# 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 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 282 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. This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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]. This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. FINITE ELEMENT EXTERIOR CALCULUS 289 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. This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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 ), This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. FINITE ELEMENT EXTERIOR CALCULUS 293 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. This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 294 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 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. FINITE ELEMENT EXTERIOR CALCULUS 295 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 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. FINITE ELEMENT EXTERIOR CALCULUS 297 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. This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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. This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. FINITE ELEMENT EXTERIOR CALCULUS 299 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 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 300 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]. This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. FINITE ELEMENT EXTERIOR CALCULUS 301 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 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 302 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 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. FINITE ELEMENT EXTERIOR CALCULUS 303 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. This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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). This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. FINITE ELEMENT EXTERIOR CALCULUS 305 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 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. FINITE ELEMENT EXTERIOR CALCULUS 307 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 , This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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 . This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. FINITE ELEMENT EXTERIOR CALCULUS 309 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. This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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. This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. FINITE ELEMENT EXTERIOR CALCULUS 311 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. This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. FINITE ELEMENT EXTERIOR CALCULUS 313 (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). This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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 , This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. FINITE ELEMENT EXTERIOR CALCULUS 315 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 . This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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 , This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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. This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. FINITE ELEMENT EXTERIOR CALCULUS 321 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 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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 , This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. FINITE ELEMENT EXTERIOR CALCULUS 323 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). This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. FINITE ELEMENT EXTERIOR CALCULUS 325 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. This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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. This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. FINITE