Daniele Boffi
email: daniele.boffi@unipv.it

Martin Costabel
email: martin.costabel@univ-rennes1.fr

Monique Dauge
email: monique.dauge@univ-rennes1.fr

Leszek Demkowicz
email: leszek@ices.utexas.edu

Ralf Hiptmair
email: hiptmair@sam.math.ethz.ch

Discrete compactness for the p-version of discrete differential forms

come   L'archive ouverte pluridisciplinaire

Introduction: Maxwell eigenvalue problem

Maxwell's eigenvalue problem in a closed cavity Ω ∈ R 3 with perfectly conducting walls can be written as follows by means of the Maxwell-Ampère and Faraday laws: Find the 1 resonance frequencies ω ∈ R and the electromagnetic fields (E, H) = (0, 0) such that

curl E = iωµH and curl H = -iωǫE in Ω E × n = 0 and H • n = 0 on ∂Ω, (1.1)
where ǫ and µ denote the dielectric permittivity and magnetic permeability, respectively.

The fields E and H are sought in L 2 (Ω) 3 .

For simplicity, we consider the case of homogeneous isotropic material with normalized material constants (ǫ, µ = 1) -we will come back to the general setting in Remark 6. [START_REF] Arnold | Differential complexes and numerical stability[END_REF]. In a classical way, the elimination of the magnetic field from equations (1.1) yields the Maxwell eigenvalue problem with perfectly electrically conducting (PEC) walls in variational form:

Seek u ∈ • H(curl, Ω) \ {0}, ω ∈ R + 0 such that (curl u, curl v) L 2 (Ω) = ω 2 (u, v) L 2 (Ω) ∀v ∈ • H(curl, Ω) .
(1.

2)

The elimination of the electric field would correspond to the same problem modelled through replacing

• H(curl, Ω) with H(curl, Ω) 1 . One aim of this paper is to prove the convergence of H(curl)-conforming Galerkin discretizations of Maxwell eigenvalue problem (1.2) in the framework of the p-version of the finite element method. The finite element approximation of Maxwell eigenvalues has been the object of intense investigations for more than 20 years. It was soon recognized that the H(curl)-conforming Galerkin finite element discretizations need special finite element spaces that are generally termed edge finite elements (see [START_REF] Élec | Mixed finite elements in R 3[END_REF][START_REF]A new family of mixed finite elements in R 3[END_REF][START_REF] Bossavit | Whitney forms: A class of finite elements for three-dimensional computations in electromagnetism[END_REF]).

The first attempts to analyze the discretized eigenvalue problem have been made for the h-version of edge finite elements. We mention [START_REF] Kikuchi | On a discrete compactness property for the Nédélec finite elements[END_REF] as a pioneering work on lowest order edge finite elements, where the discrete compactness property (see [START_REF] Anselone | Collectively Compact Operator Approximation Theory and Applications to Integral Equations[END_REF]) has been indicated as a key ingredient for the analysis. Other relevant works are [START_REF] Boffi | Computational models of electromagnetic resonators: Analysis of edge element approximation[END_REF][START_REF] Boffi | Fortin operator and discrete compactness for edge elements[END_REF][START_REF] Caorsi | On the convergence of Galerkin finite element approximations of electromagnetic eigenproblems[END_REF][START_REF] Monk | Discrete compactness and the approximation of Maxwell's equations in R 3[END_REF][START_REF]Theoretical analysis of Nédélec's edge elements[END_REF][START_REF]Computation of resonance frequencies for Maxwell equations in non-smooth domains[END_REF][START_REF]Approximation of eigenvalues in mixed form, discrete compactness property, and application to hp mixed finite elements[END_REF], and we refer the interested reader to [START_REF]Finite elements in computational electromagnetism[END_REF][START_REF] Monk | Finite Element Methods for Maxwell's Equations[END_REF] and to the references therein for a review on this topic.

In these references, the Maxwell eigenvalue problem is often studied using variational formulations different from (1.2), for example mixed formulations [START_REF]Approximation of eigenvalues in mixed form, discrete compactness property, and application to hp mixed finite elements[END_REF], regularized formulations [START_REF] Costabel | Weighted regularization of Maxwell equations in polyhedral domains[END_REF][START_REF] Costabel | Exponential convergence of hp-FEM for Maxwell's equations with weighted regularization in polygonal domains[END_REF] or mixed regularized formulations [START_REF]Finite element exterior calculus: from Hodge theory to numerical stability[END_REF][START_REF] Buffa | Solving electromagnetic eigenvalue problems in polyhedral domains with nodal finite elements[END_REF]. With the exception of the method of weighted regularization [START_REF] Costabel | Weighted regularization of Maxwell equations in polyhedral domains[END_REF][START_REF] Costabel | Exponential convergence of hp-FEM for Maxwell's equations with weighted regularization in polygonal domains[END_REF][START_REF] Buffa | Solving electromagnetic eigenvalue problems in polyhedral domains with nodal finite elements[END_REF], where H 1 -conforming elements can be used, these formulations use the H(curl)-conforming edge elements. In their analysis, special conditions implying convergence of the discrete eigenvalue problems are presented, for example the so-called Fortid property [START_REF] Boffi | Fortin operator and discrete compactness for edge elements[END_REF], or the GAP property [START_REF] Buffa | Remarks on the discretization of some non-positive operators with application to heterogeneous Maxwell problems[END_REF]. As explained there, these conditions are related to the discrete compactness property. Here we choose to work with the simple variational formulation (1.2) and its generalization to differential forms. The role of the discrete compactness property in this context has been discussed in detail in [START_REF] Caorsi | On the convergence of Galerkin finite element approximations of electromagnetic eigenproblems[END_REF].

The analysis presented in the references above covers the h-version for basically all known families of edge finite elements. It soon turned out, however, that the analysis of the pand hp-versions of edge finite elements needed tools different from those developed for the h-version. In [START_REF] Boffi | Discrete compactness for p and hp 2D edge finite elements[END_REF] the two-dimensional triangular case has been studied for the hp-version, but the analysis depends on a conjectured estimate which has only been demonstrated numerically. In [START_REF] Boffi | Discrete compactness for the hp version of rectangular edge finite elements[END_REF] a rigorous proof for the hp-version of 2D rectangular edge elements has been proposed (allowing for one-irregular hanging nodes) which, in particular, contains the first proof of eigenvalue/eigenfunction convergence for the pure spectral method (p-version with one element) on a rectangle.

What paved the way for a successful attack on a general p-version analysis was the regularized Poincaré lifting recently introduced in [START_REF] Costabel | On Bogovskiȋ and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains[END_REF]: it enjoys excellent continuity properties and at the same time respects discrete differential forms. In this paper we are going to show how the regularized Poincaré lifting can be combined with another recent invention, the projection based interpolation operators, see [START_REF] Demkowicz | Polynomial exact sequences and projection-based interpolation with applications to Maxwell equations[END_REF][START_REF] Demkowicz | H 1 , H(curl), and H(div)-conforming projection-based interpolation in three dimensions. Quasi-optimal p-interpolation estimates[END_REF], to clinch the analysis of the p-version of edge elements. This allows to prove the discrete compactness (and hence the convergence of the discrete eigensolutions) for a wide class of finite elements related to discrete differential forms: for (1.2) this includes, in particular, Nédélec elements on triangles and tetrahedra (first and second kind) and on parallelograms and parallelepipeds (first kind).

As already mentioned, one of the key ingredients for the convergence analysis is the discrete compactness property. Much insight can be gained from investigating it in the more general framework of discrete differential forms (see [START_REF] Arnold | Finite element exterior calculus, homological techniques, and applications[END_REF] for a lucid introduction to this subject). In this setting, the proofs are more natural and simultaneously cover, in particular, two-and three-dimensional Maxwell eigenvalue problems.

Plan of the paper. The structure of the paper is as follows. We start in Section 2 with a generalization of (1.2) to eigenvalue problems associated with the de Rham complex on differential forms. Then we define the discrete compactness property and discuss its significance in the context of Galerkin discretization: in association with two standard completeness properties, it gives a crucial sufficient condition for the convergence of eigenvalues and eigenvectors. Section 3 is the core of our paper and contains the description of our abstract assumptions. Having in mind the p-version of finite elements, we consider a fixed mesh M of a bounded Lipschitz polyhedron Ω ⊂ R d and a sequence of spaces of discrete differential forms of order ℓ (with 0 < ℓ < d) together with projection operators onto discrete spaces; we prove that our assumptions imply the validity of the discrete compactness property for such a sequence of spaces (Theorem 3.2). The abstract theory relies on the existence of suitable Poincaré lifting operators which are presented in Section 4. The mapping properties of these lifting operators allow to specify some of the function spaces appearing in our abstract assumptions. In Section 5 we recall the classical families of discrete differential forms with high degree polynomial coefficients on simplicial or tensor product elements.

Our abstract theory applies to any dimension d, but for want of suitable regularity results, embeddings, and projection operators, we can give examples satisfying all of its assumptions only in dimensions d = 2 and d = 3. This is done in Section 6, where we concretize the function spaces and recall embedding results and properties of projection based interpolation operators related to these spaces. All abstract assumptions are then satisfied, leading to the main convergence result stated in Theorem 6.1. The analysis of a p-version edge element discretization of the Maxwell eigenvalue problem (1.2) is covered as case d = 3 and ℓ = 1, see Corollary 6.2.

Differential forms and generalized Maxwell eigenvalue problem

The variational eigenvalue problem (1.2) turns out to be a member of a larger family of eigenvalue problems, when viewed from the perspective of differential forms. This more general perspective offers the benefit of a unified theoretical treatment of different kinds of eigenvalue problems, e.g., the scalar Laplace eigenproblem, Maxwell cavity eigenproblems in dimensions 2 and 3, the eigenproblem for the grad div-operator in dimension 3. This policy has had remarkable success in numerical analysis recently, cf. [START_REF] Arnold | Differential complexes and numerical stability[END_REF]. Thus, in this section we first recall some basic notions related to differential forms. We refer the interested reader to [START_REF] Arnold | Finite element exterior calculus, homological techniques, and applications[END_REF]Sect. 2] for an introduction to this subject.

Function spaces of differential forms

Given a bounded Lipschitz domain Ω ⊂ R d , we denote by C ∞ (Ω, Λ ℓ ), 0 ≤ ℓ ≤ d, the space of smooth differential forms of degree ℓ on Ω and by

d ℓ : C ∞ (Ω, Λ ℓ ) → C ∞ (Ω, Λ ℓ+1
) the exterior derivative. We rely on the Hilbert spaces

H(d ℓ , Ω) := {v ∈ L 2 (Ω, Λ ℓ ) : d ℓ v ∈ L 2 (Ω, Λ ℓ+1 )} , (2.1)
where L 2 (Ω, Λ ℓ ) is the space of differential ℓ-forms on Ω with square integrable coefficients in their canonical basis representation, see [START_REF] Costabel | On Bogovskiȋ and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains[END_REF]Sect. 2]. Its inner product can be expressed as

(u, v) 0,Ω := Ω u ∧ ⋆ v , u, v ∈ L 2 (Ω, Λ ℓ ) , (2.2)
with ⋆ the Hodge star operator induced by the Euclidean metric on R d , which maps ℓforms to (dℓ)-forms. As above, a • tags the subspaces of forms with vanishing trace tr ∂Ω on ∂Ω, which can also be obtained by the completion of compactly supported smooth ℓ-forms with respect to the H(d ℓ , Ω)-norm:

• H(d ℓ , Ω) := {v ∈ H(d ℓ , Ω) : tr ∂Ω v = 0}. (2.
3)

The subspace of closed forms is the kernel of d ℓ and is denoted by

• H(d ℓ 0, Ω): • H(d ℓ 0, Ω) := {v ∈ • H(d ℓ , Ω) : d ℓ v = 0}. (2.4)

Variational eigenvalue problems

After choosing bases for the spaces of alternating multilinear forms on R d , vector fields ("vector proxies") Ω → R ( d ℓ ) provide an isomorphic model for differential ℓ-forms on Ω. Choosing the standard "Euclidean basis", the operators ⋆, δ, tr ∂Ω are incarnated by familiar operators of classical vector analysis, different for different dimension d and degree ℓ, see Table 1 and [4, Table 2.1].

