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  1. 1. SIAM J. NUMER. ANAL. c XXXX Society for Industrial and Applied MathematicsVol. 0, No. 0, pp. 000–000 REMARKS ON THE DISCRETIZATION OF SOME NONCOERCIVE OPERATOR WITH APPLICATIONS TO HETEROGENEOUS MAXWELL EQUATIONS∗ ANNALISA BUFFA† Abstract. We aim to provide a framework for the analysis of convergence for the Galerkinapproximation for a class of noncoercive problems. We provide a sufficient condition on the finiteelement space for the convergence and optimality of the Galerkin scheme. This theory is then appliedto the study of the well-posedness and approximability of two problems in electromagnetism. Key words. finite elements, boundary elements, Maxwell equations AMS subject classifications. 65N30, 65N25, 65N38, 83N50 DOI. 10.1137/S003614290342385X 1. Introduction. The aim of this paper is to provide a general framework for theanalysis of convergence of Galerkin schemes for a class of linear continuous operatorswhich are not definite (neither positive nor negative) and has, in general, no compactinverse. More precisely, letting X be a separable Hilbert space, we consider the classof operators verifying the following assumption. Assumption 1. We assume that (a) A : X → X ′ is a linear, continuous, and injective operator. We denote by M the continuity constant; (b) there exists a stable splitting of the space X in V ⊕ W and denote by Θ the operator associated to the mapping: u = v + w → v − w; (c) there exists a compact operator T : X → X ′ and a positive α ∈ R+ such that 2 (1.1) Re (A + T )u, Θu X ≥α u X. Although the present theory could in principle be generalized to a more generalexpression of the operator Θ, we prefer to base the development of our main conceptson the above framework. This makes our exposition easier and, on the other hand,more general assumptions would be artificially complicated by the fact that we don’thave a precise application in mind. Assumption 1 can be written in a different way,which might make it more clear. We rephrase it in section 2. At the continuous level the invertibility of the operator A is an immediate con-sequence of Assumption 1. On the contrary, when we consider its Galerkin approxi-mation, some care has to be devoted to the analysis of stability and convergence ofthe associated Galerkin scheme. The standard requirement [18] on the family of finitedimensional spaces {Xh }h>0 is that h Xh = X, but this turns out to be insufficientfor ensuring the well-posedness of the associated Galerkin projection. Several papersexist on this subject and the most recent ones are devoted to the discretization ofMaxwell equations; see, e.g., [20], [21], [5], [4], [17]. On the other hand, when concen-trating on the edge elements approximation for the Maxwell problem with constantcoefficients in a bounded domain (see section 4.1 for the definition of the Maxwell ∗ Received by the editors March 3, 2003; accepted for publication (in revised form) January 30,2004; published electronically DATE. † MATI-CNR, Sede di Pavia, Via Ferrata 1, 27100 Pavia, Italy ( 1
  2. 2. 2 ANNALISA BUFFAproblem), the precise structure of the operators is used to write exhaustive, but ded-icated, results which cannot be used for similar problems. For example, when tryingto tackle the integral formulation of electromagnetic wave propagation, we find thatthe problem under consideration is mathematically more intricate (being of nonlocal,integrodifferential type) but reveals “almost” the same structure. This will be one ofthe applications we treat in this paper. It is then natural to try to extend the known results to a general class of operatorsin order to fit all of them into the same framework. This is partially possible, andthe aim of this paper is to collect some results in this direction. Thus, in section 3we consider a condition on the finite element discretization (that we call the (GAP)property) and try to extract some consequences. The idea behind this condition goesback to Kato [28] and is the basis of the theory developed recently in the context ofintegral equations; see [14], [17], [8], [26]. We analyze further the consequences of thiscondition and attempt to give a comprehensive theory. We also approach the problemof deducing from (GAP) “spectral correctness” of the approximation in the sense of[20], but the results in this direction are incomplete. Finally, the question of whether,and/or when, (GAP) is a necessary condition for well-posedness is discussed but notanswered. Section 4 is devoted to applications. More precisely, in section 4.1, we show thatthe present theory easily permits the extension of the known results about the ap-proximation of the Maxwell operator to the case of general bounded coefficients. Insection 4.2, we consider a similar physical phenomenon, but one which has a moreintricate mathematical formulation: the boundary integral formulation for electro-magnetic wave propagation for piecewise homogeneous dielectric scatterers. Here, theeffort to set up the problem is quite major, and it is a nontrivial application of thetheory developed in this paper. We try our best to emphasize the structure of theproblem. We dedicate a lot of room to this application because it shows the generalityof the approach and, moreover, it is relevant in itself. Boundary integral discretizationof electromagnetic problems is widely used in the engineering community, but math-ematics had failed until now to prove the well-posedness of some related boundaryelement schemes. Finally, in section 4 we inform the reader that we do not pretend that the sectionis self-contained; if