CHARACTERISTIC CONDITIONS FOR SPURIOUS-FREE FINITE ELEMENT APPROXIMATIONS OF ELECTROMAGNETIC EIGENPROBLEMS

Transcription

1 European Congress on Computational Metods in Applied Sciences and Engineering ECCOMAS 000 Barcelona, -4 September 000 ECCOMAS CHARACTERISTIC CONDITIONS FOR SPURIOUS-FREE FINITE ELEMENT APPROXIMATIONS OF ELECTROMAGNETIC EIGENPROBLEMS Salvatore Caorsi, Paolo Fernandes* and Mirco Raffetto Dipartimento di Elettronica, Università di Pavia, ia Ferrata, I-700 Pavia, Italy * Istituto per la Matematica Applicata del Consiglio Nazionale delle Ricerce, ia De Marini 6, Torre di Francia, I-649 Genova, Italy Web page: ttp://wwwimagecnrit Dipartimento di Ingegneria Biofisica ed Elettronica, Università di Genova, ia Opera Pia a, I-645 Genova, Italy Web page: ttp://wwwdibeunigeit Key words: Cavity Resonators, Electromagnetic Eigenproblems, Galerkin Finite Element Approximations, Convergence of Spectral Approximations, Spurious Modes, Nedelec's Edge Elements Abstract A compreensive teory explaining te origin and te beaviour of spurious solutions in Galerkin finite element approximation of electromagnetic eigenproblems modelling cavity resonators is reported Tis teory olds under very general assumptions allowing inomogeneous, anisotropic and discontinuous material properties, topologically nontrivial problem domains and mixed boundary conditions arising from symmetry exploitation A precise definition of spurious-free approximation is proposed and conditions necessary and sufficient to obtain an approximation wic is spurious-free in te sense just defined are given Te spurious-free caracter of an approximation proves to be independent from te properties of materials possibly filling te cavity resonator Hence, it is sufficient to prove it for an empty cavity and te result immediately carry over te case in wic te same cavity is loaded in a general way Tis is very elpful since, at present, none of te known convergence proofs still old wen discontinuous material properties are assumed

2 INTRODUCTION It is well known tat Galerkin finite element approximations of eigenproblems modelling cavity resonators may be plagued by spurious modes, depending on te particular finite element used Tis may appen only because a divergence-free constraint (wose numerical implementation is difficult) as been dropped In so doing, just a single extraneous zero frequency eigensolution as been introduced, wic we could tink to discard after calculation However, as te extraneous eigenspace is infinite dimensional, te original compactness of te operator associated to te eigenproblem as been lost and tis is actually te main issue In fact, in te noncompact case, stronger conditions tan in te compact one must be satisfied to ensure tat a Galerkin finite element approximation converges Oterwise, te wole spectrum may be polluted by spurious eigenvalues interspersed wit te pysically meaningful ones Several elements ave been proposed tat prove to be spurious-free in practice and convergence proofs for some of tem ave been worked out, but a satisfactory and compreensive teory of spectral pollution in electromagnetic eigenproblems as not been developed, yet In fact, a good definition of spurious-free approximation is still lacking and te reasons for te successful beaviour of some elements and te disastrous beaviour of some oters are not completely understood, yet In order to develop suc a teory, first of all we will propose a matematical definition of spurious-free approximation of an electromagnetic eigenproblem, wic interprets and makes precise wat practicioners intuitively means by tat To be more specific, we will call spurious-free any approximation of te modified noncompact problem tat converges in te sense defined in Reference and, moreover, satisfies a specific condition ensuring tat, on any finite triangulation, all te approximations of eigenpairs extraneous to te original compact problem are actually exact, so tat tey can be readily identified and discarded Ten, we will give two sets of conditions tat are necessary and sufficient to obtain a spurious-free finite element approximation (in te sense of te previously given definition), togeter wit te powerful result tat if a finite element approximation is spurious-free for a problem involving only one isotropic omogeneous medium ten te same approximation is spurious-free also for te same problem wit different anisotropic inomogeneous materials filling te cavity resonator Let us point out tat our teory olds under very general assumptions tat, besides inomogeneous, anisotropic and discontinuous material properties, also allow topologically nontrivial problem domains and mixed boundary conditions arising from symmetry exploitation, so covering most of te situations occurring in real-life applications As a matter of fact, by exploiting it, we ave been able to prove, for te first time under so general assumptions, te spurious-free caracter of approximations based on Nedelec's tetraedral edge elements Finally, we will use te developed teory to explain te observed beaviour of known finite element approximations

