Finite Element Exterior Calculus: From Hodge Theory to Numerical Stability

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 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 well-posedness 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 infinite dimensional 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 have 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.

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 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 unfamiliar with the finite element method, we introduce some basic ideas by considering the approximation of the simplest two-point boundary value problem

Weak solutions to this problem are sought in the Sobolev space $H^1(-1,\; 1)$ consisting of functions in $L^2(-1,\; 1)$ whose first derivative also belong to $L^2(-1,\; 1)$. Indeed, the solution $u$ can be characterized as the minimizer of the energy functional

over the space $\backslash dot\{H\}^1(-1,\; 1)$ (which consists of $H^1(-1,\; 1)$ functions vanishing at $\backslash pm\; 1$), or, equivalently, as the solution of the weak problem: Find $u\; \backslash in\; \backslash dot\{H\}^1(-1,\; 1)$ such that

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 $V\_h$ denote a finite dimensional subspace of $\backslash hat\{H\}^1(-1,\; 1)$, called the trial space, we may define an approximate solution $u\_h\; \backslash in\; V\_h$ as the minimizer of the functional $J$ over the trial space (the classical Ritz method), or, equivalently by Galerkin's method, in which $u\_h\; \backslash in\; V\_h$ is determined by requiring that the variation given in the weak problem hold only for functions in $V\_h$, i.e., by the equations

By choosing a basis for the trial space $V\_h$, the Galerkin method reduces to a linear system of algebraic equations for the coefficients of the expansion of $u\_h$ in terms of the basis functions. More specifically, if we write $u\_h\; =\; \backslash sum\_\{j=1\}^M\; c\_j\; \backslash phi\_j$, where the functions $\backslash phi\_j$ 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 $A\_\{ij\}\; =\; \backslash int\_\{-1\}^1\; \backslash phi\_j\text{'}\; \backslash phi\_i\text{'}\; dx$ and $b\_i\; =\; \backslash int\_\{-1\}^1\; f\; \backslash phi\_i\; dx$. Since this is a square system of linear equations, it is nonsingular if and only if the only solution for $f\; =\; 0$ is $u\_h\; =\; 0$. This follows immediately by choosing $v\; =\; u\_h$.

The simplest finite element method is obtained applying Galerkin's method with the trial space $V\_h$ consisting of all element of $\backslash hat\{H\}^1(-1,\; 1)$ which are linear on each subinterval of some chosen mesh of the domain $(-1,\; 1)$. Figure 1.1 compares the exact solution $u\; =\; \backslash cos(\backslash pi\; x/2)$ and the finite element solution $u\_h$ in the case of a uniform mesh with $N\; =\; 14$ subintervals. The derivatives are compared as well. For this simple problem, $u\_h$ is simply the orthogonal projection of $u$ into $V\_h$ with respect to the inner product defined by the left hand side of (3), and the finite element method gives 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.

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

$-u\text{'}\text{'}\; =\; f$ as the first order system

The pair $(\backslash sigma,\; u)$ can then be characterized variationally as the unique critical point of the functional

over $H^1(-1,\; 1)\; \backslash times\; L^2(-1,\; 1)$. Equivalently, the pair is the solution of the weak formulation: Find $\backslash sigma\; \backslash in\; H^1(-1,\; 1)$, $u\; \backslash in\; L^2(-1,\; 1)$ satisfying

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: $I(\backslash sigma,\; v)\; \backslash leq\; I(\backslash sigma,\; u)\; \backslash leq\; I(\backslash tau,\; u)$ for $\backslash tau\; \backslash in\; H^1(-1,\; 1)$, $v\; \backslash in\; L^2(-1,\; 1)$.

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 $\backslash Sigma\_h\; \backslash subset\; H^1(-1,\; 1)$ and $V\_h\; \backslash subset\; L^2(-1,\; 1)$, 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 $\backslash Sigma\_h$ and $V\_h$ leads to a singular linear system. The choice of continuous piecewise linear functions for $\backslash Sigma\_h$ and piecewise constants for $V\_h$ leads to stable discretization and good accuracy. However choosing piecewise quadratics for $\backslash Sigma\_h$ and piecewise constants for $V\_h$ 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.

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 $r\; \backslash geq\; 1$, the combination of continuous piecewise polynomials of degree at most $r$ for $\backslash sigma$ 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 in Section 2.3.1 below. In particular, we shall see that the choice of continuous piecewise linear functions for $\backslash sigma$ 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 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 $L^2$ de Rham complex of a Riemannian manifold, but it is a far more general object with other important realizations. For example, it allows 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 heart of finite element exterior calculus and the reason for its name. In this section, we construct finite element spaces of differential forms—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 $L^2$ 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 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]. (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 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.

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.

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 [8].

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 $\backslash Omega\; \backslash subset\; \backslash mathbb\{R\}^n$:

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 $V\_h\; \backslash subset\; \backslash dot\{H\}^1(\backslash Omega)$ and defining the approximate solution $u\_h\; \backslash in\; V\_h$ by Galerkin's method:

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 $\backslash Omega$, which are continuous and vanish on $\backslash partial\backslash Omega$. 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.

More generally, a finite element method is a Galerkin method for which the trial space $V\_h$ 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 $\backslash mathcal\{T\}\_h$ and, for each element $T\; \backslash in\; \backslash mathcal\{T\}\_h$, 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 $p\; \backslash mapsto\; p(x)$, 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 $p\; \backslash mapsto\; p(x)$ is associated to the vertex $x$. Given the triangulation, shape functions, and degrees of freedom, the finite element space $V\_h$ is defined as the set of functions on $\backslash Omega$ (possibly multivalued on the element boundaries) whose restriction to any $T\; \backslash in\; \backslash mathcal\{T\}\_h$ 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, $V\_h$ 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 $H^1(\backslash Omega)$. 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 $p\; \backslash mapsto\; p(x)$, $x$ a vertex of $T$, and $p\; \backslash mapsto\; \backslash int\_e\; p\; ds$, $e$ an edge of $T$. The resulting assembled finite element space is the space of all continuous piecewise quadratics. The finite element assembly process 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, $B\; :\; V\; \backslash times\; V\; \backslash rightarrow\; \backslash mathbb\{R\}$ a bounded bilinear form, and $F\; :\; V\; \backslash rightarrow\; \backslash mathbb\{R\}$ a bounded linear form. We assume the problem to be solved can be stated in the form: Find $u\; \backslash in\; V$ such that

This problem is called well-posed if for each $F\; \backslash in\; V^*$, there exists a unique solution $u\; \backslash in\; V$ and the mapping $F\; \backslash mapsto\; u$ is bounded, or, equivalently, if the operator $L\; :\; V\; \backslash rightarrow\; V^*$ given by $\backslash langle\; Lu,\; v\; \backslash rangle\; =\; B(u,\; v)$ is an isomorphism. For the Dirichlet problem for Poisson's equation,

A generalized Galerkin method for the abstract problem begins with a finite-dimensional normed space $V\_h$ (not necessarily a subspace of $V$), a bilinear form $B\_h\; :\; V\_h\; \backslash times\; V\_h\; \backslash rightarrow\; \backslash mathbb\{R\}$, and a linear form $F\_h\; :\; V\_h\; \backslash rightarrow\; \backslash mathbb\{R\}$, and defines $u\_h\; \backslash in\; V\_h$ by

A Galerkin method is the special case of a generalized Galerkin method for which $V\_h$ is a subspace of $V$ and the forms $B\_h$ and $F\_h$ 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 $V\_h$ is not a subspace of $V$. Although we do not treat approximations such as numerical integration in this paper, for the fundamental discretization method we study, namely the mixed method for the abstract Hodge Laplacian introduced in Section 3.4, the trial space $V\_h$ 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 $L\_h\; u\_h\; =\; F\_h$ where $L\_h\; :\; V\_h\; \backslash rightarrow\; V\_h^*$ is given by $\backslash langle\; L\_h\; u,\; v\; \backslash rangle\; =\; B\_h(u,\; v)$, $u,\; v\; \backslash in\; V\_h$. If the finite-dimensional problem is nonsingular, then we define the norm of the discrete solution operator, $\backslash |L\_h^\{-1\}\backslash |\_\{\backslash mathcal\{L\}(V\_h^*,\; V\_h)\}$, 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 $V\_h$, $B\_h$, and $F\_h$, we intend that the space $V\_h$ in some sense approximates $V$ and that the discrete forms $B\_h$ and $F\_h$ in some sense approximate $B$ and $F$. This is the essence of consistency. Our goal is to prove that the discrete solution $u\_h$ 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 $V\_h$. To this end, we suppose that there is a restriction operator $\backslash pi\_h\; :\; V\; \backslash rightarrow\; V\_h$, so that $\backslash pi\_h\; u$ is thought to be close to $u$. Then the consistency error is simply $L\_h\; \backslash pi\_h\; u\; -\; F\_h$ and the error in the generalized Galerkin method which we wish to control is $\backslash pi\_h\; u\; -\; u\_h$. We immediately get a relation between the error and the consistency error

and so the norm of the error is bounded by the product of the stability constant and the norm of the consistency error:

Stated in terms of the bilinear form $B\_h$, the norm of the consistency error can be written

As for stability, the finite dimensional problem is nonsingular if and only if

and the stability constant is then given by $\backslash gamma\_h^\{-1\}$.

Often we consider a sequence of spaces $V\_h$ and forms $B\_h$ and $F\_h$ where we think of $h\; >\; 0$ as an index accumulating at 0. The corresponding generalized Galerkin method is consistent if the $V\_h$ norm of the consistency error tends to zero with $h$ and it is stable if the stability constant $\backslash gamma\_h^\{-1\}$ is uniformly bounded. For a consistent, stable generalized Galerkin method, $\backslash |\backslash pi\_h\; u\; -\; u\_h\backslash |\_\{V\_h\}$ tends to zero, i.e., the method is convergent.

In the special case of a Galerkin method, we can bound the consistency error

In this case it is natural to choose the restriction $\backslash pi\_h$ to be the orthogonal projection onto $V\_h$, 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

Combining this with the triangle inequality, we obtain the basic error estimate for Galerkin methods

(In fact, in this Hilbert space setting, the quantity in parentheses can be replaced with $\backslash gamma\_h^\{-1\}\; \backslash |B\backslash |$, see [89].) Note that a Galerkin method is consistent as long as the sequence of subspaces $V\_h$ is approximating in $V$ in the sense that

A consistent, stable Galerkin method converges, and the approximation given by the method is quasioptimal, 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., there is a positive constant $\backslash alpha$ for which

and so $\backslash gamma\_h\; \backslash geq\; \backslash alpha$. 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.

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 $L^2$ 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 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 $L^2$ 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 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 $d^k$, called the differentials:

with $d^k\; \backslash circ\; d^\{k-1\}\; =\; 0$. Equivalently, we may think of such a complex as the graded vector space $V\; =\; \backslash bigoplus\; V^k$, equipped with a graded linear operator $d\; :\; V\; \backslash rightarrow\; V$ of degree $+1$ satisfying $d\; \backslash circ\; d\; =\; 0$. 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 $\backslash mathfrak\{Z\}^k\; =\; \backslash mathfrak\{Z\}^k(V,\; d)$ of $d^k$ are called the $k$-cocycles and the elements of the range $\backslash mathfrak\{B\}^k\; =\; \backslash mathfrak\{B\}^k(V,\; d)$ of $d^\{k-1\}$ the $k$-coboundaries. The $k$th cohomology group is defined to be the quotient space $\backslash mathfrak\{Z\}^k\; /\; \backslash mathfrak\{B\}^k$.

Given two cochain complexes $(V,\; d)$ and $(V\text{'},\; d\text{'})$, a set of linear maps $f^k\; :\; V^k\; \backslash rightarrow\; V\text{'}^k$ satisfying $d\text{'}^k\; f^k\; =\; f^\{k+1\}\; d^k$ (i.e., a graded linear map $f\; :\; V\; \backslash rightarrow\; V\text{'}$ of degree 0 satisfying $d\text{'}f\; =\; fd$) 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 $\backslash bar\{f\}$ on the cohomology spaces. This map is functorial, i.e., it respects composition.

Let $(V,\; d)$ be a cochain complex and $(V\_h,\; d)$ a subcomplex. In other words, $V\_h^k$ is a subspace of $V^k$ and $d^k\; V\_h^k\; \backslash subset\; V\_h^\{k+1\}$. Then the inclusion $i\_h\; :\; V\_h\; \backslash rightarrow\; V$ is a cochain map and so induces a map of cohomology. If there exists a cochain projection of $V$ onto $V\_h$, i.e., a cochain map $\backslash pi\_h$ such that $\backslash pi\_h^k\; :\; V^k\; \backslash rightarrow\; V\_h^k$ leaves the subspace $V\_h^k$ invariant, then $\backslash pi\_h\; \backslash circ\; i\_h\; =\; \backslash text\{id\}\_\{V\_h\}$, so $\backslash pi\_h\; \backslash circ\; i\_h\; =\; \backslash text\{id\}\_\{\backslash mathfrak\{Z\}\_h^k\; /\; \backslash mathfrak\{B\}\_h^k\}$ (where $\backslash mathfrak\{Z\}\_h^k\; :=\; \backslash mathfrak\{Z\}^k(V\_h,\; d)$ and similarly for $\backslash mathfrak\{B\}\_h^k$). We conclude that in this case $i\_h$ is injective and $\backslash pi\_h$ 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 $\backslash \{(x,\; Tx)\; \backslash mid\; x\; \backslash in\; V\backslash \}$ is closed in $X\; \backslash times\; Y$. We endow the domain $V$ with the graph norm inner product,

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 elements of its domain which 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 $y\; \backslash in\; Y$ for which there exists $x\; \backslash in\; X$ with

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., , then the range is closed [60, Lemma 19.1.1].

3.1.3. Hilbert complexes. A Hilbert complex is a sequence of Hilbert spaces $W^k$ and closed, densely-defined linear operators $d^k$ from $W^k$ to $W^\{k+1\}$ such that the range of $d^k$ is contained in the domain of $d^\{k+1\}$ and $d^\{k+1\}\; \backslash circ\; d^k\; =\; 0$. A Hilbert complex is bounded if, for each $k$, $d^k$ 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 $d^k$ is closed in $W^\{k+1\}$. A Fredholm complex is a Hilbert complex for which the range of $d^k$ is finite codimensional in the null space of $d^\{k+1\}$ (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 study their compactness properties and Gromov and Shubin [55] 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 $d^k$, endowed with the inner product associated to the graph norm:

Then $d^k$ 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 $\backslash mathfrak\{Z\}^k$ and $\backslash mathfrak\{B\}^k$. Utilizing the inner product, we define the space of harmonic forms $\backslash mathfrak\{H\}^k\; =\; \backslash mathfrak\{Z\}^k\; \backslash cap\; \backslash mathfrak\{B\}^\{k\backslash perp\}$, the orthogonal complement of $\backslash mathfrak\{B\}^k$ in $\backslash mathfrak\{Z\}^k$. It is isomorphic to the reduced cohomology space $\backslash mathfrak\{Z\}^k/\backslash overline\{\backslash mathfrak\{B\}^k\}$ or, for a closed complex, to the cohomology space $\backslash mathfrak\{Z\}^k/\backslash mathfrak\{B\}^k$. For a closed Hilbert complex, we immediately obtain the Hodge decomposition

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

The third summand, $\backslash mathfrak\{Z\}^\{k\backslash perp\; v\}\; =\; \backslash mathfrak\{Z\}^\{k\backslash perp\; w\}\; \backslash cap\; V^k$.

