer conjecture at
about the same time.) Other spaces of finite element differential forms have appeared in geometry as well, especially the differential graded algebra of piecewise
polynomial forms on a simplicial complex introduced by Sullivan [83, 84]. Baker
[15] calls these Sullivan–Whitney forms, and, in an early paper bringing finite element analysis techniques to bear on geometry, gives a numerical analysis of their
accuracy for approximating the eigenvalues of the Hodge Laplacian.
Independently of the work of the geometers, during the 1970s and 1980s numerical analysts and computational engineers reinvented various special cases of
the Whitney forms and developed new variants of them to use for the solution of
partial differential equations on two- and three-dimensional domains. In this work,
naturally, implementational issues, rates of convergence, and sharp estimates played
a more prominent role than in the geometry literature. The pioneering paper of
Raviart and Thomas [76], presented at a finite element conference in 1975, proposed
the first stable finite elements for solving the scalar Laplacian in two dimensions
using the mixed formulation. The mixed formulation involves two unknown fields,
the scalar-valued solution, and an additional vector-valued variable representing its
gradient. Raviart and Thomas proposed a family of pairs of finite element spaces,
one for each polynomial degree. As was realized later, in the lowest degree case the
space they constructed for the vector-valued variable is just the space of Whitney
1-forms, while they used piecewise constants, which are Whitney 2-forms, for the
scalar variable. For higher degrees, their elements are the higher-order Whitney
forms. In three dimensions, the introduction of Whitney 1- and 2-forms for finite
element computations and their higher-degree analogues was made by Nédélec [72]
in 1980, while the polynomial mixed elements which can be viewed as Sullivan–
Whitney forms were introduced as finite elements by Brezzi, Douglas, and Marini
[26] in 1985 in two dimensions, and then by Nédélec [72] in 1986 in three dimensions.
In 1988 Bossavit, in a paper in the IEE Transactions on Magnetics [21], made
the connection between Whitney’s forms used by geometers and some of the mixed
finite element spaces that had been proposed for electromagnetics, inspired in part
by Kotiuga’s Ph.D. thesis in electrical engineering [66]. Maxwell’s equations are
naturally formulated in terms of differential forms, and the computational electromagnetics community developed the connection between mixed finite elements and
Hodge theory in a number of directions. See, in particular, [17, 37, 57, 58, 59, 70].
The methods we derive here are examples of compatible discretization methods, which means that at the discrete level they reproduce, rather than merely
approximate, certain essential structures of the continuous problem. Other examples of compatible discretization methods for elliptic PDEs are mimetic finite
difference methods [16, 27] including covolume methods [74] and the discrete exterior calculus [38]. In these methods, the fundamental object used to discretize
a differential k-form is typically a simplicial cochain; i.e., a number is assigned to
each k-dimensional face of the mesh representing the integral of the k-form over the
face. This is more of a finite difference, rather than finite element, point of view,
recalling the early work of Dodziuk on combinatorial Hodge theory. Since the space
of k-dimensional simplicial cochains is isomorphic to the space of Whitney k-forms,
there is a close relationship between these methods and the simplest methods of
the finite element exterior calculus. In some simple cases, the methods even coincide. In contrast to the finite element approach, these cochain-based approaches do
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FINITE ELEMENT EXTERIOR CALCULUS
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not naturally generalize to higher-order methods. Discretizations of exterior calculus and Hodge theory have also been used for purposes other than solving partial
differential equations. For example, discrete forms which are identical or closely
related to cochains or the corresponding Whitney forms play an important role in
geometric modeling, parameterization, and computer graphics. See for example
[50, 54, 56, 87].
1.4. Highlights of the finite element exterior calculus. We close this introduction by highlighting some of the features that are unique or much more
prominent in the finite element exterior calculus than in earlier works.
• We work in an abstract Hilbert space setting that captures the fundamental
structures of the Hodge theory of Riemannian manifolds, but applies more
generally. In fact, the paper proceeds in two parts, first the abstract theory
for Hilbert complexes, and then the application to the de Rham complex
and Hodge theory and other applications.
• Mixed formulations based on saddle point variational principles play a
prominent role. In particular, the algorithms we use to approximate the
Hodge Laplacian are based on a mixed formulation, as is the analysis of the
algorithms. This is in contrast to the approach in the geometry literature,
where the underlying variational principle is a minimization principle. In
the case of the simplest elements, the Whitney elements, the two methods are equivalent. That is, the discrete solution obtained by the mixed
finite element method using Whitney forms, is the same as that obtained
by Dodziuk’s combinatorial Laplacian. However, the different viewpoint
leads naturally to different approaches to the analysis. The use of Whitney
forms for the mixed formulation is obviously a consistent discretization,
and the key to the analysis is to establish stability (see the next section
for the terminology). However, for the minimization principle, it is unclear
whether Whitney forms provide a consistent approximation, because they
do not belong to the domain of the exterior coderivative, and, as remarked
in [40], this greatly complicates the analysis. The results we obtain are
both more easily proven and sharper.
• Our analysis is based on two main properties of the subspaces used to discretize the Hilbert complex. First, they can be formed into subcomplexes,
which is a key assumption in much of the work we have discussed. Second,
there exist a bounded cochain projection from the Hilbert complex to the
subcomplex. This is a new feature. In previous work, a cochain projection
often played a major role, but it was not bounded, and the existence of
bounded cochain projections was not realized. In fact, they exist quite generally (see Theorem 3.7), and we review the construction for the de Rham
complex in Section 5.5.
• Since we are interested in actual numerical computations, it is important
that our spaces be efficiently implementable. This is not true for all piecewise polynomial spaces. As explained in the next section, finite element
spaces are a class of piecewise polynomial spaces that can be implemented
efficiently by local computations thanks to the existence of degrees of freedom, and the construction of degrees of freedom and local bases is an
important part of the finite element exterior calculus.
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290
D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER
• For the same reason, high-order piecewise polynomials are of great importance, and all the constructions and analysis of finite element exterior
calculus carries through for polynomials of arbitrary degree.
• A prominent aspect of the finite element exterior calculus is the role of
two families of spaces of polynomial differential forms, Pr Λk and Pr− Λk ,
where the index r ≥ 1 denotes the polynomial degree and k ≥ 0 the form
degree. These are the shape functions for corresponding finite element
spaces of differential k-forms which include, as special cases, the Lagrange
finite element family, and most of the stable finite element spaces that
have been used to define mixed formulations of the Poisson or Maxwell’s
equations. The space P1− Λk is the classical space of Whitney k-forms. The
finite element spaces based on Pr Λk are the spaces of Sullivan–Whitney
forms. We show that for each polynomial degree r, there are 2n−1 ways to
form these spaces in de Rham subcomplexes for a domain in n dimensions.
The unified treatment of the spaces Pr Λk and Pr− Λk , particularly their
connections via the Koszul complex, is new to the finite element exterior
calculus.
The finite element exterior calculus unifies many of the finite element methods
that have been developed for solving PDEs arising in fluid and solid mechanics,
electromagnetics, and other areas. Consequently, the methods developed here have
been widely implemented and applied in scientific and commercial programs such
as GetDP [42], FEniCS [44], EMSolve [45], deal.II [46], Diffpack [61], Getfem++
[77], and NGSolve [78]. We also note that, as part of a recent programming effort
connected with the FEniCS project, Logg and Mardal [69] have implemented the
full set of finite element spaces developed in this paper, strictly following the finite
element exterior framework as laid out here and in [8].
2. Finite element discretizations
In this section we continue the introduction to the finite element method begun
above. We move beyond the case of one dimension and consider not only the formulation of the method, but also its analysis. To motivate the theory developed
later in this paper, we present further examples that illustrate how for some problems, even rather simple ones, deriving accurate finite element methods is not a
straightforward process.
2.1. Galerkin methods and finite elements. We consider first a simple problem, which can be discretized in a straightforward way, namely the Dirichlet problem for Poisson’s equation in a polyhedral domain Ω ⊂ Rn :
−∆u = f in Ω,
(4)
u = 0 on ∂Ω.
This is the generalization to n dimensions of the problem (1) discussed in the
introduction, and the solution may again be characterized as the minimizer of an
energy functional analogous to (2) or as the solution of a weak problem analogous to
(3). This leads to discretization just as for the one-dimensional case, by choosing a
trial space Vh ⊂ H̊ 1 (Ω) and defining the approximate solution uh ∈ Vh by Galerkin’s
method:
grad uh (x) · grad v(x) dx =
Ω
f (x)v(x) dx,
v ∈ Vh .
Ω
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FINITE ELEMENT EXTERIOR CALCULUS
291
As in one dimension, the simplest finite element method is obtained by using
the trial space consisting of all piecewise linear functions with respect to a given
simplicial triangulation of the domain Ω, which are continuous and vanish on ∂Ω.
A key to the efficacy of this finite element method is the existence of a basis for the
trial space consisting of functions which are locally supported, i.e., vanish on all
but a small number of the elements of the triangulation. See Figure 2.1. Because
of this, the coefficient matrix of the linear system is easily computed and is sparse,
and so the system can be solved efficiently.
Figure 2.1. A piecewise linear finite element basis function.
More generally, a finite element method is a Galerkin method for which the trial
space Vh is a space of piecewise polynomial functions which can be obtained by
what is called the finite element assembly process. This means that the space can
be defined by specifying the triangulation Th and, for each element T ∈ Th , a space
of polynomial functions on T called the shape functions, and a set of degrees of
freedom. By degrees of freedom on T , we mean a set of functionals on the space
of shape functions, which can be assigned values arbitrarily to determine a unique
shape function. In other words, the degrees of freedom form a basis for the dual
space of the space of shape functions. In the case of piecewise linear finite elements,
the shape functions are of course the linear polynomials on T , a space of dimension
n + 1, and the degrees of freedom are the n + 1 evaluation functionals p → p(x),
where x varies over the vertices of T . For the finite element assembly process, we
also require that each degree of freedom be associated to a face of some dimension
of the simplex T . For example, in the case of piecewise linear finite elements, the
degree of freedom p → p(x) is associated to the vertex x. Given the triangulation,
shape functions, and degrees of freedom, the finite element space Vh is defined as
the set of functions on Ω (possibly multivalued on the element boundaries) whose
restriction to any T ∈ Th belongs to the given space of shape functions on T ,
and for which the degrees of freedom are single-valued in the sense that when two
elements share a common face, the corresponding degrees of freedom take on the
same value. Returning again to the example of piecewise linear functions, Vh is
the set of functions which are linear polynomials on each element, and which are
single-valued at the vertices. It is easy to see that this is precisely the space of
continuous piecewise linear functions, which is a subspace of H 1 (Ω). As another
example, we could take the shape functions on T to be the polynomials of degree
at most 2, and take as degrees of freedom the functions p → p(x), x a vertex of T ,
and p → e p ds, e an edge of T . The resulting assembled finite element space is the
space of all continuous piecewise quadratics. The finite element assembly process
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292
D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER
insures the existence of a computable locally supported basis, which is important
for efficient implementation.
2.2. Consistency, stability, and convergence. We now turn to the important
problem of analyzing the error in the finite element method. To understand when
a Galerkin method will produce a good approximation to the true solution, we
introduce the standard abstract framework. Let V be a Hilbert space, B : V × V →
R a bounded bilinear form, and F : V → R a bounded linear form. We assume the
problem to be solved can be stated in the form: Find u ∈ V such that
B(u, v) = F (v),
v ∈ V.
This problem is called well-posed if for each F ∈ V ∗ , there exists a unique solution
u ∈ V and the mapping F → u is bounded, or, equivalently, if the operator L :
V → V ∗ given by Lu, v = B(u, v) is an isomorphism. For the Dirichlet problem
for Poisson’s equation,
grad u(x) · grad v(x) dx, F (v) =
f (x)v(x) dx.
(5) V = H̊ 1 (Ω), B(u, v) =
Ω
Ω
A generalized Galerkin method for the abstract problem begins with a finitedimensional normed space Vh (not necessarily a subspace of V ), a bilinear form
Bh : Vh × Vh → R, and a linear form Fh : Vh → R, and defines uh ∈ Vh by
(6)
Bh (uh , v) = Fh (v),
v ∈ Vh .
A Galerkin method is the special case of a generalized Galerkin method for which Vh
is a subspace of V and the forms Bh and Fh are simply the restrictions of the forms
B and F to the subspace. The more general setting is important since it allows
the incorporation of additional approximations, such as numerical integration to
evaluate the integrals, and also allows for situations in which Vh is not a subspace
of V . Although we do not treat approximations such as numerical integration in
this paper, for the fundamental discretization method that we study, namely the
mixed method for the abstract Hodge Laplacian introduced in Section 3.4, the trial
space Vh is not a subspace of V , since it involves discrete harmonic forms which
will not, in general, belong to the space of harmonic forms.
The generalized Galerkin method (6) may be written Lh uh = Fh where Lh :
Vh → Vh∗ is given by Lh u, v = Bh (u, v), u, v ∈ Vh . If the finite-dimensional
problem is nonsingular, then we define the norm of the discrete solution operator,
∗
L−1
h L(Vh ,Vh ) , as the stability constant of the method.
Of course, in approximating the original problem determined by V , B, and F ,
by the generalized Galerkin method given by Vh , Bh , and Fh , we intend that the
space Vh in some sense approximates V and that the discrete forms Bh and Fh
in some sense approximate B and F . This is the essence of consistency. Our
goal is to prove that the discrete solution uh approximates u in an appropriate
sense (convergence). In order to make these notions precise, we need to compare a
function in V to a function in Vh . To this end, we suppose that there is a restriction
operator πh : V → Vh , so that πh u is thought to be close to u. Then the consistency
error is simply Lh πh u − Fh and the error in the generalized Galerkin method which
we wish to control is πh u − uh . We immediately get a relation between the error
and the consistency error:
πh u − uh = L−1
h (Lh πh u − Fh ),
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FINITE ELEMENT EXTERIOR CALCULUS
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and so the norm of the error is bounded by the product of the stability constant
and the norm of the consistency error:
∗
∗
πh u − uh Vh ≤ L−1
h L(Vh ,Vh ) Lh πh u − Fh Vh .
Stated in terms of the bilinear form Bh , the norm of the consistency error can be
written as
Bh (πh u, v) − Fh (v)
Lh πh u − Fh Vh∗ = sup
.
vVh
0=v∈Vh
As for stability, the finite-dimensional problem is nonsingular if and only if
γh :=
inf
sup
0=u∈Vh 0=v∈Vh
Bh (u, v)
> 0,
uVh vVh
and the stability constant is then given by γh−1 .
Often we consider a sequence of spaces Vh and forms Bh and Fh , where we think
of h > 0 as an index accumulating at 0. The corresponding generalized Galerkin
method is consistent if the Vh norm of the consistency error tends to zero with h
and it is stable if the stability constant γh−1 is uniformly bounded. For a consistent,
stable generalized Galerkin method, πh u − uh Vh tends to zero; i.e., the method
is convergent.
In the special case of a Galerkin method, we can bound the consistency error
sup
0=v∈Vh
Bh (πh u, v) − Fh (v)
B(πh u − u, v)
= sup
≤ Bπh u − uV .
vV
vV
0=v∈Vh
In this case it is natural to choose the restriction πh to be the orthogonal projection
onto Vh , and so the consistency error is bounded by the norm of the bilinear form
times the error in the best approximation of the solution. Thus we obtain
πh u − uh V ≤ γh−1 B inf u − vV .
v∈Vh
Combining this with the triangle inequality, we obtain the basic error estimate for
Galerkin methods
(7)
u − uh V ≤ (1 + γh−1 B) inf u − vV .
v∈Vh
(In fact, in this Hilbert space setting, the quantity in parentheses can be replaced
with γh−1 B; see [89].) Note that a Galerkin method is consistent as long as the
sequence of subspaces Vh is approximating in V in the sense that
(8)
lim inf u − vV = 0,
h→0 v∈Vh
u ∈ V.
A consistent, stable Galerkin method converges, and the approximation given by
the method is quasi-optimal; i.e., up to multiplication by a constant, it is as good
as the best approximation in the subspace.
In practice, it can be quite difficult to show that the finite-dimensional problem
is nonsingular and to bound the stability constant, but there is one important case
in which it is easy, namely when the form B is coercive, i.e., when there is a positive
constant α for which
B(v, v) ≥ αv2V , v ∈ V,
and so γh ≥ α. The bilinear form (5) for Poisson’s equation is coercive, as follows
from Poincaré’s inequality. This explains, and can be used to prove, the good
convergence behavior of the method depicted in Figure 1.1.
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D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER
2.3. Computational examples.
2.3.1. Mixed formulation of the Laplacian. For an example of a problem that fits
in the standard abstract framework with a noncoercive bilinear form, we take the
mixed formulation of the Dirichlet problem for Poisson’s equation, already introduced in one dimension in Section 1.1. Just as there, we begin by writing Poisson’s
equation as the first-order system
σ = − grad u,
(9)
div σ = f.
The pair (σ, u) can again be characterized variationally as the unique critical point
(a saddle point) of the functional
1
I(σ, u) = ( σ · σ − u div σ) dx +
f u dx
Ω 2
Ω
over H(div; Ω) × L2 (Ω), where H(div; Ω) = {σ ∈ L2 (Ω) : div σ ∈ L2 (Ω)}. Equivalently, it solves the weak problem: Find σ ∈ H(div; Ω), u ∈ L2 (Ω) satisfying
σ · τ dx −
u div τ dx = 0, τ ∈ H(div; Ω),
Ω
Ω
div σv dx =
f v dx, v ∈ L2 (Ω).
Ω
Ω
This mixed formulation of Poisson’s equation fits in the abstract framework if
we define V = H(div; Ω) × L2 (Ω),
B(σ, u; τ, v) =
σ · τ dx −
u div τ dx +
div σv dx, F (τ, v) =
f v dx.
Ω
Ω
Ω
Ω
In this case the bilinear form B is not coercive, and so the choice of subspaces and
the analysis is not so simple as for the standard finite element method for Poisson’s
equation, a point we already illustrated in the one-dimensional case.
Finite element discretizations based on such saddle point variational principles
are called mixed finite element methods. Thus a mixed finite element for Poisson’s
equation is obtained by choosing subspaces Σh ⊂ H(div; Ω) and Vh ⊂ L2 (Ω) and
seeking a critical point of I over Σh × Vh . The resulting Galerkin method has the
form: Find σh ∈ Σh , uh ∈ Vh satisfying
σh · τ dx −
uh div τ dx = 0, τ ∈ Σh ,
div σh v dx =
f v dx, v ∈ Vh .
Ω
Ω
Ω
Ω
This again reduces to a linear system of algebraic equations.
Since the bilinear form is not coercive, it is not automatic that the linear system
is nonsingular, i.e., that for f = 0, the only solution is σh = 0, uh = 0. Choosing
τ = σh and v = uh and adding the discretized variational equations, it follows
immediately that
when f = 0, σh = 0. However, uh need not vanish unless the
condition that Ω uh div τ dx = 0 for all τ ∈ Σh implies that uh = 0. In particular,
this requires that dim(div Σh ) ≥ dim Vh . Thus, even nonsingularity of the approximate problem depends on a relationship between the two finite-dimensional spaces.
Even if the linear system is nonsingular, there remains the issue of stability, i.e., of
a uniform bound on the inverse operator.
As mentioned earlier, the combination of continuous piecewise linear elements for
σ and piecewise constants for u is not stable in two dimensions. The simplest stable
elements use the piecewise constants for u, and the lowest-order Raviart-Thomas
elements for σ. These are finite elements defined with respect to a triangular
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FINITE ELEMENT EXTERIOR CALCULUS
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mesh by shape functions of theform (a + bx1 , c + bx2 ) and one degree of freedom
for each edge e, namely σ → e σ · n ds. We show in Figure 2.2 two numerical
computations that demonstrate the difference between an unstable and a stable
choice of elements for this problem. The stable method accurately approximates
the true solution u = x(1 − x)y(1 − y) on (0, 1) × (0, 1) with a piecewise constant,
while the unstable method is wildly oscillatory.
0.25
0.0625
0.05
0.0312
−0.15
0
Figure 2.2. Approximation of the mixed formulation for Poisson’s equation using piecewise constants for u and for σ using
either continuous piecewise linears (left), or Raviart–Thomas elements (right). The plotted quantity is u in each case.
This problem is a special case of the Hodge Laplacian with k = n as discussed
briefly in Section 4.2; see especially Section 4.2.4. The error analysis for a variety of
finite element methods for this problem, including the Raviart–Thomas elements, is
thus a special case of the general theory of this paper, yielding the error estimates
in Section 5.6.
2.3.2. The vector Laplacian on a nonconvex polygon. Given the subtlety of finding
stable pairs of finite element spaces for the mixed variational formulation of Poisson’s equation, we might choose to avoid this formulation, in favor of the standard
formulation, which leads to a coercive bilinear form. However, while the standard
formulation is easy to discretize for Poisson’s equation, additional issues arise already if we try to discretize the vector Poisson equation. For a domain Ω in R3
with unit outward normal n, this is the problem
(10) − grad div u + curl curl u = f, in Ω,
u · n = 0,
curl u × n = 0,
on ∂Ω.
The solution of this problem can again be characterized as the minimizer of an
appropriate energy functional,
1
2
2
(11)
J(u) =
(| div u| + | curl u| ) dx −
f · u dx,
2 Ω
Ω
but this time over the space H(curl; Ω) ∩ H̊(div; Ω), where H(curl; Ω) = {u ∈
L2 (Ω) | curl u ∈ L2 (Ω)} and H̊(div; Ω) = {u ∈ H(div; Ω) | u · n = 0 on ∂Ω} with
H(div; Ω) defined above. This problem is associated to a coercive bilinear form,
but a standard finite element method based on a trial subspace of the energy space
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296
D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER
H(curl; Ω) ∩ H̊(div; Ω), e.g., using continuous piecewise linear vector functions,
is very problematic. In fact, as we shall illustrate shortly, if the domain Ω is a
nonconvex polyhedron, for almost all f the Galerkin method solution will converge
to a function that is not the true solution of the problem! The essence of this
unfortunate situation is that any piecewise polynomial subspace of H(curl; Ω) ∩
H̊(div; Ω) is a subspace of H 1 (Ω) ∩ H̊(div; Ω), and this space is a closed subspace of
H(curl; Ω) ∩ H̊(div; Ω). For a nonconvex polyhedron, it is a proper closed subspace
and generally the true solution will not belong to it, due to a singularity at the
reentrant corner. Thus the method, while stable, is inconsistent. For more on this
example, see [35].
An accurate approximation of the vector Poisson equation can be obtained from
a mixed finite element formulation, based on the system:
σ = − div u,
grad σ + curl curl u = f in Ω,
u · n = 0,
curl u × n = 0 on ∂Ω.
Writing this system in weak form, we obtain the mixed formulation of the problem:
find σ ∈ H 1 (Ω), u ∈ H(curl; Ω) satisfying
στ dx −
u · grad τ dx = 0, τ ∈ H 1 (Ω),
Ω
Ω
grad σ · v dx +
curl u · curl v dx =
f · v dx, v ∈ H(curl; Ω).
Ω
Ω
Ω
In contrast to a finite element method based on minimizing the energy (11), a
finite element approximation based on the mixed formulation uses separate trial
subspaces of H 1 (Ω) and H(curl; Ω), rather than a single subspace of the intersection
H(curl; Ω) ∩ H̊(div; Ω).
We now illustrate the nonconvergence of a Galerkin method based on energy
minimization and the convergence of one based on the mixed formulation, via computations in two space dimensions (so now the curl of a vector u is the scalar
∂u2 /∂x1 − ∂u1 /∂x2 ). For the trial subspaces we make the simplest choices: for
the former method we use continuous piecewise linear functions and for the mixed
method we use continuous piecewise linear functions to approximate σ ∈ H 1 (Ω)
and a variant of the lowest-order Raviart–Thomas elements, for which the shape
functions are the infinitesimal rigid motions
(a − bx2 , c + bx1 ) and the degrees of
freedom are the tangential moments u → e u · s ds for e an edge. The discrete solutions obtained by the two methods for the problem when f = (−1, 0) are shown
in Figure 2.3. As we shall show later in this paper, the mixed formulation gives an
approximation that provably converges to the true solution, while, as can be seen
from comparing the two plots, the first approximation scheme gives a completely
different (and therefore inaccurate) result.
