On variational eigenvalue approximation
of semidefinite operators
Abstract
Eigenvalue problems for semidefinite operators with infinite dimensional kernels appear for instance in electromagnetics. Variational discretizations with edge elements have long been analyzed in terms of a discrete compactness property. As an alternative, we show here how the abstract theory can be developed in terms of a geometric property called the vanishing gap condition. This condition is shown to be equivalent to eigenvalue convergence and intermediate between two different discrete variants of Friedrichs estimates. Next we turn to a more practical means of checking these properties. We introduce a notion of compatible operator and show how the previous conditions are equivalent to the existence of such operators with various convergence properties. In particular the vanishing gap condition is shown to be equivalent to the existence of compatible operators satisfying an AubinNitsche estimate. Finally we give examples demonstrating that the implications not shown to be equivalences, indeed are not.
MSC classes : 65J10, 65N25, 65N30.
1 Introduction
The basic eigenvalue problem in electromagnetics reads:
(1.1) 
for some vector field satisfying boundary conditions in a bounded domain in Euclidean space. The matrix coefficients and characterize electromagnetic properties of the medium. The eigenvalue is expressed here in terms of the angular frequency . The operator on the left is semidefinite with an infinite dimensional kernel. The variational approximation of such eigenvalue problems reads: find and such that for all there holds:
(1.2) 
In Galerkin discretizations, is in some finite dimensional space and the above equation should hold for all in . One has an integer parameter and looks at convergence properties as increases.
The above eigenvalue problem is of the following general form: find and such that for all :
(1.3) 
where on the right we have the scalar product of a Hilbert space (typically an space) and on the left a symmetric semipositive bilinear form on a Hilbert space . In our setting, will be dense and continuously embedded in , but not necessarily compactly so, so that the standard theory [BabOsb91] does not directly apply. The important hypothesis is instead that the orthogonal complement of the kernel of in is compactly embedded in . In the context of electromagnetics, such compactness results are included in the discussion of [Cos90].
The most successful Galerkin spaces for (1.2) are those of [Ned80] and the convergence of discrete eigenvalues has been shown to follow essentially from a discrete compactness (DC) property of these spaces, as defined in [Kik89]. This notion has been related to that of collectively compact operators [MonDem01]. Under some assumptions, eigenvalue convergence is actually equivalent to discrete compactness [CaoFerRaf00]. Necessary and sufficient conditions for eigenvalue convergence are also obtained in [BofBreGas97] via mixed formulations. Natural looking finite element spaces, which satisfy standard infsup conditions, equivalent to a discrete Friedrichs estimate (DF), but nevertheless yielding spurious eigenvalues, have been exhibited [BofBreGas00]. For reviews see [Hip02, Mon03, Bof10].
Eigenpair convergence has long been expressed in terms of gaps between discrete and continuous eigenspaces. In [Chr04MC] the analysis of some surface integral operators was based upon another type of gap, which, transposed to the above problem, concerns the distance from discrete divergence free vector fields to truly divergence free ones. In other words, one considers the gap, in its unsymmetrized form, between the discrete and continuous spaces spanned by the eigenvectors attached to nonzero eigenvalues, from the former to the latter. The general framework was further developed in [BufChr03], to the effect that discrete infsup conditions for noncoercive operators naturally followed from a vanishing gap (VG) condition. A variant can be found in [Chr02p]. In [Buf05] this theory was applied to the source problem for (1.2) in anisotropic media. Contrary to DC, VG has a clear visual interpretation, expressing a geometric property of abstract discrete Hodge decompositions.
That VG implies DC is immediate. The converse was noted in [BufPer06]. Thus VG is, via the above cited results, equivalent to eigenvalue convergence. Moreover DC implies DF, though [BofBreGas00] shows that the converse is not true. However we shall show that VG follows from an optimal version of DF, referred to as ODF henceforth. While unimportant for eigenvalue convergence, the ODF, in the form of a negative norm estimate, was used in [Chr05] (under the name uniform norm equivalence) to prove a discrete divcurl lemma. It was also remarked that a local version of the ODF is implied by the discrete divcurl lemma. A common underlying assumption is that we have approximating discrete kernels (ADK). In section §3 we detail the relationships between ADK, DF, DC (several variants), VG and ODF. We also show how to deduce eigenvalue convergence from VG. While VG uses the norm of to compute the gap, we relate ODF to a gap property in . The results on ODF are new. Most of the other results should be considered known in principle, but the ordering and brevity of the arguments might be of interest. A summary is provided in diagram (3.46).