Continuing to use the Hilbert space structure, we define the dual complex $(W,\; d^*)$, which is a Hilbert chain complex rather than cochain complex. The dual complex uses the same spaces $W^i$, with the differential $d\_k^*$ being the adjoint of $d^\{k-1\}$, so $d\_k^*$ is a closed, densely-defined operator from $W^k$ to $W^\{k-1\}$, whose domain we denote by $V\_k^*$. The dual complex is closed or bounded if and only if the original complex is. We denote by $\backslash mathfrak\{Z\}\_k^*\; =\; \backslash mathfrak\{B\}\_k^\{k\backslash perp\; w\}$ the null space of $d\_k^*$, and by $\backslash mathfrak\{B\}\_k^*$ the range of $d\_\{k+1\}^*$. Thus $\backslash mathfrak\{H\}^k\; =\; \backslash mathfrak\{Z\}^k\; \backslash cap\; \backslash mathfrak\{Z\}\_k^*$ is the space of harmonic forms both for the original complex and the dual complex. Since $\backslash mathfrak\{Z\}^\{k\backslash perp\; w\}\; =\; \backslash mathfrak\{B\}\_k^*$, the Hodge decomposition (14) can be written

We henceforth simply write $\backslash mathfrak\{Z\}^\{k\backslash perp\}$ for $\backslash mathfrak\{Z\}^\{k\backslash perp\; v\}$.

Let $(W,\; d)$ be a closed Hilbert complex with domain complex $(V,\; d)$. Then $d^k$ is a bounded bijection from $\backslash mathfrak\{Z\}^\{k\backslash perp\}$ to the Hilbert space $\backslash mathfrak\{B\}^\{k+1\}$ and hence, by Banach's bounded inverse theorem, there exists a constant $c\_P$ such that

which we refer to as a Poincaré inequality. We remark that the condition that $\backslash mathfrak\{B\}^k$ is closed is not sufficient to obtain (16), but also necessary.

The subspace $V^k\; \backslash cap\; V\_k^*$ of $W^k$ is a Hilbert space with the norm

and is continuously included in $W^k$. We say that the Hilbert complex $(W,\; d)$ has the compactness property if $V^k\; \backslash cap\; V\_k^*$ is dense in $W^k$ and the inclusion is a compact operator. Restricted to the space $\backslash mathfrak\{H\}^k$ of harmonic forms, the $V^k\; \backslash cap\; V\_k^*$ norm is equal to the $W^k$ norm (times $\backslash sqrt\{2\}$). Therefore the compactness property implies that the inclusion of $\backslash mathfrak\{H\}^k$ into itself is compact, so $\backslash mathfrak\{H\}^k$ is finite dimensional. In summary, for Hilbert complexes, compactness property $\backslash implies$ Fredholm $\backslash implies$ 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 $W^k\; \backslash rightarrow\; W^k$ 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

If $u$ solves $Lu\; =\; f$, then

Note that, in this equation, and henceforth, we use $\backslash langle\; \backslash cdot,\; \backslash cdot\; \backslash rangle$ and $\backslash |\backslash cdot\backslash |$ without subscripts, meaning the inner product and norm in the appropriate $W^k$ space.

The harmonic functions measure the extent to which the Hodge Laplace problem (17) is well-posed. The solutions to the homogeneous problem ($f\; =\; 0$) are precisely the functions in $\backslash mathfrak\{H\}^k$. Moreover, a necessary condition for a solution to exist for non-zero $f\; \backslash in\; W^k$ is that $f\; \backslash perp\; \backslash mathfrak\{H\}^k$.

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 $V^k\; \backslash cap\; V\_k^*$. 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 $f\; \backslash in\; W^k$ given, we define the mixed formulation of the abstract Hodge Laplacian as the problem of finding $(\backslash sigma,\; u,\; p)\; \backslash in\; V^\{k-1\}\; \backslash times\; V^k\; \backslash times\; \backslash mathfrak\{H\}^k$ satisfying

Remark. The equations (18) are the Euler-Lagrange equations associated to a variational problem. Namely, if we define the quadratic functional $I\; :\; V^\{k-1\}\; \backslash times\; V^k\; \backslash times\; \backslash mathfrak\{H\}^k\; \backslash rightarrow\; \backslash mathbb\{R\}$ by

then a point $(\backslash sigma,\; u,\; p)\; \backslash in\; V^\{k-1\}\; \backslash times\; V^k\; \backslash times\; \backslash mathfrak\{H\}^k$ is a critical point of $I$ if and only if (18) holds, and in this case

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 $f\; \backslash in\; W^k$, there exists a unique $(\backslash sigma,\; u,\; p)\; \backslash in\; V^\{k-1\}\; \backslash times\; V^k\; \backslash times\; \backslash mathfrak\{H\}^k$ satisfying (18). Moreover

where $c$ is a constant depending only on the Poincaré constant $c\_P$ 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 $d^*u\; =\; \backslash sigma\; \backslash in\; V^\{k-1\}$. The second equation similarly states that $du$ belongs to the domain of $d^*$ and $d^*du\; =\; f\; -\; p\; -\; d\backslash sigma$. Thus $u$ belongs to the domain $D\_L$ of $L$ and solves the abstract Hodge Laplacian equation

The harmonic form $p$ is simply the orthogonal projection $P\_\{\backslash mathfrak\{H\}\}f$ of $f$ onto $\backslash mathfrak\{H\}^k$, required for existence of a solution. Finally the third equation fixes a particular solution, through the condition $u\; \backslash perp\; \backslash mathfrak\{H\}^k$. Thus Theorem 3.1 establishes that for any $f\; \backslash in\; W^k$ there is a unique $u\; \backslash in\; D\_L$ such that $Lu\; =\; f\; -\; P\_\{\backslash mathfrak\{H\}\}f$ and $u\; \backslash perp\; \backslash mathfrak\{H\}^k$. We define $Kf\; =\; u$, so the solution operator $K\; :\; W^k\; \backslash rightarrow\; W^k$ is a bounded linear operator mapping into $D\_L$. The solution to the mixed formulation is

The mixed formulation (18) is also intimately connected to the Hodge decomposition (15). Since $d\backslash sigma\; \backslash in\; \backslash mathcal\{B\}^k$, $p\; \backslash in\; \backslash mathfrak\{H\}^k$, and $d^*du\; \backslash in\; \backslash mathfrak\{B\}\_k^*$, the expression $f\; =\; d\backslash sigma\; +\; p\; +\; d^*du$ is precisely the Hodge decomposition of $f$. In other words

where $P\_\{\backslash mathfrak\{B\}\}$ and $P\_\{\backslash mathfrak\{B\}^*\}$ are the $W^k$-orthogonal projections onto $\backslash mathfrak\{B\}^k$ and $\backslash mathfrak\{B\}\_k^*$, respectively. We also note that $K$ commutes with $d$ and $d^*$ in the sense that

Indeed, if $f\; \backslash in\; V^k$ and $u\; =\; Kf$, then $u\; \backslash in\; D\_L$, which implies that $du\; \backslash in\; V^\{k+1\}\; \backslash cap\; V\_\{k+1\}^*$. Also $d^*u\; \backslash in\; V^\{k-1\}$, so $d^*du\; =\; f\; -\; P\_\{\backslash mathfrak\{H\}\}f\; -\; dd^*u\; \backslash in\; V^k$. This shows that $du\; \backslash in\; D\_L$. Clearly

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 $\backslash mathfrak\{B\}\_k^*$ or of $\backslash mathfrak\{B\}^k$, we get two other problems which are also of great use in applications. The $\backslash mathfrak\{B\}^*$ problem. If $f\; \backslash in\; \backslash mathfrak\{B\}\_k^*$, then $u\; =\; Kf\; \backslash in\; \backslash mathfrak\{B\}\_k^*$ satisfies

while $\backslash sigma\; =\; d^*u\; =\; 0$, $p\; =\; P\_\{\backslash mathfrak\{H\}\}f\; =\; 0$. The solution $u$ can be characterized as the unique element of $\backslash mathfrak\{Z\}^\{k\backslash perp\}$ such that

and any solution to this problem is a solution of $Lu\; =\; f$, and so is uniquely determined.

The $\backslash mathfrak\{B\}$ problem. If $f\; \backslash in\; \backslash mathfrak\{B\}^k$, then $u\; =\; Kf\; \backslash in\; \backslash mathfrak\{B\}^k$ satisfies $dd^*u\; =\; f$, while $p\; =\; P\_\{\backslash mathfrak\{H\}\}f\; =\; 0$. With $\backslash sigma\; =\; d^*u$, the pair $(\backslash sigma,\; u)\; \backslash in\; V^\{k-1\}\; \backslash times\; \backslash mathfrak\{B\}^k$ is the unique solution of

and any solution to this problem is a solution of $Lu\; =\; f$, $\backslash sigma\; =\; d^*u$, 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 $B\; :\; X\; \backslash times\; X\; \backslash rightarrow\; \backslash mathbb\{R\}$ be a symmetric bounded bilinear form on a Hilbert space $X$ which satisfies the inf-sup condition

Then the problem of finding $x\; \backslash in\; X$ such that $B(x,\; y)\; =\; F(y)$ for all $y\; \backslash in\; X$ is well-posed: it has a unique solution $x$ for each $F\; \backslash in\; X^*$, and $\backslash |x\backslash |\_X\; \backslash leq\; \backslash gamma^\{-1\}\; \backslash |F\backslash |\_\{X^*\}$. [13]. The abstract Hodge Laplacian problem (18) is of this form, where $B\; :\; [V^\{k-1\}\; \backslash times\; V^k\; \backslash times\; \backslash mathfrak\{H\}^k]\; \backslash times\; [V^\{k-1\}\; \backslash times\; V^k\; \backslash times\; \backslash mathfrak\{H\}^k]\; \backslash rightarrow\; \backslash mathbb\{R\}$ denotes the bounded bilinear form

and $F(\backslash tau,\; v,\; p)\; =\; \backslash langle\; f,\; v\; \backslash rangle$.

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 $\backslash gamma\; >\; 0$, depending only on the constant $c\_P$ in the Poincaré inequality (16), such that for any $(\backslash sigma,\; u,\; p)\; \backslash in\; V^\{k-1\}\; \backslash times\; V^k\; \backslash times\; \backslash mathfrak\{H\}^k$, there exists $(\backslash tau,\; v,\; q)\; \backslash in\; V^\{k-1\}\; \backslash times\; V^k\; \backslash times\; \backslash mathfrak\{H\}^k$ with

Proof. By the Hodge decomposition, $u\; =\; u\_\{\backslash mathfrak\{B\}\}\; +\; u\_\{\backslash mathfrak\{H\}\}\; +\; u\_\{\backslash perp\}$, where $u\_\{\backslash mathfrak\{B\}\}\; =\; P\_\{\backslash mathfrak\{B\}\}u$, $u\_\{\backslash mathfrak\{H\}\}\; =\; P\_\{\backslash mathfrak\{H\}\}u$, and $u\_\{\backslash perp\}\; =\; P\_\{\backslash mathfrak\{B\}\}^\{\backslash perp\}u$. Since $u\_\{\backslash mathfrak\{B\}\}\; \backslash in\; \backslash mathfrak\{B\}^k$, $u\_\{\backslash mathfrak\{B\}\}\; =\; d\backslash rho$, for some $\backslash rho\; \backslash in\; \backslash mathfrak\{Z\}^\{k-1\backslash perp\}$. Since $du\_\{\backslash perp\}\; =\; du$, we get using (16) that

where $c\_P\; \backslash geq\; 1$ is the constant in Poincaré inequality. Let

From (21) and the orthogonality of the Hodge decomposition, we have

We also get, from a simple computation using (21) and (22), that

The theorem easily follows from this bound and (23). $\backslash square$

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 $G\; \backslash in\; (V^\{k-1\})^*$, $F\; \backslash in\; (V^k)^*$, and $R\; \backslash in\; (\backslash mathfrak\{H\}^k)^*$ (these are the dual spaces furnished with the dual norms), there exists a unique $(\backslash sigma,\; u,\; p)\; \backslash in\; V^\{k-1\}\; \backslash times\; V^k\; \backslash times\; \backslash mathfrak\{H\}^k$ satisfying

and moreover the correspondence $(\backslash sigma,\; u,\; p)\; \backslash leftrightarrow\; (F,\; G,\; R)$ is an isomorphism of $V^\{k-1\}\; \backslash times\; V^k\; \backslash times\; \backslash mathfrak\{H\}^k$ 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 $a\; :\; X\; \backslash times\; X\; \backslash rightarrow\; \backslash mathbb\{R\}$, $b\; :\; X\; \backslash times\; Y\; \backslash rightarrow\; \backslash mathbb\{R\}$ bounded bilinear forms. Let $Z\; =\; \backslash \{x\; \backslash in\; X\; \backslash mid\; b(x,\; y)\; =\; 0\; \backslash \; \backslash forall\; y\; \backslash in\; Y\backslash \}$, and suppose that there exists positive constants $\backslash alpha$ and $\backslash gamma$ such that

Then, for all $G\; \backslash in\; X^*$, $F\; \backslash in\; Y^*$, there exists a unique $u\; \backslash in\; X$, $v\; \backslash in\; Y$ such that

Moreover $\backslash |u\backslash |\_X\; +\; \backslash |v\backslash |\_Y\; \backslash leq\; c(\backslash |G\backslash |\_\{X^*\}\; +\; \backslash |F\backslash |\_\{Y^*\})$ with the constant $c$ depending only on $\backslash alpha$, $\backslash gamma$, 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 $0\; \backslash neq\; x\; \backslash in\; X$, 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 $W^1\; =\; Y$, and define $d$ as the closed linear operator from $W^0$ to $W^1$ with domain $X$ given by

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

But, using the coercivity in the kernel, the decomposition $X\; =\; Z\; +\; Z^\backslash perp$, and Poincaré's inequality, we get $\backslash |u\backslash |\_X\; \backslash leq\; C[\backslash sqrt\{a(u,\; u)\}\; +\; \backslash |du\backslash |\_Y]$, 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 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 $V\_h^k$ of $V^k$ for each $k$. We assume that $dV\_h^k\; \backslash subset\; V\_h^\{k+1\}$ so that $(V\_h,\; d)$ is a subcomplex of $(V,\; d)$. We also take $W\_h^k$ to be the same subspace $V\_h^k$ but endowed with the norm of $W^k$. In this way we obtain a closed (even bounded) Hilbert complex $(W\_h,\; d)$ with domain complex $(V\_h,\; d)$, 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 $d\_h^*\; :\; V\_h^\{k+1\}\; \backslash rightarrow\; V\_h^k$, defined by

is not the restriction of $d^*$. We use the notations $\backslash mathfrak\{B\}\_h$, $\backslash mathfrak\{Z\}\_h$, $\backslash mathfrak\{B\}\_h^*\; =\; \backslash mathfrak\{Z\}\_h^\backslash perp$, $\backslash mathfrak\{H\}\_h$, $K\_h$, with the obvious meanings. We use the term discrete when we wish to emphasize quantities associated to the subcomplex. For example, we refer to $\backslash mathfrak\{H\}\_h^k$ as the space of discrete harmonic $k$-forms, and the discrete Hodge decomposition is

The $W^k$-projections $P\_\{\backslash mathfrak\{B\}\_h\}\; :\; W^k\; \backslash rightarrow\; \backslash mathfrak\{B\}\_h^k$, $P\_\{\backslash mathfrak\{B\}\_h^*\}\; :\; W^k\; \backslash rightarrow\; \backslash mathfrak\{Z\}\_h^\{k\backslash perp\}$ satisfy $P\_\{\backslash mathfrak\{B\}\_h\}\; =\; dd\_h^*\; K\_h$ and $P\_\{\backslash mathfrak\{B\}\_h^*\}\; =\; d\_h^*\; d\; K\_h$, respectively, when restricted to $V\_h^k$. We also have that $K\_h$ commutes with both $d$ and $d\_h^*$. Note that $\backslash mathfrak\{B\}\_h^k\; \backslash subset\; \backslash mathfrak\{B\}^k$ and $\backslash mathfrak\{Z\}\_h^k\; \backslash subset\; \backslash mathfrak\{Z\}^k$, but in general $\backslash mathfrak\{H\}\_h^k$ is not contained in $\backslash mathfrak\{H\}^k$, nor is $\backslash mathfrak\{Z\}\_h^\{k\backslash perp\}$ contained in $\backslash mathfrak\{Z\}^\{k\backslash perp\}$.