Table 1: Identification between (operators on) differential forms and (operators on) Euclidean vector proxies in R 2 and R 3

Differential form

Proxy representation

d = 2 d = 3 ℓ = 0 d 0 grad grad tr ∂Ω φ φ |∂Ω φ |∂Ω • H(d 0 , Ω) • H 1 (Ω) • H 1 (Ω) ℓ = 1 d 1 curl curl tr ∂Ω u (u × n) |∂Ω (u × n) |∂Ω • H(d 1 , Ω) • H(curl, Ω) • H(curl, Ω) ℓ = 2 d 2 0 div tr ∂Ω q 0 (q • n) |∂Ω • H(d 2 , Ω) L 2 (Ω) • H(div, Ω)
Hence, the eigenvalue problem (1.2) with ǫ, µ ≡ 1 is the special case d = 3, ℓ = 1, of the following variational eigenvalue problem for differential ℓ-forms, 0 ≤ ℓ < d:

Seek u ∈ • H(d ℓ , Ω) \ {0}, ω ∈ R + 0 , such that (d ℓ u, d ℓ v) 0,Ω = ω 2 (u, v) 0,Ω ∀v ∈ • H(d ℓ , Ω) .
(2.5)

A key observation is that the bilinear form (u, v) → (d ℓ u, d ℓ v) 0,Ω has an infinite dimensional kernel

• H(d ℓ 0, Ω) comprising all closed ℓ-forms. It provides the invariant subspace associated with the essential spectrum {0} of (2.5). This essential spectrum can be identified as the main source of difficulties confronted in the Galerkin discretization of (2.5).

On the other hand, any solution u of (2.5) for ω = 0 satisfies (u, d ℓ-1 ψ) 0,Ω = 0 for all

ψ ∈ • H(d ℓ-1 , Ω).
Thus the eigenfunctions corresponding to non-zero eigenvalues belong to the subspace

• Y (d ℓ , Ω) := {v ∈ • H(d ℓ , Ω) : (v, d ℓ-1 ψ) 0,Ω = 0 ∀ψ ∈ • H(d ℓ-1 , Ω)}, (2.6)
which means they belong to the kernel of δ ℓ . This is the generalization of the divergence free constraint found for electric fields in the Maxwell case. From [START_REF] Picard | An elementary proof for a compact imbedding result in generalized electromagnetic theory[END_REF] we learn the following theorem.

Theorem 2.1 For any d ∈ N, 0 ≤ l ≤ d, the embedding of • Y (d ℓ , Ω) in L 2 (Ω, Λ ℓ ) is compact.
Thus, by restricting the eigenvalue problem to • Y (d ℓ , Ω), we can use Riesz-Schauder theory. This implies that (2.5) gives rise to an unbounded sequence of positive eigenvalues

λ k = (ω k ) 2 λ 0 = 0 < λ 1 ≤ λ 2 ≤ . . . , λ k → ∞ (k → ∞) , (2.7)
with associated finite dimensional mutually L 2 (Ω)-orthogonal eigenspaces.

Remark 2.2 Owing to the zero trace boundary conditions imposed on the functions in (2.5), it may be called a Dirichlet eigenvalue problem. Using H(d ℓ , Ω) as variational space would result in the corresponding Neumann eigenvalue problem. Its analysis runs utterly parallel to the Dirichlet case using the techniques presented below.

Approximation of the eigenvalue problem and the role of discrete compactness

In the sequel we fix the degree ℓ, 0 ≤ ℓ < d, of the differential forms. Spaces of discrete differential forms

• V ℓ p ⊂ • H(d ℓ , Ω) , dim • V ℓ p < ∞ ,
lend themselves to a straightforward discretization of (2.5). In this section, p ∈ N stands for an abstract discretization parameter, and, sloppily speaking, large values of p hint at trial/test spaces of high resolution.

Consider the approximation of the eigenvalue problem (2.5) by the Galerkin method:

Find u p ∈ • V ℓ p \ {0}, ω ∈ R + 0 , such that (d ℓ u p , d ℓ v p ) 0,Ω = ω 2 (u p , v p ) 0,Ω ∀v ∈ • V ℓ p .
(2.8)

Now, the key issue is convergence of eigenvalues and eigenvectors as p → ∞, rigorously cast into the concept of spectrally correct, spurious-free approximation [START_REF] Caorsi | On the convergence of Galerkin finite element approximations of electromagnetic eigenproblems[END_REF]Sect. 4]. Let us recall these notions in a few words for the case of self-adjoint nonnegative operators without continuous spectrum (which is the case here).

The spectral correctness of the approximation of an eigenvalue problem such as (2.5) by a sequence of finite rank eigenvalue problems (2.8) means that all eigenvalues and all eigenvectors of (2.5) are approached by the eigenvalues and eigenvectors of (2.8) as p → ∞. If (2.5) has a compact resolvent (which is the case only when ℓ = 0), the spectral correctness is an optimal notion: It implies that if {λ k } k≥1 and {λ k p } k≥1 are the increasing eigenvalue sequences of (2.5) and (2.8) (with eigenvalues repeated according to their multiplicities), then

λ k p → λ k as p → ∞ ∀k ≥ 1, (2.9)
and the gaps between eigenspaces (correctly assembled according to multiplicities of the eigenvalues of (2.5)) tend to 0 as p → ∞.

If we face an eigenvalue problem for a self-adjoint non-negative operator with an infinite dimensional kernel, and otherwise discrete positive spectrum (which is the case for (2.5) for all ℓ ≥ 1), the spectral correctness implies the same properties as above with the following modifications of the definitions: Now {λ k } k≥1 is the increasing sequence of positive eigenvalues of (2.5) (as specified in (2.7)) and, given a positive number ε < λ 1 , {λ k p } k≥1 is the increasing sequence of the eigenvalues of (2.8) larger than ε (still with repetitions according to multiplicities). With such conventions, spectral correctness still implies convergence of eigenvalues (2.9) and eigenspaces as above. In this context, spurious-free approximation means that there exists ε 0 > 0 such that all eigenvalues of (2.8) less than ε 0 are zero. Therefore, spectrally correct, spurious-free approximation implies the convergence property (2.9) and the corresponding convergence of eigenspaces, if we define {λ k p } k≥1 as the increasing sequence of the positive eigenvalues of (2.8). There exist several different ways, all well studied and summarized in the literature of the last decade, for proving the convergence of the discrete eigenvalue problem (2.8) to the continuous eigenvalue problem (2.5): One can use a reformulation as an eigenvalue problem in mixed form as analyzed in [START_REF]Approximation of eigenvalues in mixed form, discrete compactness property, and application to hp mixed finite elements[END_REF], or one can use a regularization which gives an elliptic eigenvalue problem for the Hodge-Laplace operator as analyzed in [START_REF] Arnold | Finite element exterior calculus, homological techniques, and applications[END_REF], or one can follow the arguments of [START_REF] Caorsi | On the convergence of Galerkin finite element approximations of electromagnetic eigenproblems[END_REF] and study the non-elliptic problem (2.5) directly.

Here we outline the latter approach, which employs the analysis of [START_REF] Descloux | On spectral approximation. Part I. The problem of convergence[END_REF] of the approximation of eigenvalue problems of non-compact selfadjoint operators. Since [START_REF] Caorsi | On the convergence of Galerkin finite element approximations of electromagnetic eigenproblems[END_REF] deals only with the Maxwell case, i. e. d = 3, ℓ = 1, we examine the main arguments, in order to verify that they are also valid for the general case. The proofs we give are adaptations of those of [START_REF] Caorsi | On the convergence of Galerkin finite element approximations of electromagnetic eigenproblems[END_REF] to our more general situation.

Let us define the solution operator A :

L 2 (Ω, Λ ℓ ) → • H(d ℓ , Ω)
of the source problem corresponding to the eigenvalue problem (2.5) and its discrete counterpart A p :

• V ℓ p → • V ℓ p by (d ℓ Af , d ℓ v) 0,Ω + (Af, v) 0,Ω = (f, v) 0,Ω ∀v ∈ • H(d ℓ , Ω) (d ℓ A p f, d ℓ v) 0,Ω + (A p f, v) 0,Ω = (f, v) 0,Ω ∀v ∈ • V ℓ p .
(2.10)

Note that the operators A and A p have the same eigenfunctions and the same eigenvalues (after a transformation) as the eigenvalue problems (2.5) and (2.8). Namely, (2.5) and (2.8) are equivalent to the relations

u = (ω 2 + 1)Au ; u p = (ω 2 + 1)A p u p . (2.11)
The infinite-dimensional eigenspace at ω = 0 shows that A is not a compact operator. Following [START_REF] Caorsi | On the convergence of Galerkin finite element approximations of electromagnetic eigenproblems[END_REF], three conditions are identified that together are necessary and sufficient for a spectrally correct, spurious-free approximation of A by A p or, equivalently, of the eigenvalue problem (2.5) by the discrete eigenvalue problem (2.8).

The first condition is rather natural. It states that the sequence of discrete spaces

• V ℓ p p∈N is asymptotically dense in • H(d ℓ , Ω) (compare [19, Condition (CAS) -com- pleteness of approximating subspaces]) (CAS) lim p→∞ inf vp∈ • V ℓ p v -v p H(d ℓ ,Ω) = 0 ∀v ∈ • H(d ℓ , Ω) .
(2.12)

The second condition, only relevant for ℓ > 0, states that closed forms can be well approximated by discrete closed forms (compare [19, Condition (CDK) -completeness of discrete kernels])

(CDK) lim p→∞ inf zp∈ • V ℓ p ∩ • H(d ℓ 0,Ω) z -z p L 2 (Ω) = 0 ∀z ∈ • H(d ℓ 0, Ω) . (2.13)
The third condition is the most intricate one and has been dubbed discrete compactness.

For its formulation, we introduce the orthogonal complement space of the discrete closed forms:

• Z ℓ p := {u p ∈ • V ℓ p : (u p , z p ) 0,Ω = 0 ∀z p ∈ • V ℓ p ∩ • H(d ℓ 0, Ω)}. ( 2
u p p∈N ′ ⊂ • H(d ℓ , Ω) with u p ∈ • Z ℓ p contains a subsequence that converges in L 2 (Ω, Λ ℓ ).
The convergence proof is based on two lemmas, the first of which corresponds to [START_REF] Caorsi | On the convergence of Galerkin finite element approximations of electromagnetic eigenproblems[END_REF]Theorem 4.12]. It implies, according to [32, Condition P1) and Theorems 2,4,5,6], the spectral correctness of the approximation. Lemma 2.4 If (2.12) and the discrete compactness property hold, then

lim p→∞ sup vp∈ • V ℓ p ; vp H(d ℓ ,Ω) =1 Av p -A p v p H(d ℓ ,Ω) = 0 . (2.15) Proof. Note first that for v p ∈ • V ℓ p ∩ • H(d ℓ 0, Ω) there holds Av p = v p = A p v p , so that by orthogonal decomposition of • V ℓ p one gets sup vp∈ • V ℓ p ; vp H(d ℓ ,Ω) =1 Av p -A p v p H(d ℓ ,Ω) = sup vp∈ • Z ℓ p ; vp H(d ℓ ,Ω) =1 Av p -A p v p H(d ℓ ,Ω) .
Furthermore, one has by definition of A and

A p Av p -A p v p H(d ℓ ,Ω) = inf wp∈ • V ℓ p Av p -w p H(d ℓ ,Ω) .
Assume now that (2.15) does not hold. Then there exists ε > 0, a subsequence N ′ of N and a sequence

(v p ) p∈N ′ with v p ∈ • Z ℓ p satisfying v p H(d ℓ ,Ω) = 1 and Av p -w p H(d ℓ ,Ω) ≥ ε ∀p ∈ N ′ , w p ∈ • V ℓ p . (2.16)
We can apply the discrete compactness property to the sequence (v p ) and obtain a subsequence converging in

L 2 (Ω, Λ ℓ ) to some v ∈ L 2 (Ω, Λ ℓ ). Since A : L 2 (Ω, Λ ℓ ) → • H(d ℓ , Ω) is continuous, we find Av ∈ • H(d ℓ , Ω)
, and the approximation property (2.12) provides us with a sequence (w p ) with w p ∈

• V ℓ p that converges in • H(d ℓ , Ω) to Av.
Hence for the subsequence we obtain

Av p -w p H(d ℓ ,Ω) ≤ Av p -Av H(d ℓ ,Ω) + Av -w p H(d ℓ ,Ω) → 0 , in contradiction with (2.16).
2 The second lemma corresponds to [START_REF] Caorsi | On the convergence of Galerkin finite element approximations of electromagnetic eigenproblems[END_REF]Corollary 2.20]. It gives the discrete Friedrichs inequality (in [START_REF]Approximation of eigenvalues in mixed form, discrete compactness property, and application to hp mixed finite elements[END_REF] also called "ellipticity in the discrete kernel"), and it is easy to see that this implies that ω = 0 is not a limit point of positive discrete eigenvalues, so that the spurious-free property of the approximation follows.