it were, this would make the section far too long. Instead, we givedetailed references for all results we use. 2. Setting of the problem. Let H , X be two complex separable Hilbert spacessuch that X ⊂ H with dense injection. We denote by X ′ the dual space of X whenH plays the role of pivot space. We denote by · H and · X the associated normsand by (·, ·)H and (·, ·)X the inner product of H and X, respectively. Finally, ·, · Xdenotes the duality pairing in X. First of all, we rephrase Assumption 1 in order to make more clear in which classof operators we are interested. Suppose that A : X → X ′ verifies Assumption 1. Sincethe splitting X = V ⊕ W is stable, there exists a projection ΠV : X → V such that Πis onto and ker{ΠV } = W . We denote by Π′ its adjoint with respect to the duality Vproduct ·, · X . The operator A has a natural matrix representation associated withthe splitting: AV V AV W AV V = Π′ AΠV , V AW W = (I − ΠV )′ A(I − ΠV ),A ↔ with AW V AW W AV W = Π′ A(I − ΠV ) , V AW V = (I − ΠV )′ AΠV .Equation (1.1) can be expressed now by the following statement: The operator AV V −
  3. 3. REMARKS ON DISCRETIZATION OF NONCOERCIVE OPERATOR 3AW W + (AV W − AW V ) verifies the G˚ arding inequality. In the applications we havein mind, it will happen that (AV W − AW V ) is either equal to zero (the self-adjointcase) or compact. In this case, Assumption 1 is equivalent to the requirement thatAV V : V → V ′ and −AW W : W → W ′ verify a G˚ arding inequality. Let a : X × X → C be the bilinear form associated with A, i.e., a(u, ut ) = Au, ut X . We then solve the following. Problem 1 (continuous problem). Given f ∈ X ′ , find u ∈ X : a(u, ut ) = f, ut X ∀ ut ∈ X. It is easy to see by the following that Problem 1 is well-posed. Theorem 2.1. For every f ∈ X ′ , there exists a unique solution u ∈ X of the ˜ ˜problem Au = f . Finally, there exists an isomorphism Θ : X → X such that Θ − Θis compact and ˜ Re Au, Θu ≥ α u 2 X. Proof. The fact that A is injective and A + T is invertible implies that A as ˜well as its adjoint is invertible (see, e.g., [23]). It is immediate to construct Θ as˜ = (I + (A′ )−1 T ′ )Θ, where A′ , T ′ are the adjoints of A , T .Θ Finally, we are interested in operators enjoying some further properties and showhow these have discrete counterparts for the associated Galerkin scheme. To this aim we introduce two assumptions. Assumption 2. Let X = V ⊕ W be the decomposition in Assumption 1; thenV ֒→ H is compact. Assumption 3. Let X = V ⊕ W be the decomposition in Assumption 1; then 2 Re a(v, v) ≥ β v X ∀ v ∈ V. 3. Discretization. In this section we analyze the Galerkin discretization ofProblem 1. Our aim is to provide sufficient conditions for the stability of the dis-crete problem and for the quasi optimality of the discretization scheme (in the senseof Ciarlet [18]). The structure of this section is similar to the one chosen by Caorsi, Fernandes,and Raffetto in [16], while some of the results are a revision of the approach chosenin [17] and [8]. Let {Xh }h≥0 ⊂ X be a family of finite dimensional subspace verifying the follow-ing. Complete approximation space (CAS). limh↓0 inf uh ∈Xh u − uh X = 0. When (CAS) is verified, we say that Xh is approximating in X. We denote byIh : X → Xh the projection operator defined as ((u − Ih u, v))X = 0 ∀ v ∈ Xh . Thefamily {Ih }h>0 is uniformly bounded with respect to h, and, if (CAS) holds, Ih → Ipointwise in X. The discrete variational problem reads as follows. Problem 2 (Galerkin projection). Find uh ∈ Xh such that(3.1) a(uh , ut ) = f, ut h h X ∀ ut ∈ Xh . h Gap property (GAP). We say that Xh verifies a gap property associated withAssumption 1 when there exist two subsets Vh , Wh of Xh such that δh = max{δ(Vh , V ), δ(Wh , W )} → 0 when h → 0,
  4. 4. 4 ANNALISA BUFFAwhere v − vh X δ(Vh , V ) = sup inf . vh ∈Vh v∈V vh X The (GAP) property has several consequences which are here written as theoremsand lemmas. First of all, due to (GAP), there exists a continuous operator Π : Vh → Vsuch that vh − Πvh ≤ 2δh vh X (the same for W ). This implies the following. Lemma 3.1. The fact that δ(Vh , V ) ≤ δh , δh → 0 when h → 0 implies that everycontinuous projector P0 : X → V , which is onto in V, verifies vh − P0 vh X δh vh X, vh ∈ Vh . Proof. It is enough to compute vh − P0 vh X = vh − Πvh + Πvh − P0 vh X = (I − P0 )(vh − Πvh ) X ≤ 2 I − P0 δh vh X. We then can prove the following theorem. Theorem 3.2. (GAP) implies that the splitting Xh = Vh ⊕ Wh is uniformlystable ∀ h < h1 for some h1 ∈ R+ . Proof (cf. [8]). We denote by P the projection with range V and kernel W . Itcommutes with conjugation. For any (vh , wh ) ∈ Vh × Wh we have, with uh = vh + wh ,(3.2) vh − P vh X ≤ 2 I − P δh vh X δh vh X,and similarly,(3.3) P wh X = wh − (I − P )wh X δh wh X.We use the identity vh = P (uh ) + ((I − P )wh − wh ) + (vh − P vh )and, by triangle inequality, we obtain vh X ≤ P (vh + wh ) X + (I − P )wh − wh X + P vh − vh X ≤ P uh X + 2 P δh wh X + 2 I − P δh vh X .Similarly, wh X ≤ I −P uh X + 2 I − P δh vh X + 2 P δh wh X.Adding and rearranging we obtain for h small enough(3.4) vh X + wh X ≤ 2(1 − δh max{ P , I − P })−1 max{ P , I − P } uh X. Since the splitting is stable for h sufficiently small, we denote by Ph : Xh → Xhthe associated projection operator having Vh as range and Wh as kernel. Theorem 3.3. (GAP) and (CAS) imply that Vh is approximating in V and thatWh is approximating in W, i.e.,(3.5) lim inf v − vh X = 0, lim inf w − wh X =0 ∀ v ∈ V, w ∈ W. h↓0 vh ∈Vh h↓0 wh ∈Wh
