Convergence analysis of a Galerkin boundary element method for electromagnetic resonance problems

In this paper a convergence analysis of a Galerkin boundary element method for resonance problems arising from the time harmonic Maxwell’s equations is presented. The cavity resonance problem with perfect conducting boundary conditions and the scattering resonance problem for impenetrable and penetrable scatterers are treated. The considered boundary integral formulations of the resonance problems are eigenvalue problems for holomorphic Fredholm operator-valued functions, where the occurring operators satisfy a so-called generalized Gårding’s inequality. The convergence of a conforming Galerkin approximation of this kind of eigenvalue problems is in general only guaranteed if the approximation spaces fulfill special requirements. We use recent abstract results for the convergence of the Galerkin approximation of this kind of eigenvalue problems in order to show that two classical boundary element spaces for Maxwell’s equations, the Raviart–Thomas and the Brezzi–Douglas–Marini boundary element spaces, satisfy these requirements. Numerical examples are presented, which confirm the theoretical results.


Introduction
The numerical solution of electromagnetic resonance problems is an important task in different fields of engineering and technology. In this paper we consider for a given bounded, simply connected Lipschitz domain Ω i ⊂ R 3 the cavity resonance problem and the scatter-This article is part of the topical collection "Waves 2019 -invited papers" edited by Manfred Kaltenbacher and Markus Melenk.
The author acknowledges support from the Austrian Science Fund (FWF): P31264. B Gerhard Unger gerhard.unger@tugraz.at 1 Institut für Angewandte Mathematik, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria ing resonance problem for an impenetrable and a penetrable scatterer arising from the time harmonic Maxwell's equations with a time variation of the form e −iωt . It is assumed that the exterior domain Ω e := R 3 \ Ω i is connected.
The cavity resonance problem, usually referred to as interior resonance problem, is given as follows: Find κ ∈ C and E i ∈ H(curl; Ω i ), E i = 0, such that where κ = ω √ εμ is the wavenumber, ω is the angular frequency, ε is the electric permittivity, μ is the magnetic permeability, and n is the unit normal vector field on the boundary Γ pointing into the exterior domain Ω e . For the cavity resonance problem and the resonance problem for the impenetrable scatterer we assume throughout this paper that ε > 0 and μ > 0 are constant. This assumption implies that the resonances of the interior resonance problem (1) are real [27,Thm. 4.18] and non-zero.
The scattering resonance problem for the impenetrable scatterer, which is usually referred to as exterior resonance problem, is formulated in Ω e and is given as follows: Find κ ∈ C and E e ∈ H loc (curl; Ω e ), E e = 0, such that: E e is "outgoing".
for r = x > r 0 and j ∈ {1, 2, 3}, where Y m n are the spherical harmonics. If κ is such that 0 ≤ arg κ < π, then the radiation condition (3) for a solution E ∈ H loc (curl; Ω e ) of Maxwell's equations in Ω e is equivalent to that the Cartesian components of E satisfy the Sommerfeld radiation condition [11,Thm. 2.15]. The latter condition is again for 0 ≤ arg κ < π equivalent to that E satisfies the Silver-Müller radiation condition [11,Thm. 6.8]. The Silver-Müller radiation condition is usually imposed for scattering problems for wavenumbers κ with 0 ≤ arg κ < π. But it is well known that for wavenumbers with negative imaginary part the Silver-Müller radiation condition does not correctly characterize outgoing waves, see e. g., [28,Sect. 1]. Since the resonances of the exterior resonance problem (2) have negative imaginary part, instead of the Silver-Müller radiation condition the radiation condition (3) is used.
The definition of the scattering resonance problem for the penetrable scatterer will be given in Sect. 5. For this kind of resonance problem we will allow complex and frequencydependent permittivities and permeabilities. Such configurations occur for example in the field of plasmonics or in the context of metamaterials.
Boundary integral formulations and boundary element methods have been considered for different kinds of electromagnetic resonance problems. Examples are the interior resonance problem [13,37], the interior transmission eigenvalue problem [12,22] or the resonance problem for the penetrable scatterer [24,26,35], to mention just a few. A rigorous convergence analysis of the boundary element approximations of such kind of resonance problems has not been provided so far.
The presented numerical analysis in this paper is based on the classical theory of the regular approximation of eigenvalue problems for holomorphic Fredholm operator-valued functions [17,18]. This theory has already been applied to boundary integral formulations of acoustic resonance problems [33] and to coupled FEM-BEM formulations of vibro-acoustic resonance problems [20]. For these cases sufficient conditions for the convergence of conforming Galerkin approximations follow from the fact that the occurring operators satisfy a standard Gårding's inequality. For electromagnetic resonance problems the occurring boundary integral operators satisfy only a generalized Gårding's inequality. In such a case additional properties of the approximation spaces are required in order to guarantee convergence. In [15,16] sufficient conditions for the convergence of a conforming Galerkin approximation for such kind of eigenvalue problems are derived in an abstract setting. In this paper we show that these conditions are satisfied for the Galerkin approximation of the proposed boundary integral formulations of the considered electromagnetic resonance problems when classical boundary elements of Raviart-Thomas or Brezzi-Douglas-Marini type are used.
The rest of the paper is organized as follows: in the next section we introduce the boundary integral formulation for the interior and exterior resonance problem and collect the basic properties of the occurring boundary integral operators. In Sect. 3 we provide abstract convergence results for the Galerkin approximation of eigenvalue problems for holomorphic Fredholm operator-valued functions, where we assume that the occurring operators satisfy a generalized Gårding's inequality. We specify for this case sufficient conditions such that the classical convergence results are valid. The abstract results of Sect. 3 are then applied in Sect. 4 to a Galerkin approximation of the boundary integral formulation of the interior and exterior resonance problem. In Sect. 5 a boundary integral formulation for the scattering resonance problem for a penetrable scatterer is analyzed and sufficient conditions on the material parameter are given such that convergence of boundary element methods are guaranteed. Numerical examples are presented in Sect. 6, which confirm the theoretical results.