Lemma 2.5 If (2.13) and the discrete compactness property hold, then there exists α > 0 such that for all p ∈ N

d ℓ v L 2 (Ω) ≥ α v L 2 (Ω) ∀v ∈ • Z ℓ p (2.17)
Proof. Assume that (2.17) does not hold. Then there exists a subsequence N ′ of N and a sequence

(v p ) p∈N ′ with v p ∈ • Z ℓ p satisfying v p L 2 (Ω) = 1 and lim p→∞ d ℓ v p L 2 (Ω) = 0 . (2.18)
The discrete compactness property can be applied to this sequence and gives a subsequence converging in L 2 (Ω, Λ ℓ ) to some z ∈ L 2 (Ω, Λ ℓ ). From (2.18) follows that the convergence actually takes place in

• H(d ℓ , Ω) and that z ∈ • H(d ℓ 0, Ω).
Therefore the approximation property (2.13) provides us with a sequence

(z p ) with z p ∈ • V ℓ p ∩ • H(d ℓ 0, Ω) that converges in L 2 (Ω, Λ ℓ ) to z.
Hence for the subsequence we find

v p -z p L 2 (Ω) ≤ v p -z L 2 (Ω) + z -z p L 2 (Ω) → 0 . But v p ∈ • Z ℓ p and z p ∈ • V ℓ p ∩ • H(d ℓ 0, Ω) are L 2 (Ω)-orthogonal, hence for all p v p -z p 2 L 2 (Ω) = v p 2 L 2 (Ω) + z p 2 L 2 (Ω) ≥ 1 , which leads to a contradiction.
2 To summarize, Lemmas 2.4 and 2.5 together prove the following result.

Theorem 2.6

If the completeness of approximating subspaces (2.12), the completeness of discrete kernels (2.13) and the discrete compactness property hold, then (2.8) provides a spectrally correct, spurious-free approximation of the eigenvalue problem (2.5).

Remark 2.7

The main focus of this section is on the convergence of the eigenvalues and the eigenfunctions of problem (2.8) to those of (2.5). On the other hand, when considering concrete applications it is crucial to investigate the order of convergence. In order to do so, several strategies are available. A straightforward approach which well fits the theory summarized in this section makes use of the results from [START_REF]On spectral approximation. Part II. Error estimates for the Galerkin method[END_REF]. Theorem 1 of [START_REF]On spectral approximation. Part II. Error estimates for the Galerkin method[END_REF] states in this particular situation that the error in the eigenfunctions (measured as usual by the gap of Hilbert spaces) is bounded by the best approximation, and Theorem 3(c) of [START_REF]On spectral approximation. Part II. Error estimates for the Galerkin method[END_REF] states that the eigenvalues achieve double order of convergence since our problem is symmetric. An alternative approach makes use of the equivalence of problems (2.5) and (2.8) with suitable mixed formulations [START_REF]Finite element approximation of eigenvalue problems[END_REF]Part 4]; in this case an estimate of the order of convergence can be achieved by the standard Babuška-Osborn theory for the spectral approximation of compact operators applied to the mixed formulations [10, Theorems 13.8, 13.10, 14.9, 14.11].

Abstract framework implying discrete compactness

In this section we fix a degree of differential forms ℓ ∈ {1, . . . , d -1}, and we formulate a set of hypotheses which allow us to prove the discrete compactness property. These hypotheses are organized in three groups:

1. standard assumptions related to the finite element spaces

• V ℓ p (Sect. 3.1),
2. assumptions on the existence and key properties of "lifting operators" (Sect. 3.3),

hypotheses on projections onto

• V ℓ p complying with the commuting diagram property and satisfying an approximation property (Sect. 3.4).

To state these assumptions we have to introduce intermediate spaces X and S of more regular forms

• V ℓ p ⊂ X(M, Λ ℓ ) ⊂ • H(d ℓ , Ω) and • V ℓ-1 p ⊂ S(M, Λ ℓ-1 ) ⊂ • H(d ℓ-1 , Ω) ,
allowing compact embedding arguments and precise notions of continuity of lifting and projection operators.

Discrete spaces

Our focus is on finite element spaces. For the sake of simplicity, we restrict ourselves to polyhedral Lipschitz domains Ω. We assume that the finite dimensional trial and test spaces • V ℓ p , p ∈ N, are based on a fixed finite partition M of Ω, composed of elements (cells) K:

Ω = K∈M K , K ∩ K ′ = ∅ , if K = K ′ , K, K ′ ∈ M .
For a cell K ∈ M, let F m (K) designate the set of m-dimensional facets of K: for m = 0 these are the vertices, for m = 1 the edges, for m = d -1 the faces, and

F d (K) = {K}.
We take for granted that the discrete spaces

•

V ℓ p can be assembled from local contributions in the sense that for each mesh cell K ∈ M there is a space

V ℓ p (K) ⊂ C ∞ (K, Λ ℓ ) of smooth ℓ-forms on K, such that • V ℓ p = • V ℓ p (M) := v ∈ • H(d ℓ , Ω) : v K ∈ V ℓ p (K) ∀K ∈ M . (3.1)
In other words,

• V ℓ p can be defined by specifying the local spaces V ℓ p (K) and requiring the continuity of traces across inter-element boundaries as well as boundary conditions on ∂Ω.

In the same fashion, we introduce a corresponding family

• V ℓ-1 p ⊂ • H(d ℓ-1
, Ω) of spaces of discrete (ℓ -1)-forms. We will see later on that as a consequence of further hypotheses, the local spaces V ℓ-1 p (K) and V ℓ p (K) satisfy an exact sequence property.

Spaces of more regular forms

We introduce a Hilbert space X(M, Λ ℓ ) ⊂

• H(d ℓ , Ω) that captures the extra regularity that distinguishes ℓ-forms in the space • Y (d ℓ , Ω). We can think of this space as a space of "more regular" ℓ-forms on Ω.

Assumption 1

The space

• Y (d ℓ , Ω) defined in (2.6) is continuously embedded in X(M, Λ ℓ ).
This means that with C > 0 depending only on Ω

u X(M,Λ ℓ ) ≤ C u H(d ℓ ,Ω) ∀u ∈ • Y (d ℓ , Ω) . (3.2)
On the other hand, X(M, Λ ℓ ) has to be small enough to maintain the compact embedding satisfied by

• Y (d ℓ , Ω), cf. Thm. 2.1. Assumption 2 The space X(M, Λ ℓ ) is compactly embedded in L 2 (Ω, Λ ℓ ).
As with the discrete spaces, the spaces X(M, Λ ℓ ) are built from local contributions and will therefore depend on the mesh M. We assume that for each mesh cell K ∈ M there are Hilbert spaces X(K, Λ ℓ ) so that:

X(M, Λ ℓ ) = v ∈ • H(d ℓ , Ω) : v K ∈ X(K, Λ ℓ ) ∀K ∈ M , (3.3)
and, in addition, the norm of X(M, Λ ℓ ) is defined through local contributions:

u 2 X(M,Λ ℓ ) = u 2 H(d ℓ ,Ω) + K∈M u K 2 X(K,Λ ℓ ) . (3.4)
Finally, the local spaces have to be large enough to contain the discrete forms for any value of p:

V ℓ p (K) ⊂ X(K, Λ ℓ ) . (3.5)
Correspondingly, we introduce a space S(M, Λ ℓ-1 ) ⊂

• H(d ℓ-1 , Ω) of "more regular potentials". Similar to X(M, Λ ℓ ), the spaces S(M, Λ ℓ-1 ) are mesh-dependent and allow for a characterization through local Hilbert spaces S(K, Λ ℓ-1 ), K ∈ M,

S(M, Λ ℓ-1 ) = ψ ∈ • H(d ℓ-1 , Ω) : ψ K ∈ S(K, Λ ℓ-1 ) ∀K ∈ M . (3.6)
They are endowed with the norm

φ 2 S(M,Λ ℓ-1 ) = φ 2 H(d ℓ-1 ,Ω) + K∈M φ K 2 S(K,Λ ℓ-1 ) . (3.7)
The local spaces are large enough to contain the local discrete potential spaces:

V ℓ-1 p (K) ⊂ S(K, Λ ℓ-1 ) . (3.8)
The following assumption establishes the connection between X(M, Λ ℓ ) and S(M, Λ ℓ-1 ).

Assumption 3

The exterior derivative maps S(M, Λ ℓ-1 ) continuously into X(M, Λ ℓ ):

S(M, Λ ℓ-1 ) ⊂ {φ ∈ • H(d ℓ-1 , Ω) : d ℓ-1 φ ∈ X(M, Λ ℓ )},
and the image is maximal:

d ℓ-1 S(M, Λ ℓ-1 ) = d ℓ-1 • H(d ℓ-1 , Ω) ∩ X(M, Λ ℓ ).
To conclude this subsection, note that in the case of an element K touching the boundary ∂Ω, like for the discrete spaces V ℓ p (K) and V ℓ-1 p (K), the local spaces X(K, Λ ℓ ) and S(K, Λ ℓ-1 ) are not obliged to comply with any boundary conditions.

Local liftings

A pair of linear mappings R k,K : C ∞ (K, Λ k ) → C ∞ (K, Λ k-1 ), k = ℓ, ℓ + 1, is called a lifting operator of degree ℓ if it fulfills d ℓ-1 • R ℓ,K + R ℓ+1,K • d ℓ = Id ℓ .
(3.9)

This relation characterizes a "contracting homotopy" of the de Rham complex [5, Section 5.1.2]. Besides this algebraic relationship, our approach hinges on smoothing properties of the lifting operators, expressed by means of the local spaces S(K, Λ ℓ-1 ) of more regular potentials and X(K, Λ ℓ ) of more regular forms. The next assumption summarizes the continuity expected from the lifting operator. Assumption 4 For every K ∈ M there is a lifting operator (R ℓ,K , R ℓ+1,K ) whose components can be extended to continuous mappings

R ℓ+1,K : L 2 (K, Λ ℓ+1 ) → X(K, Λ ℓ ) and R ℓ,K : X(K, Λ ℓ ) → S(K, Λ ℓ-1 ) ,
and thus identity (3.9) holds on X(K, Λ ℓ ).