The main tool used to prove DC and error estimates for finite element spaces is the construction of interpolation operators. The role of commuting diagrams they satisfy has been highlighted [Bof00]. However the interpolators deduced from the canonical degrees of freedom of edge elements are not even defined on for domains in . This has lead to great many technical hurdles, involving delicate regularity estimates. Commuting projectors that are tame, in the sense of being uniformly bounded , were proposed in [Sch08]. Another construction combining standard interpolation with a smoothing operator, obtained by cutoff and convolution on reference macroelements, was introduced in [Chr07NM]. As pointed out in [ArnFalWin06], for flat domains, quasiuniform grids and natural boundary conditions, one can simplify the construction to use only smoothing by convolution on the entire physical domain. In [ChrWin08] these restrictions were overcome by the introduction of a space dependent smoother. Such operators yield eigenvalue convergence quite easily, as well as ODF. It should also be remarked that eigenvalue convergence for the discretization of the HodgeLaplacian by Whitney forms had been obtained much earlier in [DodPat76]. For a review of the connection between finite elements and Hodge theory, see [ArnFalWin06, ArnFalWin10].
Reciprocally one would like to know if DF, VG or ODF imply the existence of commuting projections with enough boundedness, since this will indicate possible strategies for convergence proofs in more concrete settings. For instance, in §4 of [BofBreGas97], it is shown that eigenvalue convergence for mixed formulations is equivalent to the existence of a Fortin operator, converging in a certain operator norm. In §3.3 of [ArnFalWin10] it is shown, in the context of Hilbert complexes, that DF is equivalent to the existence of commuting projections that are uniformly bounded in energy norm. We introduce here a notion of compatible operator (CO) which contains both Fortin operators and commuting projections. With this notion at hand, DF is equivalent to the existence of energy bounded COs, VG is equivalent to the existence of COs satisfying an AubinNitsche estimate, and ODF is equivalent to the existence of tame COs. The equivalences of DF, VG and ODF to various properties of compatible operators are detailed in §LABEL:sec:projections.
In §LABEL:sec:add we give some complementary results. §LABEL:sec:hodge shows how tame compatible operators appear in the context of differential complexes. §LABEL:sec:eqfor studies what happens when the bilinear forms are replaced by equivalent ones, extending a result proved in [CaoFerRaf00]. §LABEL:sec:strict considers the optimality of proved implications. We show how to construct approximating subspaces satisfying ADK but not DF. We also show how one can construct spaces satisfying DF but not DC, and spaces satisfying DC but not ODF. Thus all the major proved implications that were not proved to be equivalences, are proved not to be. These results are new in this abstract form, but as already mentioned, [BofBreGas00] gives a concrete counterexample of spaces satisfying DC but not DF.
2 Setting
2.1 Continuous spaces and operators
Let be an infinite dimensional separable real Hilbert space with scalar product . Let be a dense subspace which is itself a Hilbert space with continuous inclusion . We suppose that we have a continuous symmetric bilinear form on satisfying:
(2.1) 
Moreover we suppose that the bilinear form is coercive on so that we may take it to be its scalar product. The corresponding norm on is denoted , whereas the natural one on is denoted . Thus:
(2.2)  
(2.3) 
Define:
(2.4)  
(2.5) 
so that we have a direct sum decomposition into closed subspaces:
(2.6) 
which are orthogonal with respect to both and .
Since is semidefinite we have:
(2.7) 
We do not require to be finitedimensional. Remark that is closed also in . Let be the closure of in and remark:
(2.8) 
so that we have an orthogonal splitting of :
(2.9) 
We also have:
(2.10) 
We let be the projector in with range and kernel . It is orthogonal in . It maps to so it may also be regarded as a continuous projector in . Since the splitting (2.6) is orthogonal with respect to the scalar product on defined by (2.3), is an orthogonal projector also in .
We suppose that the injection is compact (when inherits the norm topology of ). It implies the Friedrich inequality, namely that there is such that:
(2.11) 
In particular, on , is a scalar product whose associated norm is equivalent to the one inherited from , defined by (2.3). In , can be characterized by the property that for , solves:
(2.12) 
for all . This holds then for all . We will also frequently use the identity, for :
(2.13) 
Remark 1.
Let be a bounded contractible Lipschitz domain in with outward pointing normal on . Equation (1.2) leads to the following functional frameworks, depending on which boundary conditions are used. For simplicity, we take and to be scalar Lipschitz functions on that are bounded below, above zero.