This problem is again a special case of the Hodge Laplacian, now with k = 1.
See Section 4.2.2. The error analysis thus falls within the theory of this paper,
yielding estimates as in Section 5.6.
2.3.3. The vector Laplacian on an annulus. In the example just considered, the
failure of a standard Galerkin method based on energy minimization to solve the
vector Poisson equation was related to the reentrant corner of the domain and the
resulting singular behavior of the solution. A quite different failure mode for this
method occurs if we take a domain which is smoothly bounded, but not simply
connected, e.g., an annulus. In that case, as discussed below in Section 3.2, the
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FINITE ELEMENT EXTERIOR CALCULUS
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Figure 2.3. Approximation of the vector Laplacian by the standard finite element method (left) and a mixed finite element
method (right). The former method totally misses the singular
behavior of the solution near the reentrant corner.
boundary value problem (10) is not well-posed except for special values of the forcing function f . In order to obtain a well-posed problem, the differential equation
should be solved only modulo the space of harmonic vector fields (or harmonic 1forms), which is a one-dimensional space for the annulus, and the solution should
be rendered unique by enforcing orthogonality to the harmonic vector fields. If we
choose the annulus with radii 1/2 and 1, and forcing function f = (0, x), the resulting solution, which can be computed accurately with a mixed formulation falling
within the theory of this paper, is displayed on the right in Figure 2.4. However,
the standard Galerkin method does not capture the nonuniqueness and computes
the discrete solution shown on the left of the same figure, which is dominated by an
approximation of the harmonic vector field, and so is nothing like the true solution.
Figure 2.4. Approximation of the vector Laplacian on an annulus. The true solution shown here on the right is an (accurate) approximation by a mixed method. It is orthogonal to the harmonic
fields and satisfies the differential equation only modulo harmonic
fields. The standard Galerkin solution using continuous piecewise
linear vector fields, shown on the left, is totally different.
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298
D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER
2.3.4. The Maxwell eigenvalue problem. Another situation where a standard finite
element method gives unacceptable results, but a mixed method succeeds, arises in
the approximation of elliptic eigenvalue problems related to the vector Laplacian or
Maxwell’s equation. This will be analyzed in detail later in this paper, and here we
only present a simple but striking computational example. Consider the eigenvalue
problem for the vector Laplacian discussed above, which we write in mixed form
as: find nonzero (σ, u) ∈ H 1 (Ω) × H(curl; Ω) and λ ∈ R satisfying
σ · τ dx −
grad τ · u dx = 0, τ ∈ H 1 (Ω),
Ω
Ω
(12)
grad σ · v dx +
curl u · curl v dx = λ
u · v dx, v ∈ H(curl; Ω).
Ω
Ω
Ω
As explained in Section 3.6.1, this problem can be split into two subproblems. In
particular, if 0 = u ∈ H(curl; Ω) and if λ ∈ R solves the eigenvalue problem
(13)
curl u · curl v dx = λ
u · v dx, v ∈ H(curl; Ω),
Ω
Ω
and λ is not equal to zero, then (σ, u), λ is an eigenpair for (12) with σ = 0.
We now consider the solution of the eigenvalue problem (13), with two different
choices of trial subspaces in H(curl; Ω). Again, to make our point, it is enough
to consider a two-dimensional case, and we consider the solution of (13) with Ω a
square of side length π. For this domain, the positive eigenvalues can be computed
by separation of variables. They are of the form m2 + n2 with m and n integers:
1, 1, 2, 4, 4, 5, 5, 8, . . .. If we approximate (13) using the space of continuous piecewise
linear vector fields as the trial subspace of H(curl; Ω), the approximation fails badly.
This is shown for an unstructured mesh in Figure 2.5 and for a structured crisscross
mesh in Figure 2.6, where the nonzero discrete eigenvalues are plotted. Note the
very different mode of failure for the two mesh types. For more discussion of the
spurious eigenvalues arising using continuous piecewise linear vector fields on a
crisscross mesh, see [20]. By contrast, if we use the lowest-order Raviart–Thomas
approximation of H(curl; Ω), as shown on the right of Figure 2.5, we obtain a
provably good approximation for any mesh. This is a very simple case of the
general eigenvalue approximation theory presented in Section 3.6 below.
10
10
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
0
0
Figure 2.5. Approximation of the nonzero eigenvalues of (13) on
an unstructured mesh of the square (left) using continuous piecewise linear finite elements (middle) and Raviart–Thomas elements
(right). For the former, the discrete spectrum looks nothing like
the true spectrum, while for the later it is very accurate.
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FINITE ELEMENT EXTERIOR CALCULUS
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10
9
8
7
6
5
4
3
2
1
0
Figure 2.6. Approximation of the nonzero eigenvalues of (13)
using continuous piecewise linear elements on the structured mesh
shown. The first seven discrete nonzero eigenvalues converge to
true eigenvalues, but the eighth converges to a spurious value.
3. Hilbert complexes and their approximation
In this section, we construct a Hilbert space framework for finite element exterior
calculus. The most basic object in this framework is a Hilbert complex, which extracts essential features of the L2 de Rham complex. Just as the Hodge Laplacian
is naturally associated with the de Rham complex, there is a system of variational
problems, which we call the abstract Hodge Laplacian, associated to any Hilbert
complex. Using a mixed formulation we prove that these abstract Hodge Laplacian
problems are well-posed. We next consider the approximation of Hilbert complexes
using finite-dimensional subspaces. Our approach emphasizes two key properties,
the subcomplex property and the existence of bounded cochain projections. These
same properties prove to be precisely what is needed both to show that the approximate Hilbert complex accurately reproduces geometrical quantities associated
to the complex, like cohomology spaces, and also to obtain error estimates for the
approximation of the abstract Hodge Laplacian source and eigenvalue problems,
which is our main goal in this section. In the following section of the paper we will
derive finite element subspaces in the concrete case of the de Rham complex and
verify the hypotheses needed to apply the results of this section.
Although the L2 de Rham complex is the canonical example of a Hilbert complex,
there are many others. In this paper, in Section 6, we consider some variations of
the de Rham complex that allow us to treat more general PDEs and boundary
value problems. In the final section we briefly discuss the equations of elasticity,
for which a very different Hilbert complex, in which one of the differentials is a
second-order PDE, is needed. Another useful feature of Hilbert complexes is that a
subcomplex of a Hilbert complex is again such, and so the properties we establish
for them apply not only at the continuous, but also at the discrete level.
3.1. Basic definitions. We begin by recalling some basic definitions of homological algebra and functional analysis and establishing some notation.
3.1.1. Cochain complexes. Consider a cochain complex (V, d) of vector spaces, i.e.,
a sequence of vector spaces V k and linear maps dk , called the differentials:
dk−1
dk
· · · → V k−1 −−−→ V k −→ V k+1 → · · ·
with dk ◦ dk−1 = 0.Equivalently, we may think of such a complex as the graded
vector space V =
V k , equipped with a graded linear operator d : V → V of
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D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER
degree +1 satisfying d ◦ d = 0. A chain complex differs from a cochain complex
only in that the indices decrease. All the complexes we consider are nonnegative
and finite, meaning that V k = 0 whenever k is negative or sufficiently large.
Given a cochain complex (V, d), the elements of the null space Zk = Zk (V, d) of
k
d are called the k-cocycles and the elements of the range Bk = Bk (V, d) of dk−1
the k-coboundaries. The kth cohomology group is defined to be the quotient space
Zk /Bk .
Given two cochain complexes (V, d) and (V , d ), a set of linear maps f k : V k →
k
V satisfying dk f k = f k+1 dk (i.e., a graded linear map f : V → V of degree
0 satisfying d f = f d) is called a cochain map. When f is a cochain map, f k
maps k-cocycles to k-cocycles and k-coboundaries to k-coboundaries, and hence
induces a map f¯ on the cohomology spaces. This map is functorial; i.e., it respects
composition.
Let (V, d) be a cochain complex and (Vh , d) a subcomplex. In other words, Vhk is
a subspace of V k and dk Vhk ⊂ Vhk+1 . Then the inclusion ih : Vh → V is a cochain
map and so induces a map of cohomology. If there exists a cochain projection of
V onto Vh , i.e., a cochain map πh such that πhk : V k → Vhk leaves the subspace Vhk
invariant, then πh ◦ ih = idVh , so π̄h ◦ īh = idZkh /Bkh (where Zkh := Zk (Vh , d) and
similarly for Bkh ). We conclude that in this case īh is injective and π̄h is surjective.
In particular, the dimension of the cohomology spaces of the subcomplex is at most
that of the supercomplex.
3.1.2. Closed operators on Hilbert space. This material can be found in many places,
e.g., [64, Chapter III, §5 and Chapter IV, §5.2] or [90, Chapter II, §6 and Chapter
VII].
By an operator T from a Hilbert space X to a Hilbert space Y , we mean a linear
operator from a subspace V of X, called the domain of T , into Y . The operator
T is not necessarily bounded and the domain V is not necessarily closed. We say
that the operator T is closed if its graph { (x, T x) | x ∈ V } is closed in X × Y . We
endow the domain V with the graph norm inner product,
v, wV = v, wX + T v, T wY .
It is easy to check that this makes V a Hilbert space (i.e., complete) if and only if
T is closed, and moreover, that T is a bounded operator from V to Y . Of course,
the null space of T is the set of those elements of its domain that it maps to 0,
and the range of T is T (V ). The null space of a closed operator from X to Y is a
closed subspace of X, but its range need not be closed in Y (even if the operator
is defined on all of X and is bounded).
The operator T is said to be densely defined if its domain V is dense in X. In
this case the adjoint operator T ∗ from Y to X is defined to be the operator whose
domain consists of all y ∈ Y for which there exists x ∈ X with
x, vX = y, T vY ,
v ∈ V,
∗
in which case T y = x (well-defined since V is dense). If T is closed and densely
defined, then T ∗ is as well and T ∗∗ = T . Moreover, the null space of T ∗ is the
orthogonal complement of the range of T in Y . Finally, by the closed range theorem,
the range of T is closed in Y if and only if the range of T ∗ is closed in X.
If the range of a closed linear operator is of finite codimension, i.e., dim Y /T (V ) <
∞, then the range is closed [60, Lemma 19.1.1].
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3.1.3. Hilbert complexes. A Hilbert complex is a sequence of Hilbert spaces W k and
closed, densely-defined linear operators dk from W k to W k+1 such that the range
of dk is contained in the domain of dk+1 and dk+1 ◦ dk = 0. A Hilbert complex
is bounded if, for each k, dk is a bounded linear operator from W k to W k+1 . In
other words, a bounded Hilbert complex is a cochain complex in the category of
Hilbert spaces. A Hilbert complex is closed if for each k, the range of dk is closed
in W k+1 . A Fredholm complex is a Hilbert complex for which the range of dk
is finite codimensional in the null space of dk+1 (and so is closed). Hilbert and
Fredholm complexes have been discussed by various authors working in geometry
and topology. Brüning and Lesch [28] have advocated for them as an abstraction of
elliptic complexes on manifolds and applied them to spectral geometry on singular
spaces. Glotko [53] used them to define a generalization of Sobolev spaces on
Riemannian manifolds and to study their compactness properties, and Gromov
and Shubin [55] used them to define topological invariants of manifolds.
Associated to any Hilbert complex (W, d) is a bounded Hilbert complex (V, d),
called the domain complex, for which the space V k is the domain of dk , endowed
with the inner product associated to the graph norm:
u, vV k = u, vW k + dk u, dk vW k+1 .
Then dk is a bounded linear operator from V k to V k+1 , and so (V, d) is indeed a
bounded Hilbert complex. The domain complex is closed or Fredholm if and only
if the original complex (W, d) is.
Of course, for a Hilbert complex (W, d), we have the null spaces and ranges
Zk and Bk . Utilizing the inner product, we define the space of harmonic forms
Hk = Zk ∩ Bk⊥ , the orthogonal complement of Bk in Zk . It is isomorphic to
the reduced cohomology space Zk /Bk or, for a closed complex, to the cohomology
space Zk /Bk . For a closed Hilbert complex, we immediately obtain the Hodge
decomposition
(14)
W k = Bk ⊕ Hk ⊕ Zk⊥W .
For the domain complex (V, d), the null space, range, and harmonic forms are the
same spaces as for the original complex, and the Hodge decomposition is
V k = Bk ⊕ Hk ⊕ Zk⊥V ,
where the third summand Zk⊥V = Zk⊥W ∩ V k .
Continuing to use the Hilbert space structure, we define the dual complex (W, d∗ ),
which is a Hilbert chain complex rather than a cochain complex. The dual complex
uses the same spaces W i , with the differential d∗k being the adjoint of dk−1 , so d∗k is
a closed, densely-defined operator from W k to W k−1 , whose domain we denote by
Vk∗ . The dual complex is closed or bounded if and only if the original complex is.
We denote by Z∗k = Bk⊥W the null space of d∗k , and by B∗k the range of d∗k+1 . Thus
Hk = Zk ∩ Z∗k is the space of harmonic forms both for the original complex and the
dual complex. Since Zk⊥W = B∗k , the Hodge decomposition (14) can be written as
(15)
W k = Bk ⊕ Hk ⊕ B∗k .
We henceforth simply write Zk⊥ for Zk⊥V .
Let (W, d) be a closed Hilbert complex with domain complex (V, d). Then dk is
a bounded bijection from Zk⊥ to the Hilbert space Bk+1 and hence, by Banach’s
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D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER
bounded inverse theorem, there exists a constant cP such that
vV ≤ cP dk vW ,
(16)
v ∈ Zk⊥ ,
which we refer to as a Poincaré inequality. We remark that the condition that Bk
is closed is not only sufficient to obtain (16), but also necessary.
The subspace V k ∩ Vk∗ of W k is a Hilbert space with the norm
v2V ∩V ∗ = v2V + v2V ∗ = 2v2W + dk v2W + d∗k v2W
and is continuously included in W k . We say that the Hilbert complex (W, d) has
the compactness property if V k ∩ Vk∗ is dense in W k and the inclusion is a compact
operator. Restricted to the
space Hk of harmonic forms, the V k ∩ Vk∗ norm is equal
√
to the W k norm (times 2). Therefore the compactness property implies that the
inclusion of Hk into itself is compact, so Hk is finite-dimensional. In summary, for
Hilbert complexes, compactness property =⇒ Fredholm =⇒ closed.
3.2. The abstract Hodge Laplacian and the mixed formulation. Given a
Hilbert complex (W, d), the operator L = dd∗ + d∗ d is an unbounded operator
W k → W k called, in the case of the de Rham complex, the Hodge Laplacian. We
refer to it as the abstract Hodge Laplacian in the general situation. Its domain is
∗
, d∗ u ∈ V k−1 }.
DL = { u ∈ V k ∩ Vk∗ | du ∈ Vk+1
If u solves Lu = f , then
(17)
du, dv + d∗ u, d∗ v = f, v,
v ∈ V k ∩ Vk∗ .
Note that, in this equation, and henceforth, we use · , · and · without subscripts, meaning the inner product and norm in the appropriate W k space.
The harmonic functions measure the extent to which the Hodge Laplacian problem (17) is well-posed. The solutions to the homogeneous problem (f = 0) are
precisely the functions in Hk . Moreover, a necessary condition for a solution to
exist for nonzero f ∈ W k is that f ⊥ Hk .
For computational purposes, a formulation of the Hodge Laplacian based on (17)
may be problematic, even when there are no harmonic forms, because it may not be
possible to construct an efficient finite element approximation for the space V k ∩Vk∗ .
We have already seen an example of this in the discussion of the approximation of
a boundary value problem for the vector Laplacian in Section 2.3.2. Instead we
introduce another formulation, which is a generalization of the mixed formulation
discussed in Section 2 and which, simultaneously, accounts for the nonuniqueness
associated with harmonic forms. With (W, d) a Hilbert complex, (V, d) the associated domain complex, and f ∈ W k given, we define the mixed formulation of
the abstract Hodge Laplacian as the problem of finding (σ, u, p) ∈ V k−1 × V k × Hk
satisfying
σ, τ − dτ, u = 0,
(18)
τ ∈ V k−1 ,
dσ, v + du, dv + v, p = f, v, v ∈ V k ,
u, q = 0,
q ∈ Hk .
Remark. The equations (18) are the Euler–Lagrange equations associated to a
variational problem. Namely, if we define the quadratic functional I : V k−1 × V k ×
Hk → R by
1
1
I(τ, v, q) = τ, τ − dτ, v − dv, dv − v, q + f, v,
2
2
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then a point (σ, u, p) ∈ V k−1 × V k × Hk is a critical point of I if and only if (18)
holds, and in this case,
I(σ, v, q) ≤ I(σ, u, p) ≤ I(τ, u, p),
(τ, v, q) ∈ V k−1 × V k × Hk .
Thus the critical point is a saddle point.
An important result is that if the Hilbert complex is closed, then the mixed
formulation is well-posed. The requirement that the complex is closed is crucial,
since we rely on the Poincaré inequality.
Theorem 3.1. Let (W, d) be a closed Hilbert complex with domain complex (V, d).
The mixed formulation of the abstract Hodge Laplacian is well-posed. That is, for
any f ∈ W k , there exists a unique (σ, u, p) ∈ V k−1 × V k × Hk satisfying (18).
Moreover,
σV + uV + p ≤ cf ,
where c is a constant depending only on the Poincaré constant cP in (16).
We shall prove Theorem 3.1 in Section 3.2.2. First, we interpret the mixed
formulation.
3.2.1. Interpretation of the mixed formulation. The first equation states that u
belongs to the domain of d∗ and d∗ u = σ ∈ V k−1 . The second equation similarly
states that du belongs to the domain of d∗ and d∗ du = f − p − dσ. Thus u belongs
to the domain DL of L and solves the abstract Hodge Laplacian equation
Lu = f − p.
The harmonic form p is simply the orthogonal projection PH f of f onto Hk , required
for existence of a solution. Finally the third equation fixes a particular solution,
through the condition u ⊥ Hk . Thus Theorem 3.1 establishes that for any f ∈ W k
there is a unique u ∈ DL such that Lu = f − PH f and u ⊥ Hk . We define Kf = u,
so the solution operator K : W k → W k is a bounded linear operator mapping into
DL . The solution to the mixed formulation is
σ = d∗ Kf,
u = Kf,
p = PH f.
The mixed formulation (18) is also intimately connected to the Hodge decomposition (15). Since dσ ∈ B k , p ∈ Hk , and d∗ du ∈ B∗k , the expression f = dσ+p+d∗ du
is precisely the Hodge decomposition of f . In other words
PB = dd∗ K,
PB∗ = d∗ dK,
where PB and PB∗ are the W k -orthogonal projections onto Bk and B∗k , respectively. We also note that K commutes with d and d∗ in the sense that
dKf = Kdf, f ∈ V k ,
d∗ Kg = Kd∗ g, g ∈ Vk∗ .
∗
.
Indeed, if f ∈ V k and u = Kf , then u ∈ DL , which implies that du ∈ V k+1 ∩ Vk+1
∗
k−1
∗
∗
k
Also d u ∈ V
, so d du = f − PH f − dd u ∈ V . This shows that du ∈ DL .
Clearly
Ldu = (dd∗ + d∗ d)du = dd∗ du = d(dd∗ + d∗ d)u = dLu = df,
and both du and df are orthogonal to harmonic forms. This establishes that du =
Kdf , i.e., dKf = Kdf . The second equation is established similarly.
If we restrict the data f in the abstract Hodge Laplacian to an element of B∗k
or of Bk , we get two other problems which are also of great use in applications.
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304
D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER
The B∗ problem. If f ∈ B∗k , then u = Kf ∈ B∗k satisfies
d∗ du = f,
u ⊥ Zk ,
while σ = d∗ u = 0, p = PH f = 0. The solution u can be characterized as the
unique element of Zk⊥ such that
du, dv = f, v,
(19)
v ∈ Zk⊥ ,
and any solution to this problem is a solution of Lu = f , and so is uniquely
determined.
The B problem. If f ∈ Bk , then u = Kf ∈ Bk satisfies dd∗ u = f , while p = PH f =
0. With σ = d∗ u, the pair (σ, u) ∈ V k−1 × Bk is the unique solution of
(20)
σ, τ − dτ, u = 0, τ ∈ V k−1 ,
dσ, v = f, v, v ∈ Bk ,
and any solution to this problem is a solution of Lu = f , σ = d∗ u, and so is uniquely
determined.
3.2.2. Well-posedness of the mixed formulation. We now turn to the proof of Theorem 3.1. Let B : X × X → R be a symmetric bounded bilinear form on a Hilbert
space X which satisfies the inf-sup condition
γ :=
inf
sup
0=y∈X 0=x∈X
B(x, y)
> 0.
xX yX
Then the problem of finding x ∈ X such that B(x, y) = F (y) for all y ∈ X is
well-posed: it has a unique solution x for each F ∈ X ∗ , and xX ≤ γ −1 F X ∗
[13]. The abstract Hodge Laplacian problem (18) is of this form, where B : [V k−1 ×
V k × Hk ] × [V k−1 × V k × Hk ] → R denotes the bounded bilinear form
B(σ, u, p; τ, v, q) = σ, τ − dτ, u + dσ, v + du, dv + v, p − u, q,
and F (τ, v, p) = f, v.
The following theorem establishes the inf-sup condition and so implies Theorem 3.1.
Theorem 3.2. Let (W, d) be a closed Hilbert complex with domain complex (V, d).
There exists a constant γ > 0, depending only on the constant cP in the Poincaré
inequality (16), such that for any (σ, u, p) ∈ V k−1 × V k × Hk , there exists (τ, v, q) ∈
V k−1 × V k × Hk with
B(σ, u, p; τ, v, q) ≥ γ(σV + uV + p)(τ V + vV + q).
Proof. By the Hodge decomposition, u = uB + uH + u⊥ , where uB = PB u, uH =
PH u, and u⊥ = PB∗ u. Since uB ∈ Bk , uB = dρ, for some ρ ∈ Zk−1⊥ . Since
du⊥ = du, we get using (16) that
ρV ≤ cP uB ,
(21)
u⊥ V ≤ cP du,
where cP ≥ 1 is the constant in Poincaré’s inequality. Let
(22)
τ =σ−
1
ρ ∈ V k−1 ,
c2P
v = u + dσ + p ∈ V k ,
q = p − uH ∈ H k .
From (21) and the orthogonality of the Hodge decomposition, we have
(23)
τ V + vV + q ≤ C(σV + uV + p).
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FINITE ELEMENT EXTERIOR CALCULUS
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We also get, from a simple computation using (21) and (22), that
B(σ, u, p; τ, v, q)
1
1
uB 2 − 2 σ, ρ
c2P
cP
1
1
1
σ2 + dσ2 + du2 + p2 + uH 2 + 2 uB 2 − 4 ρ2
2
cP
2cP
1
1
σ2 + dσ2 + du2 + p2 + uH 2 + 2 uB 2
2
2cP
1
1
1
1
σ2 + dσ2 + du2 + p2 + uH 2 + 2 uB 2 + 2 u⊥ 2
2
2
2cP
2cP
1
(σ2V + u2V + p2 ).
2c2P
= σ2 + dσ2 + du2 + p2 + uH 2 +
≥
≥
≥
≥
The theorem easily follows from this bound and (23).
We close this section with two remarks. First we note that in fact Theorem 3.2
establishes more than the well-posedness of the problem (18) stated in Theorem 3.1.
It establishes that, for any G ∈ (V k−1 )∗ , F ∈ (V k )∗ , and R ∈ (Hk )∗ (these are
the dual spaces furnished with the dual norms), there exists a unique (σ, u, p) ∈
V k−1 × V k × Hk satisfying
σ, τ − dτ, u = G(τ ), τ ∈ V k−1 ,
dσ, v + du, dv + v, p = F (v),
v ∈ V k,
u, q = R(q),
q ∈ Hk ,
and moreover the correspondence (σ, u, p) ↔ (F, G, R) is an isomorphism of V k−1 ×
V k × Hk onto its dual space.