  5. 5. REMARKS ON DISCRETIZATION OF NONCOERCIVE OPERATOR 5 Proof. We estimate the best approximation error as follows:(3.6) inf v − vh X ≤ v − Ph Ih v X. vh ∈VhIt is easy to see that v − Ph Ih v = P (v − Ih v) + (P − Ph )(Ih v) and that, if we setIh v = vh + wh with vh ∈ Vh , wh ∈ Wh , we obtain (P − Ph )Ih v X ≤ P vh − vh X + P wh X δh Ih v X.As a consequence, inf v − vh X v − Ph Ih v X vh ∈Vh P (v − vh ) X + δh Ih (v) X v − Ih v X + δh v X.The statement is then proved since Ih is a linear and continuous operator, and (CAS)implies that (I − Ih ) is pointwise converging to 0. Remark 3.4. In the particular case in which Wh ⊂ W , we have of course thatδ(Wh , W ) = 0. This means that the condition δ(Vh , V ) → 0 for h → 0 implies thatWh is approximating in W . Theorem 3.5. (GAP) implies the following: (i) If Assumption 3 holds, then there exists an h2 such that ∀ h < h2 , it holds that β 2 Re a(vh , vh ) ≥ vh X. 2 (ii) If Assumption 2 holds, then we have the following: Let {vh }h>0 be a sequence such that vh ∈ Vh ∀ h > 0, and it verifies vh X ≤ 1 ∀ h; there exists a subsequence (denoted again by {vh }h>0 ) and a v ∈ V such that vh → v strongly in H. Proof. (i) Let vh ∈ Vh . Then Re a(vh , v h ) = Re a(Πvh , Πvh ) + Re{a(Πvh − vh , Πvh ) − a(vh , Πvh − vh )} 2 ≥ Πvh X − 2M vh X 2δh vh X.We conclude by recalling the definition of Π and fixing h2 such that β(1 − 2δh ) −4M δh (1 + δh ) ≥ β/2. (ii) Consider a sequence {vhj }j∈N , hj → 0, for j → ∞ such that vhj X ≤ 1. Wedefine vj = Πvhj ; by continuity of Π, we have vj X ≤ C. This implies that thereexists an increasing sequence jk such that vjk → v strongly in H due to Assumption3. On the other hand, (GAP) implies that vhjk − vjk X ≤ Cδhjk , where δhjk → 0,when k → ∞. Hence,(3.7) vhjk − v H ≤ vjk − v H + vhjk − vjk X → 0, k → ∞,which means vhjk → v strongly in H. Remark 3.6. Property (ii) in Theorem 3.5 is commonly called the discrete com-pactness property for Vh and has been the object of several papers concerning edgeelements approximation for Maxwell equations. See the very recent book [32] or pa-pers [29], [30], [2], [3], [33]. Further comments are due, and we postpone them tosection 4 (Remark 4.5).
  6. 6. 6 ANNALISA BUFFA We come now to the well-posedness of the discrete problem. Such a result hasbasically been proved in [14] (see also [8]). We report it here in its most general form. Theorem 3.7. Let (CAS) and (GAP) hold. There exists an h3 such that ∀ h <h3 , Problem 2 is well-posed. Moreover, let u ∈ X, uh ∈ Xh be the solution of Problems1 and 2. We have(3.8) u − uh X inf u − ξh X. ξh ∈Xh ˜ Proof. As in the proof of Theorem 2.1, we have with Θ := (I + (A′ )−1 T ′ )Θ that ∀u ∈ X : ˜ Re Au, Θu X ≥α u 2 X. ˜Moreover, Θ − Θ = (A′ )−1 T ′ Θ is compact. Then ˜ (I − Ih )Θuh ˜ ≤ (I − Ih )(Θ − Θ)uh + (I − Ih )Θuh . X X XUsing (CAS) we have ∀ U ∈ X that (I − Ih )U ˜ → 0 as h → 0. Since Θ − Θ is Xcompact we obtain that ˜ ǫh := (I − Ih )(Θ − Θ) → 0 as h → 0. X→XNow let uh ∈ Xh be arbitrary. Then uh has the decomposition uh = v +w with v ∈ V ,w ∈ W , and we have Θuh = v − w. There is also the decomposition uh = vh + whwith vh ∈ Vh , wh ∈ Wh . We have for Θuh = v − w (I − Ih )Θuh X = (I − Ih ) Θuh − (vh − wh ) X(3.9) (v − w) − (vh − wh ) X ( v − vh X + w − wh X ).Now, using the same argument as in the proof of Theorem 3.3, we have v − vh X ( P vh − vh X + P wh X) δh uh X.As w − wh = −(v − vh ) we obtain(3.10) v − vh X + w − wh X δh uh Xand we obtain ∀ uh ∈ Xh ˜ (I − Ih )Θuh ≤ (ǫh + Cδh ) uh , X Xwhich implies that for sufficiently small h and ∀ uh ∈ Xh ˜ ˜ 2 2 Re Auh , Ih Θuh ≥ Re Auh , Θuh − C(εh + Cδh ) uh X ≥ α/2 uh X . ˜Since Ih Θ : Xh → Xh is bounded independently of h, we have proved that there existα > 0 and h⋆ > 0 such that ∀h < h⋆ Re Auh , uth α(3.11) inf sup ≥ . 0=uh ∈Xh 0=ut ∈X h h uh X ut Xh 2
  7. 7. REMARKS ON DISCRETIZATION OF NONCOERCIVE OPERATOR 7It is well known that this discrete inf-sup condition implies that Problem 2 has aunique solution and that (3.8) holds. We end this section with a result concerning the correctness of the spectral ap-proximation, i.e., we want to know if the Galerkin operator Ah : Xh → X ′ is a correctspectral approximation of A. To this aim, we need to recall some definitions andintroduce some nomenclature. First of all, we consider the solution operators S : H → X, Sh : Xh → Xh definedas(3.12) a(Su, ut ) = (u, ut )H ∀ ut ∈ X , a(Sh uh , ut ) = (uh , uh )H ∀ ut ∈ Xh . h t h By Theorems 2.1 and 3.7, we know that S , Sh exist and are continuous, at leastfor h sufficiently small. Let σ(S) denote the spectrum of the operator S, and σ(Sh ) the one for theoperator Sh . Finally, following [21] (see also [20]), we define S − Sh h = inf (S − S)uh X. uh ∈Xh , uh X ≤1 It is known that if (CAS) holds and we also have that S −Sh h → 0 when h → 0,then Sh provides a correct approximation of the spectrum in the sense expressedin [20]. We report the details of this definition only in the self-adjoint case andrefer to [20] for the general case; when S and Sh are self-adjoint, we say that Sh isasymptotically spectrally correct if the following hold: 1. limh↓0 δ(λ, σ(Sh )) = 0 ∀ λ ∈ σ(Sh ). 2. If λ has multiplicity m, there are exactly m discrete eigenvalues converging to λ. 3. Let λ ∈ σ(S) with multiplicity m and Eλ (S) the associated eigenspace, and λh,i ∈ σ(Sh ), i = 1, . . . , m, the discrete approximation of λ with Eλh,i (Sh ) the corresponding eigenspace. Then δ(Eλ (S), ⊕i Eλh,i (Sh )) , δ(⊕i Eλh,i (Sh ), Eλ (S)) → 0 , h → 0. Theorem 3.8. Let Assumption 2 hold and suppose, moreover, that sup inf Swh − λh X → 0 when h → 0. wh ∈Wh , wh X ≤1 λh ∈XhThen (CAS) and (GAP) imply that Sh − S h → 0 when h → 0. Proof. Let us fix xh ∈ Xh , xh X ≤ 1. We estimate (S − Sh )xh 2 X ˜ a((S − Sh )xh , Θ(S − Sh )xh ) 2 a((S − Sh )xh , Θ(S − Sh )xh ) + ǫh (S − Sh )xh X inf a((S − Sh )xh , Θ(S − Sh )xh − Θh λh ) λh ∈Xh(3.13) 2 + ǫh (S − Sh )xh X 2 inf a((S − Sh )xh , Θ(Sxh − λh )) + ǫh (S − Sh )xh X λh ∈Xh + δ h λh X (S − Sh )xh X, ˜where we have used that Θ − Θ is compact, the Galerkin orthogonality and, finally,that (Θ − Θh )λh X δh λh X , which is an immediate consequence of (GAP). From