In order that the subspaces $V\_h^k$ can be used effectively to approximate quantities associated to the original complex, we require not only that they form a subcomplex, but of course need to know something about the approximation of $V^k$ by $V\_h^k$, i.e., an assumption that $\backslash inf\_\{v\; \backslash in\; V\_h^k\}\; \backslash |u\; -\; v\backslash |\_V$ is sufficiently small for some or all $u\; \backslash in\; V^k$. A third assumption, which plays an essential role in our analysis, is that there exists a bounded cochain projection $\backslash pi\_h$ from the complex $(V,\; d)$ to the subcomplex $(V\_h,\; d)$. Explicitly, for each $k$, $\backslash pi\_h^k$ maps $V^k$ to $V\_h^k$, leaves the subspace invariant, satisfies $d^k\; \backslash pi\_h^k\; =\; \backslash pi\_h^\{k+1\}\; d^k$, and there exists a constant $c$ such that $\backslash |\backslash pi\_h^k\; v\backslash |\_V\; \backslash leq\; c\; \backslash |v\backslash |\_V$ for all $v\; \backslash in\; V^k$. In other words, we have the following commuting diagram relating the complex $(V,\; d)$ to the subcomplex $(V\_h,\; d)$:

Note that a bounded projection gives quasioptimal approximation:

We now present two results indicating that, under these assumptions, the space $\backslash mathfrak\{H\}\_h^k$ of discrete harmonic forms provides a faithful approximation of $\backslash mathfrak\{H\}^k$. 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, $(V\_h,\; d)$ a Hilbert subcomplex, and $\backslash pi\_h$ a bounded cochain projection. Suppose that for all $k$

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 $z\; \backslash in\; \backslash mathfrak\{Z\}^k$ with $\backslash pi\_h\; z\; \backslash in\; \backslash mathfrak\{B\}\_h^k$, we must prove that $z\; \backslash in\; \backslash mathfrak\{B\}^k$. By the Hodge decomposition, $z\; =\; q\; +\; b$ with $q\; \backslash in\; \backslash mathfrak\{H\}^k$ and $b\; \backslash in\; \backslash mathfrak\{B\}^k$. We have that $\backslash pi\_h\; z\; \backslash in\; \backslash mathfrak\{B\}\_h^k$ by assumption and $\backslash pi\_h\; b\; \backslash in\; \backslash mathfrak\{B\}\_h^k$ since $b\; \backslash in\; \backslash mathfrak\{B\}^k$ and $\backslash pi\_h$ is a cochain map. Thus $\backslash pi\_h\; q\; =\; \backslash pi\_h\; z\; -\; \backslash pi\_h\; b\; \backslash in\; \backslash mathfrak\{B\}\_h^k\; \backslash subset\; \backslash mathfrak\{B\}^k$, and so $\backslash pi\_h\; q\; \backslash perp\; q$. In view of (25), this implies that $q\; =\; 0$, and so $z\; \backslash in\; \backslash mathfrak\{B\}^k$ as desired. $\backslash square$

Remark. In applications, the space of harmonic forms, $\backslash mathfrak\{H\}^k$, is a finite-dimensional space of smooth functions, and $\backslash pi\_h$ 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 $(V\_h,\; d)$ is a finite element discretization, $\backslash pi\_h$ 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 $\backslash mathfrak\{H\}^k$ and $\backslash mathfrak\{H\}\_h^k$ 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]

Theorem 3.5. Let $(V,\; d)$ be a bounded closed Hilbert complex, $(V\_h,\; d)$ a Hilbert subcomplex, and $\backslash pi\_h$ a bounded cochain projection. Then

Proof. Given $q\; \backslash in\; \backslash mathfrak\{H\}^k$, $P\_\{\backslash mathfrak\{H\}\_h\}q\; =\; P\_\{\backslash mathfrak\{H\}\_h\}q$, since $\backslash mathfrak\{Z\}\_h^k\; =\; \backslash mathfrak\{H\}\_h^k\; \backslash oplus\; \backslash mathfrak\{B\}\_h^k$ and $q\; \backslash perp\; \backslash mathfrak\{B\}^k\; \backslash supset\; \backslash mathfrak\{B\}\_h^k$. Also $\backslash pi\_h^k\; q\; \backslash in\; \backslash mathfrak\{Z\}\_h^k$, since $\backslash pi\_h$ is a cochain map. This implies (27).

If $q\; \backslash in\; \backslash mathfrak\{H\}\_h^k\; \backslash subset\; \backslash mathfrak\{Z\}\_h^k\; \backslash subset\; \backslash mathfrak\{Z\}^k$, the Hodge decomposition gives us $q\; -\; P\_\{\backslash mathfrak\{H\}\}q\; \backslash in\; \backslash mathfrak\{B\}^k$, so $\backslash pi\_h^k(q\; -\; P\_\{\backslash mathfrak\{H\}\}q)\; \backslash in\; \backslash mathfrak\{B\}\_h^k$, and so is orthogonal to both the discrete harmonic form $q$ and the harmonic form $P\_\{\backslash mathfrak\{H\}\}q$. Therefore

Finally (29) is an immediate consequence of (27) and (28). $\backslash square$

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, $(V\_h,\; d)$ a Hilbert subcomplex, and $\backslash pi\_h^k$ a bounded cochain projection. Then

where $c\_P$ is the constant appearing in the Poincaré inequality (16) and $\backslash |\backslash pi\_h^k\backslash |$ denotes the $V^k$ operator norm of $\backslash pi\_h^k$.

Proof. Given $v\; \backslash in\; \backslash mathfrak\{Z\}\_h^\{k\backslash perp\}$, define $z\; \backslash in\; \backslash mathfrak\{Z\}^\{k\backslash perp\}\; \backslash subset\; V^k$ by $dz\; =\; dv$. By (16), $\backslash |z\backslash |\; \backslash leq\; c\_P\; \backslash |dv\backslash |$, so it is enough to show that $\backslash |v\backslash |\_V\; \backslash leq\; \backslash |\backslash pi\_h\backslash |\; \backslash |z\backslash |\_V$. Now, $v\; -\; \backslash pi\_h\; z\; \backslash in\; V\_h^k$ and $d(v\; -\; \backslash pi\_h\; z)\; =\; 0$, so $v\; -\; \backslash pi\_h\; z\; \backslash in\; \backslash mathfrak\{Z\}\_h^k$. Therefore

and the result follows. $\backslash square$

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

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 $(V\_h,\; d)$ a subcomplex. Assume that

for some constants $c\_1$ and $c\_2$. Then there exists a bounded cochain projection $\backslash pi\_h$ from $(V,\; d)$ to $(V\_h,\; d)$, and the $V$ operator norm $\backslash |\backslash pi\_h\backslash |$ can be bounded in terms of $c\_1$ and $c\_2$.

Proof. As a first step of the proof we define an operator $Q\_h\; :\; V^k\; \backslash rightarrow\; \backslash mathfrak\{H\}\_h^\{k,\backslash perp\}$ by $dQ\_h\; v\; =\; P\_\{\backslash mathfrak\{B\}\_h\}\; dv$. By the first assumption, this operator is $V$-bounded since

and if $v\; \backslash in\; V\_h$ then $Q\_h\; v\; =\; P\_\{\backslash mathfrak\{B\}\_h^*\}\; v$. By the second assumption, the operator $P\_\{\backslash mathfrak\{H\}\}|\_\{\backslash mathfrak\{H\}\_h\}$ has a bounded inverse $R\_h$ mapping $\backslash mathfrak\{H\}\_h^k$ to $\backslash mathfrak\{H\}\_h^k$. Note that for any $v\; \backslash in\; \backslash mathfrak\{H\}\_h^k$, $R\_h\; P\_\{\backslash mathfrak\{H\}\}\; v\; =\; P\_\{\backslash mathfrak\{H\}\_h\}\; v$. We now define $\backslash pi\_h\; :\; V^k\; \backslash rightarrow\; V\_h^k$ by

This operator is bounded in $V^k$ and invariant on $V\_h^k$, since the three terms correspond exactly to the discrete Hodge decomposition in this case. Furthermore, $\backslash pi\_h\; dv\; =\; P\_\{\backslash mathfrak\{B\}\_h\}\; dv\; =\; dQ\_h\; v\; =\; d\backslash pi\_h\; v$, so $\backslash pi\_h$ is indeed a bounded cochain projection. $\backslash square$

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 $V\_h^k$ of the domain spaces $V^k$. Our main assumptions are those of Section 3.3: first, that $dV\_h^k\; \backslash subset\; V\_h^\{k+1\}$, so that we obtain a subcomplex

and, second, that there exists a bounded cochain projection $\backslash pi\_h$ from $(V,\; d)$ to $(V\_h,\; d)$.

Let $f\; \backslash in\; W^k$. In view of the mixed formulation (18), we take as an approximation scheme: Find $\backslash sigma\_h\; \backslash in\; V\_h^\{k-1\}$, $u\_h\; \backslash in\; V\_h^k$, $p\_h\; \backslash in\; \backslash mathfrak\{H\}\_h^k$, such that

(Recall that we use $\backslash langle\; \backslash cdot,\; \backslash cdot\; \backslash rangle$ and $\backslash |\backslash cdot\backslash |$ 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 $\backslash mathfrak\{H\}\_h^k$ is not in general a subspace of $\backslash mathfrak\{H\}^k$. We may write the solution of (31), which always exists and is unique in view of the results of Section 3.2.2, as

where $P\_h\; :\; W^k\; \backslash rightarrow\; V\_h^k$ 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 $(V\_h,\; d)$ 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 $\backslash gamma\_h\; >\; 0$, depending only on $c\_P$ and the norm of the projection operators $\backslash pi\_h$, such that for any $(\backslash sigma,\; u,\; p)\; \backslash in\; V\_h^\{k-1\}\; \backslash times\; V\_h^k\; \backslash times\; \backslash mathfrak\{H\}\_h^k$, there exists $(\backslash tau,\; v,\; q)\; \backslash in\; V\_h^\{k-1\}\; \backslash times\; V\_h^k\; \backslash times\; \backslash mathfrak\{H\}\_h^k$ with

Proof. This is just Theorem 3.2 applied to the Hilbert complex $(V\_h,\; d)$, combined with the fact that the constant in the Poincaré inequality for $V\_h^k$ is $c\_P\; \backslash |\backslash pi\_h\backslash |$ by Theorem 3.6. $\backslash square$

From this stability result, we obtain the following error estimate.

Theorem 3.9. Let $(V\_h,\; d)$ 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 $(\backslash sigma,\; u,\; p)\; \backslash in\; V^\{k-1\}\; \backslash times\; V^k\; \backslash times\; \backslash mathfrak\{H\}^k$ be the solution of problem (18) and $(\backslash sigma\_h,\; u\_h,\; p\_h)\; \backslash in\; V\_h^\{k-1\}\; \backslash times\; V\_h^k\; \backslash times\; \backslash mathfrak\{H\}\_h^k$ the solution of problem (31). Then

Proof. First observe that $(\backslash sigma,\; u,\; p)$ satisfies

Let $\backslash tau,\; v$, and $q$ be the $V$-orthogonal projections of $\backslash sigma,\; u$, and $p$ into $V\_h^\{k-1\},\; V\_h^k$, and $\backslash mathfrak\{H\}\_h^k$, respectively. Then, for any $(\backslash tau\_h,\; v\_h,\; q\_h)\; \backslash in\; V\_h^\{k-1\}\; \backslash times\; V\_h^k\; \backslash times\; \backslash mathfrak\{H\}\_h^k$, we have

Theorem 3.8 then gives

Using (27) and the boundedness of the projection $\backslash pi\_h$ we have

Next we show that

Now $u\; \backslash perp\; \backslash mathfrak\{H\}^k$, so $u\; =\; u\_\{\backslash mathfrak\{B\}\}\; +\; u\_\{\backslash perp\}$, with $u\_\{\backslash mathfrak\{B\}\}\; \backslash in\; \backslash mathfrak\{B\}^k$ and $u\_\{\backslash perp\}\; \backslash in\; \backslash mathfrak\{Z\}^\{k\backslash perp\}$. Since $\backslash mathfrak\{H\}\_h^k\; \backslash subset\; \backslash mathfrak\{Z\}^k$, $P\_\{\backslash mathfrak\{H\}\_h\}\; u\_\{\backslash perp\}\; =\; 0$, and since $\backslash pi\_h\; u\_\{\backslash mathfrak\{B\}\}\; \backslash in\; \backslash mathfrak\{B\}\_h^k$, $P\_\{\backslash mathfrak\{H\}\_h\}\; \backslash pi\_h\; u\_\{\backslash mathfrak\{B\}\}\; =\; 0$. Let $q\; =\; P\_\{\backslash mathfrak\{H\}\_h\}\; u\; /\; \backslash |P\_\{\backslash mathfrak\{H\}\_h\}\; u\backslash |\; \backslash in\; \backslash mathfrak\{H\}\_h^k$. By Theorem 3.5, there exists $r\; \backslash in\; \backslash mathfrak\{H\}^k$ (and so $r\; \backslash perp\; \backslash mathfrak\{B\}^k$) with $\backslash |r\backslash |\; \backslash leq\; 1$ and

Therefore

since $\backslash pi\_h$ is a bounded projection. The theorem follows from (32)–(34) and the triangle inequality. $\backslash square$

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 are 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 $(\backslash sigma\_h,\; u\_h)\; \backslash in\; V\_h^\{k-1\}\; \backslash times\; V\_h^k$ such that

Then $(\backslash sigma\_h,\; u\_h)$ is a solution if and only if $\backslash sigma\_h\; =\; 0$ and $u\_h\; \backslash in\; \backslash mathfrak\{H\}\_h^k$.

Proof. Clearly $(0,\; u\_h)$ is a solution if $u\_h\; \backslash in\; \backslash mathfrak\{H\}\_h^k$. On the other hand, if $(\backslash sigma\_h,\; u\_h)$ is a solution, by taking $\backslash tau\; =\; \backslash sigma\_h$, $v\; =\; u\_h$, and combining the two equations, we find that $\backslash |\backslash sigma\_h\backslash |^2\; +\; \backslash |du\_h\backslash |^2\; =\; 0$, so that $\backslash sigma\_h\; =\; 0$ and $du\_h\; =\; 0$. Then the first equation implies that $\backslash langle\; d\backslash tau,\; u\_h\; \backslash rangle\; =\; 0$ for all $\backslash tau\; \backslash in\; V\_h^\{k-1\}$, so indeed $u\_h\; \backslash in\; \backslash mathfrak\{H\}\_h^k$. $\backslash square$

3.5. Improved error estimates. Suppose we have a family of subcomplexes $(V\_h,\; d)$ of the domain complex $(V,\; d)$ of a closed Hilbert complex, parametrized by $h$ with uniformly bounded cochain projections $\backslash pi\_h$. Assuming also that the subspaces $V\_h^k$ are approximating in $V^k$ in the sense of (8), we can conclude from Theorem 3.9 that $\backslash sigma\_h\; \backslash rightarrow\; \backslash sigma$, $u\_h\; \backslash rightarrow\; u$, and $p\_h\; \backslash rightarrow\; p$ as $h\; \backslash rightarrow\; 0$ (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 $V\_h^k$, the particular component considered ($\backslash sigma\_h$, $u\_h$, or $p\_h$), 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 $V\_h^k$ consist of piecewise polynomial differential forms with respect to a triangulation $\backslash mathcal\{T\}\_h$ of the domain with mesh size $h$. The space $W^k$ is the space of $L^2$ 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 $\backslash mathcal\{P\}\_\{r+1\}\backslash Lambda^\{k-1\}(\backslash mathcal\{T\}\_h)$ and $\backslash mathcal\{P\}\_r\backslash Lambda^k(\backslash mathcal\{T\}\_h)$. The space $\backslash mathcal\{P\}\_r\backslash Lambda^k(\backslash mathcal\{T\}\_h)$, 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, application of Theorem 3.9 will give, in this case,

Approximation theory tells us that this rate is the best possible for $\backslash |u\; -\; u\_h\backslash |\_V$, but we might hope for a faster rate for $\backslash |u\; -\; u\_h\backslash |$ and for $\backslash |\backslash sigma\; -\; \backslash sigma\_h\backslash |\_V$ and $\backslash |\backslash sigma\; -\; \backslash sigma\_h\backslash |$.