Boundary integral formulation of the interior and exterior resonance problem
In this section we introduce and analyze the boundary integral formulation for the interior and exterior resonance problem (1) and (2). The main references for the definitions and properties of the occurring boundary integral operators are [8,9]. Note that the notations in the present paper only partially coincides with the notations in [8,9].

Trace spaces
In this subsection we summarize the properties of the trace spaces which we need for the analysis of the boundary integral formulations of the resonance problems. For a detailed presentation of the trace spaces related to Maxwell's equations for Lipschitz domains we refer to [7].
where V s γ and V s π are endowed with the norms The dual space of V s γ and V s π , s ∈ (0, 1), are denoted by V −s γ and V −s π , respectively. For s = 0 we set V 0 γ = V 0 π := L 2 τ (Γ ) := {u ∈ L 2 (Γ ) : u · n = 0}, where L 2 τ (Γ ) acts as pivot space for V −s γ and V s γ as well as for V −s π and V s π . Finally, we define the space . The space is a Hilbert space and the tangential trace operators γ i τ and γ e τ can be extended such that , are continuous, surjective and possess a continuous right inverse [7,Thm. 4.1]. In the sequel we will use the shorthand notation The antisymmetric pairing u, s τ := Γ (u × n) · s ds, u, s ∈ L 2 t (Γ ) can be extended to V such that V becomes its own dual [9,Thm. 3.3], i.e., there exists a linear and isometric isomorphism J × : V → V such that u, s τ = (J × u)(s) for all u, s ∈ V.

Derivation and analysis of the boundary integral formulation
The boundary integral formulation of the resonance problems (1) and (2) where is the Maxwell single layer potential and where is the Maxwell double layer potential. Here, A (κ) and V (κ) are the vectorial and scalar single layer potentials related to the Helmholtz equation, which have the integral representations Let (κ i , E i ) be an eigenpair of (1) and let us extend E i in Ω e by zero, then the Stratton-Chu representation formula (5) gives If (κ e , E e ) is an eigenpair of (2) and if we extend E e by zero in Ω i , then we have We consider a boundary integral formulation of the eigenvalue problems (1) and (2) in terms of the single layer boundary integral operator S(κ) which is defined by Here V (κ) is the single layer operator of the Helmholtz equation and the pairing ·, · ∓ 1 2 denotes the duality pairing of H − 1 2 (Γ ) and H 1 2 (Γ ). By applying the tangential trace to (6) and (7) we see that (κ i , E i ) and (κ e , E e ) satisfy the following boundary integral equation As boundary integral formulation of the eigenvalue problems (1) and (2) we consider the following eigenvalue problem: Find κ ∈ C \ {0} and u ∈ V \ {0} such that: Note that this eigenvalue problem is nonlinear with respect to the eigenvalue parameter κ. The eigenvalue problem (10) is referred to as eigenvalue problem for the operator-valued function S : C \ {0} → B(V, V). Here B(V, V) denotes the space of linear and bounded operators mapping from V into V. The equivalence of the eigenvalue problem (10) with the interior and exterior resonance problem (1) and (2) is specified next.