  8. 8. 8 ANNALISA BUFFA(3.13), for h small enough, and using the continuity of the bilinear form a, we candeduce that(3.14) (S − Sh )xh X inf Sxh − λh X + δ h λh X, λh ∈Xhwhich is “almost” a pure approximation property. Since Sxh 1, inf Sxh − λh X = Sxh − Ih Sxh X = inf Sxh − λh X. λh ∈Xh λh ∈Xh λh X 1It is easy to see that, ∀ xh such that xh X ≤ 1, (3.14) implies(3.15) (S − Sh )xh X δh + inf Sxh − λh X. λh ∈XhNow, select xh ∈ Vh . Assumption 2 ensures that there exists a v ∈ V such that, up ˜to extractions, xh − v H → 0 when h → 0. Thus ˜ (S − Sh )xh X δh + xh − v ˜ H + inf S˜ − λh v X. λh ∈XhHence, (CAS) allows us to conclude that sup (S − Sh )xh → 0 when h → 0. xh ∈Vh , Vh X ≤1 The statement is proved since 2 2 sup (S − Sh )uh X ≤ sup (S − Sh )vh X uh ∈Xh , uh X ≤1 vh ∈Vh , vh X 1(3.16) 2 + sup (S − Sh )wh X, wh ∈wh , wh X 1and the second term in the right-hand side is converging to 0 by assumption. Corollary 3.9. If S|W : W → X is either compact or a multipication by asufficiently regular function, then Sh is asymptotically spectrally correct. Proof. It is a consequence of (3.15) and (3.16). Remark 3.10. This theorem is not completely satisfactory. We expect the state-ment to hold under much weaker conditions on S|W . In the case of Maxwell equationsin bounded domains, the discrete compactness property, which is a consequence of(GAP), turns out to be a sufficient condition for the associated Galerkin approxi-mation to be spectrally correct [2] and also spurious free in the sense given in [16].Finally, the following reasonable question remains open: “Letting (GAP) hold, underAssumptions 1, 2, can we prove that at least a part of the spectrum is well approxi-mated for h sufficiently small?” 4. Applications. We will present here two applications of this theory. They con-cern two different problems in electromagnetism: (i) Compute solutions of Maxwellequations in a cavity characterized by variable magnetic and electric properties. (ii)Compute the electromagnetic diffraction due to a heterogeneous/piecewise homoge-neous dielectric material. Note that this section will not be self-contained in the sense that we will not recall(with precise statements) all the known properties about the finite elements we use;we provide instead a list of references and we try, when possible, to refer to the recentbook [32] or to the review paper [24].
  9. 9. REMARKS ON DISCRETIZATION OF NONCOERCIVE OPERATOR 9 Let D (∂D, nD , resp.) denote a bounded connected Lipschitz domain (its bound-ary and the unit outer normal to D on ∂D, resp.) and Dc denote its complement.We set ǫ to be the electric permittivity and µ to be the magnetic permeability. E and H denote the electric and magnetic fields, respectively, and we assumethey satisfy the linear time-harmonic Maxwell equations(4.1) curl E − iωµH = 0 , curl H + iωǫE = J in D,where ω ∈ R+ is a fixed frequency and J is an imposed current density. Let us introduce some Sobolev spaces and some operators which will be usedthroughout this section. We define H(curl, D) = {u ∈ L2 (D)3 : curl u ∈ L2 (D)3 }; Hloc (curl, Dc ) = {u ∈ L2 (Dc )3 : curl u ∈ L2 (Dc )3 }; loc loc H(curl2 , D) = { u ∈ H(curl, D) | curl µ−1 curl u ∈ L2 (D)3 }. We denote by γD the tangential trace operator mapping u in nD × u|∂D , u ∈ 1/2 −1/2C (D). Let H× (∂D) = γD {H 1 (D)3 } and H× (∂D) be its dual with respect ∞to the natural duality pairing b(λ, ξ) = ∂D λ · (ξ × n). Note that the injection 1/2H× (∂D) ֒→ Lt (∂D) := {u ∈ L2 (Γ)3 : u · nD = 0} is compact. 2 −1/2 We set X(∂D) := {λ ∈ H× (∂D) : divΓ λ ∈ H −1/2 (∂D)}. See [7], [9], [10],and also [14] for definitions and details. The idea to keep in mind is that vectors inX(∂D) are tangential vector fields of Sobolev regularity −1/2, with surface divergenceof Sobolev regularity −1/2, and that the related definitions for nonsmooth boundariescan be given as extensions of the same well-known definitions for regular manifolds. c It is known that γD : H(curl, D) → X(∂D) and γD : Hloc (curl, Dc ) → X(∂D)are linear continuous and admit a right inverse [12], [7]. Finally, we denote by γN theNeumann trace operator associated with the mapping u → γD (µ−1 curl u). It turnsout [11], [14] that γN : H(curl2 , D) → X(∂D) is linear, continuous, and admits aright inverse. Finally, we set H0 (curl, D) = {u ∈ L2 (D)3 : curl u ∈ L2 (D)3 , γD (u) = 0}. 4.1. Maxwell interior problem. Let Ω be a Lipschitz bounded polyhedron inR3 . In this section we consider (4.1) on Ω together with a perfect conductor boundarycondition, i.e., γD E = 0 on ∂Ω. We suppose that ǫ , µ ∈ L∞ (Ω), 0 < c0 ≤ ǫ(x) , µ(x) ≤C0 , for almost all x ∈ Ω. Eliminating the field H, defining f := iωJ, and integrating by parts, we obtainthe following (well-known) variational formulation. Problem 3. Given f ∈ L2 (Ω)3 , find u ∈ H0 (curl, Ω) such that ∀ v ∈ H0 (curl, Ω)(4.2) µ−1 curl u curl v − ω 2 ǫu · v = f · v. Ω Ω Ω The following theorem is well known and has been proved, e.g., in [40] (see also[39] for the fundamental compactness result which is the basis of this theorem). Theorem 4.1. Problem 3 admits a unique solution u ∈ H0 (curl, Ω) except forω ⊂ {0} ∪ {ωj }j>0 , where {ωj } is a positive increasing sequence diverging to +∞. Ifdiv J = 0, then div(ǫu) = 0.