Second, we note that the above result bears some relation to a fundamental
result in the theory of mixed finite element methods, due to Brezzi [24], which we
state here.
Theorem 3.3. Let X and Y be Hilbert spaces and a : X × X → R, b : X × Y → R
bounded bilinear forms. Let Z = { x ∈ X | b(x, y) = 0 ∀y ∈ Y }, and suppose that
there exist positive constants α and γ such that
(1) (coercivity in the kernel) a(z, z) ≥ αz2X , z ∈ Z,
b(x, y)
(2) (inf-sup condition) inf
sup
≥ γ.
0=y∈Y 0=x∈X xX yY
Then, for all G ∈ X ∗ , F ∈ Y ∗ , there exists a unique u ∈ X, v ∈ Y such that
(24)
a(u, x) + b(x, v) = G(x),
b(u, y) = F (y),
x ∈ X,
y ∈ Y.
Moreover, uX + vY ≤ c(GX ∗ + F Y ∗ ) with the constant c depending only
on α, γ, and the norms of the bilinear forms a and b.
If we make the additional assumption (usually satisfied in applications of this
theorem) that the bilinear form a is symmetric and satisfies a(x, x) > 0 for all
0 = x ∈ X, then this theorem can be viewed as a special case of Theorem 3.2.
In fact, we define W 0 as the completion of X in the inner product given by a, let
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306
D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER
W 1 = Y , and define d as the closed linear operator from W 0 to W 1 with domain
X given by
dx, yY = b(x, y), x ∈ X, y ∈ Y.
In this way we obtain a Hilbert complex (with just two spaces W 0 and W 1 ). The
inf-sup condition of Theorem 3.3 implies that d has closed range, so it is a closed
Hilbert complex, and so Poincaré’s inequality holds. The associated abstract Hodge
Laplacian is just the system (24), and from Theorem 3.2 and the first remark above,
we have
a(u, u) + duY + vY ≤ c(GX ∗ + F Y ∗ ).
⊥
But, using the coercivity in the kernel,
the decomposition X = Z + Z , and
Poincaré’s inequality, we get uX ≤ C[ a(u, u)+duY ], which gives the estimate
from Brezzi’s theorem. Finally, we mention that we could dispense with the extra
assumption about the symmetry and positivity of the bilinear form a, but this
would require a slight generalization of Theorem 3.2, which we do not consider
here.
3.3. Approximation of Hilbert complexes. The remainder of this section will
be devoted to the approximation of quantities associated to a Hilbert complex, such
as the cohomology spaces, harmonic forms, and solutions to the Hodge Laplacian,
by quantities associated to a subcomplex.
Let (W, d) be a Hilbert complex with domain complex (V, d), and suppose we
choose a finite-dimensional subspace Vhk of V k for each k. We assume that dVhk ⊂
Vhk+1 so that (Vh , d) is a subcomplex of (V, d). We also take Whk to be the same
subspace Vhk but endowed with the norm of W k . In this way we obtain a closed
(even bounded) Hilbert complex (Wh , d) with domain complex (Vh , d), and all the
results of Sections 3.1.3 and 3.2 apply to this subcomplex. Although the differential
for the subcomplex is just the restriction of d, and so does not need a new notation,
its adjoint d∗h : Vhk+1 → Vhk , defined by
d∗h u, v = u, dv,
u ∈ Vhk+1 , v ∈ Vhk ,
is not the restriction of d∗ . We use the notation Bh , Zh , B∗h = Z⊥
h , Hh , Kh , with the
obvious meanings. We use the term discrete when we wish to emphasize quantities
associated to the subcomplex. For example, we refer to Hkh as the space of discrete
harmonic k-forms, and the discrete Hodge decomposition is
Vhk = Bkh ⊕ Hkh ⊕ Zk⊥
h .
satisfy PBh = dd∗h Kh
The W k -projections PBh : W k → Bkh , PB∗h : W k → Zk⊥
h
∗
k
and PB∗h = dh dKh , respectively, when restricted to Vh . We also have that Kh
commutes with both d and d∗h . Note that Bkh ⊂ Bk and Zkh ⊂ Zk , but in general
k⊥
Hkh is not contained in Hk , nor is Zk⊥
.
h contained in Z
k
In order that the subspaces Vh can be used effectively to approximate quantities
associated to the original complex, we require not only that they form a subcomplex,
but of course we need to know something about the approximation of V k by Vhk ,
i.e., an assumption that inf v∈Vhk u−vV is sufficiently small for some or all u ∈ V k .
A third assumption, which plays an essential role in our analysis, is that there exists
a bounded cochain projection πh from the complex (V, d) to the subcomplex (Vh , d).
Explicitly, for each k, πhk maps V k to Vhk , leaves the subspace invariant, satisfies
dk πhk = πhk+1 dk , and there exists a constant c such that πhk vV ≤ cvV for all
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FINITE ELEMENT EXTERIOR CALCULUS
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v ∈ V k . In other words, we have the following commuting diagram relating the
complex (V, d) to the subcomplex (Vh , d):
d
0 → V 0 −−−−→
⏐
⏐π
h
d
d
d
d
V 1 −−−−→ · · · −−−−→
⏐
⏐π
h
d
Vn →0
⏐
⏐π
h
0 → Vh0 −−−−→ Vh1 −−−−→ · · · −−−−→ Vhn → 0.
Note that a bounded projection gives quasi-optimal approximation:
u − πh uV = inf (I − πh )(u − v)V ≤ c inf u − vV .
v∈Vhk
v∈Vhk
We now present two results indicating that, under these assumptions, the space
Hkh of discrete harmonic forms provides a faithful approximation of Hk . In the first
result we show that a bounded cochain projection into a subcomplex of a bounded
closed Hilbert complex which satisfies a rather weak approximability assumption
(namely (25) below), induces, not only a surjection, but an isomorphism on cohomology.
Theorem 3.4. Let (V, d) be a bounded closed Hilbert complex, (Vh , d) a Hilbert
subcomplex, and πh a bounded cochain projection. Suppose that for all k,
(25)
q − πhk qV < qV ,
0 = q ∈ Hk .
Then the induced map on cohomology is an isomorphism.
Proof. We already know that the induced map is a surjection, so it is sufficient to
prove that it is an injection. Thus, given z ∈ Zk with πh z ∈ Bkh , we must prove
that z ∈ Bk . By the Hodge decomposition, z = q + b with q ∈ Hk and b ∈ Bk . We
have that πh z ∈ Bkh by assumption and πh b ∈ Bkh since b ∈ Bk and πh is a cochain
map. Thus πh q = πh z − πh b ∈ Bkh ⊂ Bk , and so πh q ⊥ q. In view of (25), this
implies that q = 0, and so z ∈ Bk , as desired.
Remark. In applications, the space of harmonic forms, Hk , is a finite-dimensional
space of smooth functions, and πh is a projection operator associated to a triangulation with mesh size h. The estimate (25) will then be satisfied for h sufficiently
small. However, in the most important application, in which (V, d) is the de Rham
complex and (Vh , d) is a finite element discretization, πh induces an isomorphism on
cohomology not only for h sufficiently small, but in fact for all h. See Section 5.6.
The second result relating Hk and Hkh is quantitative in nature, bounding the distance, or gap, between these two spaces. Recall that the gap between two subspaces
E and F of a Hilbert space V is defined [64, Chapter IV, §2.1] by
(26)
gap(E, F ) = max sup inf u − vV , sup inf u − vV .
u∈E v∈F
u=1
v∈F u∈E
v=1
Theorem 3.5. Let (V, d) be a bounded closed Hilbert complex, (Vh , d) a Hilbert
subcomplex, and πh a bounded cochain projection. Then
(27)
(I − PHh )qV ≤ (I − πhk )qV ,
(28)
(I − PH )qV ≤ (I − πhk )PH qV ,
q ∈ Hk ,
q ∈ Hkh ,
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308
D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER
gap Hk , Hkh ≤ sup (I − πhk )qV .
(29)
q∈Hk
q=1
Proof. Given q ∈ Hk , PHh q = PZh q, since Zkh = Hkh ⊕ Bkh and q ⊥ Bk ⊃ Bkh . Also
πhk q ∈ Zkh , since πh is a cochain map. This implies (27).
If q ∈ Hkh ⊂ Zkh ⊂ Zk , the Hodge decomposition gives us q − PH q ∈ Bk , so
k
πh (q − PH q) ∈ Bkh , and so is orthogonal to both the discrete harmonic form q and
the harmonic form PH q. Therefore
q − PH qV ≤ q − PH q − πh (q − PH q)V = (I − πhk )PH qV .
Finally (29) is an immediate consequence of (27) and (28).
Next we deduce another important property of a Hilbert subcomplex with a
bounded cochain projection. Since the complex (V, d) is closed and bounded, the
Poincaré inequality (16) holds (with the W and V norms coinciding). Now we
obtain the Poincaré inequality for the subcomplex with a constant that depends
only on the Poincaré constant for the supercomplex and the norm of the cochain
projection. In the applications, we will have a sequence of such subcomplexes
related to a decreasing mesh size parameter, and this theorem will imply that the
discrete Poincaré inequality is uniform with respect to the mesh parameter, an
essential step in proving stability for numerical methods.
Theorem 3.6. Let (V, d) be a bounded closed Hilbert complex, (Vh , d) a Hilbert
subcomplex, and πhk a bounded cochain projection. Then
vV ≤ cP πhk dvV ,
v ∈ Zk⊥
h ,
where cP is the constant appearing in the Poincaré inequality (16) and πhk denotes
the V k operator norm of πhk .
k⊥
Proof. Given v ∈ Zk⊥
⊂ V k by dz = dv. By (16), z ≤ cP dv, so
h , define z ∈ Z
it is enough to show that vV ≤ πh zV . Now, v −πh z ∈ Vhk and d(v −πh z) = 0,
so v − πh z ∈ Zkh . Therefore
v2V = v, πh zV + v, v − πh zV = v, πh zV ≤ vV πh zV ,
and the result follows.
We have established several important properties possessed by a subcomplex of a
bounded closed Hilbert complex with bounded cochain projection. We also remark
that from (28) and the triangle inequality, we have
qV ≤ cPH qV ,
(30)
q ∈ Hkh .
We close this section by presenting a converse result. Namely we show that if
the discrete Poincaré inequality and the bound (30) hold, then a bounded cochain
projection exists.
Theorem 3.7. Let (V, d) be a bounded closed Hilbert complex and (Vh , d) a subcomplex. Assume that
vV ≤ c1 dvV ,
v ∈ Zk⊥
h ,
and
qV ≤ c2 PH qV ,
q ∈ Hkh ,
for some constants c1 and c2 . Then there exists a bounded cochain projection πh
from (V, d) to (Vh , d), and the V operator norm πh can be bounded in terms of
c1 and c2 .
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FINITE ELEMENT EXTERIOR CALCULUS
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Proof. As a first step of the proof we define an operator Qh : V k → Zk⊥
by
h
dQh v = PBh dv. By the first assumption, this operator is V -bounded since
Qh vV ≤ c1 PBh dv ≤ c1 vV ,
and if v ∈ Vh , then Qh v = PB∗h v. By the second assumption, the operator PH |Hh has
a bounded inverse Rh mapping Hk to Hkh . Note that for any v ∈ Zkh , Rh PH v = PHh v.
We now define πh : V k → Vhk by
πh = PBh + Rh PH (I − Qh ) + Qh .
This operator is bounded in V k and invariant on Vhk , since the three terms correspond exactly to the discrete Hodge decomposition in this case. Furthermore,
πh dv = PBh dv = dQh v = dπh v, so πh is indeed a bounded cochain projection. 3.4. Stability and convergence of the mixed method. Next we consider a
closed Hilbert complex (W, d) and the Galerkin discretization of its Hodge Laplacian using finite-dimensional subspaces Vhk of the domain spaces V k . Our main
assumptions are those of Section 3.3: first, that dVhk ⊂ Vhk+1 , so that we obtain a
subcomplex
d
d
d
0 → Vh0 −
→ Vh1 −
→ ··· −
→ Vhn → 0,
and, second, that there exists a bounded cochain projection πh from (V, d) to (Vh , d).
Let f ∈ W k . In view of the mixed formulation (18), we take as an approximation
scheme: find σh ∈ Vhk−1 , uh ∈ Vhk , ph ∈ Hkh , such that
σh , τ − dτ, uh = 0,
(31)
τ ∈ Vhk−1 ,
dσh , v + duh , dv + v, ph = f, v,
uh , q = 0,
v ∈ Vhk ,
q ∈ Hkh .
(Recall that we use · , · and · without subscripts for the W inner product
and norm.) The discretization (31) is a generalized Galerkin method as discussed
in Section 2.2. In the case that there are no harmonic forms (and therefore no
discrete harmonic forms), it is a Galerkin method, but in general not, since Hkh is
not in general a subspace of Hk . We may write the solution of (31), which always
exists and is unique in view of the results of Section 3.2.2, as
uh = Kh Ph f,
σh = d∗h uh ,
ph = PHh f,
where Ph : W k → Vhk is the W k -orthogonal projection.
As in Section 2.2, we will bound the error in terms of the stability of the discretization and the consistency error. We start by establishing a lower bound on
the inf-sup constant, i.e., an upper bound on the stability constant.
Theorem 3.8. Let (Vh , d) be a family of subcomplexes of the domain complex (V, d)
of a closed Hilbert complex, parametrized by h and admitting uniformly V -bounded
cochain projections. Then there exists a constant γh > 0, depending only on cP and
the norm of the projection operators πh , such that for any (σ, u, p) ∈ Vhk−1 ×Vhk ×Hkh ,
there exists (τ, v, q) ∈ Vhk−1 × Vhk × Hkh with
B(σ, u, p; τ, v, q) ≥ γh (σV + uV + p)(τ V + vV + q).
Proof. This is just Theorem 3.2 applied to the Hilbert complex (Vh , d), combined
with the fact that the constant in the Poincaré inequality for Vhk is cP πh by
Theorem 3.6.
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310
D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER
From this stability result, we obtain the following error estimate.
Theorem 3.9. Let (Vh , d) be a family of subcomplexes of the domain complex (V, d)
of a closed Hilbert complex, parametrized by h and admitting uniformly V -bounded
cochain projections, and let (σ, u, p) ∈ V k−1 × V k × Hk be the solution of problem
(18) and (σh , uh , ph ) ∈ Vhk−1 × Vhk × Hkh the solution of problem (31). Then
σ − σh V + u − uh V + p − ph ≤ C( inf
τ ∈Vhk−1
σ − τ V + inf u − vV + inf p − qV + µ inf PB u − vV ),
v∈Vhk
q∈Vhk
v∈Vhk
where µ = µkh = sup (I − πhk )r.
r∈Hk
r=1
Proof. First observe that (σ, u, p) satisfies
B(σ, u, p; τh , vh , qh ) = f, vh − u, qh ,
(τh , vh , qh ) ∈ Vhk−1 × Vhk × Hkh .
Let τ , v, and q be the V -orthogonal projections of σ, u, and p into Vhk−1 , Vhk , and
Hkh , respectively. Then, for any (τh , vh , qh ) ∈ Vhk−1 × Vhk × Hkh , we have
B(σh − τ, uh − v, ph − q; τh , vh , qh )
= B(σ − τ, u − v, p − q; τh , vh , qh ) + u, qh = B(σ − τ, u − v, p − q; τh , vh , qh ) + PHh u, qh ≤ C(σ − τ V + u − vV + p − q + PHh u)(τh V + vh V + qh ).
Theorem 3.8 then gives
(32) σh − τ V + uh − vV + ph − q
≤ C(σ − τ V + u − vV + p − q + PHh u).
Using (27) and the boundedness of the projection πh we have
p − q ≤ (I − πh )p ≤ C inf p − qV .
(33)
q∈Vhk
Next we show that
PHh u ≤ µ(I − πh )uB V .
Now u ⊥ H , so u = uB + u⊥ , with uB ∈ Bk and u⊥ ∈ Zk⊥ . Since Hkh ⊂ Zk ,
PHh u⊥ = 0, and since πh uB ∈ Bkh , PHh πh uB = 0. Let q = PHh u/PHh u ∈ Hkh .
By Theorem 3.5, there exists r ∈ Hk (and so r ⊥ Bk ) with r ≤ 1 and
k
q − r ≤ (I − πh )r ≤ sup (I − πh )r.
r∈Hk
r=1
Therefore
(34) PHh u = uB − πh uB , q − r
≤ (I − πh )uB sup (I − πh )r ≤ c µ inf PB u − vV ),
r∈Hk
r=1
v∈Vhk
since πh is a bounded projection. The theorem follows from (32)–(34) and the
triangle inequality.
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To implement the discrete problem, we need to be able to compute the discrete
harmonic forms. The following lemma shows one way to do this; namely, it shows
that the discrete harmonic forms can be computed as the elements of the null space
of a matrix. For finite element approximations of the de Rham sequence, which
is the most canonical example of this theory and which will be discussed below, it
is often possible to compute the discrete harmonic forms more directly. See, for
example, [2].
Lemma 3.10. Consider the homogeneous linear system: find (σh , uh ) ∈ Vhk−1 ×Vhk
such that
σh , τ = dτ, uh ,
τ ∈ Vhk−1 ,
dσh , v + duh , dv = 0,
v ∈ Vhk .
Then (σh , uh ) is a solution if and only if σh = 0 and uh ∈ Hkh .
Proof. Clearly (0, uh ) is a solution if uh ∈ Hkh . On the other hand, if (σh , uh ) is a
solution, by taking τ = σh , v = uh , and combining the two equations, we find that
σh 2 + duh 2 = 0, so that σh = 0 and duh = 0. Then the first equation implies
that dτ, uh = 0 for all τ ∈ Vhk−1 , so indeed uh ∈ Hkh .
3.5. Improved error estimates. Suppose we have a family of subcomplexes
(Vh , d) of the domain complex (V, d) of a closed Hilbert complex, parametrized by h
with uniformly bounded cochain projections πh . Assuming also that the subspaces
Vhk are approximating in V k in the sense of (8), we can conclude from Theorem 3.9
that σh → σ, uh → u, and ph → p as h → 0 (in the norms of V k−1 and V k ). In
other words, the Galerkin method for the Hodge Laplacian is convergent.
The rate of convergence will depend on the approximation properties of the
subspaces Vhk , the particular component considered (σh , uh , or ph ), the norm in
which we measure the error (e.g., W or V ), as well as properties of the data f and the
corresponding solution. For example, in Section 5, we will consider approximation
of the de Rham complex using various subcomplexes for which the spaces Vhk consist
of piecewise polynomial differential forms with respect to a triangulation Th of the
domain with mesh size h. The space W k is the space of L2 differential k-forms
in this case. One possibility we consider for the solution of the Hodge Laplacian
for k-forms using the mixed formulation is to take subspaces Pr+1 Λk−1 (Th ) and
Pr Λk (Th ). The space Pr Λk (Th ), which is defined in Section 5.2, consists piecewise
of all k-forms of polynomial degree at most r. Assuming that the solution u to the
Hodge Laplacian is sufficiently smooth, an application of Theorem 3.9 will give, in
this case,
σ − σh V + u − uh V + p − ph = O(hr ).
Approximation theory tells us that this rate is the best possible for u − uh V , but
we might hope for a faster rate for u − uh and for σ − σh V and σ − σh .
In order to obtain improved error estimates, we make two additional assumptions, first that the complex (W, d) satisfies the compactness property introduced
at the end of Section 3.1, and second, that the cochain projection is bounded not
only in V but in W :
• The intersection V k ∩ Vk∗ is a dense subset of W k with compact inclusion.
• The cochain projections πhk are bounded in L(W k , W k ) uniformly with
respect to h.
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312
D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER
The second property implies that πh extends to a bounded linear operator W k →
Vhk . Since the subspaces Vhk are approximating in W k as well as in V k (by density),
it follows that πh converges pointwise to the identity in W k . Finally, note that if
we are given a W -bounded cochain projection mapping W k → Vhk , the restrictions
to V k define a V -bounded cochain projection.
Next, we note that on the Hilbert space V k ∩ Vk∗ , the inner product given by
u, vV ∩V ∗ := d∗ u, d∗ v + du, dv + PH u, PH v
is equivalent to the usual intersection inner product, which is the sum of the inner
products for V k and Vk∗ . This can be seen by Hodge decomposing u as PB u +
PH u + PB∗ u, and using the Poincaré bound PB∗ u ≤ cdPB∗ u = cdu and the
analogous bound PB u ≤ cd∗ PB u = cd∗ u. Now K maps W k boundedly into
V k ∩ Vk∗ and satisfies
Kf, vV ∩V ∗ = f, v − PH v,
f ∈ W k , v ∈ V k ∩ Vk∗ .
In other words, K ∈ L(W k , V k ∩ Vk∗ ) is the adjoint of the operator (I − PH )I ∈
L(V k ∩ Vk∗ , W k ), where I ∈ L(V k ∩ Vk∗ , W k ) is the compact inclusion operator.
Hence K is a compact operator W k → V k ∩ Vk∗ and, a fortiori, compact as an
operator from W k to itself. As an operator on W k , K is also selfadjoint, since
f, Kg = f − PH f, Kg = dd∗ Kf + d∗ dKf, Kg = d∗ Kf, d∗ Kg + dKf, dKg
for all f, g ∈ W k . Furthermore, if we follow K by one of the bounded operators
d : V k → W k+1 or d∗ : Vk∗ → W k−1 , the compositions dK and d∗ K are also
compact operators from W k to itself. Since we have assumed the compactness
property, dim Hk < ∞, and so PHk is also a compact operator on W k . Define
δ = δhk = (I − πh )KL(W k ,W k ) ,
µ = µkh = (I − πh )PH L(W k ,W k ) ,
η = ηhk = max [(I − πh )dKL(W k−j ,W k−j+1 ) , (I − πh )d∗ KL(W k+j ,W k+j−1 ) ].
j=0,1
(Note that µ already appeared in Theorem 3.9.) Recalling that composition on the
right with a compact operator converts pointwise convergence to norm convergence,
we see that η, δ, µ → 0 as h → 0. In the applications in Sections 5 and 6, the
spaces Λkh will consist of piecewise polynomials. We will then have η = O(h), δ =
O(hmin(2,r+1) ), and µ = O(hr+1 ), where r denotes the largest degree of complete
polynomials in the space Λkh .
In Theorem 3.9, the error estimates were given in terms of the best approximation
error afforded by the subspaces in the V norm. The improved error estimates will
be in terms of the best approximation error in the W norm, for which we introduce
the notation
E(w) = Ehk (w) = inf w − v, w ∈ W k .
v∈Vhk
The following theorem gives the improved error estimates. Its proof incorporates
a variety of techniques developed in the numerical analysis literature in recent
decades, e.g., [48, 41, 8].
Theorem 3.11. Let (V, d) be the domain complex of a closed Hilbert complex (W, d)
satisfying the compactness property, and let (Vh , d) be a family of subcomplexes
parametrized by h and admitting uniformly W -bounded cochain projections. Let
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FINITE ELEMENT EXTERIOR CALCULUS
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(σ, u, p) ∈ V k−1 × V k × Hk be the solution of problem (18) and (σh , uh , ph ) ∈
Vhk−1 ×Vhk ×Hkh the solution of problem (31). Then for some constant C independent
of h and (σ, u, p), we have
(35)
d(σ − σh ) ≤ CE(dσ),
(36)
σ − σh ≤ C[E(σ) + ηE(dσ)],
(37)
p − ph ≤ C[E(p) + µE(dσ)],
(38)
d(u − uh ) ≤ C(E(du) + η[E(dσ) + E(p)]),
(39)
u − uh ≤ C(E(u) + η[E(du) + E(σ)] + (η 2 + δ)[E(dσ) + E(p)] + µE(PB u)]).
We now develop the proof of Theorem 3.11 in a series of lemmas.