As a consequence, for each cell K ∈ M, we have the exact sequence

S(K, Λ ℓ-1 ) d ℓ-1 ---→ X(K, Λ ℓ ) d ℓ ---→ L 2 (K, Λ ℓ+1 ).
(3.10)

Finally, the local liftings have to be compatible with the local spaces of discrete differential forms: Assumption 5 The local operators R ℓ+1,K , when applied to exact local discrete (ℓ + 1)forms, yield local discrete ℓ-forms, i.e.,

R ℓ+1,K • d ℓ : V ℓ p (K) → V ℓ p (K) .

Local projectors

As usual in methods based on discrete commuting diagrams we need projection operators π k p,K onto discrete spaces for (ℓ -1)-forms and ℓ-forms. For degree ℓ -1, our local spaces S(K, Λ ℓ-1 ) of more regular potentials can play the role of domains for the projectors π ℓ-1 p,K . For the degree ℓ, by generalization of what we actually need in the case of dimension d = 2 and d = 3 for Maxwell, we define our projectors π ℓ p,K on smaller spaces than X(K, Λ ℓ ). We denote these new spaces by S(K, Λ ℓ ) and require that they contain for all p the p-dependent subspaces

X p (K, Λ ℓ ) = {u ∈ X(K, Λ ℓ ) : d ℓ u ∈ d ℓ V ℓ p (K)} . (3.11)
On the same model as (3.6)-(3.7), we define the corresponding global spaces S(M, Λ ℓ ) and

X p (M, Λ ℓ ) = {u ∈ X(M, Λ ℓ ) : d ℓ u ∈ d ℓ V ℓ p } (3.12)
and we have the continuous embeddings

X p (M, Λ ℓ ) ֒→ S(M, Λ ℓ ) ֒→ X(M, Λ ℓ ) . ( 3

.13)

Assumption 6 There are local continuous linear projections π ℓ-1 p,K : S(K, Λ ℓ-1 ) → V ℓ-1 p (K) and π ℓ p,K : S(K, Λ ℓ ) → V ℓ p (K) for all mesh cells K ∈ M.

The standard commuting diagram property is as follows.

Assumption 7

The projectors π ℓ-1 p,K and π ℓ p,K are compatible with the exterior derivative in the sense that the diagram

S(K, Λ ℓ-1 ) d ℓ-1 ---→ S(K, Λ ℓ ) π ℓ-1 p,K     π ℓ p,K V ℓ-1 p (K) d ℓ-1 ---→ V ℓ p (K) , commutes for every K ∈ M.
Let us note that, as a consequence of Assumptions 4 and 7, we find that the sequence

V ℓ-1 p (K) d ℓ-1 ---→ V ℓ p (K) d ℓ ---→ d ℓ V ℓ p (K) is exact.
Besides, the local projections acting on (ℓ -1)-forms are supposed to enjoy a crucial approximation property using the Hilbert space norms • S(K,Λ ℓ-1 ) .

Assumption 8 There is a function ε

ℓ-1 : N → R + with lim p→∞ ε ℓ-1 (p) = 0 so that d ℓ-1 (φ -π ℓ-1 p,K φ) L 2 (K,Λ ℓ ) ≤ ε ℓ-1 (p) φ S(K,Λ ℓ-1 ) ∀φ ∈ S(K, Λ ℓ-1
) .

Finally we assume for the projections π ℓ p,K a natural condition of conformity: For all u ∈ X p (K, Λ ℓ )

tr F u = 0 ⇒ tr F π ℓ p,K u = 0 ∀F ∈ F m (K) , ℓ ≤ m ≤ d , (3.14)
and the corresponding condition for the projections π ℓ-1 p,K . This makes it possible to define global linear projections

π ℓ p : S(M, Λ ℓ ) → • V ℓ p and π ℓ-1 p : S(M, Λ ℓ ) → • V ℓ-1 p
by patching together the local operators

(π ℓ p u) K := π ℓ p,K (u K ) and (π ℓ-1 p φ) K := π ℓ-1 p,K (φ K ) ∀K ∈ M . ( 3
S(M, Λ ℓ-1 ) d ℓ-1 ---→ S(M, Λ ℓ ) π ℓ-1 p     π ℓ p • V ℓ-1 p d ℓ-1 ---→ • V ℓ p .
(3.16)

Proof of the discrete compactness property

The estimate of Assumption 8 on "potentials" carries over to ℓ-forms with a discrete exterior derivative, that is, the elements of the space X p (M, Λ ℓ ), see (3.12).

Lemma 3.1 (Global projection error estimate) Making Assumptions 4 through 8, the estimate

u -π ℓ p u L 2 (Ω,Λ ℓ ) ≤ Cε ℓ-1 (p) u X(M,Λ ℓ ) ∀u ∈ X p (M, Λ ℓ ) holds true, with a constant C > 0 independent of p.
Proof. Pick any u ∈ X p (M, Λ ℓ ). The locality of the projector π ℓ p , cf. (3.15), and (3.4) allow purely local considerations. Single out one cell K ∈ M, still write u = u K ∈ X p (K, Λ ℓ ), and split u on K using (3.9) from Assumption 4:

u = d ℓ-1 R ℓ,K u + R ℓ+1,K d ℓ u = d ℓ-1 φ + R ℓ+1,K d ℓ u .
(3.17)

with φ := R ℓ,K u. The continuity of R ℓ,K from Assumption 4 reveals that

φ S(K,Λ ℓ-1 ) ≤ C u X(K,Λ ℓ ) , (3.18)
where here and below C will denote constants (possibly different at different occurrences) which depend neither on u nor on p. Thanks to identity (3.17) and the commuting diagram property from Assumption 7, we have

π ℓ p,K u = d ℓ-1 π ℓ-1 p,K φ + π ℓ p,K R ℓ+1,K d ℓ u . (3.19)
Recall that u ∈ X p (K, Λ ℓ ) belongs to the domain of π ℓ p,K by Assumption 6. Further, as u ∈ X p (K, Λ ℓ ), from Assumption 5 we infer that

R ℓ+1,K d ℓ u ∈ V ℓ p (K) . (3.20)
Thus, owing to the identities (3.17), (3.19) and the projector property of π ℓ p,K , the task is reduced to an interpolation estimate for π ℓ-1 p,K :

(Id -π ℓ p,K )u = d ℓ-1 (Id -π ℓ-1 p,K )φ + (Id -π ℓ p,K )R ℓ+1,K d ℓ u
=0 by (3.20) .

(3.21)

As a consequence, invoking Assumption 8,

(Id -π ℓ p,K )u L 2 (K,Λ ℓ ) (3.21) = d ℓ-1 (Id -π ℓ-1 p,K )φ L 2 (K,Λ ℓ ) ≤ ε ℓ-1 (p) φ S(K,Λ ℓ-1 ) (3.18) ≤ Cε ℓ-1 (p) u X(K,Λ ℓ ) , (3.22)
which furnishes a local version of the estimate. This estimate is uniform in K ∈ M because M is finite. Due to (3.4), squaring (3.22) and summing over all cells finishes the proof. 2 We are now in the position to prove the main result of this section.

Theorem 3.2 (Discrete compactness) Under Assumptions 1 through 8, the discrete compactness property of Definition 2.3 holds for the family

• V ℓ p p∈N of subspaces of • H(d ℓ , Ω).
Proof. The proof resorts to the "standard policy" for tackling the problem of discrete compactness, introduced by Kikuchi [START_REF] Kikuchi | On a discrete compactness property for the Nédélec finite elements[END_REF][START_REF]Theoretical analysis of Nédélec's edge elements[END_REF] for analyzing the h-version of Whitney-1forms. It forms the core of most papers considering the issue of discrete compactness, see [START_REF] Boffi | Discrete compactness for p and hp 2D edge finite elements[END_REF]Thm. 2], [START_REF] Boffi | Discrete compactness for the hp version of rectangular edge finite elements[END_REF]Thm. 11], [START_REF]Finite elements in computational electromagnetism[END_REF]Thm. 4.9], [31, Thm. 2], etc.

Let us introduce the discrete analogue of the space

• Y (d ℓ , Ω): • Y ℓ p := {v p ∈ • V ℓ p : v p , d ℓ-1 ψ p 0,Ω = 0 ∀ψ p ∈ • V ℓ-1 p }. (3.23)
The space

(i) u p ∈ • V ℓ p , (3.24)
(ii)

u p , d ℓ-1 ψ p 0,Ω = 0 ∀ψ p ∈ • V ℓ-1 p , (3.25) (iii) u p H(d ℓ ,Ω) ≤ 1 ∀p ∈ N ′ . (3.26)
We have to confirm that it possesses a subsequence that converges in L 2 (Ω, Λ ℓ ).

We start with the

L 2 (Ω, Λ ℓ )-orthogonal projection of u p into • Y (d ℓ , Ω) and parallel to d ℓ-1 • H(d ℓ-1 , Ω): let u p be the unique vector field in • H(d ℓ , Ω) with u p = u p + d ℓ-1 φ p , φ p ∈ • H(d ℓ-1 , Ω) , (3.27) ( u p , d ℓ-1 ψ) 0,Ω = 0 ∀ψ ∈ • H(d ℓ-1 , Ω) . (3.28)
Obviously, the latter condition implies

u p ∈ • Y (d ℓ , Ω) . (3.29)
Hence, by virtue of Assumption 1, the fact that d ℓ u p = d ℓ u p , and (3.12), u p satisfies

u p ∈ X p (M, Λ ℓ ), u p X(M,Λ ℓ ) ≤ C u p H(d ℓ ,Ω) , (3.30)
where C > 0 does not depend on p. Since d ℓ-1 φ p = u pu p ∈ X(M, Λ ℓ ), Assumption 3 implies that we may assume that φ p ∈ S(M, Λ ℓ-1 ).

Thus we can use Nédélec's trick [START_REF] Élec | Mixed finite elements in R 3[END_REF] to obtain

u p -u p 2 L 2 (Ω,Λ ℓ ) = u p -u p , u p -π ℓ p u p + π ℓ p u p -u p 0,Ω = u p -u p , u p -π ℓ p u p 0,Ω . (3.31)
This holds because from (3.27) and the projector property of π ℓ p we know

π ℓ p u p -u p = π ℓ p u p + π ℓ p d ℓ-1 φ p -u p = π ℓ p d ℓ-1 φ p ,
u p -u p , π ℓ p u p -u p 0,Ω = u p -u p , d ℓ-1 π ℓ-1 p φ p 0,Ω = 0 . (3.32)
Hence, appealing to Lemma 3.1, with C > 0 independent of p, we get

u p -u p L 2 (Ω,Λ ℓ ) ≤ u p -π ℓ p u p L 2 (Ω,Λ ℓ ) ≤ Cε ℓ-1 (p) u p X(M,Λ ℓ ) (3.30) ≤ Cε ℓ-1 (p) u p X(M,Λ ℓ ) → 0 for p → ∞ . (3.33)
From (3.30) we conclude that the sequence ( u p ) p∈N ′ is uniformly bounded in X(M, Λ ℓ ). By Assumption 2 it has a convergent subsequence in L 2 (Ω, Λ ℓ ). Owing to (3.33), the same subsequence of (u p ) p∈N ′ will converge in L 2 (Ω, Λ ℓ ). 2

Approximation of the eigenvalue problem

As discussed in Section 2.3, the discrete compactness property is the cornerstone of the proof of the convergence of the discrete generalized Maxwell eigenvalue problem (2.8).

Corollary 3.3

In addition to the hypotheses of Theorem 3.2, namely Assumptions 1 to 8, assume that property (CAS) (2.12) holds and that the space

X(M, Λ ℓ ) ∩ • H(d ℓ 0, Ω) is dense in • H(d ℓ 0, Ω). Then (2.

8) provides a spectrally correct, spurious-free approximation of the eigenvalue problem (2.5).