  10. 10. 10 ANNALISA BUFFA Now we decompose the space X := H0 (curl, Ω) as X = V ⊕ W: 1(4.3) V = {u ∈ H0 (curl, Ω) : div u = 0} , W = ∇H0 .It is known (see, e.g., [1]) that there exists a positive σ such that V ֒→ H 1/2+σ (Ω)3 .This means in particular that Ω ǫv · w is a compact bilinear form (V ֒→ L2 (Ω)is compact). Calling Θ the mapping u = v + w → v − w associated with thedecomposition (4.3), and a(·, ·) the bilinear form of the left-hand side of (4.2), wehave 2 a(u, Θu) ≥ α u H(curl,Ω) − c(u, Θu),where c(·, ·) : X×X → C is a compact bilinear form. Thus, when ω is not an eigenvalueof Problem 3, i.e., ω ∈ {0} ∪ {ωj }j>0 , Problem 3 fits exactly into Assumption 1 and /also Assumption 2. Now we pass to the discretization, and we consider the family of conformingfinite dimensional spaces {Xh }h>0 generated by N´d´lec finite elements of the first e efamily of degree k fixed. The family {Xh }h>0 corresponds to a family of triangulations{Th (Ω)}h>0 of the domain Ω which, we assume for simplicity, to be made of tetrahedra.We defer the reader to [35], [22] (see also [31]), or again [24] for a precise definition.Here we list only the properties we need: 1. The space Xh constructed by N´d´lec elements of degree k is approximating e e in H(curl, Ω), i.e., (CAS) is verified. 1 2. Let Ph be the H0 -conforming finite element space generated by piecewise polynomials of degree k on Th (Ω). Then, ∇Ph = {uh ∈ Xh : curl uh = 0}. 3. Denote by Πk the N´d´lec interpolant, we have that e e (a) Πk is well defined on the space Vh = {v ∈ V : curl v ⊂ curl Xh } and continuous as an operator from Vh to Xh ; (b) ∀ v ∈ Vh , curl v = curl Πk v, and v H(curl,Ω) ≈ Πk v H(curl,Ω) . Property 3 is a nontrivial property of N´d´lec finite elements, which is basically e edue to V. Girault, and was first used in [19]. We refer to [24] for its proof andsome comments. We set Vh = Πk Vh and Wh = ∇Ph . We need to prove thatXh = Vh + Wh . Letting uh ∈ Xh ⊂ X, it can be decomposed as uh = v + ∇p,v ∈ V , ∇p ∈ W. Since curl uh = curl v, we deduce v ∈ Vh . On the other hand,uh = Πk uh = Πk v + Πk ∇p = Πk v + ∇ph , ph ∈ Ph . Thus, ∇ph ∈ Wh , Πk v ∈ Vh . It is immediate to see that Wh ⊂ W and also Xh = Vh ⊕ Wh . We need onlyprove the following. Theorem 4.2. δ(Vh , V) , δ(Wh , W) → 0 when h → 0, i.e., (GAP) holds. Proof. The building block of this proof basically exists in several papers; see, e.g.,[24]. First of all, δ(Wh , W) = 0. Let vh ∈ Vh . We know that vh = Πk v for some v ∈ Vh . Moreover, curl v =curl vh and the regularity results proved in [1] ensure that there exists σ > 0 suchthat v ∈ H 1/2+σ (Ω) with the continuity estimate v H 1/2+σ (Ω)3 curl vh L2 (Ω)3 . Using the approximation properties of Πk on Vh , we have that ∀ v ∈ Vh , v − Πk v L2 (Ω)3 h1/2+σ ( v H 1/2+σ (Ω)3 + curl vh L2 (Ω)3 ) h1/2+σ vh H(curl,Ω) .