Define:
(2.14) (2.15) (2.16) (2.17) Then we have:
(2.18) (2.19) 
As a variant, replace in the above definitions:
(2.20) Then we get:
(2.21) (2.22)
In both examples, compactness of is guaranteed by results in [Cos90]. If the topology of is more complicated, can contain, in addition to gradients, a nontrivial albeit finite dimensional, space of harmonic vectorfields. The characterization of should then be modified accordingly.
For the case of more general coefficients and , see §LABEL:sec:eqfor.
The dual of with as pivot space is denoted . The duality pairing on is thus denoted . We let denote the subspace of consisting of elements such that for all .
Let be the continuous operator defined as follows. For all , satisfies:
(2.23) 
If , equation (2.23) holds for all . One sees that is identically on and induces an isomorphism .
Since is compact, is compact. As an operator , is selfadjoint. We deduce that is compact and a fortiori . We are interested in the following eigenvalue problem. Find and such that:
(2.24) 
When it is the eigenspace associated with the eigenvalue . A positive is an eigenvalue of iff is a positive eigenvalue of ; the eigenspaces are the same. The nonzero eigenvalues of form an increasing positive sequence diverging to infinity, each eigenspace being a finite dimensional subspace of . The direct sum of these eigenspaces is dense in .
2.2 Discretization
Given the above setting, consider a sequence of finite dimensional subspaces of . We will later formulate conditions to the effect that elements of can be approximated by sequences such that . In the meantime we state some algebraic definitions.
We consider the following discrete eigenproblems. Find and such that:
(2.25) 
We decompose:
(2.26)  
(2.27) 
so that:
(2.28) 
Thus but it’s crucial, for the points we want to make, that need not be a subspace of .
Notice that is the eigenspace and that for nonzero we have:
(2.29) 
Define as follows. For all , satisfies:
(2.30) 
Define also by, for all , satisfies:
(2.31) 
which then holds for all .
For we have, for all :
(2.32) 
So that:
(2.33) 
If, on the other hand , so , but iff is orthogonal to . If there are elements in not orthogonal to .^{1}^{1}1The (crucial) point that and need not coincide on was overlooked in [Chr09AWM].
The nonzero discrete eigenvalues of are the inverses of the nonzero eigenvalues of .
A minimal assumption for reasonable convergence properties is:
 AS