Lemma 3.12. Let vh ∈ Zk⊥
h and v = PB∗ vh . Then
v − vh ≤ (I − πhk )v ≤ ηdvh .
Proof. Since πh v−vh ∈ Zkh ⊂ Zk , vh −v ⊥ πh v−vh , so, by the Pythagorean theorem,
vh − v ≤ (I − πh )v. The second inequality holds since v = d∗ Kdvh .
Lemma 3.13. The estimates (35) and (36) hold. Moreover,
PBh (u − uh ) ≤ C[ηE(σ) + (η 2 + δ)E(dσ)].
(40)
Proof. Since dσh = PBh f = PBh PB f = PBh dσ, we have
d(σ − σh ) = (I − PBh )dσ ≤ (I − πh )dσ ≤ CE(dσ),
giving (35). To prove (36) we write
σ = d∗ Kdσ = d∗ K(I − PBh )dσ + d∗ KPBh dσ =: σ 1 + σ 2 .
Taking τ = πh σ 2 − σh in (18) and (31), we obtain
σ − σh , πh σ2 − σh = d(πh σ2 − σh ), u − uh = 0.
Hence,
σ − σh ≤ σ − πh σ 2 ≤ (I − πh )σ + πh σ 1 ≤ CE(σ)| + Cσ 1 .
Since σ 1 ∈ B∗k ,
σ 1 2 = Kdσ 1 , dσ 1 = Kdσ 1 , (I − PBh )dσ = (I − PBh )Kdσ 1 , (I − PBh )dσ
≤ (I − πh )Kdσ 1 (I − πh )dσ ≤ Cησ 1 E(dσ).
Then (36) follows from the last two estimates.
Let e = PBh (u − uh ). To estimate e, we set w = Ke, φ = d∗ w, wh = Kh e,
φh = d∗h wh . Then dφ = dπh φ = dφh = e, so πh φ − φh ∈ Zkh , and so is orthogonal
to φ − φh . Thus φ − φh ≤ (I − πh )φ = (I − πh )d∗ Ke. Then
e2 = dφh , e = dφh , u − uh = σ − σh , φh = σ − σh , φh − φ + d(σ − σh ), w
= σ − σh , φh − φ + (I − PBh )dσ, (I − PBh )w
≤ σ − σh φh − φ + (I − πh )dσ(I − πh )w
≤ σ − σh (I − πh )d∗ Ke + CE(dσ)(I − πh )Ke
≤ [ησ − σh + CE(dσ)δ]e.
Combining with (36) we get (40).
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314
D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER
Lemma 3.14. Estimate (37) holds and, moreover, PHh u ≤ CµE(PBu).
Proof. The second estimate is just (34). Using the Hodge decomposition of f and
the fact that PHh B∗k = 0, we have PHh p − ph = PHh PH f − PHh f = PHh fB , where
fB = PB f . Therefore,
p − ph = p − PHh p + PHh fB ≤ CE(p) + PHh fB ,
by (27). Applying (28) we get
PHh fB 2 = fB − πh fB , PHh fB = fB − πh fB , PHh fB − PH PHh fB ≤ C(I − πh )fB (I − πh )PH PHh fB ≤ CE(dσ)µPHh fB .
Combining these results completes the proof of the lemma.
Lemma 3.15. The estimate (38) holds.
Proof. From (18) and (31),
d(σ − σh ), vh + d(u − uh ), dvh + vh , p − ph = 0,
(41)
vh ∈ Vhk .
Choose vh = PB∗h (πh u − uh ) in (41) and set v = d∗ Kdvh = PB∗ vh .
Lemma 3.12 and (27), we get
Using
d(u − uh ), d(πh u − uh ) = d(u − uh ), dvh = −d(σ − σh ) + (p − ph ), vh = −d(σ − σh ) + (p − PHh p), vh − v ≤ [d(σ − σh ) + p − PHh p]vh − v
≤ C[E(dσ) + E(p)]ηdvh ≤ Cη[E(dσ) + E(p)]d(πh u − uh ).
Hence,
d(πh u − uh )2 = d(πh u − u), d(πh u − uh ) + d(u − uh ), d(πh u − uh )
≤ {d(πh u − u) + Cη[E(dσ) + E(p)]}d(πh u − uh )
≤ C{E(du) + η[E(dσ) + E(p)]}d(πh u − uh ).
The result follows by the triangle inequality.
Lemma 3.16. PB∗h (u − uh ) ≤ C{E(u) + ηE(du) + (η 2 + δ)[E(dσ) + E(p)]}.
Proof. Again letting vh = PB∗h (πh u − uh ) and v = PB∗ vh , we observe that
PB∗h (u − uh ) ≤ PB∗h (u − πh u) + vh ≤ u − πh u + vh ≤ CE(u) + vh .
Next,
vh 2 = vh − v, vh + v, πh u − uh = vh − v, vh + v, πh u − u + v, u − uh .
Then using Lemmas 3.12 and 3.15, we get
vh − v, vh + v, πh u − u ≤ vh − vvh + v(I − πh )u
≤ [ηdvh + (I − πh )u]vh ≤ [ηd(πh u − uh ) + CE(u)]vh .
We next estimate the term v, u − uh . Since dv = dvh and Kv ∈ Zk⊥ , we get
v, u − uh = Kdvh , d(u − uh ) = Kdv, d(u − uh ) = dKv, d(u − uh )
= (I − πh )dKv, d(u − uh ) + dπh Kv, d(u − uh )
= (I − πh )dKv, d(u − uh ) + dPB∗h πh Kv, d(u − uh ).
Now
(I − πh )dKv, d(u − uh ) ≤ ηvd(u − uh ) ≤ Cηd(u − uh )vh ,
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FINITE ELEMENT EXTERIOR CALCULUS
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while
dPB∗h πh Kv, d(u − uh ) = −d(σ − σh ) + (p − ph ), PB∗h πh Kv − Kv
= −d(σ − σh ) + (p − PHh p), PB∗h πh Kv − Kv
≤ [d(σ − σh ) + p − PHh p]PB∗h πh Kv − Kv.
But
PB∗h πh Kv − Kv2 = PB∗h πh Kv − Kv, πh Kv − Kv
≤ PB∗h πh Kv − Kv(I − πh )Kv ≤ PB∗h πh Kv − Kvδv.
Hence,
PB∗h πh Kv − Kv ≤ Cδvh .
Combining these results, and using the previous lemmas, we obtain
vh ≤ C(ηd(πh u − uh ) + ηd(u − uh ) + E(u) + δ[d(σ − σh ) + p − PHh p])
≤ C{E(u) + ηE(du) + (η 2 + δ)[E(dσ) + E(p)]}.
The final result of the lemma follows immediately.
It is now an easy matter to prove (39) and so complete the proof Theorem 3.11.
We write
u − uh = (u − Ph u) + PBh (u − uh ) + PHkh u + PB∗h (u − uh ),
and so (39) follows from Lemmas 3.13, 3.14, and 3.16.
To get a feeling for these results, we return to the example mentioned earlier,
where Vhk−1 = Pr+1 Λk−1 (Th ) and Vhk = Pr Λk (Th ) are used to approximate the
k-form Hodge Laplacian with some r ≥ 1. If the domain is convex, we may apply
elliptic regularity to see that Kf belongs to the Sobolev space H 2 Λk for f ∈ L2 Λk ,
so dKf ∈ H 1 Λk+1 and d∗ Kf ∈ H 1 Λk−1 , and then standard approximation theory
shows that η = O(h), δ = O(h2 ), and µ = O(h2 ). From Theorem 3.11, we then
obtain that
σ − σh + hd(σ − σh ) + hu − uh + hp − ph + h2 d(u − uh ) = O(hr+2 ),
assuming the solution u is sufficiently smooth. That is, all components converge
with the optimal order possible given the degree of the polynomial approximation.
Finally we note a corollary of Theorem 3.11, which will be useful in the analysis
of the eigenvalue problem.
Corollary 3.17. Under the assumptions of Theorem 3.11, there exists a constant
C such that
K − Kh Ph L(W k ,W k ) ≤ C(η 2 + δ + µ),
dK − dKh Ph L(W k ,W k+1 ) + d∗ K − d∗h Kh Ph L(W k ,W k−1 ) ≤ Cη.
Therefore all three operator norms converge to zero with h.
Proof. Let f ∈ W k and set u = Kf , σ = d∗ Kf , p = PH f , and uh = Kh Ph f ,
σh = d∗h Kh Ph f . The desired bounds on (K − Kh Ph )f = u − uh , d(K − Kh Ph )f =
d(u − uh ), and (d∗ K − d∗h Kh Ph )f = σ − σh follow from Theorem 3.11, since
E(u) ≤ δf ,
E(du) + E(σ) ≤ ηf ,
E(dσ) + E(p) + E(PB u) ≤ f .
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316
D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER
We end this section with an estimate of the difference between the true Hodge
decomposition and the discrete Hodge decomposition of an element of Vhk . While
this result is not needed in our approach, such an estimate was central to the
estimation of the eigenvalue error for the Hodge Laplacian using Whitney forms
made in [40].
Lemma 3.18. Let PB u + PH u + PB∗ u and PBh u + PHh u + PB∗h u denote the Hodge
and discrete Hodge decompositions of u ∈ Vhk . Then
PB∗ u − PB∗h u + PH u − PHh u + PB u − PBh u ≤ C(η + µ)uV .
Proof. Let vh = PB∗h u and v = PB∗ PBh u = PB∗ u. The first estimate follows
immediately from Lemma 3.12. To obtain the second estimate, we write
PH u − PHh u = PH (PHh u + PB∗h u) − PHh u = (PH − I)PHh u + PH (PB∗h u − PB∗ u).
The second estimate now follows directly from (28) and the first estimate, and the
final estimate follows from the first two by the triangle inequality.
Applied to the case of Whitney forms, both η and µ are O(h), and so this result
improves the O(h| log h|) estimate of [40, Theorem 2.10].
3.6. The eigenvalue problem. The purpose of this section is to study the eigenvalue problem associated to the abstract Hodge Laplacian (18). As in the previous
section, we will assume that (W, d) is a Hilbert complex satisfying the compactness
property and that the cochain projections πhk are bounded in L(W k , W k ), uniformly
in h. A pair (λ, u) ∈ R×V k , where u = 0, is referred to as an eigenvalue/eigenvector
of the problem (18) if there exists (σ, p) ∈ V k−1 × Hk such that
σ, τ − dτ, u = 0,
(42)
τ ∈ V k−1 ,
dσ, v + du, dv + v, p = λu, v, v ∈ V k ,
u, q = 0,
q ∈ Hk .
In operator terms, u = λKu, σ = d∗ u, p = 0. Note that it follows from this system
that
λu2 = du2 + d∗ u2 > 0,
so Ku = λ−1 u. Since the operator K ∈ L(W k , W k ) is compact and selfadjoint, we
can conclude that the problem (42) has at most a countable set of eigenvalues, each
of finite multiplicity. We denote these by
0 < λ1 ≤ λ2 ≤ · · · ,
where each eigenvalue is repeated according to its multiplicity. Furthermore, when
W k is infinite-dimensional, we have limj→∞ λj = ∞. We denote by {vi } a corresponding orthonormal basis of eigenvectors for W k .
The corresponding discrete eigenvalue problem takes the form
σh , τ − dτ, uh = 0,
(43)
dσh , v + duh , dv + v, ph = λh uh , v,
uh , q = 0,
τ ∈ Vhk−1 ,
v ∈ Vhk ,
q ∈ Hkh ,
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FINITE ELEMENT EXTERIOR CALCULUS
317
where λh ∈ R, and (σh , uh , ph ) ∈ Vhk−1 × Vhk × Hkh , with uh = 0. As above, we can
conclude that ph = 0, λh > 0, and that λ−1
h is an eigenvalue for the operator Kh ,
u
.
We
denote
by
i.e. Kh uh = λ−1
h
h
0 < λ1,h ≤ λ2,h ≤ · · · ≤ λNh ,h ,
Nh = dim Vhk , the eigenvalues of problem (43), repeated according to multiplicity,
and by {vi,h } a corresponding W k -orthonormal eigenbasis for Vhk .
Next, we will study how the discrete eigenvalue problem (43) converges to the
eigenvalue problem (42), i.e., how the eigenvalues and eigenvectors of the operator
Kh converge to those of K. We let Ei and Ei,h denote the one-dimensional spaces
spanned by vi and vi,h . For every positive integer j, let m(j) denote the number of
eigenvalues less than or equal to the jth distinct eigenvalue of the Hodge–Laplace
m(j)
problem (42). Thus i=1
Ei is the space spanned by the eigenspaces associated to
the first j distinct eigenvalues and does not depend on the choice of the eigenbasis.
The discrete eigenvalue problem (43) is said to converge to the exact eigenvalue
problem (42) if, for any > 0 and integer j > 0, there exists a mesh parameter
h0 > 0 such that, for all h ≤ h0 , we have
m(j)
m(j)
Ei ,
Ei,h ≤ ,
(44)
max |λi − λi,h | ≤ and gap
1≤i≤m(j)
i=1
i=1
where the gap between two subspaces E and F of a Hilbert space is defined by
(26). The motivation for this rather strict concept of convergence is that it not only
implies that each eigenvalue is approximated by the appropriate number of discrete
eigenvalues, counting multiplicities, and that the eigenspace is well-approximated
by the corresponding discrete eigenspaces, but it also rules out spurious discrete
eigenvalues and eigenvectors. In particular, this rules out the behavior exemplified
in Figure 2.6 in Section 2.
A key result in the perturbation theory of linear operators is that, for eigenvalue
problems of the form we consider, corresponding to the bounded compact selfadjoint
Hilbert space operators K and Kh , convergence of the eigenvalue approximation
holds if the operators Kh Ph converge to K in L(W k , W k ). This result, widely
used in the theory of mixed finite element eigenvalue approximation, essentially
follows from the contour integral representation of the spectral projection and can
be extracted from [64, Chapters III, IV] or [14, Section 7]. For a clear statement,
see [18]. In fact, as observed in [20], this operator norm convergence is sufficient as
well as necessary for obtaining convergence of the eigenvalue approximations in the
sense above. As a consequence of Corollary 3.17, we therefore obtain the following
theorem.
Theorem 3.19. Let (V, d) be the domain complex of a closed Hilbert complex (W, d)
satisfying the compactness property, and let (Vh , d) be a family of subcomplexes
parametrized by h and admitting uniformly W -bounded cochain projections. Then
the discrete eigenvalue problems (43) converge to the problem (42), i.e., (44) holds.
It is also possible to use the theory developed above to obtain rates of convergence
for the approximation of eigenvalues, based on the following result (Theorem 7.3
of [14]). Here, we consider only the case of a simple eigenvalue, but with a small
modification, the results extend to eigenvalues of positive multiplicity.
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318
D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER
Lemma 3.20. If λ is a simple eigenvalue and u a normalized eigenvector, then
2
|λ−1 − λ−1
h | ≤ C{|(K − Kh Ph )u, u| + (K − Kh Ph )uV }.
Theorem 3.21. Let (V, d) be the domain complex of a closed Hilbert complex (W, d)
satisfying the compactness property, and let (Vh , d) be a family of subcomplexes
parametrized by h and admitting uniformly W -bounded cochain projections. Let λ
be a simple eigenvalue, u the corresponding eigenvector, and λh be the corresponding
discrete eigenvalue. Let w = Ku, φ = d∗ w denote the solution of the mixed formulation of the Hodge Laplacian problem with source term u, and let wh = Kh Ph u,
φh = d∗h wh denote the corresponding discrete solution. Then
2
2
|λ−1 − λ−1
h | ≤ C(w − wh V + φ − φh + |d(φ − φh ), w − wh |).
Proof. To estimate the first term in Lemma 3.20, we write
(K − Kh Ph )u, u = Ku, u − Kh Ph u, Ph u
= w, (d∗ d + dd∗ )w − wh , (d∗h d + dd∗h )wh = dw2 + φ2 − dwh 2 − φh 2
= d(w − wh )2 + 2d(w − wh ), dwh − φ − φh 2 + 2φ − φh , φ
= d(w − wh )2 − φ − φh 2 − 2d(φ − φh ), wh + 2d(φ − φh ), w
= d(w − wh )2 − φ − φh 2 + 2d(φ − φh ), w − wh ,
where we have used the orthogonality condition
d(φ − φh ), v + d(w − wh ), dv = 0,
v ∈ Vhk
in the second to last line above. The theorem then follows from this estimate and
the fact that (K − Kh Ph )u2V = w − wh 2V .
Order of convergence estimates now follow directly from Theorem 3.11, with f
replaced by u, u replaced by w, and σ replaced by φ. In the example following
Theorem 3.11, i.e., Vhk−1 = Pr+1 Λk−1 (Th ) and Vhk = Pr Λk (Th ), we saw that
φ − φh + hd(φ − φh ) + hw − wh + h2 d(w − wh ) = O(hr+2 ),
as long as the domain is convex and the solution w = Ku is sufficiently smooth.
Inserting these results into Theorem 3.21, we find that the eigenvalue error |λ−λh | =
O(h2r ), which is double the rate achieved for the source problem. As another
example, one can check that for the Whitney forms, Vhk−1 = P1− Λk−1 (Th ), Vhk =
P1− Λk (Th ), we get |λ − λh | = O(h2 ), improving on the O(h| log h|) estimate of [40].
Remark. It has long been observed that for the mixed finite element approximation of eigenvalue problems, stability and approximability alone, while sufficient
for convergence of approximations of the source problem, are not sufficient for
convergence of the eigenvalue problem. An extensive literature was developed in
order to obtain eigenvalue convergence, and a wide variety of additional properties of the finite element spaces has been defined and hypothesized. In particular,
the first convergence results for the important case of electromagnetic eigenvalue
problems were obtained by Kikuchi based on the discrete compactness property
[65, 19], and the more recent approach by Boffi and collaborators emphasized the
Fortid property [18]. In our context, the discrete compactness property says that
is uniformly bounded in V k , then there exists a sequence hi
whenever vh ∈ Zk⊥
h
converging to zero such that vhi converges in W k , and the Fortid property says that
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FINITE ELEMENT EXTERIOR CALCULUS
319
limh→0 I − πhk L(V k ∩Vk∗ ,W k ) = 0. These additional properties play no role in the
theory as presented here. We prove eigenvalue convergence under the same sort of
assumptions we use to establish stability and convergence for the source problem,
namely subcomplexes, bounded cochain projections, and approximability, but we
require boundedness in W for eigenvalue convergence, while for stability we only
require boundedness in V . We believe that subcomplexes with bounded cochain
projections provide an appropriate framework for the analysis of the eigenvalue
problem, since, as far as we know, these properties hold in all examples where
eigenvalue convergence has been obtained by other methods. Moreover, it is easy
to show that the discrete compactness property and Fortid property hold whenever
there exist W -bounded cochain projections.
3.6.1. Related eigenvalue problems. Recall that the source problem for the Hodge
Laplacian, given by (18), can be decomposed into the problems (19) and (20), referred to as the B∗ and the B problem, respectively. More precisely, these problems
arise if the right-hand side f of (18) is in B∗k or Bk , respectively. In a similar way,
the eigenvalue problem (42) can be decomposed into a B∗ and a B problem. To
see this, note that if (λ, u) ∈ R × V k is an eigenvalue/eigenvector for (42), then
u ∈ Hk⊥ , and therefore u = uB + u⊥ , where uB ∈ Bk and u⊥ ∈ B∗k ∩ V k = Zk⊥ .
It is straightforward to check that u⊥ satisfies the B∗k eigenvalue problem given by
(45)
du⊥ , dv = λu⊥ , v,
v ∈ Zk⊥ .
Here u⊥ can be equal to zero even if u = 0 is an eigenvector. On the other hand,
if the pair (λ, u⊥ ) ∈ R × Zk⊥ is an eigenvalue/eigenvector for (45), then (λ, u⊥ ) is
also an eigenvalue/eigenvector for (42), where σ and p are both zero. Furthermore,
for the extended eigenvalue problem
(46)
du⊥ , dv = λu⊥ , v,
v ∈ V k,
where u⊥ is sought in V k , the nonzero eigenvalues correspond precisely to the
eigenvalues of (45), while the eigenvalue λ = 0 has the eigenspace Zk .
The pair (λ, uB ) satisfies the corresponding Bk eigenvalue problem given by
(47)
σ, τ − dτ, uB = 0,
τ ∈ V k−1 ,
dσ, v = λuB , v, v ∈ Bk ,
where σ = d∗ uB ∈ B∗k−1 ∩ V k−1 = Z(k−1)⊥ . Furthermore, any solution (λ, uB )
of (47), where uB = 0, corresponds to an eigenvalue/eigenvector of the full Hodge
Laplacian (42). In fact, any eigenvalue of (42) is an eigenvalue of either the B∗ or
B problem, or both, and all eigenvalues of the B∗ and B problems correspond to
an eigenvalue of (42). In short, the eigenvalue problem (42) can be decomposed into
the two problems (45) and (47).
Also, note that if (λ, uB ) is an eigenvalue/eigenvector for the Bk problem (47),
and σ = d∗ uB ∈ Z(k−1)⊥ , then σ = 0, and by taking v = dτ in (47), we obtain
(48)
dσ, dτ = λσ, τ ,
τ ∈ Z(k−1)⊥ ,
which is a B∗k−1 problem. Hence, we conclude that any eigenvalue of the Bk
problem is an eigenvalue of the B∗k−1 problem. The converse is also true. To see
this, let (λ, σ) ∈ R × Z(k−1)⊥ , σ = 0, be a solution of (48), and define uB ∈ Bk such
that d∗ uB = σ. Since d∗ : Bk → Z(k−1)⊥ is a bijection, this determines uB uniquely,
and (λ, uB ) is an eigenvalue/eigenvector for the Bk problem (47). Therefore, the
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320
D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER
two problems (47) and (48) are equivalent, with the same eigenvalues, and with
eigenvectors related by d∗ uB = σ. In particular, the eigenvalues of the full Hodge
Laplacian problem (42) correspond precisely to the eigenvalues of the B∗k−1 and B∗k
problems.
In the special case of the de Rham complex, the B∗1 problem is closely related to
Maxwell’s equations. Discretizations of this problem have therefore been intensively
studied in the literature (e.g., see [19] and the references therein). However, it is
usually not straightforward to compute the eigenvalues and eigenvectors of the
discrete B∗k problem from the formulation (45) since we do not, a priori, have a
k
basis available for the corresponding space Zk⊥
h ⊂ Vh . Usually, we will only have a
k
basis for the space Vh at our disposal, and a direct computation of a basis for Zk⊥
h
from this basis is costly. A better alternative is therefore to solve the discrete version
of the extended eigenvalue problem (46) and observe that the positive eigenvalues,
and the corresponding eigenvectors, are precisely the solutions of the corresponding
problem (45). Alternatively, we can solve the full Hodge Laplacian problem (43),
and then throw away all eigenvalues corresponding to σh = 0.
4. Exterior calculus and the de Rham complex
We next turn to the most important example of the preceding theory, in which
the Hilbert complex is the de Rham complex associated to a bounded domain Ω
in Rn . We begin by a quick review of basic notions from exterior calculus. Details
can be found in many references, e.g., [8, 12, 22, 49, 62, 67, 85].
4.1. Basic notions from exterior calculus. For a vector space V and a nonnegative integer k, we denote by Altk
V the space of real-valued k-linear forms on
V . If dim V = n, then dim Altk V = nk . The wedge product ω ∧ η ∈ Altj+k V of
ω ∈ Altj V and η ∈ Altk V is given by
(ω ∧ η)(v1 , . . . , vj+k ) =
(sign σ)ω(vσ(1) , . . . , vσ(j) )η(vσ(j+1) , . . . , vσ(j+k) ),
σ
where the sum is over all permutations σ of {1, . . . , j + k}, for which σ(1) < σ(2) <
· · · < σ(j) and σ(j + 1) < σ(j + 2) < · · · < σ(j + k). An inner product on V induces
an inner product on Altk V :
ω, η =
ω(eσ(1) , . . . , eσ(k) )η(eσ(1) , . . . , eσ(k) ), ω, η ∈ Altk V,
σ
where the sum is over increasing sequences σ : {1, . . . , k} → {1, . . . , n} and e1 , . . .,
en is any orthonormal basis (the right-hand side being independent of the choice
of orthonormal basis). If we orient V by assigning a positive orientation to some
particular ordered basis (thereby assigning a positive or negative orientation to all
ordered bases, according to the determinant of the change of basis transformation),
then we may define a unique volume form vol in Altn V , n = dim V , characterized by
vol(e1 , . . . , en ) = 1 for any positively oriented ordered orthonormal basis e1 , . . . , en .