Proof. We use Theorem 2.6 from Section 2.3. Considering that the discrete compactness property is provided by Theorem 3.2, and that we assume the approximation property (CAS) (2.12), we only need to show the approximation property (CDK) (2.13), which concerns the approximation of closed forms by closed discrete forms.

Since we assumed the density of

X(M, Λ ℓ ) ∩ • H(d ℓ 0, Ω) in • H(d ℓ 0, Ω), it is sufficient to prove (CDK) for z ∈ X(M, Λ ℓ ) ∩ • H(d ℓ 0, Ω). Such z belongs to X p (M, Λ ℓ
), and we can therefore apply Lemma 3.1, which shows that π ℓ p z → z in L 2 (Ω, Λ ℓ ). We will have accomplished to show (CDK) with z p = π ℓ p z, as soon as we show that

d ℓ z p = 0. Keeping in mind that z p ∈ • V ℓ p ⊂ H(d ℓ , Ω)
, we see that it is sufficient to show the local relation d ℓ z p = 0 in K for every K ∈ M. This follows finally as in (3.19) in the proof of Lemma 3.1, because d ℓ z = 0 implies

π ℓ p,K z = d ℓ-1 π ℓ-1 p,K R ℓ,K z . Hence d ℓ z p = d ℓ π ℓ p,K z = d ℓ d ℓ-1 π ℓ-1 p,K R ℓ,K z = 0, which ends the proof. 2
Remark 3. [START_REF] Arnold | Finite element exterior calculus, homological techniques, and applications[END_REF] The abstract theory developed in this section can be applied to the h-version of discrete differential forms, if the dependence of the constants on the size of the cell K is made explicit by means of scaling arguments. Here, we forgo this extra technicality and refer the reader to [START_REF]Finite elements in computational electromagnetism[END_REF]Sect. 4.4].

Regularized Poincaré lifting

In this section we describe the construction of a local lifting operator R ℓ that will satisfy Assumptions 4 and 5 in Section 3.3 for suitable spaces X(K, Λ ℓ ), S(K, Λ ℓ ) and V ℓ p (K). We follow the presentation in [START_REF] Costabel | On Bogovskiȋ and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains[END_REF], where these operators are analyzed and where it is shown in particular that they are pseudodifferential operators of order -1.

Definition

We consider a bounded domain D ⊂ R d that is star-shaped with respect to some subdomain B ⊂ D, that is,

∀a ∈ B, x ∈ D : {(1 -t)a + tx, 0 < t < 1} ⊂ D . (4.1)
For a ∈ B and 1 ≤ ℓ ≤ d, we define the Poincaré operator R ℓ,a , acting on a differential form u ∈ C ∞ (D, Λ ℓ ), by the path integral R ℓ,a u(x) = (xa)

1 0 t ℓ-1 u a + t(x -a) dt , x ∈ D . (4.2)
Here the symbol denotes the contraction (also called "interior product") of the vector field x → (x-a) with the ℓ-form u.

It is clear that R ℓ,a maps C ∞ (D, Λ ℓ ) to C ∞ (D, Λ ℓ-1 )
and it has been shown (see [START_REF] Demkowicz | Quasioptimality of some spectral mixed methods[END_REF] for proofs in the case d = 3) that it can be extended to a bounded operator from L 2 (D, Λ ℓ ) to L 2 (D, Λ ℓ-1 ). In order to define the regularized Poincaré operator R ℓ , we choose a function

θ ∈ C ∞ 0 (R d ) , supp θ ⊂ B , B θ(a) da = 1 ,
and set

R ℓ u(x) = B θ(a)R ℓ,a u(x) da . (4.3)

Regularity

The substitution y = a + t(xa), τ = 1/(1t) transforms the double integral in (4.2), (

R ℓ u(x) = R d ∞ 1 (τ -1) ℓ-1 τ d-ℓ θ x + τ (y -x) (x -y) u(y) dτ dy = R d k(y, y -x) u(y) dy , 4.3) into
where the kernel k(y, z) has an expansion into quasi-homogeneous terms:

k(y, z) = -z ∞ 0 s ℓ-1 (s + 1) d-ℓ θ y + sz ds = - d-ℓ j=0 d-ℓ j z |z| d-j ∞ 0 r d-j-1 θ y + r z |z| dr . (4.5)
The operator R ℓ is therefore a weakly singular integral operator. In [26, Section 3.3], the following result is shown.

Proposition 4.1 For 1 ≤ ℓ ≤ d, the operator R ℓ is a pseudodifferential operator of order -1 on R d . It is well defined on C ∞ (D, Λ ℓ ), it maps C ∞ (D, Λ ℓ ) to C ∞ (D, Λ ℓ-1 ) and C ∞ (D, Λ ℓ ) to C ∞ (D, Λ ℓ-1
), and for any s ∈ R it has an extension as a bounded operator

R ℓ : H s (D, Λ ℓ ) → H s+1 (D, Λ ℓ-1 )) .
Here, H s (D, Λ ℓ ) is the Sobolev space of ℓ-forms on D of order s.

Lifting property

The lifting property (3.9) is a consequence of the following identity, which is a special case of "Cartan's magic formula" for Lie derivatives and for a flow field generated by the dilations with center a.

d dt (t ℓ u a + t(x -a) = d ℓ-1 t ℓ-1 (x -a) u a + t(x -a) + t ℓ (x -a) d ℓ u a + t(x -a) (4.6)
Here u is an ℓ-form. The result is

d ℓ-1 R ℓ u + R ℓ+1 d ℓ u = u (1 ≤ ℓ ≤ d -1) ; R 1 d 0 u = u -θ, u 0,D (ℓ = 0) ; d d-1 R d u = u (ℓ = d) . (4.7)
These relations are valid for all u ∈ C ∞ 0 (R d , Λ ℓ ) and by extension for all u ∈ H s (D, Λ ℓ ), s ∈ R.

The perfect match of (4.7) with (3.9) from Assumption 4 suggests that the regularized Poincaré lifting R ℓ provides suitable local liftings as stipulated in Assumption 4. To this end, we can choose as local spaces of "more regular forms"

X(K, Λ ℓ ) := H(d ℓ , K) ∩ H r (K, Λ ℓ ) , S(K, Λ ℓ-1 ) := H r (d ℓ-1 , K) and S(K, Λ ℓ ) := H r (d ℓ , K) , (4.8)
for some 0 < r ≤ 1, where we denote by H r (d k , K) the space

H r (d k , K) := {v ∈ H r (K, Λ k ) : d k v ∈ H r (K, Λ k+1 )} .
All these spaces are equipped with the natural Hilbert space norms. Also keep in mind that the global spaces X(M, Λ ℓ ), S(M, Λ ℓ-1 ) and S(M, Λ ℓ ) are determined by their local definition on the mesh cells K, cf. (3.3) and (3.6). For the particular choice (4.8) an assumption of Corollary 3.3 can be verified.

Lemma 4.2 For X(M, Λ ℓ ) arising from (4.8) the space

X(M, Λ ℓ ) ∩ • H(d ℓ 0, Ω) is dense in • H(d ℓ 0, Ω).
Proof. By [26, Thm. 4.9(c)] we have a direct decomposition

• H(d ℓ 0, Ω) = d ℓ-1 • H 1 (Ω, Λ ℓ-1 ) ⊕ C ℓ , C ℓ ⊂ C ∞ Ω (R n , Λ ℓ ) , (4.9)
where C ∞ Ω (R d , Λ ℓ ) is the space of compactly supported, smooth ℓ-forms on R d with support contained in Ω or, equivalently, the space of all smooth ℓ-forms on Ω that vanish on ∂Ω together with all their derivatives. Since

C ∞ Ω (R d , Λ ℓ-1 ) is dense in • H 1 (Ω, Λ ℓ-1
), we deduce:

C ∞ Ω (R d , Λ ℓ ) ∩ d ℓ-1 • H 1 (Ω, Λ ℓ-1 ) is dense in d ℓ-1 • H 1 (Ω, Λ ℓ-1 )
As every u ∈ C ∞ Ω (R d , Λ ℓ ) belongs to X(M, Λ ℓ ), the assertion follows. 2 We point out that the choice of r in (4.8) is determined by Assumption 1. Also note that whenever we opt for (4.8), Rellich's theorem ensures Assumption 2, because the mesh is kept fixed.

The construction of R ℓ entails a constraint on the cell shapes. This is satisfied for standard finite element meshes, where the cells usually are convex polyhedra. Assumption 9 Every cell K ∈ M is a star-shaped polyhedron. Lemma 4.3 Assumption 9, the choice (4.8) for spaces X(K, Λ ℓ ) and S(K, Λ ℓ-1 ) imply Assumptions 2, 3 and 4.

Proof. The only fact remaining to be proved is the maximality relation in Assumption

3 d ℓ-1 S(M, Λ ℓ-1 ) = d ℓ-1 • H(d ℓ-1 , Ω) ∩ X(M, Λ ℓ ).
The inclusion ⊂ holds by definition. Let us prove the converse inclusion. [START_REF] Costabel | On Bogovskiȋ and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains[END_REF]Cor. 4.7] we obtain that there exists ψ ∈ • H 1 (Ω, Λ ℓ-1 ) such that u = d ℓ-1 ψ. In particular, ψ K belongs to H r (K, Λ ℓ-1 ) for all K and, since u K belongs to H r (K, Λ ℓ ), we finally find that ψ K ∈ H r (d ℓ-1 , K). 2

Let u ∈ d ℓ-1 • H(d ℓ-1 , Ω) ∩ X(M, Λ ℓ ). Thus u = d ℓ-1 φ with φ ∈ • H(d ℓ-1 , Ω). Since u ∈ L 2 (Ω, Λ ℓ ), using

Preservation of polynomial forms

Fundamental in finite element methods is the notion of polynomial differential forms. For an ordered ℓ-tuple I = (i 1 , . . . , i ℓ ), i 1 < i 2 < . . . < i ℓ , {i 1 , . . . , i ℓ } ⊂ {1, . . . , d}, let

dx I := dx i 1 ∧ • • • ∧ dx i ℓ ,
where dx j , j = 1, . . . , d, are the co-ordinate 1-forms in Euclidean space R d . The space P p (Λ ℓ ) of polynomial ℓ-forms on R d is defined as

P p (Λ ℓ ) := u = I u I dx I : u I ∈ P p (R d ) ,
where I indicates summation over all ordered ℓ-tuples, and P p (R d ) is the space of dvariate polynomials of total degree ≤ p. We remark that for d ∈ {2, 3} polynomial forms possess polynomial vector proxies.

From the definition (4.2) it is clear that the Poincaré operator R ℓ,a maps differential forms with polynomial coefficients to differential forms with polynomial coefficients. The same holds for the regularized Poincaré operator R ℓ by (4.3). If we want R ℓ to map a space P (Λ ℓ ) of differential forms of order ℓ (e.g., with polynomial coefficients) into a space P (Λ ℓ-1 ) of differential forms of order ℓ -1, it is sufficient to require the following two properties, see [START_REF] Costabel | On Bogovskiȋ and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains[END_REF]Proposition 4.2]. Proposition 4.4 Assume that P (Λ ℓ ) and P (Λ ℓ-1 ) are finite-dimensional spaces of differential forms satisfying (i) The space P (Λ ℓ ) is invariant with respect to dilations and translations, that is For any t ∈ R, a ∈ R n : if u ∈ P (Λ ℓ ), then x → u(tx + a) ∈ P (Λ ℓ ) .

(ii) The interior product x : u → x u maps P (Λ ℓ ) to P (Λ ℓ-1 ). Then R ℓ maps P (Λ ℓ ) into P (Λ ℓ-1 ).

edge element projections and can be referred to as moment based projection operators. They are suitable for the h-version of finite elements in dimensions 2 and 3. In higher dimensions some of them (for low degree forms) require a higher regularity than H 2 to be defined. In [START_REF] Arnold | Finite element exterior calculus, homological techniques, and applications[END_REF][START_REF]Finite element exterior calculus: from Hodge theory to numerical stability[END_REF], they are modified by an extension-regularization procedure in order to be defined and bounded on L 2 . However, such operators cannot be used for the p version of finite elements, because no estimates (stability or error bounds) are known with respect to the polynomial degree p.