In order to obtain improved error estimates, we make two addition 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 second property implies that $\backslash pi\_h$ extends to a bounded linear operator $W^k\; \backslash rightarrow\; V\_h^k$. Since the subspaces $V\_h^k$ are approximating in $W^k$ as well as in $V^k$ (by density), it follows that $\backslash pi\_h$ converges pointwise to the identity in $W^k$. Finally, note that if we are given a $W$-bounded cochain projection mapping $W^k\; \backslash rightarrow\; V\_h^k$, the restrictions to $V^k$ define a $V$-bounded cochain projection.

Next, we note that on the Hilbert space $V^k\; \backslash cap\; V\_k^*$, the inner product given by

is equivalent to the usual intersection inner product which is the sum of the inner products for $V^k$ and $V\_k^*$. This can be seen by Hodge decomposing $u$ as $P\_\{\backslash mathfrak\{B\}\}u\; +\; P\_\{\backslash mathfrak\{H\}\}u\; +\; P\_\{\backslash mathfrak\{B\}^\backslash perp\}u$, and using the Poincaré bound $\backslash |P\_\{\backslash mathfrak\{B\}^\backslash perp\}u\backslash |\; \backslash leq\; c\backslash |dP\_\{\backslash mathfrak\{B\}^\backslash perp\}u\backslash |\; =\; c\backslash |du\backslash |$ and the analogous bound $\backslash |P\_\{\backslash mathfrak\{B\}\}u\backslash |\; \backslash leq\; c\backslash |d^*P\_\{\backslash mathfrak\{B\}\}u\backslash |\; =\; c\backslash |d^*u\backslash |$. Now $K$ maps $W^k$ boundedly into $V^k\; \backslash cap\; V\_k^*$ and satisfies

In other words, $K\; \backslash in\; \backslash mathcal\{L\}(W^k,\; V^k\; \backslash cap\; V\_k^*)$ is the adjoint of the operator $(I\; -\; P\_\{\backslash mathfrak\{H\}\})\backslash mathcal\{I\}\; \backslash in\; \backslash mathcal\{L\}(V^k\; \backslash cap\; V\_k^*,\; W^k)$ where $\backslash mathcal\{I\}\; \backslash in\; \backslash mathcal\{L\}(V^k\; \backslash cap\; V\_k^*,\; W^k)$ is the compact inclusion operator. Hence $K$ is a compact operator $W^k\; \backslash rightarrow\; V^k\; \backslash cap\; V\_k^*$ and, a fortiori, compact as an operator from $W^k$ to itself. As an operator on $W^k$, $K$ is also self-adjoint, since

for all $f,\; g\; \backslash in\; W^k$. Furthermore, if we follow $K$ by one of the bounded operators $d\; :\; V^k\; \backslash rightarrow\; W^\{k+1\}$ or $d^*\; :\; V\_k^*\; \backslash rightarrow\; W^\{k-1\}$, the compositions $dK$ and $d^*K$ are also compact operators from $W^k$ to itself. Since we have assumed the compactness property, , and so $P\_\{\backslash mathfrak\{H\}^k\}$ is also a compact operator on $W^k$. Define

(Note that $\backslash mu$ 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 $\backslash eta,\; \backslash delta,\; \backslash mu\; \backslash rightarrow\; 0$ as $h\; \backslash rightarrow\; 0$. In the applications in Sections 5 and 6, the spaces $\backslash Lambda\_h^k$ will consist of piecewise polynomials. We will then have $\backslash eta\; =\; O(h)$,

$\backslash delta\; =\; O(h^\{\backslash min(2,r+1)\})$, and $\backslash mu\; =\; O(h^\{r+1\})$ where $r$ denotes the largest degree of complete polynomials in the space $\backslash Lambda\_\{h^*\}^k$.

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

The following theorem gives improved the improved error estimates. Its proof incorporates a variety of techniques developed in the numerical analysis literature in the last 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 $(V\_h,\; d)$ a family of subcomplexes parametrized by $h$ and admitting uniformly $W$-bounded cochain projections. Let $(\backslash sigma,\; u,\; p)\; \backslash in\; V^\{k-1\}\; \backslash times\; V^k\; \backslash times\; S^k$ be the solution of problem (18) and $(\backslash sigma\_h,\; u\_h,\; p\_h)\; \backslash in\; V\_h^\{k-1\}\; \backslash times\; V\_h^k\; \backslash times\; S\_h^k$ the solution of problem (31). Then for some constant $C$ independent of $h$ and $(\backslash sigma,\; u,\; p)$, we have

We now develop the proof of Theorem 3.11 in a series of lemmas.

Lemma 3.12. Let $v\_h\; \backslash in\; \backslash mathfrak\{Z\}\_h^\{k\backslash perp\}$ and $v\; =\; P\_\{\backslash mathfrak\{B\}^*\}v\_h$. Then

Proof. Since $\backslash pi\_h\; v\; -\; v\_h\; \backslash in\; \backslash mathfrak\{Z\}\_h^k\; \backslash subset\; \backslash mathfrak\{Z\}^k$, $v\_h\; -\; v\; \backslash perp\; \backslash pi\_h\; v\; -\; v\_h$, so, by the Pythagorean theorem, $\backslash |v\_h\; -\; v\backslash |\; \backslash leq\; \backslash |(I\; -\; \backslash pi\_h)v\backslash |$. The second inequality holds since $v\; =\; d^*Kdv\_h$. $\backslash square$

Lemma 3.13. The estimates (35) and (36) hold. Moreover

Proof. Since $d\backslash sigma\_h\; =\; P\_\{\backslash mathfrak\{B\}\_h\}f\; =\; P\_\{\backslash mathfrak\{B\}\_h\}P\_\{\backslash mathfrak\{B\}\}f\; =\; P\_\{\backslash mathfrak\{B\}\_h\}d\backslash sigma$, we have

giving (35). To prove (36) we write

Taking $\backslash tau\; =\; \backslash pi\_h\backslash sigma^2\; -\; \backslash sigma\_h$ in (18) and (31), we obtain

Hence,

Since $\backslash sigma^1\; \backslash in\; \backslash mathfrak\{B\}\_k^*$,

Then (36) follows from the last two estimates.

Let $e\; =\; P\_\{\backslash mathfrak\{B\}\_h\}(u\; -\; u\_h)$. To estimate $e$, we set $w\; =\; Ke$, $\backslash phi\; =\; d^*w$, $w\_h\; =\; K\_h\; e$, $\backslash phi\_h\; =\; d\_h^*\; w\_h$. Then $d\backslash phi\; =\; d\backslash pi\_h\; \backslash phi\; =\; d\backslash phi\_h\; =\; e$, so $\backslash pi\_h\; \backslash phi\; -\; \backslash phi\_h\; \backslash in\; \backslash mathfrak\{Z\}\_h^k$, and so is orthogonal to $\backslash phi\; -\; \backslash phi\_h$. Thus $\backslash |\backslash phi\; -\; \backslash phi\_h\backslash |\; \backslash leq\; \backslash |(I\; -\; \backslash pi\_h)\backslash phi\backslash |\; =\; \backslash |(I\; -\; \backslash pi\_h)d^*Ke\backslash |$. Then

Combining with (36) we get (40). $\backslash square$

Lemma 3.14. Estimate (37) holds and, moreover, $\backslash |P\_\{\backslash mathfrak\{S\}\_h\}u\backslash |\; \backslash leq\; C\backslash mu\; E(P\_\{\backslash mathfrak\{B\}\}u)$.

Proof. The second estimate is just (34). Using the Hodge decomposition of $f$ and the fact that $P\_\{\backslash mathfrak\{S\}\_h\}\backslash mathfrak\{B\}\_k^*\; =\; 0$, we have $P\_\{\backslash mathfrak\{S\}\_h\}p\; -\; p\_h\; =\; P\_\{\backslash mathfrak\{S\}\_h\}P\_\{\backslash mathfrak\{S\}\}f\; -\; P\_\{\backslash mathfrak\{S\}\_h\}f\; =\; P\_\{\backslash mathfrak\{S\}\_h\}f\_\{\backslash mathfrak\{B\}\}$, where $f\_\{\backslash mathfrak\{B\}\}\; =\; P\_\{\backslash mathfrak\{B\}\}f$. Therefore,

by (27). Applying (28) we get

Combining these results completes the proof of the lemma. $\backslash square$

Lemma 3.15. The estimate (38) holds.

Proof. From (18) and (31),

Choose $v\_h\; =\; P\_\{\backslash mathfrak\{B\}\_h^*\}(\backslash pi\_h\; u\; -\; u\_h)$ in (41) and set $v\; =\; d^*Kdv\_h\; =\; P\_\{\backslash mathfrak\{B\}\}v\_h$. Using Lemmas 3.12 and (27), we get

Hence,

The result follows by the triangle inequality. $\backslash square$

Lemma 3.16. $\backslash |P\_\{\backslash mathfrak\{B\}\_h^*\}(u\; -\; u\_h)\backslash |\; \backslash leq\; C\backslash \{E(u)\; +\; \backslash eta\; E(du)\; +\; (\backslash eta^2\; +\; \backslash delta)[E(d\backslash sigma)\; +\; E(p)]\backslash \}.$

Proof. Again letting $v\_h\; =\; P\_\{\backslash mathfrak\{B\}\_h^*\}(\backslash pi\_h\; u\; -\; u\_h)$ and $v\; =\; P\_\{\backslash mathfrak\{B\}^*\}\; v\_h$, we observe that

Next

Then using Lemmas 3.12 and 3.15, we get

We next estimate the term $\backslash langle\; v,\; u\; -\; u\_h\; \backslash rangle$. Since $dv\; =\; dv\_h$ and $Kv\; \backslash in\; \backslash mathfrak\{Z\}^\{k\backslash perp\}$, we get

Now

while

But

Hence,

Combining these results, and using the previous lemmas, we obtain

The final result of the lemma follows immediately. $\backslash square$

It is now an easy matter to prove (39) and so complete the proof Theorem 3.11. We write

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 $V\_h^\{k-1\}\; =\; \backslash mathcal\{P\}\_\{r+1\}\; \backslash Lambda^\{k-1\}(\backslash mathcal\{T\}\_h)$ and $V\_h^k\; =\; \backslash mathcal\{P\}\_r\; \backslash Lambda^k(\backslash mathcal\{T\}\_h)$ are used to approximate the $k$-form Hodge Laplacian with some $r\; \backslash geq\; 1$. If the domain is convex, we may apply elliptic regularity to see that $Kf$ belongs to the Sobolev space $H^2\; \backslash Lambda^k$ for $f\; \backslash in\; L^2\; \backslash Lambda^k$, so $dKf\; \backslash in\; H^1\; \backslash Lambda^\{k+1\}$ and $d^*Kf\; \backslash in\; H^1\; \backslash Lambda^\{k-1\}$, and then standard approximation theory shows that $\backslash eta\; =\; O(h)$, $\backslash delta\; =\; O(h^2)$, and $\backslash mu\; =\; O(h^2)$. From Theorem 3.11, we then obtain that

assuming the solution $u$ is sufficiently smooth. That is, all components converge with the optimal order possible given the degree of 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

Therefore all three operator norms converge to zero with $h$.

Proof. Let $f\; \backslash in\; W^k$ and set $u\; =\; Kf$, $\backslash sigma\; =\; d^*\; Kf$, $p\; =\; P\_\{\backslash mathfrak\{H\}\}\; f$, and $u\_h\; =\; K\_h\; P\_h\; f$, $\backslash sigma\_h\; =\; d\_h^*\; K\_h\; P\_h\; f$. The desired bounds on $(K\; -\; K\_h\; P\_h)f\; =\; u\; -\; u\_h$, $d(K\; -\; K\_h\; P\_h)f\; =\; d(u\; -\; u\_h)$, and $(d^*\; K\; -\; d\_h^*\; K\_h\; P\_h)f\; =\; \backslash sigma\; -\; \backslash sigma\_h$ follow from Theorem 3.11, since

We end this section with an estimate of the difference between the true Hodge decomposition and the discrete Hodge decomposition of an element of $V\_h^k$. 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 $P\_\{\backslash mathfrak\{B\}\}\; u\; +\; P\_\{\backslash mathfrak\{H\}\}\; u\; +\; P\_\{\backslash mathfrak\{B\}^*\}\; u$ and $P\_\{\backslash mathfrak\{B\}\_h\}\; u\; +\; P\_\{\backslash mathfrak\{H\}\_h\}\; u\; +\; P\_\{\backslash mathfrak\{B\}\_h^*\}\; u$ denote the Hodge and discrete Hodge decompositions of $u\; \backslash in\; V\_h^k$. Then

Proof. Let $v\_h\; =\; P\_\{\backslash mathfrak\{B\}\_h^*\}\; u$ and $v\; =\; P\_\{\backslash mathfrak\{B\}^*\}\; P\_\{\backslash mathfrak\{B\}\_h\}\; u\; =\; P\_\{\backslash mathfrak\{B\}^*\}\; u$. The first estimate follows immediately from Lemma 3.12. To obtain the second estimate, we write

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. $\backslash square$

Applied to the case of Whitney forms, both $\backslash eta$ and $\backslash mu$ are $O(h)$, and so this result improves the $O(h|\backslash log\; h|)$ 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 $(W,\; d)$ is a Hilbert complex satisfying the compactness property and that the cochain projections $\backslash pi\_k^k$ are bounded in $\backslash mathcal\{L\}(W^k,\; W^k)$, uniformly in $h$. A pair $(\backslash lambda,\; u)\; \backslash in\; \backslash mathbb\{R\}\; \backslash times\; V^k$, where $u\; \backslash neq\; 0$, is referred to as an eigenvalue/eigenvector of the problem (18) if there exists $(\backslash sigma,\; p)\; \backslash in\; V^\{k-1\}\; \backslash times\; \backslash mathfrak\{H\}^k$ such that

In operator terms, $u\; =\; \backslash lambda\; Ku$, $\backslash sigma\; =\; d^*\; u$, $p\; =\; 0$. Note that it follows from this system that

so $Ku\; =\; \backslash lambda^\{-1\}u$. Since the operator $K\; \backslash in\; \backslash mathcal\{L\}(W^k,\; W^k)$ is compact and self-adjoint, we can conclude that the problem (42) has at most a countable set of eigenvalues, each of finite multiplicity. We denote these by

where each eigenvalue is repeated according to its multiplicity. Furthermore, when $W^k$ is infinite dimensional, we have $\backslash lim\_\{j\; \backslash rightarrow\; \backslash infty\}\; \backslash lambda\_j\; =\; \backslash infty$. We denote by $\backslash \{v\_i\backslash \}$ a corresponding orthonormal basis of eigenvectors for $W^k$.