The Hodge star operator is an isometry of Altk V onto Altn−k V given by
ω ∧ µ = ω, µvol,
ω ∈ Altk V, µ ∈ Altn−k V.
Given a smooth manifold Ω, possibly with boundary, a differential k-form ω is
a section of the k-alternating bundle, i.e., a map which assigns to each x ∈ Ω an
element ωx ∈ Altk Tx Ω, where Tx Ω denotes the tangent space to Ω at x. We write
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FINITE ELEMENT EXTERIOR CALCULUS
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C m Λk (Ω) for the space of m times continuously differentiable k-forms, i.e., forms
for which
x → ωx (v1 (x), . . . , vn (x))
belongs to C m (Ω) for any smooth vector fields vi . Similarly, we may define C ∞ Λk ,
Cc∞ Λk (smooth forms with compact support contained in the interior of Ω), etc. If
Ω is a Riemannian manifold, and so has a measure defined on it, we may similarly
define Lebesgue spaces Lp Λk and Sobolev spaces Wpm Λk and H m Λk = W2m Λk . The
spaces H m Λk are Hilbert spaces. In particular, the inner product in L2 Λk = H 0 Λk
is given by
ωx , ηx dx.
ω, η = ω, ηL2 Λk =
Ω
We also write Λk or Λk (Ω) for the space of all smooth differential k-forms, or at
least sufficiently smooth, as demanded by the context.
For any smooth manifold Ω and ω ∈ Λk (Ω), the exterior derivative dω is a
(k + 1)-form, which itself has vanishing exterior derivative: d(dω) = 0. On an
oriented n-dimensional piecewise smooth manifold, a differential n-form (with, e.g.,
compact support) can be integrated, without recourse to a measure or metric.
A smooth map φ : Ω → Ω between manifolds induces a pullback of differential
forms from Ω to Ω . Namely, if ω ∈ Λk (Ω), the pullback φ∗ ω ∈ Λk (Ω ) is defined by
(φ∗ ω)x (v1 , . . . , vk ) = ωφ(x) Dφx (v1 ), . . . , Dφx (vk ) , x ∈ Ω , v1 , . . . , vk ∈ Tx Ω .
The pullback respects exterior products and exterior derivatives:
φ∗ (ω ∧ η) = φ∗ ω ∧ φ∗ η,
φ∗ (dω) = dφ∗ ω.
If φ is an orientation-preserving diffeomorphism of oriented n-dimensional manifolds, and ω is an n-form on Ω, then
φ∗ ω =
ω.
Ω
Ω
If Ω is a submanifold of Ω, then the pullback of the inclusion map is the trace
map trΩ,Ω , written simply trΩ or tr when the manifolds can be inferred from the
context. We recall the trace theorem which states that if Ω is a submanifold of
codimension 1, then the trace map extends to a bounded operator from H 1 Λk (Ω) to
L2 Λk (Ω ), or, more precisely, to a bounded surjection of H 1 Λk (Ω) onto H 1/2 Λk (Ω ).
A particularly important situation is when Ω = ∂Ω, in which case Stokes’s theorem
relates the integrals of the exterior derivative and trace of an (n − 1)-form ω on an
oriented n-dimensional manifold-with-boundary Ω:
dω =
tr ω.
Ω
∂Ω
(A common abuse of notation is to write ∂Ω ω for the ∂Ω tr ω.) Applying Stokes’s
theorem to the differential form ω ∧ η with ω ∈ Λk−1 , η ∈ Λn−k , and using the
Leibniz rule d(ω ∧ η) = dω ∧ η + (−1)k−1 ω ∧ dη, we obtain the integration-by-parts
formula for differential forms:
dω ∧ η = (−1)k
ω ∧ dη +
tr ω ∧ tr η, ω ∈ Λk−1 , η ∈ Λn−k .
(49)
Ω
Ω
∂Ω
On an oriented n-dimensional Riemannian manifold, there is a volume form
vol ∈ Λn (Ω) which at each point x of the manifold is equal to the volume form on
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322
D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER
Tx Ω. Consequently, the Hodge star operation takes ω ∈ Λk (Ω) to ω ∈ Λn−k (Ω)
satisfying
ω ∧ µ = ω, µL2 Λn−k ,
Ω
for all µ ∈ Λn−k (Ω). Introducing the coderivative operator δ : Λk → Λk−1 defined
by
δω = (−1)k d ω,
(50)
and setting η = µ, the integration-by-parts formula becomes
tr ω ∧ tr µ, ω ∈ Λk−1 , µ ∈ Λk .
(51)
dω, µ = ω, δµ +
∂Ω
Since the Hodge star operator is smooth and an isometry at every point, every
property of k-forms yields a corresponding property for (n−k)-forms. For example,
the Hodge star operator maps the spaces C m Λk , Wpm Λk , etc., isometrically onto
C m Λn−k , Wpm Λn−k , etc. By definition, δ : Λn−k → Λn−k−1 corresponds to d :
Λk → Λk+1 via the Hodge star isomorphism. Thus for every property of d there is
a corresponding property of δ.
In case Ω is a domain in Rn , we may (but usually will not) use the standard
coordinates of Rn to write a k-form as
ω=
aσ dxσ1 ∧ · · · ∧ dxσk ,
1≤σ1 <···<σk ≤n
where the aσ ∈ L (Ω) and dxj : Rn → R is the linear form which associates to a
vector its jth coordinate. In this case, the exterior derivative is given by the simple
formula
n
∂aσ
dxj ∧ dxσ1 ∧ · · · ∧ dxσk ,
d(aσ dxσ1 ∧ · · · ∧ dxσk ) =
∂xj
j=1
2
n
which is extended by linearity to a sum of such terms. If Ω is a domain in R , then
(a dx1 ∧ · · · ∧ dxσn ) has the value suggested by the notation. In the case of a
Ω
domain in Rn , the volume form is simply dx1 ∧ · · · ∧ dxn . We remark that the
exterior derivative and integral of differential forms can be computed on arbitrary
manifolds from the formulas on subdomains on Rn and pullbacks through charts.
4.2. The de Rham complex as a Hilbert complex. Henceforth we restrict
attention to the case that Ω is a bounded domain in Rn with a piecewise smooth,
Lipschitz boundary. In this section we show that the de Rham complex is a Hilbert
complex which satisfies the compactness property, and so the abstract theory of
Section 3 applies. We then interpret the results in the case of the de Rham complex.
We begin by indicating how the exterior derivative d can be viewed as a closed
densely-defined operator from W k = L2 Λk (Ω) to W k+1 = L2 Λk+1 (Ω). Let ω ∈
L2 Λk (Ω). In view of (51), we say that η ∈ L2 Λk+1 (Ω) is the weak exterior derivative
of ω if
ω, δµ = η, µ, µ ∈ Cc∞ Λk+1 .
The weak exterior derivative of ω, if one exists, is unique and we denote it by dω.
In analogy with the definition of Sobolev spaces (cf., e.g., [47, Section 5.2.2]), we
define HΛk to be the space of forms in L2 Λk with a weak derivative in L2 Λk+1 .
With the inner product
ω, ηHΛk := ω, ηL2 Λk + dω, dηL2 Λk+1 ,
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FINITE ELEMENT EXTERIOR CALCULUS
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this is easily seen to be a Hilbert space and clearly d is a bounded map from HΛk
to L2 Λk+1 . A standard smoothing argument, as in [47, Section 5.3], implies that
C ∞ Λk (Ω̄) is dense in HΛk . We take HΛk (Ω) as the domain V k of the exterior
derivative, which is thus densely defined in L2 Λk (Ω). Since HΛk is complete, d is
a closed operator.
Thus the spaces L2 Λk (Ω) and the exterior derivative operators d form a Hilbert
complex with the associated domain complex
d
d
d
→ HΛ1 (Ω) −
→ ··· −
→ HΛn (Ω) → 0.
0 → HΛ0 (Ω) −
(52)
This is the L2 de Rham complex. We shall see below that it satisfies the compactness property.
To proceed, we need to identify the adjoint operator d∗ and its domain Vk∗ .
Using the surjectivity of the trace operator from H 1 Λl (Ω) onto H 1/2 Λl (∂Ω) and
the integration-by-parts formula (49), we can show that (cf., [8, page 19]) the trace
operator extends boundedly from HΛk (Ω) to H −1/2 Λk (∂Ω), and that (49) holds
for ω ∈ HΛk−1 , η ∈ H 1 Λn−k where the integral on the boundary is interpreted
via the pairing of H −1/2 (∂Ω) and H 1/2 (∂Ω). Equivalently, we have an extended
version of (51):
tr ω ∧ tr µ, ω ∈ HΛk−1 , µ ∈ H 1 Λk .
(53)
dω, µ = ω, δµ +
∂Ω
Of course, there is a corresponding result obtained by the Hodge star isomorphism
which interchanges d and δ. After reindexing, this is nothing but the fact that (53)
holds also for ω ∈ H 1 Λk−1 , µ ∈ H ∗ Λk , where
H ∗ Λk := (HΛn−k ).
(54)
Note that H ∗ Λk consists of those differential forms in L2 Λk for which a weak
coderivative exists in L2 Λk−1 , where the weak exterior coderivative is defined in
exact analogy to the weak exterior derivative. Its inner product is
ω, ηH ∗ Λk := ω, ηL2 Λk + δω, δηL2 Λk−1 .
∗
The space H Λ is isometric to HΛn−k via the Hodge star, but is quite different
from HΛk .
We also make use of the trace defined on HΛk to define the subspace with
vanishing trace:
H̊Λk (Ω) = { ω ∈ HΛk (Ω) | tr∂Ω ω = 0 }.
Correspondingly, for ω ∈ H ∗ Λk , the quantity tr ω is well defined, and we have
(55)
k
H̊ ∗ Λk (Ω) := H̊Λn−k = { ω ∈ H ∗ Λk (Ω) | tr∂Ω ω = 0 }.
From (53), we have
(56)
dω, µ = ω, δµ,
ω ∈ HΛk−1 , µ ∈ H̊ ∗ Λk .
(We certainly have (56) with the stronger condition ω ∈ H 1 Λk−1 , but then we can
extend to all ω ∈ HΛk−1 by continuity and density.) Of course, the corresponding
result, where ω ∈ H̊Λk−1 , µ ∈ H ∗ Λk , holds as well.
Theorem 4.1. Let d be the exterior derivative viewed as an unbounded operator
L2 Λk−1 → L2 Λk with domain HΛk . Then the adjoint d∗ , as an unbounded operator
L2 Λk → L2 Λk−1 , has H̊ ∗ Λk as its domain and coincides with the operator δ defined
in (50).
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324
D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER
Proof. We must show that for µ ∈ L2 Λk , there exists ω ∈ L2 Λk−1 such that
µ, dν = ω, ν,
(57)
ν ∈ HΛk−1 ,
if and only if µ ∈ H̊ ∗ Λk and ω = δµ. The if direction is immediate from (56).
Conversely, if (57) holds, then µ has a weak exterior coderivative in L2 , namely
δµ = ω. Thus µ ∈ H ∗ Λk . Integrating by parts we have
tr ν ∧ tr µ = µ, dν − ω, ν = 0, ν ∈ H 1 Λk−1 ,
∂Ω
which implies that tr µ = 0, i.e., µ ∈ H̊ ∗ Λk .
As a corollary, we obtain a concrete characterization of the harmonic forms:
(58)
Hk = { ω ∈ HΛk ∩ H̊ ∗ Λk | dω = 0, δω = 0 }.
In other words, a k-form is harmonic if it satisfies the differential equations dω = 0
and δω = 0 together with the boundary conditions tr ω = 0.
If the boundary of Ω is smooth, then HΛk ∩ H̊ ∗ Λk is contained in H 1 Λk [52],
and hence, by the Rellich theorem, we obtain the compactness property of Section 3.1.3. For a general Lipschitz boundary, e.g., for a polygonal domain, the
inclusion of HΛk ∩ H̊ ∗ Λk in H 1 Λk need not hold, but the compactness property
remains valid [75]. Thus all the results of Section 3 apply to the de Rham complex.
In particular, we have the Hodge decomposition of L2 Λk and of HΛk , the Poincaré
inequality, well-posedness of the mixed formulation of the Hodge Laplacian, and all
the approximation results established in Sections 3.3–3.6. We now interpret these
results a bit more concretely in the present setting.
First of all, the cohomology groups associated to the complex (52) are the
de Rham cohomology groups, whose dimensions are the Betti numbers of the domain. Turning next to the Hodge Laplacian problem given in the abstract case by
(18), we get that (σ, u, p) ∈ HΛk−1 × HΛk × Hk is a solution if and only if
(59)
σ = δu, dσ + δdu = f − p
(60)
tr u = 0, tr du = 0
(61)
in Ω,
on ∂Ω,
u⊥H .
k
The first differential equation and the first boundary condition are implied by the
first equation in (18), and the second differential equation and second boundary
condition by the second equation in (18), while the third equation in (18) is simply
the side condition u ⊥ Hk . Note that both boundary conditions are natural in this
variational formulation: they are implied but not imposed in the spaces where the
solution is sought. Essential boundary conditions could be imposed instead. We
discuss this in Section 6.2.
To make things more concrete, we now restrict to a domain Ω ⊂ R3 , and consider
the Hodge Laplacian for k-forms, k = 0, 1, 2, and 3. We also discuss the B∗ and B
problems given by (19) and (20) for each k. We shall encounter many of the most
important partial differential equations of mathematical physics: the Laplacian, the
vector Laplacian, div-curl problems, and curl-curl problems. These PDEs arise in
manifold applications in electromagnetism, solid mechanics, fluid mechanics, and
many other fields.
We begin by noting that, on any oriented Riemannian manifold of dimension n,
we have a natural way to view 0-forms and n-forms as real-valued functions, and
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FINITE ELEMENT EXTERIOR CALCULUS
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1-forms and (n − 1)-forms as vector fields. In fact, 0-forms are real-valued functions
and 1-forms are covector fields, which can be identified with vector fields via the
inner product. The Hodge star operation then carries these identifications to nforms and (n − 1)-forms. In the case of a three-dimensional domain in R3 , via these
identifications all k-forms can be viewed as either scalar or vector fields (sometimes
called proxy fields). With these identifications, the Hodge star operation becomes
trivial in the sense if a certain vector field is the proxy for, e.g., a 1-form ω, the
exact same vector field is the proxy for the 2-form ω. Via proxy fields, the exterior
derivatives coincide with standard differential operators of calculus:
d0 = grad, d1 = curl, d2 = div,
and the de Rham complex (52) is realized as
grad
curl
div
0 → H 1 (Ω) −−−→ H(curl; Ω) −−→ H(div; Ω) −−→ L2 (Ω) → 0,
where
H(curl; Ω) = { u : Ω → R3 | u ∈ L2 , curl u ∈ L2 },
H(div; Ω) = { u : Ω → R3 | u ∈ L2 , div u ∈ L2 }.
The exterior coderivatives δ become, of course, − div, curl, and − grad, when acting
on 1-forms, 2-forms, and 3-forms, respectively. The trace operation on 0-forms is
just the restriction to the boundary, and the trace operator on 3-forms vanishes
(since there are no nonzero 3-forms on ∂Ω). The trace operator from 1-forms on
Ω to 1-forms on the boundary takes a vector field u on Ω to a tangential vector
field on the boundary, namely at each boundary point x, (tr u)x is the tangential
projection of ux . For a 2-form u, the trace corresponds to the scalar u · n (with n
the unit normal) at each boundary point.
4.2.1. The Hodge Laplacian for k = 0. For k = 0, the boundary value problem
(59)–(61) is the Neumann problem for the ordinary scalar Laplacian. The space
HΛ−1 is understood to be 0, so σ vanishes. The harmonic form space H0 consists
of the constant functions (we assume Ω is connected; otherwise H0 would consist
of functions which are constant on each connected component), and p is just the
average of f . The first differential equation of (59) vanishes, and the second gives
Poisson’s equation
− div grad u = f − p in Ω.
Similarly, the first boundary condition in (60) vanishes, while the second is the
Neumann condition
grad u · n = 0 on ∂Ω.
The side condition (61) specifies a unique solution by requiring its average value to
be zero.
Nothing additional is obtained by considering the split into the B∗ and B subproblems, since the latter is trivial. Furthermore, the eigenvalue problem (42) is
precisely the corresponding eigenvalue problem for the scalar Laplacian, with the
0 eigenspace H0 filtered out.
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326
D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER
4.2.2. The Hodge Laplacian for k = 1. In this case, the differential equations and
boundary conditions are
(62)
σ = − div u, grad σ + curl curl u = f − p
u · n = 0, curl u × n = 0 on ∂Ω,
in Ω,
which is a formulation of a boundary value problem for the vector Laplacian
curl curl − grad div. (Here we have used the fact that the vanishing of the tangential component of a vector is equivalent to the vanishing of its cross product
with the normal.) The solution is determined uniquely by the additional condition that u be orthogonal to H1 , which in this case consists of those vector fields
satisfying
curl p = 0, div p = 0 in Ω, p · n = 0 on ∂Ω.
The dimension of H1 is equal to the first Betti number, i.e., the number of handles,
of the domain (so H1 = 0 if the domain is simply connected).
The B∗1 problem (19) is defined for L2 vector fields f which are orthogonal to
both gradients and the vector fields in H1 . In that case, the solution to (62) has
σ = 0 and p = 0, while u satisfies
curl curl u = f, div u = 0 in Ω,
u · n = 0, curl u × n = 0 on ∂Ω.
The orthogonality condition u ⊥ H1 again determines the solution uniquely.
Next we turn to the B1 problem. For source
functions of the form f = grad F for
some F ∈ H 1 , which we normalize so that Ω F = 0, (62) reduces to the problem
of finding σ ∈ H 1 and u ∈ B1 = grad H 1 such that:
σ = − div u, grad σ = f in Ω,
u · n = 0 on ∂Ω.
The differential equations may be simplified to − grad div u = f , and the condition
that u ∈ B1 can be replaced by the differential equation
curl
u = 0, together
with
orthogonality to H1 . Now grad(σ − F ) = 0 and Ω σ = − ∂Ω u · n = 0 = Ω F , so
σ = F , and we may rewrite the system as
− div u = F, curl u = 0 in Ω,
u · n = 0 on ∂Ω,
which, again, has a unique solution subject to orthogonality to H1 .
The eigenvalue problem (42) is the corresponding eigenvalue problem for the
vector Laplacian with the boundary conditions of (62), and with the eigenspace
H1 of the eigenvalue λ = 0 filtered out. As mentioned above, in this case, the
B∗1 eigenvalue problem, given by (45), is important for models based on Maxwell’s
equations. This problem takes the form
(63) curl curl u = λu, div u = 0 in Ω,
u · n = 0, curl u × n = 0 on ∂Ω, u ⊥ H1 .
4.2.3. The Hodge Laplacian for k = 2. The differential equations and boundary
conditions are
σ = curl u, curl σ − grad div u = f − p in Ω,
(64)
u × n = 0, div u = 0 on ∂Ω.
This is again a formulation of a boundary value problem for the vector Laplacian
curl curl − grad div, but with different boundary conditions than for (62), and this
time stated in terms of two vector variables, rather than one vector and one scalar.
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FINITE ELEMENT EXTERIOR CALCULUS
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This time uniqueness is obtained by imposing orthogonality to H2 , the space of
vector fields satisfying
p × n = 0 on ∂Ω,
curl p = 0, div p = 0 in Ω,
which has dimension equal to the second Betti number, i.e., the number of voids in
the domain.
The B∗2 problem arises for source functions of the form f = grad F for some
F ∈ H̊ 1 (H̊ 1 is the function space corresponding to H̊ ∗ Λ3 ). We find σ = 0, and u
solves
− div u = F, curl u = 0 in Ω, u × n = 0 on ∂Ω,
i.e., the same differential equation as for B1 , but with different boundary conditions, an extra assumption on F , and, of course, now uniqueness is determined by
orthogonality to H2 .
If div f = 0 and f ⊥ H2 , we get the B2 problem for which the differential
equations are σ = curl u, curl σ = f and the condition div u = 0 arising from the
membership of u in B2 . Thus u solves
curl curl u = f, div u = 0 in Ω,
u × n = 0 on ∂Ω,
B∗1 ,
but with different boundary conditions.
the same differential equation as for
The eigenvalue problem (42) is the corresponding eigenvalue problem for the
vector Laplacian with the boundary conditions of (64), and with the eigenspace H2
of the eigenvalue λ = 0 filtered out, while the corresponding B2 problem, of the
form (47), takes the form
curl curl u = λu, div u = 0 in Ω,
u × n = 0 on ∂Ω, u ⊥ H2 .
Note that this is the same problem as the B∗1 eigenvalue problem (63), but with
different boundary conditions. However, if we define σ = curl u, then it is straightforward to check that the pair (λ, σ) will indeed solve the B∗1 problem (63). This
is an instance of the general equivalence between the Bk problem, given by (47),
and the corresponding B∗k−1 problem, which was pointed out in Section 3.6.1.
4.2.4. The Hodge Laplacian for k = 3. In this case the Hodge Laplacian problem,
which coincides with the B3 problem, is
σ = − grad u, div σ = f in Ω,
u = 0 on ∂Ω,
which is the Dirichlet problem for Poisson’s equation. There are no nonzero harmonic forms, and the problem has a unique solution. Furthermore, the eigenvalue
problem (42) is the corresponding eigenvalue problem for the scalar Laplacian with
Dirichlet boundary conditions.
5. Finite element approximation of the de Rham complex
Our goal in this section is to discretize the de Rham complex so that we may
apply the abstract results on approximation of Hilbert complexes from Section 3.
Hence we need to construct finite-dimensional subspaces Λkh of HΛk (Ω). As we
saw in Section 3.4, the key properties these spaces must possess is, first, that
dΛkh ⊂ Λk+1
so they form a subcomplex (Λh , d) of the de Rham complex, second,
h
that there exist uniformly bounded cochain projections πh from (L2 Λ, d) to (Λh , d),
and third, good approximation properties. We may then use these spaces in a
Galerkin method based on the mixed formulation, as described in Section 3.4,
and the error estimates given in Theorems 3.9 and 3.11 bound the error in the
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328
D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER
Galerkin solution in terms of the approximation error afforded by the subspaces. In
this section we will construct the spaces Λkh as spaces of finite element differential
forms and show that they satisfy all these requirements and can be efficiently
implemented.
As discussed in Section 2, a finite element space is a space of piecewise polynomials which is specified by the finite element assembly process, i.e., by giving
a triangulation of the domain, a finite-dimensional space of polynomial functions
(or, in our case, differential forms) on each element of the triangulation, called the
shape functions, and a set of degrees of freedom for the shape functions associated
with the faces of various dimensions of the elements, which will be used to determine the degree of interelement continuity. Spaces constructed in this way can be
implemented very efficiently, since they admit basis functions with small support
and so lead to sparse algebraic systems.
In Section 5.1, we will discuss the spaces of polynomial differential forms which
we will use as shape functions. In Section 5.2, we specify degrees of freedom for
these spaces on simplices and study the resulting finite element spaces of differential
forms. In particular, we show that they can be combined in a variety of ways to
form subcomplexes of the de Rham complex. In Section 5.3, we briefly describe
recent work related to the implementation of such finite elements with explicit local
bases. In Sections 5.4 and 5.5, we construct L2 -bounded cochain projections into
these spaces and obtain error estimates for them. Finally Section 5.6 is simply a
matter of collecting our results in order to obtain error estimates for the resulting
approximations of the Hodge Laplacian. Many of the results of this section have
appeared previously, primarily in [8], and therefore many proofs are omitted.