3 MODELLING CAITY RESONATORS Topological and regularity ypoteses Since we want to work in te functional framework defined in Reference 4, we assume te same ypoteses Te problem domain Ω R (representing eiter te cavity resonator wit an ideally conducting boundary or a symmetry cell of it) is an open connected and bounded Lipscitz domain suc tat every curl free vector as a global potential in te pseudo-lipscitz domain Ω = Ω \ Σ obtained by removing from Ω te union Σ = n Σ i= i of a finite number n of mutually disjoints orientable cuts Σ i i =,, n Eac cut Σ i is te interior of a compact and connected two-dimensional Lipscitz manifold Σ i wit boundary Σi, suc tat Σ i Ω and Σ i Γ = Ω All te n cuts must be removed to acieve te goal tat every curl free vector as a global potential in Ω In general, te problem boundary Γ = Ω is split into two parts Γ and Γ, but te limit cases Γ =, Γ = Γ and Γ = Γ, Γ = are allowed If we are not in eiter limit case, Γ and Γ are assumed to be compact Lipscitz submanifolds of Γ, wit or witout boundary, wose union is Γ and wose nonempty and disjoint interiors are Γ and Γ, respectively If Γ and Γ are manifold wit boundary, ten te common boundary Γ = Γ is a Lipscitz one-dimensional submanifold of Γ Bot Γ \ Σ and Γ \ Σ ave a finite number of connected components and te closure of eac of tem is a Lipscitz two-dimensional submanifold of Γ Material properties are introduced as two second order real symmetric tensor fields ε and µ, defined in Ω, wit bounded coefficients satisfying te ellipticity conditions and µ ijξiξ j µ ξ almost everywere in Ω, * i, j= µ * We also assume tat Ω can be split in open Lipscitz subsets k εijξiξ j ε i, j= ξ R, for two positive constants ε * and Ω k=,,r suc tat all te coefficients of ε and µ (wic may be globally discontinuous) are Lipscitz continuous in eac of tem Te tensor caracter of te material properties allows for anisotropy, wile teir piecewise continuity permits configurations were different inomogeneous materials abut (eac of tem filling one or more Ω k sub-domains) As for implications of te regularity assumptions about boundaries, it is worty to notice tat corners, edges and vertices are allowed, wile slits are not Topologically nontrivial problem domains are permitted, but te assumption tat cuts Σ i are mutually disjoint excludes suc domains, for instance, as te complement of two enlaced tori to a bounded region Notice, owever, tat tere are no topological restrictions on * ξ

4 te Ω k sub-domains Splitting of Γ as to do wit mixed boundary conditions, as we will consider function spaces wit tangential and normal traces vanising on Γ and Γ, respectively Connected components of Γ and Γ may be multiply connected Functional framework Let us define te following function spaces, scalar products and norms H = L ( Ω ) () = H 0, Γ ( curl; Ω) = { v L ( Ω) curlv L ( Ω), v n = 0} Γ 0 H = ε curl( H 0, Γ ( curl; Ω ) H 0, Γ ( div ; Ω; ε) = = v L ( Ω) div ( εv) = 0, ( εv) n = 0 { } 0 0 = H0, ( curl Ω) = { v curlv = 0} Γ ; = H Te inner product and natural norm of L ( Ω ) will be indicated by (, ) 0 and, 0 respectively H and are endowed wit te following inner products and norms, wic make tem Hilbert spaces a ( u, v) H ( εu, v) 0 ( v, v) H/ Γ = u H, v H () v = v H H ( u, v) ( µ curlu, curlv) 0 = u, v ( u, v) ( εu, v) 0 + a( u, v) ( v, v) / = u, v v = v proves to be te ortogonal complement of 0 in Continuous and finite element models Te electric field u inside te cavity resonator and te angular frequency ω of te resonant modes are determined by te following eigenproblem in variational form, u 0 R a u, v u, v Find ( ω ) suc tat ( ) ω ( ) H = v, (P) wic is equivalent to te Maxwell system (Notice tat, if we give to u te meaning of magnetic field and we excange µ and ε, ten (P) still models a cavity resonator and te teory in te following applies, as well) A conforming finite element approximation of problem (P), owever, would require a 4