  11. 11. REMARKS ON DISCRETIZATION OF NONCOERCIVE OPERATOR 11 We are now ready to define the Galerkin approximation of Problem 3. Problem 4 (Galerkin). Given f ∈ L2 (Ω)3 , let ω ∈ R+ be such that Problem 3 iswell-posed. Find uh ∈ Xh such that ∀ v ∈ Xh ,(4.4) µ−1 curl uh · curl vh − ω 2 ǫ uh · vh = f · vh . Ω Ω Ω The general setting developed in section 3 (and Theorem 3.7) provides the fol-lowing statement as a corollary of Theorem 4.2. Corollary 4.3. There exists an h⋆ such that ∀ h < h⋆ Problem 4 is wellposed.Moreover, u − uh X ≤ C inf u − vh X. vh ∈Xh Remark 4.4. Thanks to our general setting, the proof of well-posedness for Prob-lem 4 is completely independent of the fact that we treat general coefficients (notethat the numerical method is proved convergent under the most general assumptionof ǫ , µ ∈ L∞ (Ω), 0 < c0 ≤ ǫ(x) , µ(x) < C0 ). Remark 4.5 (spectral correctness). The relevant eigenvalue problem associatedwith Problem 3 is, of course, the one of computing the frequency ω for which the prob-lem is not well-posed. This fits into the theory developed in section 3 when choosingas space H the space L2 (Ω)3 endowed with the inner product (u, ut )H = Ω ǫ u · ut ,u , ut ∈ L2 (Ω)3 . In this case, it is immediate to see that the associated solution 1operator S (see (3.12)) coincides with − ω2 I on W, i.e., Theorem 3.8 and Corollary3.9 can be applied and ensure asymptotic spectral correctness. This statement is notnew for Maxwell equations and has been proved in [16]. Actually, there it was provedthat the discrete compactness property (see Remark 3.6) together with the fact thatWh is approximating in W is a necessary and sufficient condition for the spuriousfree asymptotic spectral correctness (see [16] for definitions). Nonetheless, as before,the result expressed in our framework is completely independent of the fact that wetreat general coefficients. Finally, it is easy to see that for this particular application,the discrete compactness property is equivalent to (GAP) and δ(Wh , W) = 0. 4.2. Maxwell transmission problem. We suppose that the space is filled withdifferent magnetic materials; i.e., the electric permittivity and the magnetic perme-ability ǫ and µ are piecewise positive constants on a fixed nonoverlapping polyhedralpartition P of R3 , R3 = J Ωj . Moreover, we suppose that ΩJ is the only unbounded j=1element of the partition and call Ω = R3 ΩJ . √ We define the piecewise constant function k := k(x) := ω ǫµ and denote by JΣ the set of interfaces, i.e., Σ = j=1 ∂Ωi . Note that according to the notation Jintroduced, R3 = j=1 Ωj ∪ Σ. The problem we want to solve is the following: Findu ∈ Hloc (curl, R3 Σ) verifying (i)(4.5a) curl curl u − k 2 u = 0 in R3 Σ;(ii) Silver–M¨ller condition at infinity: u r 1(4.5b) curl u(r) × − iku = o , |r| → ∞; |r| |r|
  12. 12. 12 ANNALISA BUFFA(iii) suitable transmission conditions on the set of interfaces Σ. In order to makeprecise the transmission conditions, we need to introduce suitable notation. First ofall, for any vector v defined almost everywhere in R3 , vj always denotes its restrictionto Ωj . Let Γj = ∂Ωj and nj be the unit outer normal to Ωj , j = 1, . . . , J; we have atour disposal the space X(Γj ) defined at the beginning of the section. We can thenconstruct(4.6) X := X(Γ1 ) × X(Γ2 ) × · · · × X(ΓJ ) , ξ = (ξ 1 , . . . , ξ J ),endowed with the product norm ξ X = j ξj X(Γj ) . On such a space, we definethe jump operator [·] as the mapping(4.7) [ξ] : [ξ]|Γij = ξ i + ξ j ∀ i, j = 1, . . . , J s.t. Γj ∩ Γi = ∅.Now, if u solves (4.5a), we can construct the set of its Cauchy data as follows: Let γDξ j = γN uj and ξ = (ξ 1 , . . . , ξ J ).1 Then, ξ ∈ X × X . Applying the jump operator ujto each of the two lines of ξ, we impose the transmission condition as(4.8) [ξ] = f for some fixed f , f ∈ [X ]2 . Remark 4.6. We have to be careful in the definition of transmission conditionsbecause the tangential trace operators depend on the orientation. Hence, the trans-mission condition cannot be defined directly on Σ, because Σ is never orientable. A uniqueness result is available as follows. Theorem 4.7. The problem (4.5)–(4.8) admits at most one solution. Proof. This is a direct consequence of Rellich’s theorem (see M¨ller [34] for a uproof). 4.3. Statement of the problem. In order to show existence and to discretizethe problem (4.5)–(4.8), we need to formulate it in terms of boundary integral equa-tions on the interfaces Σ. We adopt a construction of the system of integral equationsinspired by [14]. Other derivations are possible; see, e.g., [27]. We first introduce the 1-dielectric problem: Find E ∈ Hloc (curl, Ωj ∪ Ωc ) such jthat: 2 curl curl E − kj E = 0 in Ωj ∪ Ωc , j(4.9) [γD ]E = m; [γN ]E = j on Γj , Silver–M¨ller radiation condition at ∞, uwhere kj ∈ R+ , kj = k(x)|Ωj , and m , j are imposed transmission conditions. Thereexists an explicit representation for the solution E of (4.9), which reads for almost allx ∈ Ωj ∪ Ωc as j E = −Ψj (j) − ΨDL (m), SL j 1 Note that the operators γ D and γN are not indexed to keep the notation shorter. If applied toa field defined on Ωj for some j, they represent the tangential trace operators on the boundary ofΩj , i.e., Γj . Moreover, if they are applied to a field u ∈ Hloc (curl, R3 ), γD u stands for the vector(γD u1 , . . . , γD uJ ), γN stands for (γN u1 , γN u2 , . . . , γN uJ ).