5.1. Polynomial differential forms and the Koszul complex. In this section,
we consider spaces of polynomial differential forms, which lead to a variety of subcomplexes of the de Rham complex. These will be used in the next section to construct finite element spaces of differential forms. The simplest spaces of polynomial
differential k-forms are the spaces Pr Λk (Rn ) consisting of all differential k-forms
on Rn whose coefficients are polynomials of degree at most r. In addition to these
spaces, we will use another family of polynomial form spaces, denoted Pr− Λk (Rn ),
which will be constructed and analyzed using the Koszul differential and the associated Koszul complex. Spaces taken from these two families can be combined into
polynomial subcomplexes of the de Rham complex in numerous ways (there are
essentially 2n−1 such subcomplexes associated to each polynomial degree). These
will lead to finite element de Rham subcomplexes, presented in Section 5.2. Some
of these have appeared in the literature previously, with the systematic derivation
of all of them first appearing in [6].
5.1.1. Polynomial differential forms. Let Pr (Rn ) and Hr (Rn ) denote the spaces of
polynomials in n variables of degree at most r and of homogeneous polynomial
functions of degree r, respectively. We interpret these spaces to be the zero space
if r < 0. We can then define spaces of polynomial differential forms, Pr Λk (Rn ),
Hr Λk (Rn ), etc., as those differential forms which, when applied to a constant vector
field, have the indicated polynomial dependence. For brevity, we will at times
suppress Rn from the notation and write simply Pr , Hr , Pr Λk , etc.
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FINITE ELEMENT EXTERIOR CALCULUS
329
The dimensions of these spaces are easily calculated:
dim Pr Λk (Rn ) = dim Pr (Rn ) · dim Altk Rn
n+r
n
r+k
n+r
=
=
,
n
k
r
n−k
and dim Hr Λk (Rn ) = dim Pr Λk (Rn−1 ).
For each polynomial degree r ≥ 0, we get a homogeneous polynomial subcomplex
of the de Rham complex:
(65)
d
d
d
→ Hr−1 Λ1 −
→ ··· −
→ Hr−n Λn → 0.
0 → Hr Λ0 −
We shall verify below the exactness of this sequence. More precisely, the cohomology vanishes if r > 0 and also for r = 0 except in the lowest degree, where
the cohomology space is R (reflecting the fact that the constants are killed by the
gradient).
Taking the direct sum of the homogeneous polynomial de Rham complexes over
all polynomial degrees gives the polynomial de Rham complex:
(66)
d
d
d
→ Pr−1 Λ1 −
→ ··· −
→ Pr−n Λn → 0,
0 → Pr Λ0 −
for which the cohomology space is R in the lowest degree and vanishes otherwise.
5.1.2. The Koszul complex. Let x ∈ Rn . Since there is a natural identification of Rn
with the tangent space T0 Rn at the origin, there is a vector in T0 Rn corresponding
to x. (The origin is chosen for convenience here, but we could use any other point
instead.) Then the translation map y → y + x induces an isomorphism from T0 Rn
to Tx Rn , and so there is an element X(x) ∈ Tx Rn corresponding to x. (Essentially
X(x) is the vector based at x which points opposite to the origin and whose length is
|x|.) Contraction with the vector field X defines a map κ from Λk (Rn ) to Λk−1 (Rn )
called the Koszul differential :
(κω)x (v1 , . . . , vk−1 ) = ωx X(x), v1 , . . . , vk−1 .
It is easy to see that κ is a graded differential, i.e.,
κ◦κ=0
and
κ(ω ∧ η) = (κω) ∧ η + (−1)k ω ∧ (κη),
ω ∈ Λk , η ∈ Λl .
In terms of coordinates, if ωx = a(x) dxσ1 ∧ · · · ∧ dxσk , then
(κω)x =
k
σ(i) ∧ · · · ∧ dxσ ,
(−1)i+1 a(x) xσ(i) dxσ1 ∧ · · · ∧ dx
k
i=1
σ means that the term is omitted. Note that κ maps Hr Λk
where the notation dx
i
k−1
to Hr+1 Λ
; i.e., κ increases polynomial degree and decreases form degree, the
exact opposite of the exterior derivative d.
The Koszul differential gives rise to the homogeneous Koszul complex [68, Chapter 3.4.6],
(67)
κ
κ
κ
→ Hr−n+1 Λn−1 −
→ ··· −
→ Hr Λ0 → 0.
0 → Hr−n Λn −
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330
D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER
We show below that this complex is exact for r > 0. Adding over polynomial
degrees, we obtain the Koszul complex (for any r ≥ 0),
κ
κ
κ
→ Pr−n+1 Λn−1 −
→ ··· −
→ Pr Λ0 → 0,
0 → Pr−n Λn −
for which all the cohomology spaces vanish, except the rightmost, which is equal to
R.
To prove the exactness of the homogeneous polynomial de Rham and Koszul
complexes, we note a key connection between the exterior derivative and the Koszul
differential. In the language of homological algebra, this says that the Koszul
differential is a contracting homotopy for the homogeneous polynomial de Rham
complex.
Theorem 5.1.
(68)
(dκ + κd)ω = (r + k)ω,
ω ∈ Hr Λk .
Proof. This result can be established by a direct computation or by using the
homotopy formula of differential geometry (Cartan’s magic formula). See [8] for
the details.
As a simple consequence of (68), we prove the injectivity of d on the range of κ
and vice versa.
Theorem 5.2. If dκω = 0 for some ω ∈ PΛ, then κω = 0. If κdω = 0 for some
ω ∈ PΛ, then dω = 0.
Proof. We may assume that ω ∈ Hr Λk for some r, k ≥ 0. If r = k = 0, the result
is trivial, so we may assume that r + k > 0. Then (r + k)κω = κ(dκ + κd)ω = 0
if dκω = 0, so κω = 0 in this case. Similarly, (r + k)dω = d(dκ + κd)ω = 0 if
κdω = 0.
Another easy application of (68) is to establish the claimed cohomology of the
Koszul complex and polynomial de Rham complex. Suppose that ω ∈ Hr Λk for
some r, k ≥ 0 with r + k > 0, and that κω = 0. From (68), we see that ω = κη with
η = dω/(r + k) ∈ Hr−1 Λk+1 . This establishes the exactness of the homogeneous
Koszul complex (67) (except when r = 0 and the sequence reduces to 0 → R → 0).
A similar argument establishes the exactness of (65).
Another immediate but important consequence of (68) is a direct sum decomposition of Hr Λk for r, k ≥ 0 with r + k > 0:
(69)
Hr Λk = κHr−1 Λk+1 ⊕ dHr+1 Λk−1 .
Indeed, if ω ∈ Hr Λk , then η = dω/(r + k) ∈ Hr−1 Λk+1 and µ = κω/(r + k) ∈
Hr+1 Λk−1 and ω = κη + dµ, so Hr Λk = κHr−1 Λk+1 + dHr+1 Λk−1 . Also, if
κ = 0), and so,
ω ∈ κHr−1 Λk+1 ∩ dHr+1 Λk−1 , then dω = κω = 0 (since d ◦ d = κ ◦
r
by (68), ω = 0. This shows that the sum is direct. Since Pr Λk = j=0 Hj Λk , we
also have
Pr Λk = κPr−1 Λk+1 ⊕ dPr+1 Λk−1 .
The exactness of the Koszul complex can be used to compute the dimension of
the summands in (69) (cf. [8]).
Theorem 5.3. Let r ≥ 0, 1 ≤ k ≤ n, for integers r, k, and n. Then
n+r
r+k−1
k
n
k−1
n
(70)
dim κHr Λ (R ) = dim dHr+1 Λ
(R ) =
.
n−k
k−1
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5.1.3. The space Pr− Λk . Let r ≥ 1. Obviously, Pr Λk = Pr−1 Λk + Hr Λk . In view
of (69), we may define a space of k-forms intermediate between Pr−1 Λk and Pr Λk
by
Pr− Λk = Pr−1 Λk + κHr−1 Λk+1 = Pr−1 Λk + κPr−1 Λk+1 .
Note that the first sum is direct, while the second need not be. An equivalent
definition is
Pr− Λk = { ω ∈ Pr Λk | κω ∈ Pr Λk−1 }.
Note that Pr− Λ0 = Pr Λ0 and Pr− Λn = Pr−1 Λn , but for 0 < k < n, Pr− Λk is
contained strictly between Pr−1 Λk and Pr Λk . For r ≤ 0, we set Pr− Λk = 0.
From (70), we have
dim Pr− Λk (Rn ) = dim Pr−1 Λk + dim κHr−1 Λk+1
n+r−1 n
n+r−1
r+k−1
=
+
n
k
n−k−1
k
r+k−1
n+r
=
,
k
n−k
where the last step is a simple identity.
We also note the following simple consequences of Lemma 5.2.
Theorem 5.4. If ω ∈ Pr− Λk and dω = 0, then ω ∈ Pr−1 Λk . Moreover, dPr Λk =
dPr− Λk .
Proof. Write ω = ω1 + κω2 with ω1 ∈ Pr−1 Λk and ω2 ∈ Pr−1 Λk+1 . Then
dω = 0 =⇒ dκω2 = 0 =⇒ κω2 = 0 =⇒ ω ∈ Pr−1 Λk ,
showing the first result. For the second it suffices to note that Pr Λk = Pr− Λk +
dPr+1 Λk−1 .
Remark. We defined the Koszul differential as contraction with the vector field X,
where X(x) is the translation to x of the vector pointing from the origin in Rn to
x. The choice of the origin as a base point is arbitrary; any point in Rn could be
used. That is, if y ∈ Rn , we can define a vector field Xy by assigning to each point
x the translation to x of the vector pointing from y to x, and then define a Koszul
differential κy by contraction with Xy . It is easy to check that for ω ∈ Pr−1 Λk+1
and any two points y, y ∈ Rn , the difference κy ω − κy ω ∈ Pr−1 Λk . Hence the
space
Pr− Λk = Pr−1 Λk + κy Pr−1 Λk+1
does not depend on the particular choice of the point y. This observation is important, because it allows us to define Pr− Λk (V ) for any affine subspace V of Rn . We
simply set
Pr− Λk (V ) = Pr−1 Λk (V ) + κy Pr−1 Λk+1 (V ),
where y is any point of V . Note that if ω ∈ Pr− Λk (Rn ), then the trace of ω on V
belongs to Pr− Λk (V ).
Remark. The spaces Pr− Λk (Rn ) are affine-invariant; i.e., if φ : Rn → Rn is an
affine map, then the pullback φ∗ maps this space into itself. Of course, the full
polynomial space Pr Λk (Rn ) is affine-invariant as well. In [8, Section 3.4], all the
finite-dimensional affine-invariant spaces of polynomial differential forms are determined. These are precisely the spaces in the P and P − families together with one
further family of spaces which is of less interest to us.
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332
D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER
5.1.4. Exact sequences of polynomial differential forms. We have seen that the polynomial de Rham complex (66) is a subcomplex of the de Rham complex on Rn for
which cohomology vanishes except for the constants at the lowest order. In other
words, for any r ≥ 0, the sequence
(71)
d
d
d
R → Pr Λ0 −
→ Pr−1 Λ1 −
→ ··· −
→ Pr−n Λn → 0
is exact, i.e., is a resolution of R.
As we shall soon verify, the complex
(72)
R → Pr− Λ0 −
→ Pr− Λ1 −
→ ··· −
→ Pr− Λn → 0
d
d
d
is another resolution of R, for any r > 0. Note that in this complex, involving the
Pr− Λk spaces, the polynomial degree r is held fixed, while in (71), the polynomial
degree decreases as the form order increases. Recall that the 0th-order spaces
in these complexes, Pr Λ0 and Pr− Λ0 , coincide. In fact, the complex (71) is a
subcomplex of (72), and these two are the extreme cases of a set of 2n−1 different
resolutions of R, each a subcomplex of the next, and all of which have the space Pr Λ0
in the 0th-order. To exhibit them, we begin with the inclusion R → Pr Λ0 . We may
continue the complex with either the map d : Pr Λ0 → Pr−1 Λ1 or d : Pr Λ0 → Pr− Λ1 ,
the former being a subcomplex of the latter. With either choice, the cohomology
vanishes at the first position. Next, if we made the first choice, we can continue
−
Λ2 . Or,
the complex with either d : Pr−1 Λ1 → Pr−2 Λ2 or d : Pr−1 Λ1 → Pr−1
− 1
if we made the second choice, we can continue with either d : Pr Λ → Pr−1 Λ2
or d : Pr− Λ1 → Pr− Λ2 . In the first case, we may use the exactness of (71) to
see that the second cohomology space vanishes. In the second case, this follows
from Lemma 5.4. Continuing in this way at each order, k = 1, . . . , n − 1, we have
two choices for the space of k-forms (but only one choice for k = n, since Pr−1 Λn
coincides with Pr− Λn ), and so we obtain 2n−1 complexes. These form a totally
ordered set with respect to subcomplex inclusion. For r ≥ n these are all distinct
(but for small r some coincide because the later spaces vanish).
In the case n = 3, the four complexes so obtained are:
d
d
d
R → Pr Λ0 −−−−→ Pr−1 Λ1 −−−−→ Pr−2 Λ2 −−−−→ Pr−3 Λ3 → 0,
−
Λ2 −−−−→ Pr−2 Λ3 → 0,
R → Pr Λ0 −−−−→ Pr−1 Λ1 −−−−→ Pr−1
d
d
d
d
d
d
d
d
d
R → Pr Λ0 −−−−→ Pr− Λ1 −−−−→ Pr−1 Λ2 −−−−→ Pr−2 Λ3 → 0,
R → Pr Λ0 −−−−→ Pr− Λ1 −−−−→ Pr− Λ2 −−−−→ Pr−1 Λ3 → 0.
5.2. Degrees of freedom and finite element differential forms. Having introduced the spaces of polynomial differential forms Pr Λk (Rn ) and Pr− Λk (Rn ), we
now wish to create finite element spaces of differential forms. We begin with the
notation for spaces of polynomial differential forms on simplices. If f is a simplex
(of any dimension) in Rn , we define
Pr Λk (f ) = trRn ,f Pr Λk (Rn ),
P̊r Λk (f ) = { ω ∈ Pr Λk (f ) | trf,∂f ω = 0 }.
The spaces Pr− Λk (f ) and P̊r− Λk (f ) are defined similarly.
Now let Ω be a bounded polyhedral domain which is triangulated, i.e., partitioned into a finite set T of n-simplices determining a simplicial decomposition
of Ω. This means that the union of the elements of T is the closure of Ω, and
the intersection of any two is either empty or a common subsimplex of each. By
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FINITE ELEMENT EXTERIOR CALCULUS
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way of notation, for any simplex T we denote by ∆d (T ) the set of subsimplices
of T of dimension
d, and by ∆(T ) the
set of all subsimplices of T . We also set
∆d (T ) := T ∈T ∆d (T ) and ∆(T ) := T ∈T ∆(T ).
Corresponding to the P and P − families of spaces of polynomial differential
forms, we will define two families of spaces of finite element differential forms with
respect to the triangulation T , denoted Pr Λk (T ) and Pr− Λk (T ). We shall show
that these are subspaces of HΛk (Ω) and can be collected, in various ways, into
subcomplexes of the de Rham complex.
The spaces Pr Λk (T ) and Pr− Λk (T ) will be obtained by the finite element assembly process. For each T ∈ T , we will choose the corresponding polynomial
space Pr Λk (T ) or Pr− Λk (T ) to be used as shape functions. The other ingredient
needed to define the finite element space is a set of degrees of freedom for the shape
function spaces, that is, a basis for the dual space, in which each degree of freedom
is associated with a particular subsimplex. When a subsimplex is shared by more
than one simplex in the triangulation, we will insist that the degrees of freedom
associated with that subsimplex be single-valued in a sense made precise below,
and this will determine the interelement continuity.
The degrees of freedom for Pr Λk (T ) and Pr− Λk (T ), which we shall associate to
a d-dimensional subsimplex f of T , will be
of the following form: for some (d − k)form η on f , the functional will be ω → f trT,f ω ∧ η. The span of all the degrees
of freedom associated to f is a subspace of the dual space of the shape function
space, and so we obtain a decomposition of the dual space of the shape functions
on T into a direct sum of subspaces indexed by the subsimplices of T . It is really
this geometric decomposition of the dual space that determines the interelement
continuity rather than the particular choice of degrees of freedom, since we may
choose any convenient basis for each space in the decomposition and obtain the
same assembled finite element space.
The geometric decompositions of the dual spaces of Pr Λk (T ) or Pr− Λk (T ) are
given specifically in the following theorem, which is proven in [8, Sections 4.5 and
4.6].
Theorem 5.5. Let r, k, and n be integers with 0 ≤ k ≤ n and r > 0, and let T be
an n-simplex in Rn .
1. To each f ∈ ∆(T ), associate a space Wrk (T, f ) ⊂ Pr Λk (T )∗ :
−
dim f −k
trT,f ω ∧ η η ∈ Pr+k−dim
Λ
(f
)
.
Wrk (T, f ) = ω →
f
f
Then
Wrk (T, f )
−
dim f −k
∼
(f ) via the obvious correspondence, and
= Pr+k−dim
fΛ
Wrk (T, f ).
Pr Λk (T )∗ =
f ∈∆(T )
2. To each f ∈ ∆(T ), associate a space Wrk− (T, f ) ⊂ Pr− Λk (T )∗ :
trT,f ω ∧ η η ∈ Pr+k−dim f −1 Λdim f −k (f ) .
Wrk− (T, f ) = ω →
f
Then
Wrk− (T, f )
∼
= Pr+k−dim f −1 Λdim f −k (f ) via the obvious correspondence, and
Wrk− (T, f ).
Pr− Λk (T )∗ =
f ∈∆(T )
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334
D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER
Note that the spaces Wrk (T, f ) and Wrk− (T, f ) vanish if dim f < k. Note also
that the dual space of Pr Λk (T ) is expressed in terms of spaces in the P − family, and
vice versa. This intimate connection between the P and P − families of spaces of
polynomial differential forms is most clearly seen in the following algebraic proposition, which is closely related to Theorem 5.5, and is also proved in [8, Sections 4.5
and 4.6].
Lemma 5.6. With r, k, n as above,
−
P̊r Λk (T )∗ ∼
Λn−k (T )
= Pr+k−n
and
P̊r− Λk (T )∗ ∼
= Pr+k−n−1 Λn−k (T ).
With the decompositions in Theorem 5.5, we can define the finite element spaces.
Thus Pr Λk (T ) consists of all forms ω ∈ L2 Λk (Ω) such that ω|T belongs
to the shape
function space Pr Λk (T ) for all T ∈ T , and for which the quantities f trf ω ∧ η are
−
dim f −k
single-valued for all f ∈ ∆(T ) and all η ∈ Pr+k−dim
(f ). More precisely,
fΛ
this means that if f is a common face of T1 , T2 ∈ T , then
trT1 ,f (ω|T1 ) ∧ η =
trT2 ,f (ω|T2 ) ∧ η
f
f
−
for all such f and η. The P family of spaces is defined analogously.
The degrees of freedom determine the amount of interelement continuity enforced
on the finite element space. Of course we need to know that the assembled spaces
belong to HΛk (Ω). In fact, the degrees of freedom we imposed enforce exactly
the continuity needed, as shown in the following theorem. This is proved in [8,
Section 5.1], where the equations below are taken as definitions, and it is shown
that the assembly process leads to the same spaces.
Theorem 5.7.
Pr Λk (T ) = {ω ∈ HΛk (Ω) | ω|T ∈ Pr Λk , T ∈ T },
Pr− Λk (T ) = {ω ∈ HΛk (Ω) | ω|T ∈ Pr− Λk , T ∈ T }.
Next we note that these spaces of finite element differential forms can be collected in subcomplexes of the de Rham complex. In view of Theorem 5.7, we
have dPr Λk (T ) ⊂ Pr−1 Λk+1 (T ) and dPr− Λk (T ) ⊂ Pr− Λk+1 (T ). Corresponding
to the resolutions defined in Section 5.1.4, we obtain 2n−1 de Rham subcomplexes for each value of r and each mesh T . Each complex begins with the space
Pr Λ0 (T ) = Pr− Λ0 (T ). The maps making up the subcomplexes are all one of the
following types:
⎧
⎧ − k
⎫
⎫
k−1
(T )⎪
⎪
⎪
⎨Ps+1 Λ
⎨Ps+1 Λ (T )⎪
⎬
⎬
d
or
(73)
or
−−−−→
⎪
⎪
⎪
⎪
⎩ − k−1
⎩
⎭
⎭
Ps Λk (T )
(T )
Ps+1 Λ
for some s and some k. In Section 5.5, we shall show that all the subcomplexes
admit bounded cochain projections. These 2n−1 complexes, which are distinct for
r ≥ n, are linearly ordered by inclusion. The maximal complex is
0 → Pr− Λ0 (T ) −
→ Pr− Λ1 (T ) −
→ ··· −
→ Pr− Λn (T ) → 0.
d
d
d
The spaces Pr− Λk (T ) in this complex are referred to as the higher-order Whitney
forms, since for r = 1, this is exactly the complex introduced by Whitney [88]. The
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FINITE ELEMENT EXTERIOR CALCULUS
335
minimal complex with this starting space is the complex of polynomial differential
forms
d
d
d
→ Pr−1 Λ1 (T ) −
→ ··· −
→ Pr−n Λn (T ) → 0.
0 → Pr Λ0 (T ) −
This complex was used extensively by Sullivan [83, 84] and is sometimes referred
to as the complex of Sullivan–Whitney forms [15]. It was introduced into the finite
element literature in [37]. The intermediate complexes involve both higher-order
Whitney spaces and full polynomial spaces.
Finally, we note that the degrees of freedom used to define the space determine
a canonical projection Ih : CΛk (Ω) → Pr Λk (T ). Namely, Ih ω ∈ Pr Λk (T ) is
determined by
−
dim f −k
trT,f (ω − Ih ω) ∧ η = 0, η ∈ Pr+k−dim
(f ), f ∈ ∆(T ).
fΛ
f
Similar considerations apply to Pr− Λk (T ). The canonical projection can be viewed
as a map from the (sufficiently) smooth de Rham complex to any one of the complexes built from the maps (73). The degrees of freedom were chosen exactly so that
the canonical projection is a cochain map, i.e., commutes with d. This is verified
using Stokes’s theorem. See [8, Theorem 5.2].
Remark. Given a simplicial triangulation T , we have defined, for every form degree
k and every polynomial degree r, two finite element spaces of differential k-forms
of degree at most r: Pr Λk (T ) and Pr− Λk (T ). In n = 2 and n = 3 dimensions,
we may use proxy fields to identify these spaces of finite element differential forms
with finite element spaces of scalar and vector functions. The space Pr Λ0 (T ) corresponds to the Lagrange elements [33], and the spaces Pr Λn (T ) and Pr− Λn (T )
correspond to the space of discontinuous piecewise polynomials of degree ≤ r and
≤ r − 1, respectively. When n = 2, the spaces Pr Λ1 (T ) and Pr− Λ1 (T ) correspond to the Brezzi–Douglas–Marini H(div) elements of degree ≤ r, introduced in
[26] and the Raviart–Thomas H(div) elements of degree ≤ r − 1 introduced in [76].
These spaces were generalized to three dimensions by Nédélec [72], [73]. The spaces
Pr Λ1 (T ), Pr Λ2 (T ), Pr− Λ1 (T ), and Pr− Λ2 (T ) then correspond to the Nédélec 2nd
kind H(curl) and H(div) elements of degree ≤ r and the Nédélec 1st kind H(curl)
and H(div) elements of degree ≤ r − 1, respectively.