The proper projection operators for p-version approximation are so-called projection based interpolation operators, see [START_REF] Demkowicz | p interpolation error estimates for edge finite elements of variable order in two dimensions[END_REF][START_REF] Demkowicz | H 1 , H(curl), and H(div)-conforming projection-based interpolation in three dimensions. Quasi-optimal p-interpolation estimates[END_REF][START_REF] Cao | Optimal error estimate of a projection based interpolation for the p-version approximation in three dimensions[END_REF][START_REF] Demkowicz | Polynomial exact sequences and projection-based interpolation with applications to Maxwell equations[END_REF][START_REF] Demkowicz | Projection-based interpolation and automatic hpadaptivity for finite element discretizations of elliptic and Maxwell problems[END_REF]. Variants for any ℓ and d are available and they are designed to commute in the sense of Assumption 7 [START_REF]Finite elements in computational electromagnetism[END_REF]Sect. 3.5].

At this point we have to abandon the framework of general ℓ and d, because both regularity results (Assumption 1) and the analysis of projection operators (Assumption 8) are not presently available for general ℓ and d. We have to discuss them for special choices of ℓ and d separately, relying on a wide array of sophisticated results from the literature. Theorem 6.1 (Convergence of Galerkin approximations) For d = 2, 3, and 0 ≤ ℓ < d, the Galerkin discretization of (2.5) on a Lipschitz polyhedron based on any of the families of discrete differential forms introduced in Sect. 5 offers a spectrally correct, spurious-free approximation.

Proof. We skip the case ℓ = 0, for which the standard Galerkin approximation theory for operators with compact resolvent can be applied, see [START_REF] Knyazev | New a priori FEM error estimates for eigenvalues[END_REF].

To begin with, we focus on the discrete compactness property and verify the assumptions 1, 6, 7, and 8 for d = 2 and d = 3 separately. • d = 2, ℓ = 1: in terms of vector proxies we find the correspondence

• Y (d 1 , Ω) ∼ • H(curl, Ω) ∩ H(div 0, Ω) . (6.1)
Regularity theorems for boundary value problems for -∆ on the polygon confirm the existence of δ = δ(Ω) > 0 such that

• H(curl, Ω) ∩ H(div 0, Ω) ⊂ H δ+1/2 (Ω) , (6.2)
in the sense of continuous embedding, see [START_REF] Girault | Finite element methods for Navier-Stokes equations[END_REF]Sect. 3.2]. This suggests to choose r = δ + 1/2 in (4.8) and Assumption 1 will hold true. Hence, we deal with the concrete spaces

X(M, Λ 1 ) = • H(curl, Ω) ∩ K∈M (H δ+1/2 (K)) 2 , (6.3) S(M, Λ 0 ) = • H 1 (Ω) ∩ K∈M H δ+3/2 (K) . (6.4)
Commuting local projection based interpolation operators π 1 p,K and π 0 p,K have been proposed for triangles and for quadrilaterals in [START_REF] Demkowicz | p interpolation error estimates for edge finite elements of variable order in two dimensions[END_REF]. With the choice (6.3) and (6.4) they live up to Assumptions 6 and 7. Assumption 8 holds with ε 0 (p) = Cp -1/2 and C > 0 depending only on the shape-regularity of the cells, cf. [START_REF] Demkowicz | Polynomial exact sequences and projection-based interpolation with applications to Maxwell equations[END_REF]Thm. 4.3] and [START_REF] Bespalov | Optimal error estimation for H(curl)conforming p-interpolation in two dimensions[END_REF]Thm. 4.1]. Finally, these interpolation operators satisfy the natural condition of conformity (3.14) by construction, which makes they meet all our requirements, cf. Sect. 3.4.

• d = 3, ℓ = 1, 2: we have the vector proxy incarnation

• Y (d ℓ , Ω) ∼ • H(curl, Ω) ∩ H(div 0, Ω) for ℓ = 1 , • H(div, Ω) ∩ H(curl 0, Ω) for ℓ = 2 . ( 6.5)
Citing results from [START_REF] Amrouche | Vector potentials in three-dimensional nonsmooth domains[END_REF] and [START_REF]Finite elements in computational electromagnetism[END_REF]Sect. 4.1], we find δ = δ(Ω) ∈]0, 1 2 ] and continuous embeddings

• H(curl, Ω) ∩ H(div, Ω), • H(div, Ω) ∩ H(curl, Ω) ⊂ H δ+1/2 (Ω) . (6.6)
Therefore, using the construction (4.8) with r = δ + 1/2, Assumption 1 is satisfied for ℓ ∈ {1, 2}. The relevant spaces of more regular forms now read

X(M, Λ 1 ) = • H(curl, Ω) ∩ K∈M (H δ+1/2 (K)) 3 , (6.7)
X(M, Λ 2 ) = • H(div, Ω) ∩ K∈M (H δ+1/2 (K)) 3 , (6.8)
S(M, Λ 0 ) = • H 1 (Ω) ∩ K∈M H δ+3/2 (K) , (6.9
)

S(M, Λ 1 ) = • H(curl, Ω) ∩ K∈M H δ+1/2 (curl, K) . (6.10)
The essential commuting local projection based interpolation operators π m p,K , m = 0, 1, 2, have been introduced in [START_REF] Demkowicz | H 1 , H(curl), and H(div)-conforming projection-based interpolation in three dimensions. Quasi-optimal p-interpolation estimates[END_REF] for tetrahedral meshes and in [START_REF] Demkowicz | p interpolation error estimates for edge finite elements of variable order in two dimensions[END_REF] for meshes comprising parallelepipeds. By construction they comply with Assumption 7. Assumption 6 for the spaces S(M, Λ 0 ) and S(M, Λ 1 ) from (6.9) and (6.10), respectively, and r = δ + 1/2 is a consequence of Sobolev embedding theorems. Relying on [START_REF] Demkowicz | Polynomial exact sequences and projection-based interpolation with applications to Maxwell equations[END_REF]Th.5.3] we obtain like in the 2D case that in Lemma 3.1 we can take ε m (p) = Cp -1/2 for m = 0 and m = 1.

• Finally, we appeal to Lemmas 5.

Remark 6.4

The restriction on the families of elements mentioned in the Corollary is essentially due to the availability of published results about suitable interpolation operators. Thus, for example, as soon as a generalization of the p version error estimates of [START_REF] Demkowicz | H 1 , H(curl), and H(div)-conforming projection-based interpolation in three dimensions. Quasi-optimal p-interpolation estimates[END_REF][START_REF] Demkowicz | Polynomial exact sequences and projection-based interpolation with applications to Maxwell equations[END_REF] for projection-based interpolants to meshes containing prismatic or more general polyhedral elements becomes available, our result about the approximation of the Maxwell eigenvalue problem will apply to such meshes, too. Remark 6.5 Several obstacles prevent us from establishing the assumptions of the abstract theory for d > 3. On the one hand, continuity properties of projection based interpolation operators have not been investigated for d > 3. Also, regularity results along the lines of (6.6) are have not been published for polyhedra in higher dimensions.

On the other hand, the innocuously looking requirement (3.14) for the projection operators -corresponding to the requirement that the global projection operators are constructed elementwise from local degrees of freedom -entails that the trace of forms in S(K, Λ ℓ-1 ) onto ℓ-1-dimensional facets in F ℓ-1 (K) must make sense. However, we cannot expect more than H 2 regularity for the space S(K, Λ ℓ-1 ). Hence, by trace theorems for Sobolev spaces, the spaces S(K, Λ ℓ-1 ) allow for traces on m-facets for m > d 2 -2 at best, which means that ℓ > d 2 -1 is required to allow for the construction of a local projection based interpolation complying with Assumptions 6 and 8. Perhaps, an analysis in L p -spaces as in [START_REF] Amrouche | Vector potentials in three-dimensional nonsmooth domains[END_REF]Lemma 4.7] can make possible an extension of the theory to higher dimensions, but this is beyond the scope of the present article. Remark 6.6 Our approach does not cover hp-refinement, for various reasons. One reason is that there exist many variants of hp refinements in 3D, and covering them would in any case require a much longer paper than the present one. Another reason is technical: The existing convergence proof of the hp approximation of the Maxwell eigenvalue problem in [START_REF] Boffi | Discrete compactness for the hp version of rectangular edge finite elements[END_REF] -while also based on the proof of the discrete compactness property -uses a different technical tool, namely an estimate of the L 2 stability of a certain projection operator. This kind of estimate is currently only available for intervals in 1D and for rectangles in 2D. The technique used in the present paper is based on the regularized Poincaré lifting, and adjusting this to variable polynomial degree poses formidable technical challenges. Only in 2D these could be mastered so far, as was demonstrated in [START_REF] Bespalov | Convergence of hp-BEM for the electric field integral equation on polyhedral surfaces[END_REF] in the context of boundary element analysis.

Conclusion

In this paper we have proved that the p-version of finite elements based on generalized Nédélec edge elements provides a spurious-free spectrally correct approximation of the Maxwell eigenvalue problem. The essential point was the proof of the discrete compactness property. We showed that this property follows from a set of rather natural assumptions about the family of finite element spaces and interpolation operators, and in addition we showed that these assumptions are implied by recently found results on lifting operators and on projection-based interpolants.

In the approach pursued in [START_REF] Arnold | Finite element exterior calculus, homological techniques, and applications[END_REF][START_REF]Finite element exterior calculus: from Hodge theory to numerical stability[END_REF] the discrete compactness property is not addressed directly: in the framework of the h-version for differential forms, modified momentbased projection operators are used. These new operators satisfy the strong property of being uniformly bounded in L 2 and are constructed by means of a delicate extensionregularization procedure, see also [START_REF] Christiansen | Smoothed projections in finite element exterior calculus[END_REF][START_REF] Christiansen | Foundations of finite element methods for wave equations of Maxwell type[END_REF].

On the one hand this uniform boundedness property is stronger than our assumption 6 and replaces in a certain way the discrete compactness property. But on the other hand, it is currently not known whether a construction of projection operators by extensionregularization could also be employed in the case of the p-version of finite elements, or whether the construction of a p-uniformly L 2 -bounded family of cochain projections is even possible.

. 14 ) 2 . 3

 1423 Definition Let us choose ℓ ∈ {1, . . . , d-1}. The discrete compactness property holds for a family • V ℓ p p∈N of finite dimensional subspaces of • H(d ℓ , Ω), if for any subsequence N ′ of N, any bounded sequence

. 15 ) 1 p

 151 As a consequence of Assumption 7 and (3.15), the global projectors π ℓ-and π ℓ p inherit the global commuting diagram property

  and thanks to the commuting diagram property (3.16) (deduced from Assumption 7) combined with the orthogonality conditions (3.25) and (3.28), we find

  We consider a subsequence N ′ of N and a H(d ℓ , Ω)-bounded sequence (u p ) p∈N ′ with members in

	• Y ℓ p contains	• Z ℓ p as a subspace.

• Z ℓ p . Thus u p belongs in particular to • Y ℓ p and the sequence (u p ) p∈N ′ satisfies

  1, 5.2 together with Lemma 4.2 and apply the abstract theory of Sect. 3 in the form of Corollary 3.3 to conclude the proof of the theorem.2 \ {0}, ω ∈ R + 0 such that µ -1 curl u, curl v L 2 (Ω) = ω 2 (ǫu, v) L 2 (Ω) ∀v ∈with uniformly positive material tensors µ = µ(x), ǫ = ǫ(x). The same edge element discretizations listed in Corollary 6.2 provide spectrally correct, spurious-free approximations of this problem. This generalization of Corollary 6.2 can be achieved with standard tools (see, in particular, Propositions 2.25, 2.26, and 2.27 of[START_REF] Caorsi | On the convergence of Galerkin finite element approximations of electromagnetic eigenproblems[END_REF], and[START_REF]Discrete compactness for p-version of tetrahedral edge elements[END_REF] Sect. 6],[START_REF]Finite elements in computational electromagnetism[END_REF] Thm. 4.9]).