5 finite element space exactly satisfying te constraint div ( ε v) = 0, wic is difficult to obtain, even for constant ε Hence, problem (P) is invariably substituted by te modified problem Find ( ω, u 0) R suc tat a( u, v) ω ( u, v) H = v, (MP) wic is ten approximated by a finite element problem Denote by I a denumerable and bounded set of strictly positive indexes aving zero as te only limit point and introduce a family of triangulations { Τ } I of Ω By te usual abuse of notation, let ave te meaning of te maximum diameter of all te elements of te triangulation Τ Ten, by introducing a specific finite element on te family of triangulations { Τ } I, we define a family { } I of finite element spaces We restrict ourselves to consider te approximations of (MP) tat can be written as a Galerkin finite element problem of te form:, u 0 R a u, v u, v Find ( ω ) suc tat ( ) ω ( ) H = v, (FEP) wit I In order tat a Galerkin finite element approximation can be written in te form of problem (FEP), it must satisfy te following constraints: (i) (ii) (iii) (iv) te problem domain is exactly covered by te finite element triangulations; te essential boundary conditions are exactly imposed; elements conforming in H(curl) are used; te integrals involved in te bilinear forms are exactly evaluated In fact, in a finite element approximation of (MP) tat does not satisfy conditions (i)-(iv) eiter or te sesquilinear forms are not exactly te same of (MP), but rater approximations of tem depending on Hence, our analysis does not cover non-conforming elements and te most general situation easily manageable can be described as follows: te domain Ω is a Lipscitz polyedron; te common boundary of Γ and Γ is te union of continuous piecewise straigt lines (wose pieces do not necessarily coincide wit te edges of Ω); te material properties are globally discontinuous µ and ε tensor fields tat are piecewise polynomials on Lipscitz polyedral subdomains 4 A nonstandard situation Let us stress tat we ave te following peculiar situation: Problem (MP) as te same solutions of problem (P) plus te infinite dimensional eigenspace 0 belonging to ω = 0 Problem (FEP) is an approximation of problem (MP) and some of its solutions will be 5

6 ω 0 Our aim is obtaining approximate solutions of (P) from (FEP) approximations of ( = 0, ) In spite of teir similarity, problems (P) and (MP) are very different in caracter Tis difference, wic is revealed by te presence of an infinite dimensional eigenspace in problem (MP), originates from te fact tat is compactly embedded in H, wile is not As a consequence te operator associated to problem (P) is compact, wile te operator associated to problem (MP) is only continuous Hence, by substituting problem (MP) for (P), a difficulty (te divergence-free constraint) as been eluded, but new ones ave been introduced: te extraneous solution ( ω = 0, 0 ) and te noncompactness of te underlying operator Tis latter entails tat stronger conditions must be satisfied to ensure a good numerical beaviour of (FEP) SPURIOUS-FREE FINITE ELEMENT APPROXIMATIONS: DEFINITION In order tat (FEP) can be really used to approximately solve (P), it is now clear tat we need two tings: (FEP) must be a good approximation of (MP) in spite of te noncompactness of te underlying operator All te solutions of (FEP) approximating ( ω = 0, 0 ) must be easily identified to be discarded Approximating te noncompact eigenproblem Te natural way to satisfy te first requirement,5 is to ask tat, as 0, te operator A associated to (FEP) converges to te operator A associated to (MP) in te sense of te spectral approximation of general continuous operators developed in Reference Te aforementioned operators are defined by A : ( A v, w) = ( v, w) H v, w () : A v, w = v, w v, w A ( ) ( ) H and ave exactly te same eigenspaces as (MP) and (FEP), respectively Teir eigenvalues λ and λ are linked to te corresponding ones of (MP) and (FEP) by ω = and λ ω = λ, respectively T Let us indicate by σ ( ) and σ p ( T) generic T operator and by ( T), respectively, te spectrum and te point spectrum of a E λ te eigenspace associated to its eigenvalue ( T) λ σ p and 6