5.3. Computational bases. This subsection relates to the implementation of the
P and P − families of finite element differential forms. It is not essential to the rest of
the paper. Because the spaces Pr− Λk (T ) and Pr Λk (T ) were constructed through a
finite element assembly procedure (shape functions and degrees of freedom), we have
at hand a basis for their dual spaces. Consider the space Pr− Λk (T ), for example,
with reference to the decomposition of the dual space of the shape function space
Pr− Λk (T ) given in Theorem 5.5. Choose any f ∈ ∆(T ) of dimension d ≥ k, and
choose any convenient basis for Pr+k−d Λd−k (f ). For each of the basis functions η,
we obtain an element of Pr− Λk (T )∗ :
ω →
trf ω ∧ η, ω ∈ Pr− Λk (T ).
f
(This is meaningful since the integral is, by construction, single-valued.) Taking
the union over f ∈ ∆(T ) of the sets of elements of Pr− Λk (T )∗ obtained in this
way gives a basis for that space. An interesting case is that of the Whitney forms
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336
D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER
P1− Λk (T ). Inthis case there is exactly one dual basis function for each f ∈ ∆k (T ),
namely ω → f trf ω.
For computation with finite elements, we also need a basis for the finite element
space itself, not only for the dual space. One possibility which is commonly used
is to use the dual basis to the basis for the dual space just discussed. Given an
element of the basis of the dual space, associated to some f ∈ ∆(T ), it is easy to
check that the corresponding basis function of the finite element space vanishes on
all simplices T ∈ T which do not contain f . Thus we have a local basis, which is
very efficient for computation.
In the case of the Whitney space P1− Λk (T ), the dual basis can be written down
very easily. We begin with the standard dual basis for P1 Λ0 (Th ): the piecewise
linear function λi associated to the vertex xi is determined by λi (xj ) = δij (after
picking some numbering x1 , . . . , xN of the vertices). Then given a k-face f with
vertices xσ(0) , . . . , xσ(k) we define the Whitney form [88, p. 229]
φf =
k
(−1)i λσ(i) dλσ(0) ∧ · · · ∧ dλ
σ(i) ∧ · · · ∧ dλσ(k) .
i=0
For higher degree finite element spaces it does not seem possible to write down
the dual basis explicitly, and it must be computed. The computation comes down
to inverting a matrix of size d × d, where d is the dimension of the space of shape
functions. This can be carried out once on a single reference simplex and the result
transferred to any simplex via an affine transformation.
An alternative to the dual basis, which is often preferred, is to use a basis for the
finite element space which can be written explicitly in terms of barycentric coordinates. In particular, for the Lagrange finite element space Pr Λ0 (T ) consisting of
continuous piecewise polynomials of degree at most r, one often uses the Bernstein
basis, defined piecewise by monomials in the barycentric coordinates, instead of the
dual or Lagrange basis. It turns out that explicit bases analogous to the Bernstein
basis can be given for all the finite element spaces in the P and P − families, as was
shown in [10]. Here we content ourselves with displaying a few typical cases.
Bases for the spaces Pr− Λ1 (T ) and Pr Λ2 (T ) are summarized in Tables 5.1 and
5.2, respectively, for n = 3 dimensions and polynomial degrees r = 1, 2, and 3. To
explain the presentation, we interpret the second line of Table 5.1. We are assuming
that T is a triangulation of a three-dimensional polyhedron. The table indicates
that for the space P2− Λ1 (T ), there are two basis functions associated to each edge
of T and two basis functions associated to each two-dimensional face of T . If an
edge has vertices xi and xj with i < j, then on a simplex T containing the edge,
the corresponding basis functions are given by λi φij and λj φij , where λi and λj
are the barycentric coordinates functions on T equal to 1 at the vertices xi and xj ,
respectively, and
φij = λi dλj − λj dλi
is the Whitney form associated to the edge. Similarly, if T contains a face with
vertices xi , xj , xk , i < j < k, then on T the two basis functions associated to the
face are given by λk φij and λj φik .
5.4. Approximation properties. To apply the general theory for approximation
of Hilbert complexes to the finite element exterior calculus, outlined above, we need
to construct bounded cochain projections from L2 Λk (Ω) onto the various finite
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FINITE ELEMENT EXTERIOR CALCULUS
337
Table 5.1. Basis for the spaces Pr− Λ1 , n = 3.
r
Edge [xi , xj ]
Face [xi , xj , xk ]
1
φij
2
{λi , λj }φij
λk φij , λj φik
3
{λ2i , λ2j , λi λj }φij
{λi , λj , λk }λk φij
{λi , λj , λk }λj φik
Tet [xi , xj , xk , xl ]
λk λl φij , λj λl φik ,
λj λk φil
Table 5.2. Basis for the spaces Pr Λ2 , n = 3.
r
Face [xi , xj , xk ]
Tet [xi , xj , xk , xl ]
1
λk dλi ∧ dλj , λj dλi ∧ dλk , λi dλj ∧ dλk
2
λ2k dλi ∧ dλj , λj λk dλi ∧ d(λk − λj )
λ2j dλi ∧ dλk , λi λj d(λj − λi ) ∧ dλk
λ2i dλj ∧ dλk , λi λk dλj ∧ d(λk − λi )
λk λl dλi ∧ dλj , λj λl dλi ∧ dλk
λj λk dλi ∧ dλl , λi λl dλj ∧ dλk
λi λk dλj ∧ dλl , λi λj dλk ∧ dλl
3
λ3k dλi ∧ dλj , λ3j dλi ∧ dλk , λ3i dλj ∧ dλk
λ2j λk dλi ∧ d(2λk − λj ), λj λ2k dλi ∧ d(λk − 2λj )
λ2i λj d(2λj − λi ) ∧ dλk , λ2i λk dλj ∧ d(2λk − λi )
λi λ2j d(λj − 2λi ) ∧ dλk , λi λ2k dλj ∧ d(λk − 2λi )
λi λj λk d(2λj − λi − λk ) ∧ d(2λk − λi − λj )
{λk , λl }λk λl dλi ∧ dλj
{λj , λk , λl }λj λl dλi ∧ dλk
{λj , λk , λl }λj λk dλi ∧ dλl
{λi , λj , λk , λl }λi λl dλj ∧ dλk
{λi , λj , λk , λl }λi λk dλj ∧ dλl
{λi , λj , λk , λl }λi λj dλk ∧ dλl
element spaces. The error estimates given in Theorems 3.9 and 3.11 bound the
error in the finite element solution in terms of the approximation error afforded by
the subspaces. In this subsection, we show that the spaces Pr Λk (T ) and Pr− Λk (T )
provide optimal order approximation of differential k-forms as the mesh size tends
to zero, where optimal order means that the rate of convergence obtained is the
highest possible given the degree of the piecewise polynomials and the smoothness
of the form being approximated. In the next subsection, we shall construct L2 bounded cochain projections which attain the same accuracy.
Let {Th } be a family of simplicial triangulations of Ω ⊂ Rn , indexed by decreasing
values of the mesh parameter h given as h = maxT ∈Th diam T . We will assume
throughout that the family {Th } is shape regular, i.e., that the ratio of the volume
of the circumscribed to the inscribed ball associated to any element T is bounded
uniformly for all the simplices in all the triangulations of the family. Parts of the
construction below simplify if we assume, in addition, that the family {Th } is quasiuniform, i.e., that the ratio h/ diam T is bounded for all T ∈ Th , uniformly over
the family. However, we do not require quasi-uniformity, only shape-regularity.
will denote a subspace of
Throughout this and the following subsection, Λk−1
h
HΛk−1 (Ω) and Λkh a subspace of HΛk (Ω). Motivated by Section 5.1.4, for each
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338
D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER
r = 0, 1, . . . , we consider the following possible pairs of spaces:
⎧
⎧
⎫
⎫
−
k−1
(T )⎪
Pr+1
Λk (T )
⎪
⎪
⎪
⎨Pr+1 Λ
⎨
⎬
⎬
k−1
k
or
or
, Λh =
.
(74)
Λh =
⎪
⎪
⎪
⎪
⎩ − k−1
⎩
⎭
⎭
k
Pr Λ (T ) (if r > 0)
(T )
Pr+1 Λ
Note that Λkh contains all polynomials of degree r, but not r + 1 in each case. As we
have seen in Section 5.2, the degrees of freedom define canonical projection operators Ih = Ihk mapping CΛk (Ω) boundedly to Λkh and commuting with the exterior
derivative. However, the operators Ih do not extend boundedly to all of L2 Λk (Ω),
or even to HΛk (Ω) and so do not fulfill the requirements of bounded cochain projections of the abstract theory developed in Section 3. In this subsection, we will
describe the Clément interpolant, which is a modification of the canonical projection which is bounded on L2 Λk and can be used to establish the approximation
properties of the finite element space. However, the Clément interpolant is not
a projection and does not commute with the exterior derivative. So in the next
subsection we construct a further modification which regains these properties and
maintains the approximation properties of the Clément interpolant.
The difficulty with the canonical interpolation operators Ih is that they make use
of traces onto lower-dimensional simplexes, and as a consequence, they cannot be
extended boundedly to L2 Λk (Ω) (except for k = n). The Clément interpolant [34]
is a classical tool of finite element theory developed to overcome this problem in the
case of 0-forms. To define this operator (for general k), we need some additional
notation. For any f ∈ ∆(Th ), we let Ωf ⊂ Ω be the union of elements containing
f:
Ωf = {T | T ∈ Th , f ∈ ∆(T )},
and let Pf : L2 Λk (Ωf ) → Pr Λk (Ωf ) be the L2 projection onto polynomial k-forms
of degree at most r. For ω ∈ L2 Λk (Ω), we determine the Clément interpolant
I˜h ω := I˜hk ω ∈ Λkh by specifying φ(I˜h ω) for each degree of freedom φ of the space
Λkh ; see Section 5.2. Namely, if φ is a degree of freedom for Λkh associated to
f ∈ ∆(Th ), we take
φ(I˜h ω) = φ(Pf ωf ),
where ωf = ω|Ωf .
˜
The Clément interpolant is local
in the sense that for any T ∈ Th , Ih ω|T is
∗
determined by ω|T ∗ , where T = {Ωf | f ∈ ∆(T )}. In fact, by scaling, as in [8,
Section 5.3], for example, it can be seen that
I˜h ωL2 Λk (T ) ≤ c0
Ph ωL2 Λk (Ωf ) ≤ c1 ωL2 Λk (T ∗ ) ,
f ∈∆(T )
where the constants c0 , c1 may depend on the polynomial degree r and the dimension n, but are independent of T ∈ Th and h, thanks to the shape regularity assumption. Therefore, the Clément interpolant is uniformly bounded in
L(L2 Λk (Ω), L2 Λk (Ω)):
I˜h ωL2 Λk (Ω) ≤ c2 ωL2 Λk (Ω)
(with the constant independent of h).
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FINITE ELEMENT EXTERIOR CALCULUS
339
Another key property of the Clément interpolant is that it preserves polynomials
locally in the sense that I˜h ω|T = ω|T if ω ∈ Pr Λk (T ∗ ). It is then standard, using
the Bramble–Hilbert lemma [23] and scaling, to obtain the following error estimate.
−
Theorem 5.8. Assume that Λkh is either Pr+1
Λk (Th ), or, if r ≥ 1, Pr Λk (Th ).
Then there is a constant c, independent of h, such that the Clément interpolant
I˜hk : L2 Λk (Ω) → Λkh satisfies the bound
ω − I˜hk ωL2 Λk (Ω) ≤ chs |ω|H s Λk (Ω) ,
ω ∈ H s Λk (Ω),
for 0 ≤ s ≤ r + 1.
Note that the estimate implies that any sufficiently smooth k-form is approximated by elements Λkh with order O(hr+1 ) in L2 . Since the polynomial spaces
used to construct Λkh contain Pr Λk but not Pr+1 Λk , this is the optimal order of
approximation.
5.5. Bounded cochain projections. The Clément interpolant I˜h is both uniformly bounded in L2 and gives optimal error bounds for smooth functions. However, it is not a bounded cochain projection in the sense of the theory of Section
3. Indeed, it is neither a projection operator—it does not leave Λkh invariant—nor
does it commute with the exterior derivative. Therefore, to construct bounded
cochain projections, we consider another modification of the canonical projection
Ih , in which the operator Ih is combined with a smoothing operator.
This construction, key ingredients of which were contributed by Schöberl [79]
and Christiansen [31], was discussed in detail in [8] (where it was called a smoothed
projection), under the additional assumption that the family of triangulations {Th }
is quasi-uniform, and then in [32] in the general shape regular case. Therefore, we
will just give a brief outline of this construction here.
−
Λk (Th ) or, if r ≥ 1, Pr Λk (Th ), and let Ih :
Let Λkh be one of the spaces Pr+1
CΛk (Ω) → Λkh be the corresponding canonical interpolant. To define an appropriate
smoothing operator, we let ρ : Rn → R be a nonnegative smooth function with
support in the unit ball and with integral equal to one. In the quasi-uniform case,
we utilize a standard convolution operator of the form
ρ(y)ω(x + δy) dy,
ω →
|y|≤1
mapping L2 Λk (Ω̃) into C ∞ Λk (Ω), where the domain Ω̃ ⊃ Ω is sufficiently large
so that the convolution operator is well defined. We set the smoothing parameter
δ = h, where the proportionality constant > 0 is a parameter to be chosen. In
the more general shape regular case, we need to generalize this operator slightly.
As in [32], we introduce a Lipschitz continuous function gh : Ω → R+ , with Lipschitz constant bounded uniformly in h, such that gh (x) is uniformly equivalent to
diam T for T ∈ Th and x ∈ T . Hence, gh |T approximates diam T . The appropriate
smoothing operator in the general case, mapping L2 Λk (Ω̃) into CΛk (Ω), is now
given by
ω →
|y|≤1
∗
ρ(y)((Φy
h ) ω)x dy,
n
n
where the map Φy
h : R → R is defined by
Φy
h (x) = x + gh (x)y.
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340
D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER
Since this smoothing operator is defined as an average of pullbacks, it will indeed
commute with the exterior derivative. By combining it with an appropriate extension operator E : L2 Λk (Ω) → L2 Λk (Ω̃), constructed such that it commutes with the
exterior derivative, we obtain an operator Rh : L2 Λk (Ω) → CΛk (Ω). The operator
Qh = Ih ◦ Rh maps L2 Λk (Ω) into Λkh and commutes with the exterior derivative.
Furthermore, for each > 0, the operators Qh are bounded in L2 , uniformly in h.
However, the operator Qh is not invariant on the subspace Λkh . The remedy to
fix this is to establish that the operators Qh |Λkh : Λkh → Λkh converge to the identity
in the L2 operator norm as tends to zero, uniformly in h. Therefore, for a fixed
> 0, taken sufficiently small, this operator has an inverse Jh : Λkh → Λkh . The
desired projection operator, πh = πhk , is now given as πh = Jh ◦ Qh . The operators
πh are uniformly bounded with respect to h as operators in L(L2 Λk (Ω), L2 Λk (Ω)).
Furthermore, since they are projections onto Λkh , we obtain
ω − πh ωL2 Λk (Ω) ≤ inf (I − πh )(ω − µ)L2 Λk (Ω) .
(75)
µ∈Λk
h
Based on these considerations, we obtain the following theorem. Cf. [8, Theorem
5.6] and [32, Corollary 5.3].
−
Λk (Th ) or, if r ≥ 1, Pr Λk (Th )
Theorem 5.9. 1. Let Λkh be one of the spaces Pr+1
k
2 k
k
and πh : L Λ (Ω) → Λh the smoothed projection operator constructed above. Then
πhk is a projection onto Λkh and satisfies
ω − πhk ωL2 Λk (Ω) ≤ chs ωH s Λk (Ω) ,
ω ∈ H s Λk (Ω),
for 0 ≤ s ≤ r + 1. Moreover, for all ω ∈ L2 Λk (Ω), πhk ω → ω in L2 as h → 0.
2. Let Λkh be one of the spaces Pr Λk (Th ) or Pr− Λk (Th ) with r ≥ 1. Then
d(ω − πhk ω)L2 Λk (Ω) ≤ chs dωH s Λk (Ω) ,
ω ∈ H s Λk (Ω),
for 0 ≤ s ≤ r.
3. Let Λk−1
and Λkh be taken as in (74) and let πhk−1 and πhk be the corresponding
h
smoothed projections. Then dπhk−1 = πhk d.
Proof. We have already established that the πhk are uniformly bounded projections
which commute with d. The error estimate in the first statement of the theorem
then follows from (75). Statement 2 follows from 1 and 3.
5.6. Approximation of the de Rham complex and the Hodge Laplacian.
Let Λnh be the space of piecewise polynomial n-forms of degree at most s for some
nonnegative integer s. We showed in Section 5.2 that there are 2n−1 distinct discrete
de Rham complexes
d
d
d
→ Λ0h −
→ ··· −
→ Λnh → 0,
0 → Λ0h −
(76)
d
−
→ Λkh being of the form (73) for an appropriate
with each of the mappings Λk−1
h
choice of s. Making use of the bounded projections, we obtain in each case a
commuting diagram
d
d
d
d
d
0 → HΛ0 (Ω) −−−−→ HΛ1 (Ω) −−−−→ · · · −−−−→ HΛn (Ω) → 0
⏐
⏐
⏐
⏐π
⏐π
⏐π
h
h
h
0→
Λ0h
d
−−−−→
Λ1h
−−−−→ · · · −−−−→
Λnh
→ 0.
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FINITE ELEMENT EXTERIOR CALCULUS
341
Thus the projections give a cochain projection from the de Rham complex to
the discrete de Rham complex and so induce a surjection on cohomology. Since
limh→0 ω −πh wHΛk (Ω) = 0, we can use Theorem 3.4 (with V = HΛ and Vh = Λh )
to see immediately that for h sufficiently small, this is in each case an isomorphism
on cohomology.
The simplest finite element de Rham complex is the complex of Whitney forms,
→ P1− Λ1 (Th ) −
→ ··· −
→ P1− Λn (Th ) → 0.
0 → P1− Λ0 (Th ) −
d
(77)
d
d
Via the basic Whitney forms φf , the space P1− Λk (Th ) is isomorphic to the space
of simplicial cochains of dimension k associated to the triangulation Th . In this
way, the complex (77) of Whitney forms is isomorphic, as a cochain complex, to
the simplicial cochain complex, and its cohomology is isomorphic to the simplicial
cohomology of Ω. Thus Theorem 3.4, when applied to the Whitney forms, gives
an isomorphism between the de Rham cohomology and simplicial cohomology for
a sufficiently fine triangulation. Since the simplicial cohomology is independent of
the triangulation (being isomorphic to the singular cohomology), for the complex
of Whitney forms, the isomorphism on cohomology given by Theorem 3.4 holds for
any triangulation, not just one sufficiently fine. This proves de Rham’s theorem on
the equality of de Rham and simplicial cohomology.
For all of the discrete complexes (76), as for the Whitney forms, the cohomology
is independent of the triangulation (and equal to the de Rham cohomology). To see
this, following [31], we consider the Whitney forms complex (77) as a subcomplex
of (76). The canonical projections Ih define cochain projections. Note that the Ih
are defined on the finite element spaces Λkh , because all of the trace moments they
require are single-valued on Λkh . From the commuting diagram
0→
Λ0h
⏐
⏐I
h
d
−−−−→
Λ1h
⏐
⏐I
h
d
d
−−−−→ · · · −−−−→
Λnh
⏐
⏐I
h
→0
0 → P1− Λ0 (Th ) −−−−→ P1− Λ1 (Th ) −−−−→ · · · −−−−→ P1− Λn (Th ) → 0,
d
d
d
we conclude that the cohomology of the top row, which we have already seen to be
an image of the de Rham cohomology, maps onto the cohomology of the bottom
row, which is isomorphic to the de Rham cohomology. Hence the dimensions of
all the corresponding cohomology groups are equal and both cochain projections
induce an isomorphism on cohomology.
Consider now the numerical solution of the Hodge Laplacian problem (59). We
, uh ∈ Λkh , and ph ∈ Hkh
approximate this by a Galerkin method: Find σh ∈ Λk−1
h
such that
τ ∈ Λk−1
,
σh , τ − dτ, uh = 0,
h
dσh , v + duh , dv + v, ph = f, v,
uh , q = 0,
v ∈ Λkh ,
q ∈ Hkh .
For the finite element spaces, we may choose Λk−1
and Λkh to be any of the pairs
h
k−1
given in (74). We have verified that d maps Λh into Λkh and this map can be
extended to a full subcomplex of the de Rham complex admitting a bounded cochain
projection. Therefore we may combine the error estimates derived in the abstract
setting in Sections 3.5 and 3.6 and approximation estimates from Section 5.5 to
obtain convergence and rates of convergence. The rates of convergence are limited
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342
D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER
by three factors: (1) the smoothness of the data f , (2) the amount of elliptic
regularity (determined by the smoothness of the domain), and (3) the degree of
complete polynomials contained in the finite element shape functions. Specifically,
from Theorem 3.11 we get the following result. Assume that the regularity estimate
uH t+2 (Ω) + pH t+2 (Ω) + duH t+1 (Ω) + σH t+1 (Ω) + dσH t (Ω) ≤ Cf H t (Ω)
holds for 0 ≤ t ≤ tmax . Then for 0 ≤ s ≤ tmax the following estimates hold:
d(σ − σh ) ≤ Chs f H s (Ω) , if s ≤ r + 1,
= Pr+1 Λk−1 (T ),
s ≤ r + 1, Λk−1
s+1
h
σ − σh ≤ Ch f H s (Ω) , if
k−1
−
Λk−1 (T ),
s ≤ r,
Λh = Pr+1
−
s ≤ r,
Λkh = Pr+1
Λk (T ),
s+1
d(u − uh ) ≤ Ch f H s (Ω) , if
s ≤ r − 1, Λkh = Pr Λk (T ),
Chf ,
Λkh = P1− Λk (T ),
u − uh + p − ph ≤
s+2
Ch f H s (Ω) if s ≤ r − 1, otherwise.
Thus, the error in each case is the optimal order allowed by the subspace if we have
sufficient elliptic regularity.
Concerning the eigenvalue problem, applying Theorem 3.21, we immediately
obtain the following error estimate:
|λ − λh | ≤ Ch2s+2 u2H s (Ω) ,
0 ≤ s ≤ r − 1.
This estimate does not apply in case Λkh = P1− Λk (T ), i.e., the Whitney forms. In
that case, we get the estimate:
Chu2 ,
|λ − λh | ≤
Ch2 uuH 1 (Ω) .
6. Variations of the de Rham complex
In Sections 4 and 5, we applied the abstract theory of Section 3 to the Hilbert
complex obtained by choosing W k to be L2 Λk (Ω) and dk to be the exterior derivative with domain V k = HΛk (Ω). This led to finite element approximations of
certain boundary value problems for the Hodge Laplacian, and a variety of related
problems (B and B∗ problems, eigenvalue problems). In this section, we show
how some small variations of these choices lead to methods for important related
problems, namely generalizations of the Hodge Laplacian with variable coefficients
and problems with essential boundary conditions.
6.1. Variable coefficients. In this section, dk is again taken to be the exterior
derivative with domain V k = HΛk (Ω) and W k is again taken to be the space L2 Λk ,
but W k is furnished with an inner product which is equivalent to, but not equal to,
the standard L2 Λk inner product. Specifically, we let ak : W k → W k be a bounded,
symmetric, positive definite operator and choose
u, vW k = ak u, vL2 Λk .
With this choice, we can check that d∗k = (dk−1 )∗ is given by
d∗k u = (ak−1 )−1 δk ak u,
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FINITE ELEMENT EXTERIOR CALCULUS
343
with the domain of d∗k given by
Vk∗ = { u ∈ L2 Λk (Ω) | ak u ∈ H̊ ∗ Λk (Ω)} = (ak )−1 H̊ ∗ Λk (Ω).
The harmonic forms are then
Hk = Zk ∩ Z∗k = { u ∈ HΛk ∩ (ak )−1 H̊ ∗ Λk | du = 0, δak u = 0 }.
The Hodge Laplacian d∗ du + dd∗ u is now realized as
d(ak−1 )−1 δak u + (ak )−1 δak+1 du.