The corresponding discrete eigenvalue problem takes the form

where $\backslash lambda\_h\; \backslash in\; \backslash mathbb\{R\}$, and $(\backslash sigma\_h,\; u\_h,\; p\_h)\; \backslash in\; V\_h^\{k-1\}\; \backslash times\; V\_h^k\; \backslash times\; \backslash mathfrak\{H\}\_h^k$, with $u\_h\; \backslash neq\; 0$. As above, we can conclude that $p\_h\; =\; 0$, $\backslash lambda\_h\; >\; 0$, and that $\backslash lambda\_h^\{-1\}$ is an eigenvalue for the operator $K\_h$, i.e. $K\_h\; u\_h\; =\; \backslash lambda\_h^\{-1\}\; u\_h$. We denote by

$N\_h\; =\; \backslash dim\; V\_h^k$, the eigenvalues of problem (43), repeated according to multiplicity, and by $\backslash \{v\_\{i,h\}\backslash \}$ a corresponding $W^k$-orthonormal eigenbasis for $V\_h^k$.

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 $K\_h$ converge to those of $K$. We let $E\_i$ and $E\_\{i,h\}$ denote the 1-dimensional spaces spanned by $v\_i$ and $v\_\{i,h\}$. For every positive integer $j$, let $m(j)$ denote the number of eigenvalues less than or equal to the $j$th distinct eigenvalue of the Hodge–Laplace problem (42). Thus $\backslash sum\_\{i=1\}^\{m(j)\}\; E\_i$ 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 $\backslash epsilon\; >\; 0$ and integer $j\; >\; 0$, there exists a mesh parameter $h\_0\; >\; 0$ such that, for all $h\; \backslash leq\; h\_0$, we have

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 self-adjoint Hilbert space operators $K$ and $K\_h$, convergence of the eigenvalue approximation holds if the operators $K\_h\; P\_h$ converge to $K$ in $\backslash mathcal\{L\}(W^k,\; W^k)$. 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 $(V\_h,\; d)$ 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.

Lemma 3.20. If $\backslash lambda$ is a simple eigenvalue and $u$ a normalized eigenvector, then

Theorem 3.21. Let $(V,\; d)$ be the domain complex of a closed Hilbert complex $(W,\; d)$ satisfying the compactness property, and $(V\_h,\; d)$ a family of subcomplexes parametrized by $h$ and admitting uniformly $W$-bounded cochain projections. Let $\backslash lambda$ be a simple eigenvalue, $u$ the corresponding eigenvector, and $\backslash lambda\_h$ be the corresponding discrete eigenvalue. Let $w\; =\; Ku$, $\backslash phi\; =\; d^*w$ denote the solution of the mixed formulation of the Hodge Laplace problem with source term $u$, and let $w\_h\; =\; K\_h\; P\_h\; u$, $\backslash phi\_h\; =\; d\_h^*\; w\_h$ denote the corresponding discrete solution. Then

Proof. To estimate the first term in Lemma 3.20, we write

where we have used the orthogonality condition

in the second last line above. The theorem then follows from this estimate and the fact that $\backslash |(K\; -\; K\_h\; P\_h)u\backslash |\_V^2\; =\; \backslash |w\; -\; w\_h\backslash |\_V^2$. $\backslash square$

Order of convergence estimates now follow directly from Theorem 3.11, with $f$ replaced by $u$, $u$ replaced by $w$, and $\backslash sigma$ replaced by $\backslash phi$. In the example following Theorem 3.11, i.e., $V\_h^\{k-1\}\; =\; \backslash mathcal\{P\}\_\{r+1\}\backslash Lambda^\{k-1\}(\backslash mathcal\{T\}\_h)$ and $V\_h^k\; =\; \backslash mathcal\{P\}\_r\backslash Lambda^k(\backslash mathcal\{T\}\_h)$, we saw that

as long as the domain is convex and the solution $w\; =\; Ku$ sufficiently smooth. Inserting these results into Theorem 3.21, we find that the eigenvalue error $|\backslash lambda\; -\; \backslash lambda\_h|\; =\; O(h^\{2r\})$, double the rate achieved for the source problem. As another example, one can check that for the Whitney forms, $V\_h^\{k-1\}\; =\; \backslash mathcal\{P\}\_1^-\backslash Lambda^\{k-1\}(\backslash mathcal\{T\}\_h)$, $V\_h^k\; =\; \backslash mathcal\{P\}\_1^-\backslash Lambda^k(\backslash mathcal\{T\}\_h)$ we get $|\backslash lambda\; -\; \backslash lambda\_h|\; =\; O(h^2)$, improving on the $O(h|\backslash log\; h|)$ estimate of [40].

Remark. It has long been observed that for mixed finite element approximation of eigenvalue problems, stability and approximability alone, while sufficient for convergence of approximations of the source problem, is not sufficient for convergence of the eigenvalue problem. An extensive literature developed in order to obtain eigenvalue convergence, and a wide variety of additional properties of the finite element spaces have been defined and hypothesized. In particular, the first convergence results for the 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 emphasize the Fortin property [18]. In our context, the discrete compactness property says that whenever $v\_h\; \backslash in\; \backslash mathfrak\{Z\}\_h^\{k\backslash perp\}$ is uniformly bounded in $V^k$, then there exists a sequence $h\_i$ converging to zero such that $v\_\{h\_i\}$ converges in $W^k$, and the Fortin property says that $\backslash lim\_\{h\; \backslash rightarrow\; 0\}\; \backslash |I\; -\; \backslash pi\_h^k\backslash |\_\{\backslash mathcal\{L\}(V^k\; \backslash cap\; V\_k^*,\; W^k)\}\; =\; 0$. 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 Fortin 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 $\backslash mathfrak\{B\}^*$ and the $\backslash mathfrak\{B\}$ problem, respectively. More precisely, these problems arise if the right hand side $f$ of (18) is in $\backslash mathfrak\{B\}\_k^*$ or $\backslash mathfrak\{B\}^k$, respectively. In a similar way the eigenvalue problem (42) can be decomposed into a $\backslash mathfrak\{B\}^*$ and the $\backslash mathfrak\{B\}$ problem. To see this, note if $(\backslash lambda,\; u)\; \backslash in\; \backslash mathbb\{R\}\; \backslash times\; V^k$ is an eigenvalue/eigenvector for (42), then $u\; \backslash in\; \backslash mathfrak\{S\}^\{k\backslash perp\}$, and therefore $u\; =\; u\_\{\backslash mathfrak\{B\}\}\; +\; u\_\{\backslash perp\}$, where $u\_\{\backslash mathfrak\{B\}\}\; \backslash in\; \backslash mathfrak\{B\}^k$ and $u\_\{\backslash perp\}\; \backslash in\; \backslash mathfrak\{B\}\_k^*\; \backslash cap\; V^k\; =\; \backslash mathfrak\{Z\}^\{k\backslash perp\}$. It is straightforward to check that $u\_\{\backslash perp\}$ satisfies the $\backslash mathfrak\{B\}\_k^*$ eigenvalue problem given by

Here $u\_\{\backslash perp\}$ can be equal to zero even if $u\; \backslash neq\; 0$ is an eigenvector. On the other hand, if the pair $(\backslash lambda,\; u\_\{\backslash perp\})\; \backslash in\; \backslash mathbb\{R\}\; \backslash times\; \backslash mathfrak\{Z\}^\{k\backslash perp\}$ is an eigenvalue/eigenvector for (45), then $(\backslash lambda,\; u\_\{\backslash perp\})$ is also an eigenvalue/eigenvector for (42), where $\backslash sigma$ and $p$ are both zero. Furthermore, for the extended eigenvalue problem

where $u\_\{\backslash perp\}$ is sought in $V^k$, the nonzero eigenvalues correspond precisely to the eigenvalues of (45), while the eigenvalue $\backslash lambda\; =\; 0$ has the eigenspace $\backslash mathfrak\{Z\}^k$.

The pair $(\backslash lambda,\; u\_\{\backslash mathfrak\{B\}\})$ satisfies the corresponding $\backslash mathfrak\{B\}^k$ eigenvalue problem given by

where $\backslash sigma\; =\; d^*u\_\{\backslash mathfrak\{B\}\}\; \backslash in\; \backslash mathfrak\{B\}\_\{k-1\}^*\; \backslash cap\; V^\{k-1\}\; =\; \backslash mathfrak\{Z\}^\{(k-1)\backslash perp\}$. Furthermore, any solution $(\backslash lambda,\; u\_\{\backslash mathfrak\{B\}\})$ of (47), where $u\_\{\backslash mathfrak\{B\}\}\; \backslash neq\; 0$, corresponds to an eigenvalue/eigenvector of the full Hodge Laplacian (42). In fact, any eigenvalue of (42) is an eigenvalue of either the $\backslash mathfrak\{B\}^*$ or

$\backslash mathfrak\{B\}$ problem, or both, and all eigenvalues of the $\backslash mathfrak\{B\}^*$ and $\backslash mathfrak\{B\}$ 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 $(\backslash lambda,\; u\_\{\backslash mathfrak\{B\}\})$ is an eigenvalue/eigenvector for the $\backslash mathfrak\{B\}^k$ problem (47), and $\backslash sigma\; =\; d^*u\_\{\backslash mathfrak\{B\}\}\; \backslash in\; \backslash mathfrak\{Z\}^\{(k-1)\backslash perp\}$, then $\backslash sigma\; \backslash neq\; 0$, and by taking $v\; =\; d\backslash tau$ in (47), we obtain

which is a $\backslash mathfrak\{B\}\_\{k-1\}^*$ problem. Hence, we conclude that any eigenvalue of the $\backslash mathfrak\{B\}^k$ problem is an eigenvalue of the $\backslash mathfrak\{B\}\_\{k-1\}^*$ problem. The converse is also true. To see this, let $(\backslash lambda,\; \backslash sigma)\; \backslash in\; \backslash mathbb\{R\}\; \backslash times\; \backslash mathfrak\{Z\}^\{(k-1)\backslash perp\}$, $\backslash sigma\; \backslash neq\; 0$, be a solution of (48), and define $u\_\{\backslash mathfrak\{B\}\}\; \backslash in\; \backslash mathfrak\{B\}^k$ such that $d^*u\_\{\backslash mathfrak\{B\}\}\; =\; \backslash sigma$. Since $d^*\; :\; \backslash mathfrak\{B\}^k\; \backslash rightarrow\; \backslash mathfrak\{Z\}^\{(k-1)\backslash perp\}$ is a bijection, this determines $u\_\{\backslash mathfrak\{B\}\}$ uniquely, and $(\backslash lambda,\; u\_\{\backslash mathfrak\{B\}\})$ is an eigenvalue/eigenvector for the $\backslash mathfrak\{B\}^k$ problem (47). Therefore, the two problems (47) and (48) are equivalent, with the same eigenvalues, and with eigenvectors related by $d^*u\_\{\backslash mathfrak\{B\}\}\; =\; \backslash sigma$. In particular, the eigenvalues of the full Hodge Laplace problem (42) correspond precisely to the eigenvalues of the $\backslash mathfrak\{B\}\_\{k-1\}^*$ and $\backslash mathfrak\{B\}\_k^*$ problems.

In the special case of the de Rham complex, the $\backslash mathfrak\{B\}\_1^*$ 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 $\backslash mathfrak\{B\}\_k^*$ problem from the formulation (45) since we do not, a priori, have a basis for the corresponding space $\backslash mathfrak\{Z\}\_h^\{k\backslash perp\}\; \backslash subset\; V\_h^k$ available. Usually, we will have only have a basis for the space $V\_h^k$ at our disposal, and a direct computation of a basis for $\backslash mathfrak\{Z\}\_h^\{k\backslash perp\}$ 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 Laplace problem (43), and then throw away all eigenvalues corresponding to $\backslash sigma\_h\; \backslash neq\; 0$.

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 $\backslash Omega$ in $\backslash mathbb\{R\}^n$. 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 non-negative integer $k$, we denote by $\backslash text\{Alt\}^k\; V$ the space of real-valued $k$-linear forms on $V$. If $\backslash dim\; V\; =\; n$, then $\backslash dim\; \backslash text\{Alt\}^k\; V\; =\; \backslash binom\{n\}\{k\}$. The wedge product $\backslash omega\; \backslash wedge\; \backslash eta\; \backslash in\; \backslash text\{Alt\}^\{j+k\}\; V$ of $\backslash omega\; \backslash in\; \backslash text\{Alt\}^j\; V$ and $\backslash eta\; \backslash in\; \backslash text\{Alt\}^k\; V$ is given by

where the sum is over all permutations $\backslash sigma$ of $\backslash \{1,\; \backslash dots,\; j+k\backslash \}$, for which and . An inner product on $V$ induces an inner product on $\backslash text\{Alt\}^k\; V$:

where the sum is over increasing sequences $\backslash sigma\; :\; \backslash \{1,\; \backslash dots,\; k\backslash \}\; \backslash rightarrow\; \backslash \{1,\; \backslash dots,\; n\backslash \}$ and $e\_1,\; \backslash dots,\; e\_n$ is any orthonormal basis (the right-hand side being independent of the choice of orthonormal basis). If we orient $V$ by assigning 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 $\backslash text\{vol\}$ in $\backslash text\{Alt\}^n\; V$, $n\; =\; \backslash dim\; V$, characterized by $\backslash text\{vol\}(e\_1,\; \backslash dots,\; e\_n)\; =\; 1$ for any positively oriented ordered orthonormal basis $e\_1,\; \backslash dots,\; e\_n$. The Hodge star operator is an isometry of $\backslash text\{Alt\}^k\; V$ onto $\backslash text\{Alt\}^\{n-k\}\; V$ given by

Given a smooth manifold $\backslash Omega$, possibly with boundary, a differential $k$-form $\backslash omega$ is a section of the $k$-alternating bundle, i.e., a map which assigns to each $x\; \backslash in\; \backslash Omega$ an element $\backslash omega\_x\; \backslash in\; \backslash text\{Alt\}^k\; T\_x\; \backslash Omega$ where $T\_x\; \backslash Omega$ denotes the tangent space to $\backslash Omega$ at $x$. We write $C^m\; \backslash Lambda^k(\backslash Omega)$ for the space of $m$ times continuously differentiable $k$-forms, i.e., forms for which

belongs to $C^m(\backslash Omega)$ for any smooth vector fields $v\_i$. Similarly, we may define $C\_c^\backslash infty\; \backslash Lambda^k$, $C\_c^\backslash infty\; \backslash Lambda^k$ (smooth forms with compact support contained in the interior of $\backslash Omega$), etc. If $\backslash Omega$ is a Riemannian manifold, and so has a measure defined on it, we may similarly define Lebesgue spaces $L\_p\; \backslash Lambda^k$ and Sobolev spaces $W\_p^m\; \backslash Lambda^k$ and $H^m\; \backslash Lambda^k\; =\; W\_2^m\; \backslash Lambda^k$. The spaces $H^m\; \backslash Lambda^k$ are Hilbert spaces. In particular, the inner product in $L\_2\; \backslash Lambda^k\; =\; H^0\; \backslash Lambda^k$ is given by

We also write $\backslash Lambda^k$ or $\backslash Lambda^k(\backslash Omega)$ for the space of all smooth differential $k$-forms, or at least sufficiently smooth, as demanded by the context.