ELEMENT EXTERIOR CALCULUS 327 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 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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. This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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 − This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. FINITE ELEMENT EXTERIOR CALCULUS 331 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. This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. FINITE ELEMENT EXTERIOR CALCULUS 333 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 ) This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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). This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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. This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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. This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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, This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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 . This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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. This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. FINITE ELEMENT EXTERIOR CALCULUS 345 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, This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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) This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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. This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 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. References 1. Scot Adams and Bernardo Cockburn, A mixed ﬁnite element method for elasticity in three dimensions, J. Sci. Comput. 25 (2005), no. 3, 515–521. MR2221175 (2006m:65251) 2. C. Amrouche, C. Bernardi, M. Dauge, and V. Girault, Vector potentials in three-dimensional non-smooth domains, Math. Methods Appl. Sci. 21 (1998), 823–864. MR1626990 (99e:35037) 3. Douglas N. Arnold, Gerard Awanou, and Ragnar Winther, Finite elements for symmetric tensors in three dimensions, Math. Comp. 77 (2008), no. 263, 1229–1251. MR2398766 (2009b:65291) 4. Douglas N. Arnold, Franco Brezzi, and Jim Douglas, Jr., PEERS: a new mixed ﬁnite element for plane elasticity, Japan J. Appl. Math. 1 (1984), 347–367. MR840802 (87h:65189) 5. Douglas N. Arnold, Jim Douglas, Jr., and Chaitan P. Gupta, A family of higher order mixed ﬁnite element methods for plane elasticity, Numer. Math. 45 (1984), 1–22. MR761879 (86a:65112) 6. Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Diﬀerential complexes and stability of ﬁnite element methods. I. The de Rham complex, Compatible Spatial Discretizations, IMA Vol. Math. Appl., vol. 142, Springer, New York, 2006, pp. 24–46. MR2249344 (2008c:65296) , Diﬀerential complexes and stability of ﬁnite element methods. II. The elasticity com7. plex, Compatible Spatial Discretizations, IMA Vol. Math. Appl., vol. 142, Springer, New York, 2006, pp. 47–67. MR2249345 (2008c:65307) , Finite element exterior calculus, homological techniques, and applications, Acta Nu8. mer. 15 (2006), 1–155. MR2269741 (2007j:58002) , Mixed ﬁnite element methods for linear elasticity with weakly imposed symmetry, 9. Math. Comp. 76 (2007), no. 260, 1699–1723 (electronic). MR2336264 , Geometric decompositions and local bases for spaces of ﬁnite element diﬀerential 10. forms, Comput. Methods Appl. Mech. Engrg. 198 (2009), 1660–1672. MR2517938 11. Douglas N. Arnold and Ragnar Winther, Mixed ﬁnite elements for elasticity, Numer. Math. 92 (2002), 401–419. MR1930384 (2003i:65103) 12. Vladimir I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1978. MR0690288 (57:14033b) 13. Ivo Babuška, Error-bounds for ﬁnite element method, Numer. Math. 16 (1970/1971), 322–333. MR0288971 (44:6166) 14. Ivo Babuška and John Osborn, Eigenvalue problems, Handbook of Numerical Analysis, Vol. II, North-Holland, Amsterdam, 1991, pp. 641–787. MR1115240 15. Garth A. Baker, Combinatorial Laplacians and Sullivan-Whitney forms, Diﬀerential geometry (College