7 state te following properties of A and A Te spectra ( A) A σ and σ ( ) are related by σ ( A) = σ p ( A) { 0} and ( A) p σ consists of λ = wit infinite multiplicity plus a denumerable set of real isolated eigenvalues wit finite multiplicities, belonging to ( 0,) and accumulating only at 0 Moreover, E ( A) = 0 For any I, σ ( A ) consists of a finite set of real isolated eigenvalues wit finite 0 In particular, if = {} 0, ten λ is an multiplicities, belonging to (,] eigenvalue and ( A ) 0 E = 0 0 Since te spectrum of A is muc simpler tan expected for a generic non-compact operator, we will exploit tis fact to write te conditions defining convergence in te sense of Reference in a simpler equivalent way, wic makes no sense in te general case, but is meaningful for A and A Since all te eigenvalues of A are isolated and real, a collection of mutually disjoint and arbitrarily small real neigbouroods of tese eigenvalues can be found and used to express condition (γ) and (δ) of Reference directly in terms of eigenspaces, witout using spectral projectors To be specific, for any ε > 0, let us associate to eac λ σ p ( A) a real neigbourood N λ λ ε, λ + ε in suc a way tat N ( µ ) N( ) = wenever µ σ A ( ) ( ) and ten define S N( λ) λ σ p ( A) N () 0 ( ε, + ε) of 0 σ( A) { } p =, p µ wit ( ) = Let us introduce also a real neigbourood in order to define Λ = N() λ Notice tat N () 0, no matter ow small is cosen, overlaps an infinite number of N ( λ) neigbouroods and tat we ave ( ) Λ σ A = Λ Finally, for any λ ( A), define σ A and, moreover, ( ) E N ( λ)( A ) ( A ) N( λ) ( A ) = E λ σ λ ε> 0 We are now in a position to give te following definition, wic is noting oter tat convergence in te sense of Reference in our particular situation: DEFINITION :We will say tat (FEP) is a spectrally correct approximation of (MP) if, for any given S and Λ defined as above, te following conditions are fulfilled: λ σ ( A) σ p Completeness of te spectrum: λ σ( A) lim 0 inf λ σ ( A ) λ λ = 0 (CS) Non-pollution of te spectrum: 0 > 0 suc tat σ( ) Λ A < 0, I (NPS) 7