As an example of the utility of this generalization, we consider the B∗1 eigenvalue
problem in three dimensions. We take a1 = ε and a2 = µ−1 , where ε and µ are
symmetric positive definite 3 × 3 matrix fields. Then the B∗1 eigenvalue problem is
curl µ−1 curl u = λεu.
This is the standard Maxwell eigenvalue problem, where ε is the dielectric tensor
of the material and µ is its magnetic permeability. These are scalar for an isotropic
material, but given by matrices in general. They are constant for a homogeneous
material, but vary if the material is not constant across the domain.
We notice that the Hilbert complex in this case is exactly the de Rham complex
(52), except that the spaces are furnished with different, but equivalent, inner products. Thus the finite element de Rham subcomplexes we constructed in Section 5
apply for this problem and satisfy the subcomplex and bounded cochain projection
properties. In other words, the same finite element spaces which were developed for
the constant coefficient Hodge Laplacian can be applied equally well for variable
coefficient problems.
6.2. The de Rham complex with boundary conditions. For another important application of the abstract theory of Section 3, we again choose W k = L2 Λk (Ω)
and dk to be the exterior derivative, but this time we take its domain V k to be
the space H̊Λk (Ω). This space is dense in W k and is a closed subspace of HΛk ,
so is complete, and hence d is again a closed operator with this domain. Thus we
obtain the following complex of Hilbert spaces, called the de Rham complex with
boundary conditions:
(78)
d
d
d
0 → H̊Λ0 (Ω) −
→ H̊Λ1 (Ω) −
→ ··· −
→ H̊Λn (Ω) → 0.
In Section 4.2, we used the fact that HΛk ∩ H̊ ∗ Λk is compactly included in L2 Λk .
Replacing k by n − k and applying the Hodge star operator, we conclude the
analogous result, that H̊Λk ∩ H ∗ Λk is compactly included in L2 Λk . Then, just
as in Section 4.2, all the results of Section 3 apply to the de Rham complex with
boundary conditions. In particular, we have the Hodge decomposition of L2 Λk and
of H̊Λk , the Poincaré inequality, well-posedness of the mixed formulation of the
Hodge Laplacian, and all the approximation results established in Sections 3.3–3.6.
We denote the spaces of k-cocycles, k-coboundaries, and harmonic k-forms as
Z̊k = { ω ∈ H̊Λk | dω = 0 }, B̊k = { dη | η ∈ H̊Λk−1 },
H̊k = { ω ∈ Z̊k | ω, µ = 0, µ ∈ B̊k }.
Now, from (53), we have that
(79)
dω, µ = ω, δµ,
ω ∈ H̊Λk−1 , µ ∈ H ∗ Λk .
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344
D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER
Therefore
B̊k⊥L2 := { ω ∈ L2 Λk | ω, dη = 0 ∀η ∈ H̊Λk−1 } = { ω ∈ H ∗ Λk | δω = 0 }.
This gives a concrete characterization of the harmonic forms analogous to (58):
(80)
H̊k = {ω ∈ H̊Λk (Ω) ∩ H ∗ Λk (Ω) | dω = 0, δω = 0}.
Now, from (54), (55), and (50), the Hodge star operator maps the set of ω ∈ HΛn−k
with dω = 0 isomorphically onto the set of µ ∈ H ∗ Λk with δµ = 0, and it maps
the set of ω ∈ H̊∗ Λn−k with δω = 0 isomorphically onto the set of µ ∈ H̊Λk with
dµ = 0. Comparing (58) and (80), we conclude that the Hodge star operator maps
Hn−k isomorphically onto H̊k . This isomorphism is called Poincaré duality. In
particular, dim H̊k is the (n − k)th Betti number of Ω.
Finally, from (79), we can characterize d∗ , the adjoint of the exterior derivative
d viewed as an unbounded operator L2 Λk−1 → L2 Λk with domain H̊Λk . Namely,
in this case, d∗ has domain H ∗ Λk and coincides with the operator δ defined in (50).
By contrast, when we took the domain of d to be all of HΛk , the domain of d∗
turned out to be the smaller space H̊ ∗ Λk .
We now consider the Hodge Laplacian problem given in the abstract case by
(18). In this case, we get that (σ, u, p) ∈ H̊Λk−1 × H̊Λk × H̊k is a solution if and
only if
(81)
σ = δu, dσ + δdu = f − p
in Ω,
(82)
tr σ = 0, tr u = 0 on ∂Ω,
(83)
u ⊥ H̊k .
Note that now both boundary conditions are essential in the sense that they are
implied by membership in the spaces H̊Λk−1 and H̊Λk where the solution is sought,
rather than by the variational formulation.
To make things more concrete, we restrict to a domain Ω ⊂ R3 and consider the
Hodge Laplacian for k-forms, k = 0, 1, 2, and 3. We also discuss the B∗ and B
problems given by (19) and (20) for each k. Again we have
d0 = grad, d1 = curl, d2 = div,
and the de Rham complex (78) is now realized as
grad
curl
div
0 → H̊ 1 (Ω) −−−→ H̊(curl; Ω) −−→ H̊(div; Ω) −−→ L2 (Ω) → 0,
where
H̊(curl; Ω) = { u : Ω → R3 | u ∈ L2 , curl u ∈ L2 , u × n = 0 on ∂Ω },
H̊(div; Ω) = { u : Ω → R3 | u ∈ L2 , div u ∈ L2 , u · n = 0 on ∂Ω }.
6.2.1. The Hodge Laplacian for k = 0. For k = 0, the boundary value problem
(81)–(83) is the Dirichlet problem for the ordinary scalar Laplacian. The space
HΛ−1 is understood to be 0, so σ vanishes. The space H̊0 of harmonic 0-forms
vanishes. The first differential equation of (81) vanishes, and the second gives
Poisson’s equation
− div grad u = f in Ω.
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FINITE ELEMENT EXTERIOR CALCULUS
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Similarly, the first boundary condition in (82) vanishes, while the second is the
Dirichlet condition u = 0 on ∂Ω. The side condition (83) is automatically satisfied. Nothing additional is obtained by considering the split into the B∗ and B
subproblems, since the latter is trivial.
6.2.2. The Hodge Laplacian for k = 1. In this case the differential equations and
boundary conditions are
σ = − div u, grad σ + curl curl u = f − p
(84)
in Ω,
σ = 0, u × n = 0 on ∂Ω,
which is a formulation of a boundary value problem for the vector Laplacian
curl curl − grad div. The solution is determined uniquely by the additional condition that u be orthogonal to H̊1 = H2 , i.e., those vector fields satisfying
curl p = 0, div p = 0 in Ω,
B∗1
p × n = 0 on ∂Ω.
2
problem (19) is defined for L vector fields f which are orthogonal to
The
both gradients of functions in H̊ 1 (Ω) and the vector fields in H̊1 . In that case, the
solution to (84) has σ = 0 and p = 0, while u satisfies
curl curl u = f, div u = 0 in Ω,
u × n = 0 on ∂Ω.
The orthogonality condition u ⊥ H̊ again determines the solution uniquely.
Next we turn to the B1 problem. For source functions of the form f = grad F for
some F ∈ H̊ 1 , (84) reduces to the problem of finding σ ∈ H̊ 1 and u ∈ B1 = grad H̊ 1
such that:
1
σ = − div u, grad σ = f in Ω.
The differential equations may be simplified to − grad div u = f and the condition
that u ∈ B1 can be replaced by the differential equation curl u = 0 and the boundary condition u × n = 0, together with orthogonality to H̊1 . Now grad(σ − F ) = 0
and σ − F = 0 on ∂Ω, so σ = F . Hence, we may rewrite the system as
− div u = F, curl u = 0 in Ω,
u × n = 0 on ∂Ω,
which, again, has a unique solution subject to orthogonality to H̊1 .
6.2.3. The Hodge Laplacian for k = 2. The differential equations and boundary
conditions are
σ = curl u, curl σ − grad div u = f − p in Ω,
σ × n = 0, u · n = 0 on ∂Ω.
This is again a formulation of a boundary value problem for the vector Laplacian
curl curl − grad div, but with different boundary conditions than for (84), and this
time stated in terms of two vector variables, rather than one vector and one scalar.
This time uniqueness is obtained by imposing orthogonality to H̊2 = H1 , the space
of vector fields satisfying
curl p = 0, div p = 0 in Ω,
p · n = 0 on ∂Ω.
The B∗2 problem arises for source functions of the form f = grad F for some
F ∈ H 1 which we may take to have integral 0. We find σ = 0, and u solves
− div u = F, curl u = 0 in Ω,
u · n = 0 on ∂Ω,
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346
D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER
i.e., the same differential equation as for B1 , but with different boundary conditions
and, of course, now uniqueness is determined by orthogonality to H̊2 .
If div f = 0, f · n = 0 on ∂Ω, and f ⊥ H̊2 , we get the B2 problem for which the
differential equations are σ = curl u, curl σ = f and the condition div u = 0 arising
from the membership of u in B2 . Thus u solves
curl curl u = f, div u = 0 in Ω,
u · n = 0 on ∂Ω,
the same differential equation as for B∗1 , but with different boundary conditions.
6.2.4. The Hodge Laplacian for k = 3. The space H̊3 consists of the constants (or, if
Ω is not connected, of functions which are constant on each connected component).
The Hodge Laplacian problem is
σ = − grad u, div σ = f − p in Ω, σ · n = 0 on ∂Ω,
u q dx = 0, q ∈ H̊3 ,
Ω
which is the Neumann problem for Poisson’s equation, where the (piecewise) constant p is required for there to exist a solution and the final condition fixes a unique
solution.
7. The elasticity complex
In this section, we present another complex, which we call the elasticity complex,
and apply it to the development of numerical methods for linear elasticity. In
contrast to the other examples presented above, the elasticity complex is not a
simple variant of the de Rham complex, and, in particular, it involves a differential
operator of second order. However, the two complexes are related in a subtle
manner, via a construction known as the Bernstein–Bernstein–Gelfand resolution,
explained for example in [8, 9, 43] and the references given there.
The equations of elasticity are of great importance in modeling solid structures,
and the need to solve them in engineering applications was probably the primary
reason for the development of the finite element method in the 1960s. The simplest finite element methods for elasticity are based on displacement approaches, in
which the displacement vector field is characterized as the minimizer of an elastic
energy functional. The design and analysis of displacement finite element methods is standard and discussed in many textbooks (e.g., [33]). However, for more
general material models, arising, for example, for incompressible materials or some
viscoelastic or plastic materials, the displacement method is either not feasible or
performs poorly, and a mixed approach, in which the elastic stress and the displacement are taken as unknowns, is the natural alternative. In fact, mixed finite
element methods for elasticity were proposed in the earliest days of finite elements
[51], and stable finite elements for the mixed formulation of elasticity have been
sought for over four decades. These proved very elusive. Indeed, one of the motivations of the pioneering work of Raviart and Thomas [76] on mixed finite elements
for the Laplacian, was the hope that the solution to this easier problem would pave
the way to such elements for elasticity, and there were many attempts to generalize
their elements to the elasticity system [4, 5, 63, 80, 81, 82]. However, it was not
until 2002 that the first stable mixed finite elements for elasticity using polynomial
shape functions were discovered by two of the present authors [11], based on techniques and insights from the then nascent finite element exterior calculus. See also
[7]. They developed and analyzed a family of methods for plane (two-dimensional)
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FINITE ELEMENT EXTERIOR CALCULUS
347
elasticity. The basic elements in the family involve a space of shape functions intermediate between quadratics and cubics for the stress, with 24 degrees of freedom,
while the displacement field consists of the complete space of piecewise linear vector
fields, without continuity constraints. It is these elements we will describe here.
The plane elasticity elements of [11] were generalized to three dimensions in
[3]. See also [1]. These elements are, however, quite complicated (162 degrees of
freedom per tetrahedron for the stress). A much more promising approach involves a
variant Hilbert complex for elasticity, the elasticity complex with weak symmetry.
In [8] and [9], stable families of finite elements are derived for this formulation
in n dimensions, again using techniques from the finite element exterior calculus.
For the lowest-order stable elements, the stress space is discretized by piecewise
linears with just 12 degrees of freedom per triangle in two dimensions, 36 per
tetrahedron in three dimensions, while the other variables are approximated by
piecewise constants.
7.1. The elasticity system. The equations of linear elasticity arise as a system
consisting of a constitutive equation and an equilibrium equation:
(85)
Aσ = u,
div σ = f
in Ω.
Here the unknowns σ and u denote the stress and displacement fields engendered by
a body force f acting on a linearly elastic body which occupies a region Ω ⊂ Rn . The
constitutive equation posits a linear relationship between the linearized deformation
or strains due to the displacement and the stresses. The equilibrium equation states
that the stresses, which measure the internal forces in the body, are in equilibrium
with the externally applied force. The stress field σ takes values in the space S :=
n
Rn×n
sym of symmetric matrices and the displacement field u takes values in V := R .
The differential operator is the symmetric part of the gradient, the div operator
is applied row-wise to a matrix, and the compliance tensor A = A(x) : S → S is a
bounded and symmetric, uniformly positive definite operator reflecting the material
properties of the body. If the body is clamped on the boundary ∂Ω of Ω, then the
appropriate boundary condition for the system (85) is u = 0 on ∂Ω. For simplicity,
this boundary condition will be assumed here.
The pair (σ, u) can alternatively be characterized as the unique critical point of
the Hellinger–Reissner functional
1
Aτ : τ + div τ · v − f · v dx.
I(τ, v) =
Ω 2
The critical point is sought among τ ∈ H(div; Ω; S), the space of square-integrable
symmetric matrix fields with square-integrable divergence, and v ∈ L2 (Ω; V), the
space of square-integrable vector fields. Equivalently, the pair (σ, u) ∈ H(div; Ω; S)
×L2 (Ω; V) is the unique solution to the weak formulation of the system (85):
(Aσ : τ + div τ · u) dx = 0,
τ ∈ H(div; Ω; S),
Ω
(86)
div σ · v dx
= Ω f · v dx,
v ∈ L2 (Ω; V).
Ω
7.2. The elasticity complex and its discretization. The discretization of the
problem (86) is closely tied to the discretization of the elasticity complex. For a
two-dimensional domain Ω, the elasticity complex takes the form
(87)
J
div
0 → H 2 (Ω) −−−−→ H(div; Ω; S) −−−−→ L2 (Ω; R2 ) → 0.
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348
D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER
The operator J is a second-order differential operator mapping scalar fields into
symmetric matrix fields, namely the rotated Hessian
Jq =
∂y2 q
−∂x ∂y q
−∂x ∂y q
∂x2 q
= O(grad grad q)O T ,
O=
0 1
.
−1 0
We now show that this complex is closed, i.e., that the range of the two operators
J and div are closed subspaces. The null space of J is just the space P1 of linear
polynomials, and we have qH 2 ≤ cJqL2 for all q ∈ H 2 (Ω) with q ⊥ P1 , which
implies that the range of J is closed. We now show that div maps H(div; Ω; S)
onto L2 (Ω; R2 ), using the fact that the scalar-valued divergence maps H 1 (Ω; R2 )
onto L2 (Ω). Given f ∈ L2 (Ω; R2 ), the latter result implies the existence of τ ∈
H 1 (Ω; R2×2 ) with div τ = f . This τ may not be symmetric. However, for any
v ∈ H 1 (Ω; R2 ), we let curl v denote the matrix with ith row equal to curl vi =
(−∂vi /∂x2 , ∂vi /∂x1 ). Then div(τ + curl v) = f and τ + curl v ∈ H(div; Ω; S) if
div v = τ21 − τ12 . Thus div : H(div; Ω; S) → L2 (Ω; R2 ) is indeed surjective. Setting
W 0 = L2 (Ω), W 1 = L2 (Ω; S), and W 2 = L2 (Ω; R2 ), we obtain a closed Hilbert
complex with domain complex (87).
Now we describe only the simplest discretization from [11]. The space Vh0 ⊂
0
V = H 2 (Ω) is the finite element space whose shape functions on each triangle
T are the quintic polynomials P5 (T ) and whose degrees of freedom are the values,
first derivatives and second derivatives at each vertex, together with the mean value
of the normal derivatives along each edge, which gives 21 = dim P5 (T ) degrees of
freedom per triangle. The resulting finite element space Vh0 is referred to in the
finite element literature as the Argyris space, and is easily seen to be contained
in C 1 (Ω), and hence in H 2 (Ω). In fact, the Argyris space is in a certain sense
the simplest finite element space contained in C 1 (Ω). However, the Argyris space
does not coincide with the space of all C 1 piecewise quintics (which is not a finite
element space, as it cannot be defined via degrees of freedom). Elements of the
Argyris space have extra smoothness at the vertices, where they are C 2 .
The space Vh1 ⊂ V 1 = H(div; Ω; S) consists of those continuous piecewise cubic symmetric matrix fields whose divergence is piecewise linear. The degrees of
freedom determining an element σ ∈ Vh1 are
• the value of σ at each vertex,
• the moments of degree 0 and 1 of the two normal components of σ on each
edge, and
• the moments of degree 0 of σ on each triangle.
Hence, the restriction of σ to a triangle T ∈ Th is uniquely determined by 24 degrees
of freedom. Finally, the space Vh2 ⊂ V 2 = L2 (Ω; R2 ) consists of all piecewise linear
vector fields with respect to Th , without any imposed continuity. For degrees of
freedom, we choose the moments of degrees 0 and 1 for each component on each
triangle T . Using the degrees of freedom, we also define projections Ihk v ∈ Vhk
defined for sufficiently smooth v ∈ V k by φ(Ihk v) = φ(v) for all degrees of freedom
φ.
It is straightforward to check that we have the subcomplex property JVh0 ⊂ Vh1
and div Vh1 ⊂ Vh2 and that the interpolation operators Ih := Ihk give a commuting
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FINITE ELEMENT EXTERIOR CALCULUS
349
diagram
J
(88)
div
0 → V 0 ∩ C 2 −−−−→ V 1 ∩ C 0 −−−−→ V 2 → 0
⏐
⏐
⏐
0⏐
1⏐
2⏐
Ih
Ih
Ih
0→
Vh0
J
−−−−→
Vh1
div
−−−−→ Vh2 → 0.
However, since Ih0 involves point values of the second derivative and Ih1 involves
point values, these operators do not extend to bounded operators on the Hilbert
spaces V 0 and V 1 .
In order to apply the general theory of approximation of Hilbert complexes
developed in Section 3, we will modify the interpolation operators Ihk in (88) to
obtain bounded cochain projections πhk : W k → Vhk for k = 0, 1, 2. As in Section 5.5,
the πhk will be obtained from Ihk via smoothing.
7.3. Bounded cochain projections for the elasticity complex. First, we define an appropriate pullback of an affine map and show that it defines a cochain map
for the elasticity sequence. Let F : Ω → Ω ⊂ R2 be an affine map, so that DF =
DF (x) is a constant 2 × 2 matrix. Let B be the matrix B = det(DF )(DF )−1 =
O(DF )T O T and define pullbacks F ∗ = Fk∗ : W k (Ω ) → W k (Ω) by
F ∗ v = v ◦ F,
v ∈ W 0 (Ω ),
F ∗ v = B(v ◦ F )B T ,
v ∈ W 1 (Ω ),
F ∗ v = det(DF )B(v ◦ F ), v ∈ W 2 (Ω ).
It is straightforward to check that, as long as F is affine, F ∗ commutes with the
differential operators J and div. In other words, we obtain the commuting diagram
J
div
J
div
0 → V 0 (Ω ) −−−−→ V 1 (Ω ) −−−−→ V 2 (Ω ) → 0
⏐
⏐
⏐
⏐
⏐
⏐
F ∗
F ∗
F ∗
0 → V 0 (Ω) −−−−→ V 1 (Ω) −−−−→ V 2 (Ω) → 0.
To define a smoothing operator which maps W k into itself, we need to make
the assumption that the domain Ω is star-shaped with respect to some point in its
interior (which we assume, without further loss of generality, to be the origin).
We then let a = maxx∈∂Ω |x|−1 and define the dilation Fδ : R2 → R2 by Fδ (x) =
x/(1 + aδ), δ > 0. Then Fδ maps the δ-neighborhood of Ω, Ωδ , into Ω and so Fδ∗
maps L2 (Ω) to L2 (Ωδ ). Composing with a standard mollification
ρ(z)v(x + δz) dz,
v →
|z|<1
we obtain a smoothing operator Rδ mapping W k into W k . Here ρ : R2 → R is
a smooth, nonnegative function supported in the unit ball and with integral equal
to one. This smoothing commutes with any differential operator with constant
coefficients, in particular with J and div.
The construction now proceeds as in Section 5.5, and we just give an outline of
it. For simplicity, we assume a quasi-uniform family of meshes, although extension
to general shape regular meshes could be made as in Section 5.5. Let > 0 be fixed.
Define operators Qkh : W k → Whk by Qkh = Ihk ◦ Rh . The operator Qkh maps W k
into Vhk . In fact, it can be seen by a scaling argument that the operators Qkh are
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350
D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER
bounded as operators in L(W k , W k ), uniformly in h. Furthermore, Qkh commutes
with J and div, i.e.,
(89)
Q1h ◦ J = J ◦ Q0h
and
Q2h ◦ div = div ◦ Q1h .
However, the operators Qkh are not projections, since they do not restrict to the
identity on Vhk . Therefore, we define the desired projections πhk : W k → Vhk as
πhk = (Qkh |Vhk )−1 Qkh . To justify this definition, we need to argue that (Qkh |Vhk )−1
exists. In fact, for sufficiently small, but fixed, (Qkh |Vhk )−1 is even uniformly
bounded in L(Whk , Whk ), where Whk = Vhk , but equipped with the norm of W k .
This follows from the fact that the operators Qkh |Vhk converge to the identity with
in L(Whk , Whk ), uniformly in h, established by a scaling argument.
We conclude that if the parameter > 0 is fixed, but sufficiently small, then
the operators πhk are well defined and uniformly bounded with respect to h in
L(W k , W k ). Furthermore, the diagram
J
div
J
div
0 → V 0 −−−−→ V 1 −−−−→ V 2 → 0
⏐
⏐
⏐
⏐
⏐
⏐
πh πh πh 0 → Vh0 −−−−→ Vh1 −−−−→ Vh2 → 0
commutes, as a consequence of (89).
Thus, under the assumption of a star-shaped domain, we have established the
existence of bounded cochain projections and so the results of Section 3.4 apply to
the elasticity complex. This means that the corresponding Hodge Laplacians are
well-posed and that the discretizations are stable. In particular, using the spaces
Vh1 and Vh2 , we obtain a stable, convergent mixed discretization of the elasticity
equations.
Remark. Stability and convergence of the mixed finite element method for elasticity
using the spaces Vh1 and Vh2 described above is proven in [11] without restricting to
star-shaped domains. However, we do not know how to construct bounded cochain
projections without this restriction.
Acknowledgments
The authors would like to thank Marie Rognes for her help with some of the
computations appearing in Section 2.3.
About the authors
Douglas Arnold is McKnight Presidential Professor of Mathematics at the University of Minnesota and President of the Society for Industrial and Applied Mathematics (SIAM). He was a plenary speaker at the ICM in Beijing in 2002, a Guggenheim Fellow in 2008–2009, and is a foreign member of the Norwegian Academy of
Science and Letters.
Richard Falk is a professor at Rutgers University. He has served both as Chair
of the Department of Mathematics and Acting Dean of the Faculty of Arts and
Sciences and of the Graduate School and has co-organized the biannual “Finite
Element Circus” meetings of researchers on finite element methods since 1996.
Ragnar Winther is a professor at and director of the Centre of Mathematics for
Applications at the University of Oslo, Norway. He is chairman of the board of the
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FINITE ELEMENT EXTERIOR CALCULUS
351
N.H. Abel Memorial Fund and was an invited speaker at the European Congress
of Mathematics in Amsterdam in 2008.
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School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
E-mail address: [email protected]
Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854
E-mail address: [email protected]
Centre of Mathematics for Applications and Department of Informatics, University
of Oslo, 0316 Oslo, Norway
E-mail address: [email protected]
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