Park, Md., 1981/1982), Progr. Math., vol. 32, Birkhäuser, Boston, Mass., 1983, pp. 1– 33. MR702525 (84m:58005) 16. Pavel B. Bochev and James M. Hyman, Principles of mimetic discretizations of diﬀerential operators, Compatible Spatial Discretizations, IMA Vol. Math. Appl., vol. 142, Springer, New York, 2006, pp. 89–119. MR2249347 (2007k:65161) 17. Daniele Boﬃ, A note on the de Rham complex and a discrete compactness property, Appl. Math. Lett. 14 (2001), 33–38. MR1793699 (2001g:65145) , Compatible discretizations for eigenvalue problems, Compatible Spatial Discretiza18. tions, IMA Vol. Math. Appl., vol. 142, Springer, New York, 2006, pp. 121–142. MR2249348 (2007i:65084) , Approximation of eigenvalues in mixed form, discrete compactness property, and 19. application to hp mixed ﬁnite elements, Comput. Methods Appl. Mech. Engrg. 196 (2007), no. 37-40, 3672–3681. MR2339993 (2008e:65339) 20. Daniele Boﬃ, Franco Brezzi, and Lucia Gastaldi, On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form, Math. Comp. 69 (2000), 121–140. MR1642801 (2000i:65175) 21. Alain Bossavit, Whitney forms: A class of ﬁnite elements for three-dimensional computations in electromagnetism, IEE Trans. Mag. 135, Part A (1988), 493–500. 22. Raoul Bott and Loring W. Tu, Diﬀerential Forms in Algebraic Topology, Graduate Texts in Mathematics, vol. 82, Springer, New York, 1982. MR658304 (83i:57016) This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 352 D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER 23. James H. Bramble and Stephen R. Hilbert, Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation, SIAM J. Numer. Anal. 7 (1970), 112–124. MR0263214 (41:7819) 24. Franco Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8 (1974), 129–151. MR0365287 (51:1540) 25. Franco Brezzi and Klaus-Jürgen Bathe, A discourse on the stability conditions for mixed ﬁnite element formulations, Comput. Methods Appl. Mech. Engrg. 82 (1990), 27–57. MR1077650 (92b:65085) 26. Franco Brezzi, Jim Douglas, Jr., and L. D. Marini, Two families of mixed ﬁnite elements for second order elliptic problems, Numer. Math. 47 (1985), 217–235. MR799685 (87g:65133) 27. Franco Brezzi, Konstantin Lipnikov, and Mikhail Shashkov, Convergence of the mimetic ﬁnite diﬀerence method for diﬀusion problems on polyhedral meshes, SIAM J. Numer. Anal. 43 (2005), no. 5, 1872–1896 (electronic). MR2192322 (2006j:65311) 28. Jochen Brüning and Matthias Lesch, Hilbert complexes, J. Funct. Anal. 108 (1992), no. 1, 88–132. MR1174159 (93k:58208) 29. Jules G. Charney, Ragnar Fjörtoft, and John von Neumann, Numerical integration of the barotropic vorticity equation, Tellus 2 (1950), 237–254. MR0042799 (13:164f) 30. Jeﬀrey Cheeger, Analytic torsion and the heat equation, Ann. of Math. (2) 109 (1979), no. 2, 259–322. MR528965 (80j:58065a) 31. Snorre H. Christiansen, Stability of Hodge decompositions in ﬁnite element spaces of diﬀerential forms in arbitrary dimension, Numer. Math. 107 (2007), no. 1, 87–106. MR2317829 (2008c:65318) 32. Snorre H. Christiansen and Ragnar Winther, Smoothed projections in ﬁnite element exterior calculus, Math. Comp. 77 (2008), no. 262, 813–829. MR2373181 (2009a:65310) 33. Philippe G. Ciarlet, The ﬁnite element method for elliptic problems, North-Holland Publishing Co., Amsterdam, 1978. MR0520174 (58:25001) 34. Philippe Clément, Approximation by ﬁnite element functions using local regularization, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge, RAIRO Analyse Numérique 9 (1975), 77–84. MR0400739 (53:4569) 35. Martin Costabel, A coercive bilinear form for Maxwell’s equations, J. Math. Anal. Appl. 157 (1991), 527–541. MR1112332 (92c:35113) 36. Richard Courant, Kurt Friedrichs, and Hans Lewy, Über die partiellen Diﬀerenzengleichungen der mathematischen Physik, Math. Ann. 100 (1928), no. 1, 32–74. MR1512478 37. Leszek Demkowicz, Peter Monk, Leon Vardapetyan, and Waldemar Rachowicz, de Rham diagram for hp ﬁnite element spaces, Comput. Math. Appl. 39 (2000), 29–38. MR1746160 (2000m:78052) 38. Mathieu Desbrun, Anil N. Hirani, Melvin Leok, and Jerrold E. Marsden, Discrete exterior calculus, 2005, available from arXiv.org/math.DG/0508341. 