We say that are Approximating Subspaces of if:
(2.34)
Proposition 1.
If AS holds we also have:
(2.35)  
(2.36)  
(2.37) 
Proof.
(i) The first property holds by density of in .
(ii) If we denote by the orthogonal projection onto and is approximated in the norm by a sequence we have:
(2.38) 
and, since is orthogonal to both and :
(2.39) 
(iii) Define as above. In it is a projection with norm . Pick and choose a sequence converging to in . Remark that is orthogonal to both and , also in . We get:
(2.40) 
This concludes the proof. ∎
Statements of the form “there exists such that for all , certain quantities and satisfy ”, will be abbreviated . Thus might depend on the choice of sequence but not on .
3 Friedrichs, gaps and eigenvalue convergence
In this section, and for the rest of this paper, we assume that the spaces have been chosen so that AS holds.
Recall that is in general not a subspace of . Below we will relate convergence of variational discretizations, to how well elements of can be approximated by those of . More generally one is interested in what properties of elements of , the elements of share.
For nonzero subspaces and of one defines the gap from to by:
(3.1) 
In what follows gaps will be computed with respect to the norm of , unless otherwise specified.
In the above setting we define:
Definition 1.
 ADK

We say that we have Approximating Discrete Kernels if:
(3.2)  DF

Discrete Friedrichs holds if there is such that:
(3.3)  DC

Discrete Compactness holds if, from any subsequence of elements which is bounded in , one can extract a subsequence converging strongly in , to an element of .
 VG

We say that Vanishing Gap holds if:
(3.4)  ODF

We say that the Optimal Discrete Friedrichs inequality holds if for some we have:
(3.5)
This section discusses first the relations between these conditions, and second their relation to the convergence of to in various senses. The first three conditions are studied in particular in [Kik89, CaoFerRaf00], see also [Bof10] §19. There are many possible variants of DC and we discuss several below. For an often overlooked subtlety in the definition of DC, see Remark 2. VG appeared in [Chr04MC, BufChr03, Buf05]. ODF is a condition we introduce here, as a counterpart to boundedness in of certain operators (mimicking so called commuting projections and Fortin operators) that will be discussed later, in §LABEL:sec:projections. Notice that ODF implies DF, due to (2.11) and (2.13). A summary of the proved implications is provided at the end of this section, in diagram (3.46).
Even ADK, which we will see is the weakest of the above hypotheses, is quite restrictive, but arises naturally in discretizations of complexes of Hilbert spaces by subcomplexes as will be explained in §LABEL:sec:hodge. For the examples of Remark 1 the most important finite element spaces are those of [Ned80]. We will see that they have all the above properties, of which ODF is the strongest.
Strictness of proved implications will be discussed in §LABEL:sec:strict below.
3.1 Internal relations
We first study the implications between these conditions.
Lemma 2.
Suppose ADK does not hold. Then there is a subsequence of elements such that:

converges weakly in to a nonzero element of .

converges strongly to in X.
Proof.
For any closed subspace of , let be the orthogonal projection onto defined by (2.3). Suppose ADK is not satisfied. Choose , and an infinite subset of such that for all :
(3.6) 
By AS we have:
(3.7) 
Since is bounded in , we may suppose in addition that it converges weakly in to some . Since then also converges weakly to , we must have .
If , converges weakly to and we can write:
(3.8)  
(3.9)  
(3.10) 
This is impossible. So .
The subsequence satisfies the stated conditions. ∎
Proposition 3.
The following are equivalent:

ADK.

For any subsequence that converges weakly in to some , we have .

For any subsequence that converges weakly in to some , we have .
Proof.
(i) Suppose ADK holds. Let converge weakly in to . Pick and choose converging, in the norm, to . Then we have:
(3.12) 
so that .
(ii) The third statement follows from the second by (2.10).
(iii) The third statement implies the first by Lemma 2. ∎
The following corresponds to Proposition 2.21 in [CaoFerRaf00].
Proposition 4.
DF implies ADK.
Proof.
Remark 2.
So far the finite element literature has been rather cavalier about taking subsequences from the outset in the formulation of DC, probably because it is rarely made clear what the set of indexes is, in the first place. See Remark 19.3 in [Bof10]. Without this precision one cannot hope to deduce eigenvalue convergence, since one cannot rule out that good Galerkin spaces are interspaced with bad ones.
In the formulation of DC some authors, including [CaoFerRaf00] and [Bof10], prefer not to impose that the limit is in . Compactness is also frequently used in the form that a weak convergence implies a strong one. For completeness we state the following two alternatives to DC:
 DC’

We say DC’ holds if, from any subsequence of elements which is bounded in , one can extract a subsequence converging strongly in .
 DC”

We say DC” holds if, for any sequence of elements which converges weakly in , the convergence is strong in .
Notice that in the last variant, we use sequences, not subsequences – though we could have done that as well. These conditions are related as follows.
Proposition 5.
The following are equivalent:

DC.

DC’ and ADK.