For any smooth manifold $\backslash Omega$ and $\backslash omega\; \backslash in\; \backslash Lambda^k(\backslash Omega)$, the exterior derivative $d\backslash omega$ is a $(k+1)$-form, which itself has vanishing exterior derivative: $d(d\backslash omega)\; =\; 0$. 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 $\backslash phi\; :\; \backslash Omega\text{'}\; \backslash rightarrow\; \backslash Omega$ between manifolds, induces a pullback of differential forms from $\backslash Omega$ to $\backslash Omega\text{'}$. Namely, if $\backslash omega\; \backslash in\; \backslash Lambda^k(\backslash Omega)$, the pullback $\backslash phi^*\; \backslash omega\; \backslash in\; \backslash Lambda^k(\backslash Omega\text{'})$ is defined by

The pullback respects exterior products and exterior derivatives:

If $\backslash phi$ is an orientation-preserving diffeomorphism of oriented $n$-dimensional manifolds, and $\backslash omega$ is an $n$-form on $\backslash Omega$, then

If $\backslash Omega\text{'}$ is a submanifold of $\backslash Omega$, then the pullback of the inclusion map is the trace map $\backslash text\{tr\}\_\{\backslash Omega,\; \backslash Omega\text{'}\}$, written simply $\backslash text\{tr\}\_\{\backslash Omega\text{'}\}$ or $\backslash text\{tr\}$ when the manifolds can be inferred from context. We recall the trace theorem which states that if $\backslash Omega\text{'}$ is a submanifold of codimension 1, then the trace map extends to a bounded operator from $H^1\; \backslash Lambda^k(\backslash Omega)$ to $L^2\; \backslash Lambda^k(\backslash Omega\text{'})$, or, more precisely, to a bounded surjection of $H^1\; \backslash Lambda^k(\backslash Omega)$ onto $H^\{1/2\}\; \backslash Lambda^k(\backslash Omega\text{'})$. A particularly important situation is when $\backslash Omega\text{'}\; =\; \backslash partial\; \backslash Omega$, in which case Stokes's theorem relates the integrals of the exterior derivative and trace of an $(n-1)$-form $\backslash omega$ on an oriented $n$-dimensional manifold-with-boundary $\backslash Omega$:

(A common abuse of notation is to write $\backslash int\_\{\backslash partial\backslash Omega\}\; \backslash omega$ for the $\backslash int\_\{\backslash partial\backslash Omega\}\; \backslash text\{tr\; \}\; \backslash omega$.) Applying Stokes's theorem to the differential form $\backslash omega\; \backslash wedge\; \backslash eta$ with $\backslash omega\; \backslash in\; \backslash Lambda^\{k-1\}$, $\backslash eta\; \backslash in\; \backslash Lambda^\{n-k\}$, and using the Leibniz rule $d(\backslash omega\; \backslash wedge\; \backslash eta)\; =\; d\backslash omega\; \backslash wedge\; \backslash eta\; +\; (-1)^\{k-1\}\; \backslash omega\; \backslash wedge\; d\backslash eta$, we obtain the integration-by-parts formula for differential forms:

On an oriented $n$-dimensional Riemannian manifold, there is a volume form $\backslash text\{vol\}\; \backslash in\; \backslash Lambda^n(\backslash Omega)$ which at each point $x$ of the manifold is equal to the volume form on $T\_x\backslash Omega$. Consequently, the Hodge star operation takes $\backslash omega\; \backslash in\; \backslash Lambda^k(\backslash Omega)$ to $\backslash star\backslash omega\; \backslash in\; \backslash Lambda^\{n-k\}(\backslash Omega)$ satisfying

for all $\backslash mu\; \backslash in\; \backslash Lambda^\{n-k\}(\backslash Omega)$. Introducing the coderivative operator $\backslash delta\; :\; \backslash Lambda^k\; \backslash rightarrow\; \backslash Lambda^\{k-1\}$ defined by

and setting $\backslash eta\; =\; \backslash star\backslash mu$, the integration-by-parts formula becomes

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 $C^m\backslash Lambda^k$, $W\_p^m\backslash Lambda^k$, etc., isometrically onto $C^m\backslash Lambda^\{n-k\}$, $W\_p^m\backslash Lambda^\{n-k\}$, etc. By definition, $\backslash delta\; :\; \backslash Lambda^\{n-k\}\; \backslash rightarrow\; \backslash Lambda^\{n-k-1\}$ corresponds to $d\; :\; \backslash Lambda^k\; \backslash rightarrow\; \backslash Lambda^\{k+1\}$ via the Hodge star isomorphism. Thus for every property of $d$ there is a corresponding property of $\backslash delta$.

In case $\backslash Omega$ is a domain in $\backslash mathbb\{R\}^n$, we may (but usually will not) use the standard coordinates of $\backslash mathbb\{R\}^n$ to write a $k$-form as

where the $a\_\{\backslash sigma\}\; \backslash in\; L^2(\backslash Omega)$ and $dx\_j\; :\; \backslash mathbb\{R\}^n\; \backslash rightarrow\; \backslash mathbb\{R\}$ is the linear form which associates to a vector its $j$th coordinate. In this case, the exterior derivative is given by the simple formula

extended by linearity to a sum of such terms. If $\backslash Omega$ is a domain in $\backslash mathbb\{R\}^n$, then $\backslash int\_\{\backslash Omega\}\; a\; dx\_1\; \backslash wedge\; \backslash dots\; \backslash wedge\; dx\_n$, has the value suggested by the notation. In the case of a domain in $\backslash mathbb\{R\}^n$, the volume form is simply $dx\_1\; \backslash wedge\; \backslash dots\; \backslash wedge\; dx\_n$. We remark that the exterior derivative and integral of differential forms can be computed on arbitrary manifolds from the formulas on subdomains on $\backslash mathbb\{R\}^n$ and pullbacks through charts.

4.2. The de Rham complex as a Hilbert complex. Henceforth we restrict attention to the case that $\backslash Omega$ is a bounded domain in $\backslash mathbb\{R\}^n$ 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 $W^k\; =\; L^2\backslash Lambda^k(\backslash Omega)$ to $W^\{k+1\}\; =\; L^2\backslash Lambda^\{k+1\}(\backslash Omega)$. Let $\backslash omega\; \backslash in\; L^2\backslash Lambda^k(\backslash Omega)$. In view of (51), we say that $\backslash eta\; \backslash in\; L^2\backslash Lambda^\{k+1\}(\backslash Omega)$ is the weak exterior derivative of $\backslash omega$ if

The weak exterior derivative of $\backslash omega$, if one exists, is unique and we denote it by $d\backslash omega$. In analogy with the definition of Sobolev spaces (cf., e.g., [47, Section 5.2.2]), we define $H\backslash Lambda^k$ to be the space of forms in $L^2\backslash Lambda^k$ with a weak derivative in $L^2\backslash Lambda^\{k+1\}$. With the inner product

this is easily seen to be a Hilbert space and clearly $d$ is a bounded map from $H\backslash Lambda^k$ to $L^2\backslash Lambda^\{k+1\}$. A standard smoothing argument, as in [47, Section 5.3], implies that $C^\backslash infty\backslash Lambda^k(\backslash bar\{\backslash Omega\})$ is dense in $H\backslash Lambda^k$. We take $H\backslash Lambda^k(\backslash Omega)$ as the domain $V^k$ of the exterior derivative, which is thus densely defined in $L^2\backslash Lambda^k(\backslash Omega)$. Since $H\backslash Lambda^k$ is complete, $d$ is a closed operator.

Thus the spaces $L^2\backslash Lambda^k(\backslash Omega)$ and the exterior derivative operators $d$ form a Hilbert complex with the associated domain complex

This is the $L^2$ 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 $V\_k^*$. Using the surjectivity of the trace operator from $H^1\backslash Lambda^l(\backslash Omega)$ onto $H^\{1/2\}\backslash Lambda^l(\backslash partial\backslash Omega)$ and the integration-by-parts formula (49), we can show that (cf., [8, page 19]) the trace operator extends boundedly from $H\backslash Lambda^k(\backslash Omega)$ to $H^\{-1/2\}\backslash Lambda^k(\backslash partial\backslash Omega)$, and that (49) holds for $\backslash omega\; \backslash in\; H\backslash Lambda^\{k-1\}$, $\backslash eta\; \backslash in\; H^1\backslash Lambda^\{n-k\}$ where the integral on the boundary is interpreted via the pairing of $H^\{-1/2\}(\backslash partial\backslash Omega)$ and $H^\{1/2\}(\backslash partial\backslash Omega)$. Equivalently, we have an extended version of (51):

Of course, there is a corresponding result obtained by the Hodge star isomorphism which interchanges $d$ and $\backslash delta$. After reindexing, this is nothing but the fact that (53) holds also for $\backslash omega\; \backslash in\; H^1\backslash Lambda^\{k-1\}$, $\backslash mu\; \backslash in\; H^*\backslash Lambda^k$ where

Note that $H^*\backslash Lambda^k$ consists of those differential form in $L^2\backslash Lambda^k$ for which a weak coderivative exists in $L^2\backslash Lambda^\{k-1\}$, where the weak exterior coderivative is defined in exact analogy to the weak exterior derivative. Its inner product is

The space $H^*\backslash Lambda^k$ is isometric to $H\backslash Lambda^\{n-k\}$ via the Hodge star, but is quite different from $H\backslash Lambda^k$.

We also make use of the trace defined on $H\backslash Lambda^k$ to define the subspace with vanishing trace:

Correspondingly, for $\backslash omega\; \backslash in\; H^*\backslash Lambda^k$, the quantity $\backslash text\{tr\}\; \backslash star\; \backslash omega$ is well defined, and we have

From (53), we have

(We certainly have (56) with the stronger condition $\backslash omega\; \backslash in\; H^1\backslash Lambda^\{k-1\}$, but then we can extend to all $\backslash omega\; \backslash in\; H\backslash Lambda^\{k-1\}$ by continuity and density.) Of course, the corresponding result, where $\backslash omega\; \backslash in\; \backslash dot\{H\}\backslash Lambda^\{k-1\}$, $\backslash mu\; \backslash in\; H^*\backslash Lambda^k$, holds as well.

Theorem 4.1. Let $d$ be the exterior derivative viewed as an unbounded operator $L^2\backslash Lambda^\{k-1\}\; \backslash rightarrow\; L^2\backslash Lambda^k$ with domain $H\backslash Lambda^k$. Then the adjoint $d^*$, as an unbounded operator $L^2\backslash Lambda^k\; \backslash rightarrow\; L^2\backslash Lambda^\{k-1\}$, has as its domain $\backslash dot\{H\}^*\backslash Lambda^k$, and coincides with the operator $\backslash delta$ defined in (50).

Proof. We must show that for $\backslash mu\; \backslash in\; L^2\backslash Lambda^k$, there exists $\backslash omega\; \backslash in\; L^2\backslash Lambda^\{k-1\}$ such that

if and only if $\backslash mu\; \backslash in\; \backslash dot\{H\}^*\backslash Lambda^k$ and $\backslash omega\; =\; \backslash delta\backslash mu$. The if direction is immediate from (56). Conversely, if (57) holds, then $\backslash mu$ has a weak exterior coderivative in $L^2$, namely $\backslash delta\backslash mu\; =\; \backslash omega$. Thus $\backslash mu\; \backslash in\; H^*\backslash Lambda^k$. Integrating by parts we have

which implies that $\backslash text\{tr\}\; \backslash star\; \backslash mu\; =\; 0$, i.e., $\backslash mu\; \backslash in\; \backslash dot\{H\}^*\backslash Lambda^k$. $\backslash square$

As a corollary, we obtain a concrete characterization of the harmonic forms:

In other words, a $k$-form is harmonic if it satisfies the differential equations $d\backslash omega\; =\; 0$ and $\backslash delta\backslash omega\; =\; 0$ together with the boundary conditions $\backslash text\{tr\}\; \backslash star\; \backslash omega\; =\; 0$.

If the boundary of $\backslash Omega$ is smooth, then $H\backslash Lambda^k\; \backslash cap\; \backslash dot\{H\}^*\backslash Lambda^k$ is contained in $H^1\backslash Lambda^k$ [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 $H\backslash Lambda^k\; \backslash cap\; \backslash dot\{H\}^*\backslash Lambda^k$ in $H^1\backslash Lambda^k$ 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 $L^2\backslash Lambda^k$ and of $H\backslash Lambda^k$, 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 $(\backslash sigma,\; u,\; p)\; \backslash in\; H\backslash Lambda^\{k-1\}\; \backslash times\; H\backslash Lambda^k\; \backslash times\; \backslash mathfrak\{H\}^k$ is a solution if and only if

The first differential equation and the first boundary condition are implied by the first equation in (18), and the second differential equation and boundary condition by the second, while the third equation in (18) is simply the side condition $u\; \backslash perp\; \backslash mathfrak\{H\}^k$. 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 $\backslash Omega\; \backslash subset\; \backslash mathbb\{R\}^3$, and consider the Hodge Laplacian for $k$-forms, $k\; =\; 0,\; 1,\; 2$, and $3$. We also discuss the $\backslash mathfrak\{B\}^*$ and $\backslash mathfrak\{B\}$ 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 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 $n$-forms and $(n-1)$-forms. In the case of a 3-dimensional domain in $\backslash mathbb\{R\}^3$, 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 $\backslash omega$, the exact same vector field is the proxy for the 2-form $\backslash star\backslash omega$. Via proxy fields, the exterior derivatives coincide with standard differential operators of calculus:

and the de Rham complex (52) is realized as

where

The exterior coderivatives $\backslash delta$ become, of course, $-\backslash text\{div\}$, $\backslash text\{curl\}$, and $-\backslash text\{grad\}$, 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 $\backslash partial\backslash Omega$). The trace operator from 1-forms on $\backslash Omega$ to 1-forms on the boundary takes a vector field $u$ on $\backslash Omega$ to a tangential vector field on the boundary, namely at each boundary point $x$, $(\backslash text\{tr\; \}\; u)\_x$ is the tangential projection of $u\_x$. For a 2-form $u$, the trace corresponds to the scalar $u\; \backslash cdot\; n$ (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 $H\backslash Lambda^\{-1\}$ is understood to be 0, so $\backslash sigma$ vanishes. The harmonic form space $\backslash mathfrak\{H\}^0$ consists of the constant functions (we assume $\backslash Omega$ is connected—otherwise $\backslash mathfrak\{H\}^0$ 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

Similarly, the first boundary condition in (60) vanishes, while the second is the Neumann condition

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 $\backslash mathfrak\{B\}^*$ and $\backslash mathfrak\{B\}$ 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 $\backslash mathfrak\{H\}^0$ filtered out.

4.2.2. The Hodge Laplacian for $k\; =\; 1$. In this case the differential equations and boundary conditions are

which is a formulation of a boundary value problem for the vector Laplacian $\backslash operatorname\{curl\}\; \backslash operatorname\{curl\}\; -\; \backslash operatorname\{grad\}\; \backslash operatorname\{div\}$. (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 $\backslash mathfrak\{H\}^1$, which in this case consists of those vector fields satisfying

The dimension of $\backslash mathfrak\{H\}^1$ is equal to the first Betti number, i.e., the number of handles, of the domain (so $\backslash mathfrak\{H\}^1\; =\; 0$ if the domain is simply-connected).

The $\backslash mathfrak\{B\}\_1^*$ problem (19) is defined for $L^2$ vector fields $f$ which are orthogonal to both gradients and the vector fields in $\backslash mathfrak\{H\}^1$. In that case, the solution to (62) has $\backslash sigma\; =\; 0$ and $p\; =\; 0$, while $u$ satisfies

The orthogonality condition $u\; \backslash perp\; \backslash mathfrak\{H\}^1$ again determines the solution uniquely.

Next we turn to the $\backslash mathfrak\{B\}^1$ problem. For source functions of the form $f\; =\; \backslash operatorname\{grad\}\; F$ for some $F\; \backslash in\; H^1$, which we normalize so that $\backslash int\_\{\backslash Omega\}\; F\; =\; 0$, (62) reduces to the problem of finding $\backslash sigma\; \backslash in\; H^1$ and $u\; \backslash in\; \backslash mathfrak\{B\}^1\; =\; \backslash operatorname\{grad\}\; H^1$ such that:

The differential equations may be simplified to $-\backslash operatorname\{grad\}\; \backslash operatorname\{div\}\; u\; =\; f$, and the condition that $u\; \backslash in\; \backslash mathfrak\{B\}^1$ can be replaced by the differential equation $\backslash operatorname\{curl\}\; u\; =\; 0$, together with orthogonality to $\backslash mathfrak\{H\}^1$. Now $\backslash operatorname\{grad\}(\backslash sigma\; -\; F)\; =\; 0$ and $\backslash int\_\{\backslash Omega\}\; \backslash sigma\; =\; -\backslash int\_\{\backslash partial\backslash Omega\}\; u\; \backslash cdot\; n\; =\; 0\; =\; \backslash int\_\{\backslash Omega\}\; F$, so $\backslash sigma\; =\; F$, and we may rewrite the system as

which, again, has a unique solution subject to orthogonality to $\backslash mathfrak\{H\}^1$.