39. Jozef Dodziuk, Finite-diﬀerence approach to the Hodge theory of harmonic forms, Amer. J. Math. 98 (1976), no. 1, 79–104. MR0407872 (53:11642) 40. Jozef Dodziuk and V. K. Patodi, Riemannian structures and triangulations of manifolds, J. Indian Math. Soc. (N.S.) 40 (1976), no. 1-4, 1–52. MR0488179 (58:7742) 41. Jim Douglas, Jr. and Jean E. Roberts, Global estimates for mixed methods for second order elliptic equations, Math. Comp. 44 (1985), 39–52. MR771029 (86b:65122) 42. Patrick Dular and Christophe Geuzaine, GetDP: A general environment for the treatment of discrete problems, http://geuz.org/getdp/. 43. Michael Eastwood, A complex from linear elasticity, Rend. Circ. Mat. Palermo (2) Suppl. (2000), no. 63, 23–29. MR1758075 (2001j:58033) 44. Anders Logg et al., The FEniCS project, http://www.fenics.org. 45. Daniel White et al., EMSolve: Unstructured grid computational electromagnetics using mixed ﬁnite element methods, https://www-eng.llnl.gov/emsolve/emsolve_home.html. 46. Wolfgang Bangerth et al., deal.II: A ﬁnite element diﬀerential equations analysis library, http://www.dealii.org. 47. Lawrence C. Evans, Partial diﬀerential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR1625845 (99e:35001) 48. Richard S. Falk and John E. Osborn, Error estimates for mixed methods, RAIRO Anal. Numér. 14 (1980), 249–277. MR592753 (82j:65076) This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. FINITE ELEMENT EXTERIOR CALCULUS 353 49. Herbert Federer, Geometric Measure Theory, Vol. 153 of Die Grundlehren der mathematischen Wissenschaften, Springer, New York, 1969. MR0257325 (41:1976) 50. Matthew Fisher, Peter Schröder, Mathieu Desbrun, and Hugues Hoppe, Design of tangent vector ﬁelds, SIGGRAPH ’07: ACM SIGGRAPH 2007 Papers (New York), ACM, 2007, paper 56. 51. Badouin M. Fraeijs de Veubeke, Displacement and equilibrium models in the ﬁnite element method, Stress Analysis (O. C. Zienkiewicz and G. S. Holister, eds.), Wiley, New York, 1965, pp. 145–197. 52. Matthew P. Gaﬀney, The harmonic operator for exterior diﬀerential forms, Proc. Nat. Acad. Sci. USA 37 (1951), 48–50. MR0048138 (13:987b) 53. N. V. Glotko, On the complex of Sobolev spaces associated with an abstract Hilbert complex, Sibirsk. Mat. Zh. 44 (2003), no. 5, 992–1014. MR2019553 (2004h:46030) 54. Steven J. Gortler, Craig Gotsman, and Dylan Thurston, Discrete one-forms on meshes and applications to 3D mesh parameterization, Comput. Aided Geom. Design 23 (2006), no. 2, 83–112. MR2189438 (2006k:65039) 55. M. Gromov and M. A. Shubin, Near-cohomology of Hilbert complexes and topology of nonsimply connected manifolds, Astérisque (1992), no. 210, 9–10, 283–294, Méthodes semiclassiques, Vol. 2 (Nantes, 1991). MR1221363 (94g:58204) 56. Xianfeng David Gu and Shing-Tung Yau, Computational conformal geometry, Advanced Lectures in Mathematics (ALM), vol. 3, International Press, Somerville, MA, 2008, With 1 CDROM (Windows, Macintosh and Linux). MR2439718 57. Ralf Hiptmair, Canonical construction of ﬁnite elements, Math. Comp. 68 (1999), 1325–1346. MR1665954 (2000b:65214) , Higher order Whitney forms, Geometrical Methods in Computational Electromag58. netics (F. Teixeira, ed.), PIER, vol. 32, EMW Publishing, Cambridge, MA, 2001, pp. 271–299. , Finite elements in computational electromagnetism, Acta Numerica, vol. 11, Cam59. bridge University Press, Cambridge, 2002, pp. 237–339. MR2009375 (2004k:78028) 60. Lars Hörmander, The analysis of linear partial diﬀerential operators. III, Classics in Mathematics, Springer, Berlin, 2007, Pseudo-diﬀerential operators, Reprint of the 1994 edition. MR2304165 (2007k:35006) 61. inuTech GmbH, Diﬀpack: Expert tools for expert problems, http://www.diffpack.com. 62. Klaus Jänich, Vector Analysis, Undergraduate Texts in Mathematics, Springer, New York, 2001, Translated from the second German (1993) edition by Leslie Kay. MR1811820 (2001m:58001) 63. Claes Johnson and Bertram Mercier, Some equilibrium ﬁnite element methods for twodimensional elasticity problems, Numer. Math. 30 (1978), 103–116. MR0483904 (58:3856) 64. Tosio Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer, Berlin, 1995, Reprint of the 1980 edition. MR1335452 (96a:47025) 65. Fumio Kikuchi, On a discrete compactness property for the Nédélec ﬁnite elements, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36 (1989), no. 3, 479–490. MR1039483 (91h:65173) 66. P. Robert Kotiuga, Hodge decompositions and computational electromagnetics, Ph.D. thesis in Electrical Engineering, McGill University, 1984. 