8 Completeness of te eigenspaces: ( A) u E ( A) λ σ p λ lim 0 inf u E N ( λ )( A ) u u = 0 (CE) Non-pollution of te eigenspaces: λ σ p ( A) lim 0 sup u E N u inf ( )( A ) u E ( A) λ λ u u = 0 (NPE) Condition (CS) says tat for any eigenvalue of (MP) we can find a sequence of eigenvalues of (FEP) converging to it as 0 In oter words, no eigenvalue of (MP) is missed by its approximation (FEP) Condition (NPS) says tat, by sufficiently reducing, we can simultaneously pus all te eigenvalues of (FEP) out of any given closed and bounded subset of te complement of te spectrum of (MP) to te real axis Tis conditions means tat no bounded sequence of eigenvalues of (FEP) can ave a nonvanising distance from te spectrum of (MP) as 0 In oter words, (FEP), as an approximation of (MP), does not introduce any eigenvalue extraneous to te original problem Conditions (CS) and (NPS) togeter say tat te set of all te accumulation points of te union of te spectra of (FEP) obtained for all I is just te spectrum of (MP) As te eigenvalues of (MP) are not necessarily simple, te remaining two conditions do not separately concern single eigenvectors, but rater wole eigenspaces Owing to (CS) and (NPS), for any given eigenvalue of (MP) and for any less tan some sufficiently small value, te eigenvalue(s) of (FEP) approximating tis specific eigenvalue of (MP) can be identified as te one(s) corresponding to λ σ( A ) N( λ) Te span of te eigenvector(s) belonging to tis (tese) eigenvalue(s) of (FEP) (ie E N( λ )( A )) will take te meaning of approximate eigenspace corresponding to te eigenspace belonging to te given eigenvalue of (MP) (ie E λ ( A) ) Condition (CE) says tat, for any element of any eigenspace of (MP) we can find a sequence of elements of te corresponding approximate eigenspace converging to it as 0 In oter words, no eigenspace of (MP) is missed by (FEP), not even partially Condition (NPE) says tat, for any eigenspace of (MP), te greatest distance a normalized element of te corresponding approximate eigenspace can ave from te eigenspace of (MP) itself vanises as 0 Tis conditions means tat no sequence tat consists of normalised eigenvectors of (FEP) corresponding to a bounded sequence of eigenvalues can ave a nonvanising distance from te union of all te eigenspaces of (MP) as 0 In oter words, (FEP), as an approximation of (MP), does not introduce any eigenvector extraneous to te original problem 8

9 One more condition to be satisfied However, if we ave just convergence in te sense of Reference (ie spectral correctness, as defined above), some approximations of ( ω = 0, 0 ), te sole eigensolution of (MP) not satisfying (P), will appear as sequences of positive eigenvalues of (FEP) converging to zero as 0 On te oter and, wen (FEP) is regarded as an approximation of (P), any ω = 0, is extraneous to te original problem In eigensolution of (FEP) approximating ( 0 ) order tat te elements of te sequences approximating ( = 0, ) ω 0 can be easily identified and discarded on any finite mes, te best situation occurs wen, for any I, all of tem are actually exact eigensolutions of (MP) (ie tey satisfy ω = 0, u 0 ) In order to enforce tis situation, besides spectral correctness, we ask tat ω = 0 is an isolated point of te union of te spectra of (FEP) obtained for any I, so ruling out any sequence of positive eigenvalues of (FEP) converging to zero Hence, we give te following definition, in wic te above condition is reprased in terms of te operator A associated to (FEP): DEFINITION : We will say tat (FEP) is a spurious-free approximation of (P) if (FEP) is a spectrally correct approximation of (MP) and, moreover, te following condition is satisfied: Isolation of te discrete kernel: = λ is an isolated point of σ ( ) I A (IDK) 4 CHARACTERISTIC CONDITIONS FOR SPURIOUS-FREE APPROXIMATIONS { = 0 w } Let us define v ( v, w ) of te irrotational subspace =, te ortogonal complement in 0 0 and notice tat, wile 0 0, we ave We are now in a position to state our main result giving caracteristic conditions for spurious-free approximations: THEOREM : Te following tree independent conditions are necessary and sufficient in order tat (FEP) is a spurious-free approximation of (P) Completeness of te approximating subspace: lim inf v v 0 v = 0 v (CAS) Completeness of te discrete kernel: lim inf 0 v 0 v v = 0 v 0 (CDK) 9