The eigenvalue problem (42) is the corresponding eigenvalue problem for the vector Laplacian with the boundary conditions of (62), and with the eigenspace $\backslash mathfrak\{H\}^1$ of the eigenvalue $\backslash lambda\; =\; 0$ filtered out. As mentioned above, in this case the

$\backslash mathfrak\{B\}\_1^*$ eigenvalue problem, given by (45), is important for models based on Maxwell's equations. This problem takes the form

4.2.3. The Hodge Laplacian for $k\; =\; 2$. The differential equations and boundary conditions are

This is again a formulation of a boundary value problem for the vector Laplacian $\backslash operatorname\{curl\}\; \backslash operatorname\{curl\}\; -\; \backslash operatorname\{grad\}\; \backslash operatorname\{div\}$, 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 time uniqueness is obtained by imposing orthogonality to $\backslash mathfrak\{H\}^2$, the space of vector fields satisfying

which has dimension equal to the second Betti number, i.e., the number of voids in the domain.

The $\backslash mathfrak\{B\}\_2^*$ problem arises for source functions of the form $f\; =\; \backslash operatorname\{grad\}\; F$ for some $F\; \backslash in\; \backslash dot\{H\}^1$ ($\backslash dot\{H\}^1$ is the function space corresponding to $\backslash dot\{H\}^*\; \backslash Lambda^3$). We find $\backslash sigma\; =\; 0$, and $u$ solves

i.e., the same differential equation as for $\backslash mathfrak\{B\}^1$, but with different boundary conditions, an extra assumption on $F$, and, of course, now uniqueness is determined by orthogonality to $\backslash mathfrak\{H\}^2$.

If $\backslash operatorname\{div\}\; f\; =\; 0$ and $f\; \backslash perp\; \backslash mathfrak\{H\}^2$, we get the $\backslash mathfrak\{B\}^2$ problem for which the differential equations are $\backslash sigma\; =\; \backslash operatorname\{curl\}\; u$, $\backslash operatorname\{curl\}\; \backslash sigma\; =\; f$ and the condition $\backslash operatorname\{div\}\; u\; =\; 0$ arising from the membership of $u$ in $\backslash mathfrak\{B\}^2$. Thus $u$ solves

the same differential equation as for $\backslash mathfrak\{B\}\_1^*$, but with different boundary conditions.

The eigenvalue problem (42) is the corresponding eigenvalue problem for the vector Laplacian with the boundary conditions of (64), and with the eigenspace $\backslash mathfrak\{H\}^2$ of the eigenvalue $\backslash lambda\; =\; 0$ filtered out, while the corresponding $\backslash mathfrak\{B\}^2$ problem, of the form (47), takes the form

Note that this is the same problem as the $\backslash mathfrak\{B\}\_1^*$ eigenvalue problem (63), but with different boundary conditions. However, if we define $\backslash sigma\; =\; \backslash operatorname\{curl\}\; u$, then it is straightforward to check that the pair $(\backslash lambda,\; \backslash sigma)$ will indeed solve the $\backslash mathfrak\{B\}\_1^*$ problem (63). This is an instance of the general equivalence between the $\backslash mathfrak\{B\}^k$ problem, given by (47), and the corresponding $\backslash mathfrak\{B\}\_\{k-1\}^*$ 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 $\backslash mathfrak\{B\}^3$ problem, is

which is the Dirichlet problem for Poisson's equation. There are no non-zero 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.

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 $\backslash Lambda\_h^k$ of $H\backslash Lambda^k(\backslash Omega)$. As we saw in Section 3.4, the key properties these spaces must possess is, first, that $d\backslash Lambda\_h^k\; \backslash subset\; \backslash Lambda\_h^\{k+1\}$ so they form a subcomplex $(\backslash Lambda\_h,\; d)$ of the de Rham complex, second, that there exist uniformly bounded cochain projections $\backslash pi\_h$ from $(L^2\backslash Lambda,\; d)$ to $(\backslash Lambda\_h,\; d)$, 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 Galerkin solution in terms of the approximation error afforded by the subspaces. In this section we will construct the spaces $\backslash Lambda\_h^k$ 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 $L^2$-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 $\backslash mathcal\{P\}\_r\backslash Lambda^k(\backslash mathbb\{R\}^n)$ consisting of all differential $k$-forms on $\backslash mathbb\{R\}^n$ whose coefficients are polynomials of degree at most $r$. In addition to these spaces, we will use another family of polynomial form spaces, denoted $\backslash mathcal\{P\}\_r^-\backslash Lambda^k(\backslash mathbb\{R\}^n)$, 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 $2^\{n-1\}$ 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 $\backslash mathcal\{P\}\_r(\backslash mathbb\{R\}^n)$ and $\backslash mathcal\{H\}\_r(\backslash mathbb\{R\}^n)$ 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 . We can then define spaces of polynomial differential forms, $\backslash mathcal\{P\}\_r\backslash Lambda^k(\backslash mathbb\{R\}^n)$, $\backslash mathcal\{H\}\_r\backslash Lambda^k(\backslash mathbb\{R\}^n)$, 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 $\backslash mathbb\{R\}^n$ from the notation and write simply $\backslash mathcal\{P\}\_r$, $\backslash mathcal\{H\}\_r$, $\backslash mathcal\{P\}\_r\backslash Lambda^k$, etc.

The dimensions of these spaces are easily calculated:

and $\backslash dim\; \backslash mathcal\{H\}\_r\backslash Lambda^k(\backslash mathbb\{R\}^n)\; =\; \backslash dim\; \backslash mathcal\{P\}\_r\backslash Lambda^k(\backslash mathbb\{R\}^\{n-1\})$.

For each polynomial degree $r\; \backslash geq\; 0$, we get a homogeneous polynomial subcomplex of the de Rham complex:

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 $\backslash mathbb\{R\}$ (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:

for which the cohomology space is $\backslash mathbb\{R\}$ in the lowest degree, and vanishes otherwise.

5.1.2. The Koszul complex. Let $x\; \backslash in\; \backslash mathbb\{R\}^n$. Since there is a natural identification of $\backslash mathbb\{R\}^n$ with the tangent space $T\_0\backslash mathbb\{R\}^n$ at the origin, there is a vector in $T\_0\backslash mathbb\{R\}^n$ corresponding to $x$. (The origin is chosen for convenience here, but we could use any other point instead.) Then the translation map $y\; \backslash mapsto\; y\; +\; x$ induces an isomorphism from $T\_0\backslash mathbb\{R\}^n$ to $T\_x\backslash mathbb\{R\}^n$, and so there is an element $X(x)\; \backslash in\; T\_x\backslash mathbb\{R\}^n$ 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 $\backslash kappa$ from $\backslash Lambda^k(\backslash mathbb\{R\}^n)$ to $\backslash Lambda^\{k-1\}(\backslash mathbb\{R\}^n)$ called the Koszul differential:

It is easy to see that $\backslash kappa$ is a graded differential, i.e.,

and

In terms of coordinates, if $\backslash omega\_x\; =\; a(x)\; dx\_\{\backslash sigma\_1\}\; \backslash wedge\; \backslash cdots\; \backslash wedge\; dx\_\{\backslash sigma\_k\}$, then

where the notation $\backslash widehat\{dx\_\{\backslash sigma\_i\}\}$ means that the term is omitted. Note that $\backslash kappa$ maps $\backslash mathcal\{H\}\_r\backslash Lambda^k$ to $\backslash mathcal\{H\}\_\{r+1\}\backslash Lambda^\{k-1\}$, i.e., $\backslash kappa$ 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],

We show below that this complex is exact for $r\; >\; 0$. Adding over polynomial degrees, we obtain the Koszul complex (for any $r\; \backslash geq\; 0$),

for which all the cohomology spaces vanish, except the rightmost, which is equal to $\backslash mathbb\{R\}$.

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.

In Sections 4 and 5, we applied the abstract theory of Section 3 to the Hilbert complex obtained by choosing $W^k$ to be $L^2\backslash Lambda^k(\backslash Omega)$ and $d^k$ to be the exterior derivative with domain $V^k\; =\; H\backslash Lambda^k(\backslash Omega)$. This led to finite element approximations of certain boundary value problems for the Hodge Laplacian, and a variety of related problems ($\backslash mathfrak\{B\}$ and $\backslash mathfrak\{B\}^*$ 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, $d^k$ is again taken to be the exterior derivative with domain $V^k\; =\; H\backslash Lambda^k(\backslash Omega)$ and $W^k$ is again taken to be the space $L^2\backslash Lambda^k$, but $W^k$ is furnished with an inner product which is equivalent to, but not equal to, the standard $L^2\backslash Lambda^k$ inner product. Specifically, we let $a^k\; :\; W^k\; \backslash rightarrow\; W^k$ be a bounded, symmetric, positive definite operator and choose

With this choice, we can check that $d\_k^*\; =\; (d^\{k-1\})^*$ is given by

with the domain of $d\_k^*$ given by

The harmonic forms are then

The Hodge Laplacian $d^*du\; +\; dd^*u$ is now realized as

As an example of the utility of this generalization, we consider the $\backslash mathfrak\{B\}\_1^*$ eigenvalue problem in 3 dimensions. We take $a^1\; =\; \backslash varepsilon$ and $a^2\; =\; \backslash mu^\{-1\}$ where $\backslash varepsilon$ and $\backslash mu$ are symmetric positive definite $3\; \backslash times\; 3$ matrix fields. Then the $\backslash mathfrak\{B\}\_1^*$ eigenvalue problem is

This is the standard Maxwell eigenvalue problem where $\backslash varepsilon$ is the dielectric tensor of the material and $\backslash mu$ 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 $W^k\; =\; L^2\backslash Lambda^k(\backslash Omega)$ and $d^k$ to be the exterior derivative, but this time we take its domain $V^k$ to be the space $\backslash dot\{H\}\backslash Lambda^k(\backslash Omega)$. This space is dense in $W^k$, and is a closed subspace of $H\backslash Lambda^k$, 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:

In Section 4.2, we used the fact that $H\backslash Lambda^k\; \backslash cap\; \backslash dot\{H\}^*\backslash Lambda^k$ is compactly included in $L^2\backslash Lambda^k$. Replacing $k$ by $n\; -\; k$ and applying the Hodge star operator, we conclude the analogous result, that $\backslash dot\{H\}\backslash Lambda^k\; \backslash cap\; H^*\backslash Lambda^k$ is compactly included in $L^2\backslash Lambda^k$. 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 $L^2\backslash Lambda^k$ and of $\backslash dot\{H\}\backslash Lambda^k$, 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

Now, from (53), we have that

Therefore

This gives a concrete characterization of the harmonic forms analogous to (58):

Now, from (54), (55), and (50), the Hodge star operator maps the set of $\backslash omega\; \backslash in\; H\backslash Lambda^\{n-k\}$ with $d\backslash omega\; =\; 0$ isomorphically onto the set of $\backslash mu\; \backslash in\; H^*\backslash Lambda^k$ with $\backslash delta\backslash mu\; =\; 0$, and maps the set of $\backslash omega\; \backslash in\; \backslash mathcal\{H\}^*\backslash Lambda^\{n-k\}$ with $\backslash delta\backslash omega\; =\; 0$ isomorphically onto the set of $\backslash mu\; \backslash in\; \backslash dot\{H\}\backslash Lambda^k$ with $d\backslash mu\; =\; 0$. Comparing (58) and (80), we conclude that the Hodge star operator maps $\backslash mathring\{\backslash mathfrak\{H\}\}^\{n-k\}$ isomorphically onto $\backslash mathring\{\backslash mathfrak\{H\}\}^k$. This isomorphism is called Poincaré duality. In particular, $\backslash dim\; \backslash mathring\{\backslash mathfrak\{H\}\}^k$ is the $(n\; -\; k)$th Betti number of $\backslash Omega$.

Finally, from (79), we can characterize $d^*$, the adjoint of the exterior derivative $d$ viewed as an unbounded operator $L^2\backslash Lambda^\{k-1\}\; \backslash rightarrow\; L^2\backslash Lambda^k$ with domain $\backslash dot\{H\}\backslash Lambda^k$. Namely, in this case $d^*$ has domain $H^*\backslash Lambda^k$ and coincides with the operator $\backslash delta$ defined in (50). By contrast, when we took the domain of $d$ to be all of $H\backslash Lambda^k$, the domain of $d^*$ turned out to be the smaller space $\backslash dot\{H\}^*\backslash Lambda^k$.

We now consider the Hodge Laplacian problem given in the abstract case by (18). In this case, we get that $(\backslash sigma,\; u,\; p)\; \backslash in\; \backslash dot\{H\}\backslash Lambda^\{k-1\}\; \backslash times\; \backslash dot\{H\}\backslash Lambda^k\; \backslash times\; \backslash mathring\{\backslash mathfrak\{H\}\}^k$ is a solution if and only if

Note that now both boundary conditions are essential in the sense that they are implied by membership in the spaces $\backslash dot\{H\}\backslash Lambda^\{k-1\}$ and $\backslash dot\{H\}\backslash Lambda^k$ where the solution is sought, rather than by the variational formulation.

To make things more concrete, we restrict to a domain $\backslash Omega\; \backslash subset\; \backslash mathbb\{R\}^3$, and consider the Hodge Laplacian for $k$-forms, $k\; =\; 0,\; 1,\; 2$, and $3$. We also discuss the $\backslash mathfrak\{B\}^*$ and $\backslash mathfrak\{B\}$ problems given by (19) and (20) for each $k$. Again we have

and the de Rham complex (78) is now realized as

where

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 $H\backslash Lambda^\{-1\}$ is understood to be 0, so $\backslash sigma$ vanishes. The space $\backslash mathring\{\backslash mathfrak\{H\}\}^0$ of harmonic 0-forms vanishes. The first differential equation of (81) vanishes, and the second gives Poisson's equation

Similarly, the first boundary condition in (82) vanishes, while the second is the Dirichlet condition $u\; =\; 0$ on $\backslash partial\backslash Omega$. The side condition (83) is automatically satisfied. Nothing additional is obtained by considering the split into the $\backslash mathfrak\{B\}^*$ and $\backslash mathfrak\{B\}$ 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

which is a formulation of a boundary value problem for the vector Laplacian $\backslash text\{curl\; curl\}\; -\; \backslash text\{grad\; div\}$. The solution is determined uniquely by the additional condition that $u$ be orthogonal to $\backslash mathring\{\backslash mathfrak\{H\}\}^1\; =\; \backslash mathfrak\{H\}^2$, i.e., those vector fields satisfying

The $\backslash mathfrak\{B\}\_1^*$ problem (19) is defined for $L^2$ vector fields $f$ which are orthogonal to both gradients of functions in $\backslash dot\{H\}^1(\backslash Omega)$ and the vector fields in $\backslash mathring\{\backslash mathfrak\{H\}\}^1$. In that case, the solution to (84) has $\backslash sigma\; =\; 0$ and $p\; =\; 0$, while $u$ satisfies

The orthogonality condition $u\; \backslash perp\; \backslash mathring\{\backslash mathfrak\{H\}\}^1$ again determines the solution uniquely.