	Remark 6.3 Instead of (1.2) we may consider the variational formulation of the more
	general Maxwell eigenvalue problem (1.1), corresponding to the case of anisotropic inho-
	mogeneous material:
	Seek u ∈	• H(curl, Ω) ,	(6.11)
	Corollary 6.2 (Approximation of the Maxwell eigenvalue problem) The p version fi-
	nite element discretization of the Maxwell eigenvalue problem (1.2) based on edge ele-
	ments from the first or second Nédélec family on triangles or on tetrahedra, or from the
	first Nédélec family on parallelograms or on parallelepipeds offers a spectrally correct,
	spurious-free approximation.

• H(curl, Ω)

By and large, we adopt the standard notations for Sobolev spaces, see[START_REF] Girault | Finite element methods for Navier-Stokes equations[END_REF] Ch.

2].

For the compatibility Assumption 5 to hold, it is therefore sufficient to make the following assumption about the local polynomial space V ℓ p (K).

Assumption 10

(i) The space V ℓ p (K) is invariant with respect to dilations and translations.

(ii) The differential operator x d ℓ : u → x d ℓ u maps V ℓ p (K) into V ℓ p (K).

To summarize:

Assumptions 9, 10, and (4.8) =⇒ Assumptions 2, 3, 4, and 5.

Discrete differential forms

Now we introduce concrete spaces of discrete differential forms. We merely summarize the constructions that have emerged from research in differential geometry (the "Whitneyforms" introduced in [START_REF] Whitney | Geometric Integration Theory[END_REF]) and finite element theory ("Raviart-Thomas elements" of [START_REF] Thomas | A Mixed Finite Element Method for Second Order Elliptic Problems[END_REF] and "Nédélec finite elements" of [START_REF] Élec | Mixed finite elements in R 3[END_REF][START_REF]A new family of mixed finite elements in R 3[END_REF]). These schemes were later combined into the concept of discrete differential forms [START_REF] Bossavit | Whitney forms: A class of finite elements for three-dimensional computations in electromagnetism[END_REF][START_REF] Hiptmair | Canonical construction of finite elements[END_REF]. Surveys and many more details can be found in [START_REF]Finite elements in computational electromagnetism[END_REF][START_REF] Arnold | Finite element exterior calculus, homological techniques, and applications[END_REF][START_REF]Finite element exterior calculus: from Hodge theory to numerical stability[END_REF][START_REF]Discretization of electromagnetic problems: The "generalized finite differences[END_REF].

Simplicial meshes

Let M be a conforming simplicial finite element mesh covering the Lipschitz polyhedron Ω ⊂ R d . As elaborated in [START_REF] Arnold | Finite element exterior calculus, homological techniques, and applications[END_REF]Sect. 3 & 4] for p ∈ N the following choices

and

of local spaces, through (3.1), gives rise to meaningful global finite elment spaces

By construction both Assumption 9 and Assumption 10 are satisfied for these spaces. The asymptotic density property also holds. Proof. It is a classical result of finite element theory that the spaces of degree p Lagrangian finite element functions

Thus the space of polynomials ℓ-forms with coefficients in

Tensor product meshes

Let M be a conforming finite element mesh of the Lipschitz polyhedron Ω whose cells are affine images of the unit hypercube K in R d : for K ∈ M the we write Φ K : K → K for the associated unique affine mapping. We generalize the construction of [START_REF] Élec | Mixed finite elements in R 3[END_REF]: on the cube we define (with notations introduced in Section 4.4)

The local spaces are obtained by affine pullback

This affine tensor product construction also complies with Assumption 9 and Assumption 10. Completely parallel to Lemma 5.1, one proves the following result.

Lemma 5.2

The requirement (2.12) is satisfied for the spaces • V p ℓ (M) spawned by (5.3).

Remark 5.3 For all the above meshes the cells are affine images of a single reference cell, the "unit simplex" or "unit hypercube". We could allow some non-affine cells: Under the assumption that the transformations are "nearly affine", see [22, §4.3], and the projection operators π ℓ p,K are defined correspondingly, all crucial estimates like Lemma 3.1 can be transferred to the reference cell using the pullback of differential forms.

Application in dimensions two and three

We adopt the discrete spaces from Sect. 5 along with the regularized Poincaré lifting from Sect. 4. We rely on the choice (4.8) for spaces X and S, with a regularity exponent r ∈ (0, 1] which has to be chosen suitably.

In order to establish the discrete compactness property from Definition 2.3, it remains to verify the regularity Assumption 1 and Assumptions 6, 7, and 8 for convenient local projectors π ℓ p,K . Local projectors which make the discrete diagram of Assumption 7 commute do exist in the general framework of differential forms of any degree. They generalize Nédélec


Lei Yu
email: lei.yu@ensea.fr

Jean Pierre Barbot

Gang Zheng
email: gang.zheng@inria.fr

Hong Sun
email: hongsun@whu.edu.cn

Toeplitz-Structured Chaotic Sensing Matrix for Compressive Sensing

Compressive Sensing (CS) is a new sampling theory which allows signals to be sampled at sub-Nyquist rate without loss of information. Fundamentally, its procedure can be modeled as a linear projection on one specific sensing matrix, which, in order to guarantee the information conservation, satisfies Restricted Isometry Property (RIP). Ordinarily, this matrix is constructed by the Gaussian random matrix or Bernoulli random matrix. In previous work, we have proved that the typical chaotic sequencelogistic map can be adopted to generate the sensing matrix for CS. In this paper, we show that Toeplitz-structured matrix constructed by chaotic sequence is sufficient to satisfy RIP with high probability. With the Toeplitz-structured Chaotic Sensing Matrix (TsCSM), we can easily build a filter with small number of taps. Meanwhile, we implement the TsCSM in compressive sensing of images.

I. INTRODUCTION

In information field, traditional information capturing paradigm always follows the famous Shannon-Nyquist sampling theory, which says that signals must be sampled larger than Nyquist rate (2 times of the frequency bandwidth of the signal) to guarantee the exact recovery [START_REF] Shannon | Communication in the presence of noise[END_REF]. This leads that, even for a 2 minutes audio record, 5MB data are needed to retain the voice undistorted. Consequently, in order to reduce the storage consumption and the transmission bandwidth, conventionally, an immediate data compression takes place after the sensing procedure. The principle of this kind of sensing-then-compression scheme, such as JPEG or JPEG 2000, is to discard the small coefficients and retain the significant ones under some fixed transformation. In reality, the data stored and transmitted is only a very small part of the original ones, while we still have to sense the whole of the samples and then discard most of them. It is extremely wasteful of power consumption and memory storage. Moreover, this sensing-then-compression scheme is not practicable if the signal is presented at a high rate or if the measurement device has limited computational resources (such as in a sensor network).

Recently, a new theory called Compressive Sensing (CS) was proposed by Candès et. al. [START_REF] Candès | An introduction to compressive sampling[END_REF] and Donoho [9]. Unlike the traditional sampling methods, CS provides an information capturing paradigm with both sampling and compression. It permits signals being sampled below the conventional Nyquist rate, meanwhile still allowing optimal reconstruction of the signal. If signals are sparse or compressible, the required measurements are far less than those of traditional methods, and even more, the measure process is non-adaptive which makes the sampling process much more universal and faster.

A. Preliminary of Compressive Sensing

Mathematically, the procedure of CS can be expressed as a linear projection

y = Φv (1)
where v ∈ R n is the original signal, Φ ∈ R m×n is the sensing matrix and y ∈ R m is the measurement. Remark 1.1: Particularly, the traditional sampling paradigm following Nyquist theory, is one special case of CS with sensing matrix Φ being identity matrix with m = n. From this point, CS improves the way of capturing signal information, typically by using random sensing matrix with m n. Consequently, we say CS samples and compresses signals simultaneously.

Since m n, CS highly reduces the length of the signal. However, it also leads equation (1) be highly underdetermined. With an additional assumption on original signal, supposing that sparsity is always fulfilled for natural signals (itself or in some basis), the solution of v for (1) can then be possibly uniquely determined.

Definition 1.2: (Sparsity) Given a vector v ∈ R n , and denote Card(v) = Card{v i = 0, i ∈ [1, n]} the cardinality of non zero entries of v, then v is called to be an s-sparsity vector, if Card(v) ≤ s n. Based on Definition 1.2, we can then define the set of sparse vector of s-sparsity Σ s as follows:

Σ s = {v ∈ R n | Card(v) ≤ s} (2)
Intuitively, equation (1) can be solved by searching the sparsest one of vectors lied on the null surface of Φ

[7], i.e. v * = arg min Φv=y v 0 (3)
where v 0 is 0 norm and denotes the counter of nonzero entries of v. However, it was shown to be a NP-Hard problem in [START_REF] Candès | An introduction to compressive sampling[END_REF], and thus practically, it was relaxed in [START_REF] Candès | An introduction to compressive sampling[END_REF] to solve the following problem.

v * = arg min Φv=y v 1 (4)
where v 1 is 1 norm and denotes the absolute sum of all entries of v. Definition 1.3 (Restricted Isometry Property [START_REF] Candès | The restricted isometry property and its implications for compressed sensing[END_REF]): For matrix Φ ∈ R m×n , define δ s > 0 be the smallest constant satisfying following inequality,

(1 -δ s ) v 2 2 ≤ Φv 2 2 ≤ (1 + δ s ) v 2 2 (5)
for all vectors v ∈ Σ s . Then matrix Φ is said to satisfy the RIP of order s with constant δ s .

Theorem 1.4 (Exact Recovery Theorem [START_REF] Candès | The restricted isometry property and its implications for compressed sensing[END_REF] ):

• If RIP of order 2s for matrix Φ satisfies δ 2s < 1, then for any vector v ∈ Σ s , there exists a unique solution for (3).

• If δ 2s < √
2 -1, then for any vector v ∈ Σ s , solutions for (3) and ( 4) are equivalent.

B. Motivation

Consequently, finding a proper sensing matrix Φ satisfying RIP with conditions in Theorem 1.4 is one of the most important problem in CS. Candès and Tao have proposed that matrix with elements drawn by Gaussian distribution or Bernoulli distribution satisfies RIP with overwhelming probability, providing that sparsity s ≤ O(m/ log n) [START_REF] Candès | Near-optimal signal recovery from random projections: Universal encoding strategies[END_REF]. And the randomly selected Fourier basis also retains RIP with overwhelming probability if sparsity s ≤ O(m/(log n) 6 ) [START_REF] Candès | Near-optimal signal recovery from random projections: Universal encoding strategies[END_REF]. Even though there exist some deterministic sensing matrices, such as Chirp Sensing Codes by L. Applebaum, et al. [START_REF] Applebaum | Chirp sensing codes: Deterministic compressed sensing measurements for fast recovery[END_REF], deterministic sensing matrix using finite fields by R. A. Devore [START_REF] Devore | Deterministic constructions of compressed sensing matrices[END_REF] and using second order Reed-Muller codes by S. Howard et al. [START_REF] Howard | A fast reconstruction algorithm for deterministic compressive sensing using second order reed-muller codes[END_REF], their RIP properties are not either fulfilled certainly. Until now, the widely used sensing matrix in practice is generated randomly.