DC” and ADK.
Proof.
(i) Suppose DC holds. DC’ trivially holds, so we focus on ADK. Consider a subsequence that converges weakly in to some . Extract from it one that converges strongly in to an element in . Since this limit must be , . This proves ADK by Proposition 3.
(ii) Let a sequence converge weakly in to say . If it does not converge strongly in to choose an and a subsequence for which . No subsequence of this subsequence can converge strongly in , since the limit would have to be . Hence DC’ implies DC”.
(iii) Suppose DC” and ADK hold. Consider a subsequence bounded in . Extract from it one that converges weakly in to say . By ADK, . For the indexes not pertaining to this subsequence, insert the best approximation of in . Using Proposition 1 we now have a sequence converging weakly in to , so by DC” it converges strongly in .
This proves DC. ∎
Remark 3.
Neither DC’ nor DC” implies ADK. Consider the case where is finite dimensional but non zero. Since both and are compactly injected into , the injection is compact, so that DC’ and DC” are automatically satisfied. But one can construct subspaces satisfying AS, whose intersection with is zero, so that ADK is not satisfied.
Corresponding to the well known fact that DC implies DF, we state:
Proposition 6.
VG implies DF.
Proof.
We have, for :
(3.14) 
which gives:
(3.15) 
For, say, we then get:
(3.16) 
which completes the proof. ∎
The following is an abstract variant of Proposition 7.13 in [BufPer06].
Proposition 7.
DC is equivalent to VG.
Proof.
(i) Suppose VG holds. Suppose a subsequence of elements is bounded in . Then is bounded in , so it has a subsequence converging strongly in and weakly in . By (2.10) the limit, call it , is in . We have:
(3.17) 
so that converges in to . Hence DC holds.
(ii) Suppose that DC holds but not VG. We get a contradiction as follows. Choose and a subsequence of elements which is bounded in but such that, for all pertaining to the subsequence:
(3.18) 
Extract a subsequence that converges strongly in to some . Then also converges to in . We can now write:
(3.19) 
which is eventually bounded below by , contradicting the convergence of . ∎
As already indicated, ODF implies DF, but actually something stronger is true:
Proposition 8.
ODF implies DC.
Proof.
Suppose ODF holds. Consider a subsequence , bounded in . Then is bounded in , so we may find a subsequence converging strongly in to some . Choose converging strongly to in (Proposition 1). Then converges to in . Using ODF we deduce:
(3.20)  
(3.21) 
Therefore converges to strongly in . ∎
One can also consider gaps defined by the norm of rather than . Explicitly, for subspaces and of we define:
(3.22) 
Proposition 9.
ODF is equivalent to the condition: there exists such that for all , .
Proof.
Remark first that for all and all nonzero , , so by compactness of the unit sphere of , .
(i) If the stated condition did not hold we would have a subsequence of elements such that and converges to . Then we have:
(3.23) 
Hence ODF does not hold.
(ii) If on the other hand the stated condition holds we have, for all and :
(3.24) 
and this gives ODF. ∎
Remark 4.
In the context of problems with natural boundary conditions in convex domains (see Remark 1), the ODF has the following equivalent formulation in terms of a negative norm:
(3.25) 
Here, is the subspace of vectorfields in that are orthogonal to discrete gradients (of functions that can be nonzero on the boundary). In this form the condition was discussed in connection with a discrete div curl lemma in [Chr05]. An advantage of the present formulation is to avoid negative norms and boundary conditions.
In [Chr09Calc] it was pointed out that for the discrete div curl lemma to hold, ODF is actually necessary, at least on periodic domains.
Remark 5.
An interpretation of results in [Chr09Calc] (§3), is that for some standard spaces satisfying ODF (the edge elements of [Ned80] for instance) we do not not have that converges to .
3.2 Convergence of operators
We now explore the relation of the conditions DF and VG to the convergence of the operators .
We first state, concerning :
Proposition 10.
The following are equivalent:

DF.

For any , converges to in .

The are uniformly bounded .

The are uniformly bounded .
Proof.
(i) Suppose DF holds, and pick . Choose converging to in (Proposition 1). We have:
(3.26)  
(3.27) 
which gives (quasioptimal) convergence of to .
(ii) Pointwise convergence of implies uniform boundedness by the principle. Uniform boundedness implies uniform boundedness , of course.
(iii) Suppose DF does not hold. Pick a subsequence such that but . Since , the sequence converges to in . On the other hand , whose norm is .
This precludes uniform boundedness of . ∎
We state similar results for , the only subtlety being the behaviour of on . This particular point will be further expanded upon in Remark 6.
Proposition 11.
The following are equivalent:

DF.