10 Discrete compactness property: Any sequence { v } I suc tat (DCP) v and v C for some constant C and any I v ) converging in H contains a sub-sequence (still denoted by { } Condition (CAS) means tat te finite element space is able to approximate te space were te solution is sougt; (CDK) means tat te finite element space contains an irrotational subspace able to approximate te kernel of te curl operator and (DCP) is te discrete counterpart of te compact embedding of in H, but is not a trivial consequence of it, because It is classical tat condition (CAS) alone is enoug to ensure convergence for source problems and eigenvalue problems for compact operators We ave also proved tat (CAS) and (DCP) togeter are sufficient to ensure tat (FEP) is a spectrally correct approximation of (MP) In te electromagnetic community, conditions (CAS) and (CDK) togeter ave been erroneously believed for long time to be sufficient in order to obtain a spurious-free approximation, owing to an unsatisfactory definition of "spurious-free 6,7,8 " Let us consider one more condition, wic is equivalent to (IDK), and state a result involving it Discrete Friedrics Inequality: α > 0 suc tat a( v, v ) α v v I (DFI) THEOREM : (CAS), (DFI) and (DCP) constitute anoter set of independent conditions necessary and sufficient in order tat (FEP) is a spurious-free approximation of (P) In spite of wat migt be superficially tougt, owever, (DFI) and (CDK) are not equivalent, in general In fact, for te sake of preciseness, we ave te following situation: (CAS) and (DFI) togeter imply (CDK), wile (DCP) and (CDK) togeter imply (DFI) Let us also point out tat if only (CAS) and (DCP) are satisfied, ten just (IDK) is not fulfilled, wile te four conditions in Definition still old true In tis case, spurious eigenvalues may occur only in a neigbourood of zero tat can be narrowed by mes refinement However, we do not know any practical finite element sowing spurious eigenvalues tat beave in tis way 5 APPLICATION UNDER REALISTIC ASSUMPTIONS Te above teory is completed by te following powerful result tat permits to apply it in its full generality : 0

11 THEOREM : For a given sequence { } and only if te same sequence { } and ε =, (FEP) is a spurious-free approximation of (P) if provides a spurious-free approximation of (P) for µ = In fact, owing to Teorem, wenever we are able to prove tat a specific finite element gives rise to a spurious-free approximation for a problem involving only one isotropic omogeneous material, te result immediately extends to any number of anisotropic and inomogeneous materials abutting Te notable importance of Teorem originates from te fact tat all te proofs of (DCP) given till now concern te case of a single isotropic omogeneous material and do not extend directly to te general case as tey make use of a s regularity ypotesis (for te sake of precision, tat problem solution belongs to H ( Ω) wit s > / ) wic is no longer satisfied wen material properties are discontinuous Te same regularity assumption is violated also if general mixed boundary conditions are present However, if mixed boundary conditions arise only from te exploitation of problem symmetries to reduce te problem domain to a symmetry cell, problem solution for a single s isotropic omogeneous material still belongs to H ( Ω) wit s > / Tis is te case, for instance, if we assume tat, wenever bot Γ and Γ, a larger domain Ω ~ Ω can be generated from Ω, witout any overlapping, by a finite number of reflections wit respect to planes containing plane parts (not necessarily connected) of eiter Γ or Γ (but not bot) in suc a way tat Ω ~ originates eiter from Γ only or from Γ only By applying our teory wit only tis restriction to te rater weak ypoteses under wic it as been developed, we ave proved in Reference, for first time in tis very general setting, tat Nedelec s edge elements 9,0 of any fixed order on regular tetraedral meses, wen used in (FEP), give rise to spurious-free approximations of (P) Let us stress once more tat tese general assumptions permit inomogeneous, anisotropic and discontinuous material properties, topologically nontrivial problem domains and mixed boundary conditions arising from symmetry exploitation, so covering most of te situations occurring in real-life applications 6 EXPLAINING OBSERED BEHAIOURS Finally, let us discuss te beaviour of some known elements in te ligt of our teory Nodal vector element approximations on general regular triangulations satisfy only (CAS) As it is well known, tese approximations sow someting like a flow of spurious eigenvalues ceaselessly entering te spectrum from above and nonuniformly converging to zero as 0 Tis beaviour violates bot (NPS) and (IDK) iolation of (NPS) implies tat te union of te spectra of (FEP) obtained for all I as some accumulation point wic is not in te spectrum of (MP), even toug tis may be difficult to observe in numerical experiments A simple abstract example of Galerkin approximation wit global basis (not a finite element one) tat satisfies only (CAS) and as spurious eigenvalues