In previous work [START_REF] Yu | Chaotic sensing matrix for compressive sensing[END_REF], we have proposed the chaotic sequence to construct such a sensing matrix, called chaotic matrix. Based on Egodicity and statistical property of chaotic sequence, we have proved that chaotic matrix also has RIP with overwhelming probability, providing that s ≤ O(m/ log (n/s)). Meanwhile, the proof shows that the probability of satisfying RIP for the proposed chaotic matrix is larger than that for Gaussian random matrix and Bernoulli random matrix [START_REF] Candès | Near-optimal signal recovery from random projections: Universal encoding strategies[END_REF].

In this paper, we aim to construct a Toeplitz matrix with chaotic sequence and prove that the Toeplitz chaotic matrix also retains the RIP property with overwhelming probability. The use of Toeplitz matrix in CS application takes the following advantages:

1) Only O(n) independent chaotic variables are needed; 2) The multiplication can be efficiently implemented using FFT; 3) Toeplitz-structured matrix arises naturally in certain applications such as filter design.

C. Paper Organization

In section II, the Toeplitz-structured Chaotic Sensing Matrix (TsCSM) is proposed for CS and its RIP property is proved. In section III, Iterative Threshold (IT) algorithm to reconstruct the original signals is introduced. In section IV, we compare the TsCSM with Chaotic Sensing Matrix (CSM) and Gaussian Sensing Matrix (GSM), meanwhile, some experiments on images are illustrated.

II. TOEPLITZ-STRUCTURED CHAOTIC SENSING

MATRIX A. Chaotic Sensing Matrix and its Johnson-Lindenstrauss (J-L) Embedding [START_REF] Johnson | Extensions of Lipschitz maps into a Hilbert space[END_REF] Property

Considering one popular discrete chaotic system, logistic map, whose difference equation can be written as below.

z

+ = rz(1 -z), with r ∈]0, 4] (6)
where

z ∈]0, 1[⊂ R is the discrete state.
It is well known that for logistic map [START_REF] Candès | Near-optimal signal recovery from random projections: Universal encoding strategies[END_REF] with parameter r = 4, the state sequence z(t) satisfies Beta distribution with parameter α = 0.5 and β = 0.5 [START_REF] Vlad | Computational measurements of the transient time and of the sampling distance that enables statistical independence in the logistic map[END_REF], with the following probability density function.

f (x; 0.5, 0.5) = 1 π (x -x 2 ) -1/2 (7)
Set z i (t) as the output sequence generated by logistic map [START_REF] Candès | Near-optimal signal recovery from random projections: Universal encoding strategies[END_REF] with initial condition z i (0), and let x i (t) denote the regularization of z i (t) as below

x i (t) = z i (t) -0.5 (8)
Approximately, x i (t) can be considered as a random variable and it satisfies the following distribution.

f (x) = 1 π (0.25 -x 2 ) -1/2 (9)
which is called here Beta-like distribution.

Then, by selecting m different initial conditions z(0 x 0 (0) . . . x 0 (n -1) . . . . . . . . .

) ∈ ]0, 1[ m ⊂ R m ,
x m-1 (0) . . . x m-1 (n -1) ⎞ ⎟ ⎠
which is called here the Beta-like matrix.

In the previous work [START_REF] Yu | Chaotic sensing matrix for compressive sensing[END_REF], it has the following theorem: Theorem 2.1 (J-L Embedding with Beta-like Matrix): Given a Beta-like matrix Φ ∈ R m×s , there exists a constant δ s > 0, such that, for any vector u ∈ R s , the following inequality

(1 -δ s ) u 2 2 ≤ Φu 2 2 ≤ (1 + δ s ) u 2 2 ( 10
)
is satisfied with overwhelming probability.

Pr Φu 2 -1 ≥ δ s ≤ 2 exp(-c(δ s )m) (11)
where c(δ s ) is a constant related to δ s . Remark 2.2: Note that we do not refer the final theorem in [START_REF] Yu | Chaotic sensing matrix for compressive sensing[END_REF], and the formation of Φ is not exactly the same. Even though, the theorem is also verified because of, statistically, the Egodicity property of the chaotic system.

Remark 2.3: This theorem is also retained for matrix constructed with Gaussian or Bernoulli matrix, but with different constant c(δ s ) [START_REF] Achlioptas | Database-friendly random projections[END_REF].

B. Toeplitz-structured Chaotic Sensing Matrix

For chaotic system [START_REF] Candès | Near-optimal signal recovery from random projections: Universal encoding strategies[END_REF], let us set one initial condition z(0) ∈ R, then generate a sequence x ∈ R n according to [START_REF] Devore | Deterministic constructions of compressed sensing matrices[END_REF]. Then we can construct a Toeplitz-structured matrix Φ ∈ R m×n of the following form

Φ = 8 m ⎛ ⎜ ⎜ ⎜ ⎝ x(n -1) x(n -2) . . . x(0) x(0)
x(n -1) . . . x(1) . . . . . . . . . . . .

x(m -2) x(m -3) . . . x(m -1) ⎞ ⎟ ⎟ ⎟ ⎠
where 8/m is for normalization.

In [3], the authors proposed to construct the Toeplitzstructured matrix with Gaussian or Bernoulli random sequences and according to the structure of Toeplitz matrix, it can be concluded that for Gaussian (or Bernoulli) Toeplitz-structured matrix, if m ≥ c 1 s 3 ln(n/s), its RIP of order s with constant δ s is verified with probability at least

1e -c2k/m 2 where c 1 and c 2 are dependant only on δ s . According to Theorem 2.1, both the TsCSM and Toeplitz-structured Gaussian or Bernoulli matrix retains the J-L embedding property with overwhelming probability. Similarly, for TsCSM, if m ≥ c 1 s 3 ln(n/s), the RIP of order s with constant δ s is verified with probability at least

1e -c 2 k/m 2 where c 1 and c 2 are dependant only on δ s .

III. SIGNAL RECONSTRUCTION

From now on, there exist many reconstruction methods for signal reconstruction in CS framework, such as Basis Pursuit (BP) [START_REF] Candès | Decoding by linear programming[END_REF], Orthogonal Matching Pursuit (OMP) [START_REF] Tropp | Signal recovery from random measurements via orthogonal matching pursuit[END_REF], CoSaMP [START_REF] Needell | Cosamp: Iterative signal recovery from incomplete and inaccurate samples[END_REF], et. al. Since this paper does not focus on the reconstruction algorithms, we just briefly introduce one iterative algorithm -Iterative Thresholding (IT) [START_REF] Daubechies | An iterative thresholding algorithm for linear inverse problems with a sparsity constraint[END_REF], to reconstruct the original signal.

In reality, the nature signals are compressible, i. e. the signals can be sparsely represented by an specific basis, such as Fourier basis, Wavelet basis. Denote the nature signals as f ∈ R n and a basis B ∈ R n×n , then

f = Bv (12)
where v ∈ Σ s is s-sparse. And the CS procedure is expressed as the following linear projection

y = Φf = ΦBv ( 13
)
IT algorithm is to solve the following problem

f * = arg min f ∈R n y -Φf 2 2 + λ B * f 1
which is a relaxion of optimization (4). And IT algorithm proceeds as follows to find the optimal solution for (13), let f (0) = 0 and use the iteration

f (i+1) = BS λ (B * f (i) + B * Φ * (y -Φf (i) )) (14)
where i represents the ith iteration, B is a basis such as discrete wavelet basis, B * is the inverse of B and S λ ( * ) is a non-linear shrinkage operator that keeps the larger coefficients, i.e.

S λ (x) = ⎧ ⎪ ⎨ ⎪ ⎩ x -λ 2 if x ≥ λ 2 0 if |x| < λ 2 x + λ 2 if x ≤ -λ 2 ( 15
)
where λ is the median of all coefficients, i.e. λ = median(x). The IT algorithm is illustrated as Algorithm 1.

Algorithm 1 Iterative Thresholding Algorithm (IT) Input: measurements y ∈ R m and measurement matrix Φ ∈ R m×n , m n; Output: The recovery image f * = f (tmax) Initial: f (0) = 0, maximum iteration t max for i = 0 to t max do Update:

f (i) = f (i) + Φ * (y -Φf (i) ); Transform: Compute the coefficients over basis B, v = B * f (i) ; Sparse promote: v = S λ (v);
Inverse transform: f (i+1) = Bv. end for

IV. NUMERICAL SIMULATIONS

A. TsCSM, Chaos Sensing Matrix (CSM) and Gaussian Sensing Matrix (GSM)

In section II, it has been proved that the TsCSM also has the RIP property with overwhelming probability, i.e. it guarantees the exact recovery for any sparse signals. To verify the property, a comparison between TsCSM and CSM (or GSM) is carried out by considering their exact recovery rates. Let us describe the experiment setup: we build a TsCSM (or CSM, GSM) with dimension 50×100, which means that there are 50 measurements, then, for each sparsity level s ∈ {1, ..., 50}, we arbitrarily generate 1000 s-sparse signals with dimension 100, we compress them following the CS procedure (1) and we use the Algorithm 1 (here the basis B is identity matrix) to recover the sparse signals. For each trial, we denote it successful if its recovery error is less than a level (here set it as 0.001) and we can finally obtain a successful rate, i.e. recovery rate, see Fig. 1.

In Fig. 1, the recovery rate for TsCSM is similar with CSM but little larger than GSM, and the reason is that the probability of J-L embedding property for GSM is smaller than that for CSM which has been demonstrated in [START_REF] Yu | Chaotic sensing matrix for compressive sensing[END_REF].

B. TsCSM for Images

Recently, the application of CS on imaging is attracted a lot researchers [START_REF] Duarte | Single-pixel imaging via compressive sampling[END_REF], [START_REF] Ma | Single-pixel remote sensing[END_REF]. In this simulation, we only analyze the performance of the TsCSM fundamentally on imaging rather than the physical implementation. Suppose TsCSM is used in the single imaging system introduced in [START_REF] Duarte | Single-pixel imaging via compressive sampling[END_REF], [START_REF] Ma | Single-pixel remote sensing[END_REF], we will give two experiments to illustrate the performance of TsCSM.

First, let's fix the sample rate at 0.5, i.e. for image with size 256 × 256, we only measure 32, 768 samples by projection with TsCSM following CS procedure (1); then, we use Algorithm 1 (with B as wavelet basis) to reconstruct the original images from the compressed data. The result are listed in Fig. 2, where Peak Signal to Noise Ratio (PSNR) is defined as

PSNR = 20 log f 2 f -f * 2 (16)
Second, let's verify the efficiency of compression. For one image with dimension 256 × 256, we measure this image with sample rate at 0.3, 0.5, 0.7 and 0.9 separately, then use Algorithm 1 (with B as wavelet basis) to reconstruct the original images, see Fig. 3. The error is the direct difference between the recovery and the original image.

V. CONCLUSION

This paper briefly introduced a new sampling theory -Compressive Sensing (CS) and proposed a method to construct a Toeplitz-structured Matrix for CS with chaotic sequence. We prove that Toeplitz-structured Chaotic Sensing Matrix (TsCSM) retains the Restricted Isometry Property (RIP) with overwhelming probability, which guarantees the exact recovery. Then Iterative Thresholding (IT) algorithm is introduced to reconstruct the original signal. After that, a simulation on 1-dimensional signal CS is implemented and the comparison between TsCSM and CSM (or GSM) shows that TsCSM has the similar (or better) performance. Meanwhile, we also simulate imaging system with TsCSM, and the results show a good perspective for TsCSM.

8

 8 one can obtain m vectors with dimension n, which enables us to construct the following matrix Φ scaled by

Fig. 1 .

 1 Fig. 1. Recovery rate for TsCSM, CSM and GSM (over 1 000 trials), with dimension 50 × 100.

Fig. 2 .

 2 Fig. 2. Reconstructions of CS with TsCSM for different types of images.

ACKNOWLEDGMENT

This work is partially supported by the "Bourses Doctorales en Alternance" and the PhD Research Foundation of Wuhan University (20082120201000043).