Next we turn to the $\backslash mathfrak\{B\}^1$ problem. For source functions of the form $f\; =\; \backslash text\{grad\; \}\; F$ for some $F\; \backslash in\; \backslash dot\{H\}^1$, (84) reduces to the problem of finding $\backslash sigma\; \backslash in\; \backslash dot\{H\}^1$ and $u\; \backslash in\; \backslash mathfrak\{B\}^1\; =\; \backslash text\{grad\; \}\; \backslash dot\{H\}^1$ such that:

The differential equations may be simplified to $-\backslash operatorname\{grad\}\; \backslash operatorname\{div\}\; u\; =\; f$ and the condition that $u\; \backslash in\; \backslash mathfrak\{B\}^1$ can be replaced by the differential equation $\backslash operatorname\{curl\}\; u\; =\; 0$ and the boundary condition $u\; \backslash times\; n\; =\; 0$, together with orthogonality to $\backslash mathring\{\backslash mathfrak\{H\}\}^1$. Now $\backslash operatorname\{grad\}(\backslash sigma\; -\; F)\; =\; 0$ and $\backslash sigma\; -\; F\; =\; 0$ on $\backslash partial\backslash Omega$, so $\backslash sigma\; =\; F$. Hence, we may rewrite the system as

which, again, has a unique solution subject to orthogonality to $\backslash mathring\{\backslash mathfrak\{H\}\}^1$.

6.2.3. The Hodge Laplacian for $k\; =\; 2$. The differential equations and boundary conditions are

This is again a formulation of a boundary value problem for the vector Laplacian $\backslash operatorname\{curl\}\; \backslash operatorname\{curl\}\; -\; \backslash operatorname\{grad\}\; \backslash operatorname\{div\}$, 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 $\backslash mathring\{\backslash mathfrak\{H\}\}^2\; =\; \backslash mathfrak\{H\}^1$, the space of vector fields satisfying

The $\backslash mathfrak\{B\}\_2^*$ problem arises for source functions of the form $f\; =\; \backslash operatorname\{grad\}\; F$ for some $F\; \backslash in\; H^1$ which we may take to have integral 0. We find $\backslash sigma\; =\; 0$, and $u$ solves

i.e., the same differential equation as for $\backslash mathfrak\{B\}^1$, but with different boundary conditions and, of course, now uniqueness is determined by orthogonality to $\backslash mathring\{\backslash mathfrak\{H\}\}^2$.

If $\backslash operatorname\{div\}\; f\; =\; 0$, $f\; \backslash cdot\; n\; =\; 0$ on $\backslash partial\backslash Omega$, and $f\; \backslash perp\; \backslash mathring\{\backslash mathfrak\{H\}\}^2$, we get the $\backslash mathfrak\{B\}^2$ problem for which the differential equations are $\backslash sigma\; =\; \backslash operatorname\{curl\}\; u$, $\backslash operatorname\{curl\}\; \backslash sigma\; =\; f$ and the condition $\backslash operatorname\{div\}\; u\; =\; 0$ arising from the membership of $u$ in $\backslash mathfrak\{B\}^2$. Thus $u$ solves

the same differential equation as for $\backslash mathfrak\{B\}\_1^*$, but with different boundary conditions.

6.2.4. The Hodge Laplacian for $k\; =\; 3$. The space $\backslash mathring\{\backslash mathfrak\{H\}\}^3$ consists of the constants (or, if $\backslash Omega$ is not connected, of functions which are constant on each connected component). The Hodge Laplacian problem is

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.

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 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 (2-dimensional) 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:

Here the unknowns $\backslash sigma$ and $u$ denote the stress and displacement fields engendered by a body force $f$ acting on a linearly elastic body which occupies a region $\backslash Omega\; \backslash subset\; \backslash mathbb\{R\}^n$. 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 $\backslash sigma$ takes values in the space $\backslash mathbb\{S\}\; :=\; \backslash mathbb\{R\}\_\{\backslash text\{sym\}\}^\{n\; \backslash times\; n\}$ of symmetric matrices and the displacement field $u$ takes values in $\backslash mathbb\{V\}\; :=\; \backslash mathbb\{R\}^n$. The differential operator $\backslash epsilon$ is the symmetric part of the gradient, the div operator is applied row-wise to a matrix, and the compliance tensor $A\; =\; A(x)\; :\; \backslash mathbb\{S\}\; \backslash rightarrow\; \backslash mathbb\{S\}$ is a bounded and symmetric, uniformly positive definite operator reflecting the material properties of the body. If the body is clamped on the boundary $\backslash partial\backslash Omega$ of $\backslash Omega$, then the appropriate boundary condition for the system (85) is $u\; =\; 0$ on $\backslash partial\backslash Omega$. For simplicity, this boundary condition will be assumed here.

The pair $(\backslash sigma,\; u)$ can alternatively be characterized as the unique critical point of the Hellinger–Reissner functional

The critical point is sought among $\backslash tau\; \backslash in\; H(\backslash operatorname\{div\};\; \backslash Omega;\; \backslash mathbb\{S\})$, the space of square-integrable symmetric matrix fields with square-integrable divergence, and $v\; \backslash in\; L^2(\backslash Omega;\; \backslash mathbb\{V\})$, the space of square-integrable vector fields. Equivalently, the pair $(\backslash sigma,\; u)\; \backslash in\; H(\backslash operatorname\{div\};\; \backslash Omega;\; \backslash mathbb\{S\})\; \backslash times\; L^2(\backslash Omega;\; \backslash mathbb\{V\})$ is the unique solution to the weak formulation of the system (85):

7.2. The elasticity complex and its discretization. The discretization of the problem (86) is closely tied to discretization of the elasticity complex. For a two dimensional domain $\backslash Omega$, the elasticity complex takes the form

The operator $J$ is a second-order differential operator mapping scalar fields into symmetric matrix fields, namely the rotated Hessian

We now show that this complex is closed, i.e., that the range of the two operators $J$ and $\backslash operatorname\{div\}$ are closed subspaces. The null space of $J$ is just the space $\backslash mathcal\{P\}\_1$ of linear polynomials and we have $\backslash |q\backslash |\_\{H^2\}\; \backslash leq\; c\backslash |Jq\backslash |\_\{L^2\}$ for all $q\; \backslash in\; H^2(\backslash Omega)$ with $q\; \backslash perp\; \backslash mathcal\{P\}\_1$, which implies that the range of $J$ is closed. We now show that $\backslash operatorname\{div\}$ maps $H(\backslash operatorname\{div\};\; \backslash Omega;\; \backslash mathbb\{S\})$ onto $L^2(\backslash Omega;\; \backslash mathbb\{R\}^2)$, using the fact that the scalar-valued divergence maps $H^1(\backslash Omega;\; \backslash mathbb\{R\}^2)$ onto $L^2(\backslash Omega)$. Given $f\; \backslash in\; L^2(\backslash Omega;\; \backslash mathbb\{R\}^2)$ the latter result implies the existence of $\backslash tau\; \backslash in\; H^1(\backslash Omega;\; \backslash mathbb\{R\}^\{2\; \backslash times\; 2\})$ with $\backslash operatorname\{div\}\; \backslash tau\; =\; f$. This $\backslash tau$ may not be symmetric. However, for any $v\; \backslash in\; H^1(\backslash Omega;\; \backslash mathbb\{R\}^2)$, we let $\backslash operatorname\{curl\}\; v$ denote the matrix with $i$th row equal to $\backslash operatorname\{curl\}\; v\_i\; =\; (-\backslash partial\; v\_i\; /\; \backslash partial\; x\_2,\; \backslash partial\; v\_i\; /\; \backslash partial\; x\_1)$. Then $\backslash operatorname\{div\}(\backslash tau\; +\; \backslash operatorname\{curl\}\; v)\; =\; f$ and $\backslash tau\; +\; \backslash operatorname\{curl\}\; v\; \backslash in\; H(\backslash operatorname\{div\};\; \backslash Omega;\; \backslash mathbb\{S\})$ if $\backslash operatorname\{div\}\; v\; =\; \backslash tau\_\{12\}\; -\; \backslash tau\_\{21\}$. Thus $\backslash operatorname\{div\}\; :\; H(\backslash operatorname\{div\};\; \backslash Omega;\; \backslash mathbb\{S\})\; \backslash rightarrow\; L^2(\backslash Omega;\; \backslash mathbb\{R\}^2)$ is indeed surjective. Setting $W^0\; =\; L^2(\backslash Omega)$, $W^1\; =\; L^2(\backslash Omega;\; \backslash mathbb\{S\})$, and $W^2\; =\; L^2(\backslash Omega;\; \backslash mathbb\{R\}^2)$, we obtain a closed Hilbert complex with domain complex (87).

Now we describe only the simplest discretization from [11]. The space $V\_h^0\; \backslash subset\; V^0\; =\; H^2(\backslash Omega)$ is the finite element space whose shape functions on each triangle $T$ are the quintic polynomials $\backslash mathcal\{P\}\_5(T)$ and whose degrees of freedom are the values, first, and second derivatives at each vertex, together with the mean value of the normal derivatives along each edge, which gives $21\; =\; \backslash dim\; \backslash mathcal\{P\}\_5(T)$ degrees of freedom per triangle. The resulting finite element space $V\_h^0$ is referred to in the finite element literature as the Argyris space, and is easily seen to be contained in $C^1(\backslash Omega)$, and hence in $H^2(\backslash Omega)$. In fact, the Argyris space is in a certain sense the simplest finite element space contained in $C^1(\backslash Omega)$. 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 $V\_h^1\; \backslash subset\; V^1\; =\; H(\backslash operatorname\{div\};\; \backslash Omega;\; \backslash mathbb\{S\})$ consists of those continuous piecewise cubic symmetric matrix fields whose divergence is piecewise linear. The degrees of freedom determining an element $\backslash sigma\; \backslash in\; V\_h^1$ are

Hence, the restriction of $\backslash sigma$ to a triangle $T\; \backslash in\; \backslash mathcal\{T\}\_h$ is uniquely determined by 24 degrees of freedom. Finally, the space $V\_h^2\; \backslash subset\; V^2\; =\; L^2(\backslash Omega;\; \backslash mathbb\{R\}^2)$ consists of all piecewise linear vector fields with respect to $\backslash mathcal\{T\}\_h$, 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 $I\_h^k\; v\; \backslash in\; V\_h^k$ defined for sufficiently smooth $v\; \backslash in\; V^k$ by $\backslash phi(I\_h^k\; v)\; =\; \backslash phi(v)$ for all degrees of freedom $\backslash phi$.

It is straightforward to check that we have the subcomplex property $JV\_h^0\; \backslash subset\; V\_h^1$ and $\backslash operatorname\{div\}\; V\_h^1\; \backslash subset\; V\_h^2$ and that the interpolation operators $I\_h\; :=\; I\_h^k$ give a commuting diagram

However, since $I\_h^0$ involves point values of the second derivative and $I\_h^1$ 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 $I\_h^k$ in (88) to obtain bounded cochain projections $\backslash pi\_h^k\; :\; W^k\; \backslash rightarrow\; V\_h^k$ for $k\; =\; 0,\; 1,\; 2$. As in Section 5.5, the $\backslash pi\_h^k$ will be obtained from $I\_h^k$ 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 $F\; :\; \backslash Omega\; \backslash rightarrow\; \backslash Omega\text{'}\; \backslash subset\; \backslash mathbb\{R\}^2$ be an affine map, so that $DF\; =\; DF(x)$ is a constant $2\; \backslash times\; 2$ matrix. Let $B$ be the matrix $B\; =\; \backslash det(DF)(DF)^\{-1\}\; =\; O(DF)^T\; O^T$ and define pullbacks $F^*\; =\; F\_k^*\; :\; W^k(\backslash Omega\text{'})\; \backslash rightarrow\; W^k(\backslash Omega)$ by

It is straightforward to check that, as long as $F$ is affine, $F^*$ commutes with the differential operators $J$ and $\backslash operatorname\{div\}$. In other words, we obtain the commuting diagram

To define a smoothing operator which maps $W^k$ into itself, we need to make the assumption that the domain $\backslash Omega$ 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 $a\; =\; \backslash max\_\{x\; \backslash in\; \backslash partial\backslash Omega\}\; |x|^\{-1\}$ and define the dilation $F\_\backslash delta\; :\; \backslash mathbb\{R\}^2\; \backslash rightarrow\; \backslash mathbb\{R\}^2$ by $F\_\backslash delta(x)\; =\; x/(1+a\backslash delta)$, $\backslash delta\; >\; 0$. Then $F\_\backslash delta$ maps the $\backslash delta$-neighborhood of $\backslash Omega$, $\backslash Omega\_\backslash delta$, into $\backslash Omega$ and so $F\_\backslash delta^*$ maps $L^2(\backslash Omega)$ to $L^2(\backslash Omega\_\backslash delta)$. Composing with a standard mollification

we obtain a smoothing operator $R\_\backslash delta$ mapping $W^k$ into $W^k$. Here $\backslash rho\; :\; \backslash mathbb\{R\}^2\; \backslash rightarrow\; \backslash mathbb\{R\}$ 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 $\backslash text\{div\}$.

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 $\backslash epsilon\; >\; 0$ be fixed. Define operators $Q\_h^k\; :\; W^k\; \backslash rightarrow\; W\_h^k$ by $Q\_h^k\; =\; I\_h^k\; \backslash circ\; R\_\{\backslash epsilon\; h\}$. The operator $Q\_h^k$ maps $W^k$ into $V\_h^k$. In fact, it can be seen by a scaling argument, that the operators $Q\_h^k$ are bounded as operators in $\backslash mathcal\{L\}(W^k,\; W\_h^k)$, uniformly in $h$. Furthermore, $Q\_h^k$ commutes with $J$ and $\backslash text\{div\}$, i.e.,

However, the operators $Q\_h^k$ are not projections, since they do not restrict to the identity on $V\_h^k$. Therefore, we define the desired projections $\backslash pi\_h^k\; :\; W^k\; \backslash rightarrow\; V\_h^k$ as $\backslash pi\_h^k\; =\; (Q\_h^k|\_\{V\_h^k\})^\{-1\}\; Q\_h^k$. To justify this definition, we need to argue that $(Q\_h^k|\_\{V\_h^k\})^\{-1\}$ exists. In fact, for $\backslash epsilon$ sufficiently small, but fixed, $(Q\_h^k|\_\{V\_h^k\})^\{-1\}$ is even uniformly bounded in $\backslash mathcal\{L\}(W\_h^k,\; W\_h^k)$, where $W\_h^k\; =\; V\_h^k$, but equipped with the norm of $W^k$. This follows from the fact that the operators $Q\_h^k|\_\{V\_h^k\}$ converge to the identity with $\backslash epsilon$ in $\backslash mathcal\{L\}(W\_h^k,\; W\_h^k)$, uniformly in $h$, established by a scaling argument.

We conclude that if the parameter $\backslash epsilon\; >\; 0$ is fixed, but sufficiently small, then the operators $\backslash pi\_h^k$ are well-defined and uniformly bounded with respect to $h$ in $\backslash mathcal\{L\}(W^k,\; W\_h^k)$. Furthermore, the diagram

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 $V\_h^1$ and $V\_h^2$, 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 $V\_h^1$ and $V\_h^2$ 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.

SCHOOL OF MATHEMATICS, UNIVERSITY OF MINNESOTA, MINNEAPOLIS, MN 55455

E-mail address: arnold@umn.edu

DEPARTMENT OF MATHEMATICS, RUTGERS UNIVERSITY, PISCATAWAY, NJ 08854

E-mail address: falk@math.rutgers.edu

CENTRE OF MATHEMATICS FOR APPLICATIONS AND DEPARTMENT OF INFORMATICS, UNIVERSITY OF OSLO, 0316 OSLO, NORWAY

E-mail address: ragnar.winther@cma.uio.no