12 sowing tis beaviour as been built (Example 75 of Reference ) It as been recently found a case (D nodal vector elements on a so called criss-cross mes, ) in wic te spurious modes exibit a different beaviour: some eigenvalues of (FEP) converge to positive values not in te spectrum of (MP) (so violating (NPS)), wile te corresponding eigenvectors do not converge Wat is te reason of tis unusual beaviour? In tis case, (CAS) and (DFI) are satisfied 4, but (DCP) is surely not, since for tis case te existence of spurious eigenvalues as been analytically proved Condition (DFI), being equivalent to (IDK), proibits any sequence of positive eigenvalues of (FEP) converging to zero Hence, te more usual beaviour of te spurious modes cannot take place On te oter and, wandering sequences of eigenvectors are permitted, as (DCP) is not satisfied It is wortwile to stress tat, as (CAS) and (DFI) togeter imply (CDK), tis is a counterexample to te aforementioned common opinion tat (CAS) and (CDK) are sufficient to avoid spurious modes,4 It sould be clear, owever, tat te unusual beaviour sown by te spurious modes in tis case is not caused by te fact tat (CDK) is satisfied, but rater by te fact tat (DFI) olds true In fact, (CDK) does not proibit sequences of positive eigenvalues converging to zero (see Example 7 of Reference ), unless corroborated by (DCP) (remember tat (DCP) and (CDK) togeter imply (DFI)) Edge element approximations on regular triangular or tetraedral meses do satisfy (CAS), (CDK), (DFI) and (DCP),,4 and are spurious-free in te sense of Reference, as expected REFERENCES [] S Caorsi, P Fernandes, and M Raffetto, On te convergence of Galerkin finite element approximations of electromagnetic eigenproblems, CNR-IMA, Genova, Italy, Tecnical Report No7/99, 999 and SIAM J Numer Anal, accepted [] J Descloux, N Nassif, and J Rappaz, On spectral approximation, Part : Te problem of convergence, RAIRO Numer Anal,, 97- (978) [] S Caorsi, P Fernandes, and M Raffetto, Approximations of electromagnetic eigenproblems: a general proof of convergence for edge finite elements of any order of bot Nedelec s families, CNR-IMA, Genova, Italy, Tecnical Report No6/99, 999 [4] P Fernandes and G Gilardi, "Magnetostatic and electrostatic problems in inomogeneous anisotropic media wit irregular boundary and mixed boundary conditions", Matematical Models and Metods in Applied Sciences, 7, (997) [5] S Caorsi, P Fernandes and M Raffetto, Towards a good caracterization of spectrally correct finite element metods in electromagnetics, COMPEL, 5, -5 (996) [6] S H Wong and Z J Cendes, Combined finite element-modal solution of treedimensional eddy current problems, IEEE Trans Magn, 4, (988) [7] S H Wong, "Stable finite element metods for eddy current analysis," PD Dissertation, Carnegie Mellon University, Pittsburg, PA, Marc 7, 989 [8] A Bossavit, Solving Maxwell's equations in a closed cavity, and te question of 'spurious modes', IEEE Trans Magn, 6, (990)

13 [9] JC Nédélec, Mixed finite elements in R, Numer Mat, 5, 5-4 (980) [0] JC Nédélec, A new family of mixed finite elements in R, Numer Mat, 50, 57-8 (986) [] P G Ciarlet, Te Finite Element Metod for Elliptic Problems, Nort-Holland, Amsterdam, 978 [] D Boffi, P Fernandes, L Gastaldi, and I Perugia, Computational models of electromagnetic resonators: analysis of edge element approximation, SIAM J Numer Anal, 6, (999) [] D Boffi, F Brezzi, and L Gastaldi, On te problem of spurious eigenvalues in te approximation of linear elliptic problems in mixed form, Mat Comp, 69, -40 (000) [4] S Caorsi, P Fernandes and M Raffetto, A counterexample to te currently accepted explanation for spurious modes and a set of conditions to avoid tem, CNR-IMA, Genova, Italy, Tecnical Report No/00, 000