67. Serge Lang, Diﬀerential and Riemannian manifolds, Graduate Texts in Mathematics, vol. 160, Springer, New York, 1995. MR1335233 (96d:53001) 68. Jean-Louis Loday, Cyclic Homology, Grundlehren der Mathematischen Wissenschaften, vol. 301, Springer, Berlin, 1992. MR1217970 (94a:19004) 69. Anders Logg and Kent-Andre Mardal, Finite element exterior calculus, http://www.fenics. org/wiki/Finite_Element_Exterior_Calculus, 2009. 70. Peter Monk, Finite element methods for Maxwell’s equations, Numerical Mathematics and Scientiﬁc Computation, Oxford University Press, New York, 2003. MR2059447 (2005d:65003) 71. Werner Müller, Analytic torsion and R-torsion of Riemannian manifolds, Adv. in Math. 28 (1978), no. 3, 233–305. MR498252 (80j:58065b) 72. Jean-Claude Nédélec, Mixed ﬁnite elements in R3 , Numer. Math. 35 (1980), 315–341. MR592160 (81k:65125) , A new family of mixed ﬁnite elements in R3 , Numer. Math. 50 (1986), 57–81. 73. MR864305 (88e:65145) This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 354 D. N. ARNOLD, R. S. FALK, AND RAGNAR WINTHER 74. Roy A. Nicolaides and Kathryn A. Trapp, Covolume discretization of diﬀerential forms, Compatible Spatial Discretizations, IMA Vol. Math. Appl., vol. 142, Springer, New York, 2006, pp. 161–171. MR2249350 (2008b:65151) 75. Rainer Picard, An elementary proof for a compact imbedding result in generalized electromagnetic theory, Math. Z. 187 (1984), 151–164. MR753428 (85k:35212) 76. Pierre-Arnaud Raviart and Jean-Marie Thomas, A mixed ﬁnite element method for 2nd order elliptic problems, Mathematical aspects of ﬁnite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975), Vol. 606 of Lecture Notes in Mathematics, Springer, Berlin, 1977, pp. 292–315. MR0483555 (58:3547) 77. Yves Renard and Julien Pommier, Getfem++, http://home.gna.org/getfem/. 78. Joachim Schöberl, NGSolve – 3D ﬁnite element solver, http://www.hpfem.jku.at/ngsolve/. 79. Joachim Schöberl, A posteriori error estimates for Maxwell equations, Math. Comp. 77 (2008), no. 262, 633–649. MR2373173 (2008m:78017) 80. Rolf Stenberg, On the construction of optimal mixed ﬁnite element methods for the linear elasticity problem, Numer. Math. 48 (1986), 447–462. MR834332 (87i:73062) , A family of mixed ﬁnite elements for the elasticity problem, Numer. Math. 53 (1988), 81. 513–538. MR954768 (89h:65192) , Two low-order mixed methods for the elasticity problem, The Mathematics of Finite 82. Elements and Applications, VI (Uxbridge, 1987), Academic Press, London, 1988, pp. 271–280. MR956898 (89j:73074) 83. Dennis Sullivan, Diﬀerential forms and the topology of manifolds, Manifolds—Tokyo 1973 (Proc. Internat. Conf., Tokyo, 1973), Univ. Tokyo Press, Tokyo, 1975, pp. 37–49. MR0370611 (51:6838) , Inﬁnitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. (1977), 84. no. 47, 269–331 (1978). MR0646078 (58:31119) 85. Michael E. Taylor, Partial Diﬀerential Equations. I: Basic Theory, Applied Mathematical Sciences, vol. 115, Springer, New York, 1996. MR1395148 (98b:35002b) 86. John von Neumann and Herman H. Goldstine, Numerical inverting of matrices of high order, Bull. Amer. Math. Soc. 53 (1947), 1021–1099. MR0024235 (9:471b) 87. Ke Wang, Weiwei, Yiying Tong, Desbrun Mathieu, and Peter Schröder, Edge subdivision schemes and the construction of smooth vector ﬁelds, ACM Trans. on Graphics 25 (2006), 1041–1048. 88. Hassler Whitney, Geometric Integration Theory, Princeton University Press, Princeton, NJ, 1957. MR0087148 (19:309c) 89. Jinchao Xu and Ludmil Zikatanov, Some observations on Babuška and Brezzi theories, Numer. Math. 94 (2003), no. 1, 195–202. MR1971217 (2004a:65160) 90. Kōsaku Yosida, Functional analysis, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the sixth (1980) edition. MR1336382 (96a:46001) 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] This is a free offprint provided to the author by the publisher. Copyright restrictions may apply.

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