For any , converges to in .

The are uniformly bounded .

The are uniformly bounded .
Proof.
(i) Suppose DF holds.
Pick . Then and , so by the preceding proposition in .
Pick . By ADK (Proposition 4) we can choose a sequence converging to in . We have:
(3.28)  
(3.29) 
Hence in .
This proves the second statement.
(ii) The second statement implies the third, which in turn implies the last.
(iv) When the last condition holds, the largest eigenvalue of is uniformly bounded, which implies DF. ∎
The above propositions show, as is well known, that DF is the key estimate for the convergence of the Galerkin method (2.30) for source problems. Turning to eigenvalue approximation, DF simply says that the smallest nonzero discrete eigenvalue of is bounded below, above zero. It is well known, since [BofBreGas00], that DF by itself is not enough to guarantee eigenvalue convergence. This has been the main reason for considering extra conditions, like DC [Kik89].
To study the convergence of the spectral attributes (such as eigenvalues) of , to those of , a number of conditions have been used [DunSch63, Kat80, Cha83], in quite general settings that include unbounded operators. The most convenient sufficient condition for us, appears to be:
 NC

We say Norm Convergence holds if .
In [Osb75], see [BabOsb91] §7, it is shown directly that if NC holds, then for any nonzero eigenvalue of , any small enough disk around in , the eigenvalues of inside it converge to , with corresponding convergence of eigenspaces in the sense of gaps. A key intermediate step is the norm convergence of the spectral projections associated with the disc, written as contour integrals. The argument easily extends to any contour in that does not meet the spectrum of or enclose .
In [BofBreGas00] §5 a reciprocal is proved, for positive and injective : a convergence of eigenvalues and spaces in the sense of gaps, implies NC.
Arguably DF is not necessary for eigenvalue convergence of the discretizations (2.25). Indeed one can imagine a scenario where some discrete nonzero eigenvalues of cluster around , arbitrarily close as , with corresponding eigenspaces close to , in the sense of gaps. This scenario need not be catastrophic in practice, for instance to someone who needs to identify resonances of an electromagnetic device. In the case where is finite dimensional this is even more patent: any approximating subspaces will yield a reasonable notion of eigenvalue convergence, even those that do not contain . For such discretizations, is not even uniformly bounded in .
However our present theory is motivated by the situation where DF, and hence ADK, holds. We then take it for granted that eigenvalue and eigenspace convergence for (2.25), properly defined in terms of gaps, is equivalent to NC.
In [CaoFerRaf00] Definition 6.1, a notion of “spurious free” approximation is introduced. It is quite easily seen to imply DF. In Theorem 6.10 of that paper, it is shown that spurious free approximation is equivalent to DC (and AS). They use results of [DesNasRap78I] which concern a weaker variant of NC, applicable when is bounded but not necessarily compact.
We now prove the analogue result for us, namely that NC is equivalent to VG.
Lemma 12.
We have:
(3.30) 
Proof.
For any we have:
(3.31)  
(3.32)  
(3.33)  
(3.34) 
We deduce:
(3.35) 
This proves the claim. ∎
Proposition 13.
If DF holds then:
(3.36) 
and:
(3.37) 
Proof.
By DF, the are projectors which converge pointwise to (Proposition 10). Since is compact, Lemma LABEL:lem:pointnorm in the appendix then gives , which gives (3.36) since and coincide on , see (2.33).
Pick . We have:
(3.38)  
(3.39)  
(3.40) 
Since we also have, by DF:
(3.41) 
we get:
(3.42) 
∎
Proposition 14.
The following are equivalent:

VG.

tends to .

NC.
Proof.
(i) If VG holds then DF holds. Applying Proposition 13 shows that tends to .
(ii) We have:
(3.43) 
3.3 Summary
The following diagram represents the main implications proved in this section. The numbers refer to propositions.
(3.46) 