  13. 13. REMARKS ON DISCRETIZATION OF NONCOERCIVE OPERATOR 13where Ψj and Ψj denote the single and double layer operators for the Maxwell SL DL eik|x−y|problems as defined in [13]. More precisely, let G(x, y) = 4π |x−y| be the standardHelmholtz kernel; then for x ∈ Ωj ∪ Ωc , Ψj and Ψj are given by j SL DL j ΨSL j(x) := G(x, y)j(y) ds(y) + k −2 ∇ G(x, y) divΓ j(y) ds(y), Γj Γj ΨDL m(x) := curl G(x, y)m(y) ds(y). Γj 1 c 1 c Let {γD } := 2 (γD + γD ), {γN } := 2 (γN + γN ). We construct the operator Ajassociated with the domain Ωj as m {γD }(4.10) Aj := (−Ψj j − Ψj m). SL DL j {γN } Finally, for each Ωj , we define the antisymmetric bilinear form Bj : X(Γj )2 ×X(Γj )2 → C acting on sets of Cauchy data as m ˜ m m ˜ m(4.11) Bj , := −bj (m, ˜) + bj (m, j) ∀  ˜ , ∈ X(Γj )2 , j ˜  j ˜ where b(·, ·) is the duality pairing defined, as at the beginning of this section, asbj (µ, λ) = Γj µ · (λ × nj ). The following theorem has been proved in [14]. Theorem 4.8. Let W(Γj ) = {λ ∈ X(Γj ) : divΓ λ = 0} and let V(Γj ) be anysupplement of W(Γj ) in X(Γj ) such that (i) V(Γj ) ֒→ L2 (Γj ) is compact; (ii) the tdecomposition X(Γj ) = V(Γj ) ⊕ W(Γj ) is stable in X(Γj ). Then call Θ : X(Γj )2 →X(Γj )2 the operator associated with the mapping u = v + w → v − w componentwise.The operator Aj is injective and verifies 2 Re B((A + T )ξ, Θ(ξ)) ≥ ξ X, T : X(Γj )2 → X(Γj )2 compact. In other words, we are within Assumptions 1 and 2. Note that the supplementV(Γj ) fulfilling the assumptions of Theorem 4.8 can be constructed in several ways[14], [25]. We are now ready to give the integral formulation associated with (4.5)–(4.8). Proposition 4.9. Let A = diag{A1 , . . . , AJ }, and J(4.12) B(ξ, λ) = Bj (ξ j , λj ), ξ , λ ∈ X 2. j=1The vector field u is a solution of (4.5)–(4.8) if and only if its Cauchy data verifiesξ = ξ 0 + ξ hom , [ξ 0 ] = f, and(4.13) 2 ξ hom ∈ Xhom : B(Aξ hom , λhom ) = B(( 1 I − A)ξ 0 , λhom ) 2 2 ∀ λhom ∈ Xhom . The proof of this result is completely equivalent to the one for the correspondingHelmholtz problem and can be found in [37] (see also [38] and [15]). The next sectionis devoted to showing that this problem fits into our framework.
  14. 14. 14 ANNALISA BUFFA 4.3.1. Verification of Assumption 1. We want to prove that the operatorA : Xhom → (Xhom )′ , (Xhom )′ being the dual of Xhom with respect to the duality 2 2 2 2product B(·, ·) fits Assumption 1. We have the following theorem. Theorem 4.10. Define Whom = {ξ ∈ Xhom , divΓ ξ j = 0 ∀ j}.Then there exists a supplement Vhom of Whom in Xhom such that(4.14) Xhom = Vhom ⊕ Whom with Vhom ֒→ L2 (Σ) compact, twhere L2 (Σ) = ⊗J Lt (Γj ). Moreover, the splitting is stable, i.e., the following t j=1 2stability estimate holds: λ ∈ X , λ = v + w, v ∈ V, w ∈ W, and(4.15) v X + w X λ X v X + w X. Proof. We assume for simplicity that each Ωj is connected and simply connected,we construct the space Vhom by the definition of a projector ΠVhom : Xhom → Xhomsuch that ker{ΠVhom } = Whom , and choose Vhom = ΠVhom (Xhom ). Let ξ ∈ Xhomand BR be a ball of radius R sufficiently large to ensure that Σ ⊂ BR . Solve thefollowing problems in each Ωj : −∆pj = 0 in Ωj ∩ BR ,(4.16) ∇pj · nj = divΓ ξ j on Γj (and, for j = J , ∇pJ · nR = 0 on ∂BR ).We denote by ϕ the function defined as ϕj = ∇pj . We deduce then that div ϕ = 0on BR since div ϕj = 0 in Ωj and ϕj · nj + ϕi · ni = 0 on Γij ∀ i, j, since ξ ∈ Xhom . Using, e.g., [1], ϕ admits a vector potential in BR , V ∈ H1 (BR ) ∩ H0 (curl, BR )verifying curl V = ϕ, div V = 0.By continuity we have V H1 (BR ) ϕ L2 (BR ) divΓ ξ −1/2,Σ . 1/2Set ΠVhom ξ = γD V. By construction ΠVhom : X → H× (Σ) and ker{ΠVhom } =Whom , and for vj = (γD V)j , divΓ vj = (curl V)|Ωj · nj = ϕj · nj = divΓ ξ j . Thus, 1/2ΠVhom is a projection and we conclude by observing that H× (Σ) is compactly em-bedded in L2 (Σ). t 2 2 2 We consider an operator Θ : Xhom → Xhom which maps ξ ∈ Xhom with Hodgedecomposition (Theorem 4.10 applied twice) ξ = v + w into Θ(ξ) = v − w. The nexttheorem follows then as an immediate consequence of Theorems 4.8 and 4.10. 2 2 Theorem 4.11. There exists a compact operator T : Xhom → Xhom and aconstant α > 0 such that 2(4.17) Re B((A + T )ξ, Θ(ξ)) ≥ α ξ X2 . Proof. We use the definitions of B and A: B(Aξ, Θ(ξ)) = Bj (Aj (vj + wj ), vj − wj ) j
  15. 15. REMARKS ON DISCRETIZATION OF NONCOERCIVE OPERATOR 15with ξ = v + w, v = (v1 , . . . , vJ ) and w = (w1 , . . . , wJ ). Applying Theorem 4.8, weknow that, for each j, there exists a compact operator Tj : X(Γj )2 → X(Γj )2 suchthat(4.18) Re Bj (Aj + Tj (vj + wj ), vj − wj ) ≥ ξ j X(Γj )2 .Summing (4.18) over j, the statement is proved. Since we know that (4.13) admits at most one solution (as a consequence ofTheorem 4.7 and Proposition 4.9), this theorem says that (4.13) is in our framework,i.e., it verifies Assumption 1. As a matter of fact, it verifies also Assumption 2. 4.3.2. Discretization and verification of the (GAP) property. We concen-trate now on the discretization of the problem (4.13). First we construct a compatiblemesh on the interfaces Σ in the following way: Consider that there exists a triangu-lation Th (BR ) of a ball BR containing Σ and such that Σ is composed only of facesof the underlying triangulation Th (BR ), i.e., Σ does not cut elements of the mesh.This automatically generates a compatible triangulation of Σ, Th (Σ) and of coursealso triangulations of Γj ’s, that we denote by Th (Γj ). Each single space X(Γj ) is discretized by means of Raviart–Thomas finite ele-ments of degree k defined on Th (Γj ), [36], [22]. We denote the discrete spaces byXh (Γj ). Now, the discrete counterpart of X is Xh = Xh (Γ1 ) × · · · × Xh (ΓJ ), and thediscrete counterpart of Xhom is(4.19) (Xh )hom = Xh ∩ Xhom .Note that, thanks to the fact that the nonempty intersections Γij = Γi ∩ Γj arediscretized by means of only one triangulation, namely, Th (Σ)|Γij , the space (Xh )homis well defined as a constrained subspace of Xh . We are then ready to state thetheorem. Theorem 4.12. There exists a splitting (Xh )hom = (Vh )hom ⊕ (Wh )hom suchthat δ((Vh )hom , Vhom ) → 0 when h → 0 and δ((Wh )hom , Whom ) = 0. Proof. We assume for the sake of simplicity that each Ωj is connected and simplyconnected. We set (Wh )hom = Whom ∩ Xh . An alternate definition is the following: SetWh (Γj ) = {λh ∈ Xh (Γj ) : divΓ λh = 0}. Wh (Γj ) is the finite elements spaceof curlΓ Ph , where Ph is the space of continuous piecewise polynomials of degree k[22], [11], i.e., characterized by degrees of freedom attached to vertices, edges, andtriangles on Γj . Then we set Wh = Wh (Γ1 ) × · · · × Wh (ΓJ ); this space has for eachvertex (or edge, or triangle) belonging to an intersection Γij two sets of independentdegrees of freedom, one defining Wh (Γi ) and the other Wh (Γj ). Now, in (Wh )hom =Wh ∩ Xhom , the degrees of freedom belonging to these two sets are constrained to beequal for each vertex, edge, or triangle. Now we construct the supplement (Vh )hom . We use two intermediate finite el-ements spaces: the N´d´lec edge elements of degree k, Xh (BR ), as introduced in e esection 4.1, but on BR (with vanishing tangential component on ∂BR ), and theRaviart–Thomas finite elements of degree k, Yh (BR ), with vanishing normal compo-nent on ∂BR . We construct local N´d´lec and Raviart–Thomas elements as Xh (Ωj ) = e e
  16. 16. 16 ANNALISA BUFFAXh (BR )|Ωj and Yh (Ωj ) := Yh (BR )|Ωj (and Xh (ΩJ ) = Xh (BR )ΩJ ∩BR , Yh (ΩJ ) :=Yh (BR )|ΩJ ∩BR ). We refer to [36], [6], [22] for suitable definitions and properties. Weconstruct the space (Vh )hom as we did for Vhom in the proof of Theorem 4.10, i.e., byconstructing a projection operator Π : (Xh )hom → (Xh )hom with ker{Π} = (Wh )hom . Then let λh ∈ (Xh )hom , λh = (λ1,h , . . . , λJ,h ), λj,h ∈ Xh (Γj ). We solve thefollowing:   div Ξj,h = 0 on Ωj , Find Ξj,h ∈ Yh (Ωj ) : Ξj,h curl χh = 0 ∀ χh ∈ Xh (Ωj ),  Ωj Ξj,h · nj = divΓ λj,h on Γj .This problem is solvable since Γj divΓ λj,h = 0; moreover, it is uniquely solvable [6].Now, since λh ∈ (Xh )hom , by construction the vector Ξh , Ξh|Ωj := Ξj,h , belongs toYh (BR ). Now we solve the discrete and continuous vector potential problem: curl Ψh = Ξh on BR , Find Ψh ∈ Xh (BR ) : BR Ψh · wh = 0 ∀ wh ∈ Xh (BR ) , curl wh = 0, curl Ψ = Ξh on BR , Find Ψ ∈ H0 (curl, BR ) : div Ψ = 0 on BR .These problems are uniquely solvable [1]; moreover, the continuity estimate J(4.20) Ψ H1 (B R) + Ψh H(curl,BR ) Ξh L2 (B R) divΓ λj H −1/2 (Γj ) j=1holds. Now the proof is basically finished. Denote by Ψj,h (Ψj , resp.) the restriction ofΨh (Ψ, resp.) to Ωj (for j = J to ΩJ ∩ BR ) ∀ j. It is enough to set Πλh = λV := (γD (Ψ1,h ), . . . , (ΨJ,h )); by construction, h divΓ λV = curl Ψj,h |Γj · nj = Ξj,h · nj = divΓ λj,h . j,hThus, ker{Π} = (Wh )hom . Set λV = (γD (Ψ1 ), . . . , γD (ΨJ )); by construction,λV ∈ Vhom . It is the candidate in Vhom that is needed to verify the (GAP) property: λV − λV h X Ψ − Ψh H(curl,BR ) Ψ − Ψh L2 (BR ) h Ψ H1 (BR ) h Ξh L2 (BR )(4.21) J h divΓ λj,h H −1/2 (Γj ) h λV h X, j=1which proves that δ((Vh )hom , Vhom ) h. Note that the forth estimate in (4.21)comes from the (GAP) property for N´d´lec finite elements, which has been proved e ein section 4.1. Corollary 4.13. The following Galerkin problem admits a unique solution whenh is sufficiently small: Find ξ h ∈ (Xh )2 such that hom(4.22) B(Aξ h , λh ) = B(( 1 I − A)ξ 0 , λh ) 2 ∀ λh ∈ (Xh )2 , hom
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