Proposition 1
The following assertions hold true: is an resonance pair either of the interior resonance problem (1) or of the exterior resonance problem (2). Then (κ, γ i/e N E) is an eigenpair of the eigenvalue problem (10). (ii) Let (κ, u) be an eigenpair of the eigenvalue problem (10). If κ is real, then it is a resonance of the interior resonance problem (1) and ( SL (κ)u) |Ω i is a corresponding resonance function. Otherwise, κ is a resonance of the exterior resonance problem (2) and ( SL (κ)u) |Ω e is a corresponding resonance function.
First we consider the case that κ ∈ R. Then E |Ω e = 0 because of the unique solvability of the related exterior boundary value problem [21,Cor. 5.63]. From this we get γ e N E = 0 and the jump relation γ e N SL (κ)u − γ i N SL (κ)u = −u [8,Thm. 7] implies E |Ω i = 0. Suppose now that κ is non-real. Then E |Ω i = 0 because otherwise (κ, E |Ω i ) would be an interior resonance pair which is not possible since all interior resonances are real [27,Thm. 4.18]. From the jump relation of the single layer potential we get γ e N E = u. Hence we have E |Ω e = 0.
For the analysis of the eigenvalue problem (10) and its Galerkin approximation it is essential that the single layer boundary integral operator S(κ) satisfies a generalized Gårding's inequality in V for all wavenumbers κ ∈ C \ {0}. This property is based on the direct sum decomposition where X and N are closed . We denote by R and Z the associated continuous projectors onto X and N, respectively. An equivalent norm in V is given by see [9,Thm. 3.4]. Further, we define the operator which is by construction an isomorphism.
Next, we show that the mapping S : Proof It is sufficient to show that the mapping κ → S(κ)u, r τ is holomorphic as mapping from C \ {0} into C for all u, r of a dense subspace of V, see Theorem III.3.12 in [19] and the remark following it. Let us choose γ i τ ((C ∞ 0 (R 3 )) 3 ) as dense subspace of V. Then we can use the following integral representation of the pairing ·, · τ and of the boundary integral operator S(κ) [8, Eq. 32]: Hence, it is sufficient to show that both terms on the right hand side in (15) are holomorphic in κ on C \ {0}. We give a proof for the first term only, since the second term can be treated analogously. We divide the proof in two steps.
Step 1: Consider for a fixed κ ∈ C the series expansion of the kernel We show that the order of integration and summation can be interchanged by using Lebesgue's dominated convergence theorem. Let Obviously N n=0 f n (κ, x, y) converges pointwise almost everywhere on Γ × Γ to the kernel on the left hand side in (16) Since g(κ, x, y) is integrable on Γ × Γ the order of integration and summation in (16) may be interchanged.
Step 2: We have where V (0) is the single layer boundary integral operator of the Laplace equation, see [32,Sect. 6.2]. Obviously, ∞ n=0 M n is convergent. Hence, by the Weierstrass M-test it follows that the series ∞ n=0 h n (κ) converges absolutely and uniformly on B K (0) to a limit function h(κ). Since N n=0 h n (κ) is a polynomial in κ and therefore holomorphic in κ the uniform convergence of N n=0 h n (κ) implies that the limit function h(κ) is holomorphic on B K (0). From the first part of the proof we get which shows that the first term on the right hand side in (15) defines a holomorphic function on C. Analogously, the holomorphy of the second term can be shown.

Galerkin approximation of eigenvalue problems for holomorphic Fredholm operator-valued functions
In this section we provide abstract convergence results for the Galerkin approximation of eigenvalue problems for holomorphic Fredholm operator-valued functions where we assume that the occurring operators satisfy a generalized Gårding's inequality. We specify for this case sufficient conditions such that the classical convergence results for the approximation of eigenvalue problems for holomorphic Fredholm operator-valued functions as given in [17,18] can be applied. Our analysis is based on results on the approximation of non-coercive operators [4] and on recent results on the regular approximation of operators which satisfy a generalized Gårding's inequality [15,16].
The results of this section build the abstract framework which we will utilize in order to show the convergence of the boundary element method for the approximation of the interior and exterior resonance problem as well as of the resonance problem for the penetrable scatterer.

Assumptions on the eigenvalue problem
Let V be a Hilbert space and suppose that V is equipped with a conjugation, i. e., with a continuous, unary operation v → v satisfying for all u, v ∈ V and α ∈ C. We denote by (·, ·) V the inner product in V and consider in addition a bilinear form ·, · V : V × V → C with the property where J ∈ B(V , V ) is a given isomorphism. Further, we assume that there exists a direct sum decomposition V = X ⊕ N which is stable, i. e., there exists a constant c > 0 such that for all v X ∈ X and v N ∈ N the inequality is satisfied. We define the operator Θ : V → V by Note that Θ is an isomorphism and that Let Λ ⊂ C be an open and connected subset of C and S : Λ → B(V , V ) be a holomorphic operator-valued function. We assume that S(λ) satisfies a generalized Gårding's inequality for all λ ∈ Λ of the following kind: there exist a compact operator C(λ) ∈ B(V , V ) and an α(λ) > 0 such that where as well as that S(λ) is a Fredholm operator of index zero [25,Thm. 2.33]. We want to mention that we do not require that the operator-valued functions C(·) and T (·) are holomorphic in Λ. We consider the eigenvalue problem for the operator-valued function S(·) of the form: find eigenvalues λ ∈ Λ and corresponding eigenelements u ∈ V \ {0} such that If (λ, u) satisfy (21), then we say that (λ, u) is an eigenpair of the operator-valued function S(·).

Notations and properties of eigenvalue problems for holomorphic Fredholm operator-valued functions
We briefly summarize basic results of the theory of eigenvalue problems for holomorphic Fredholm operator-valued functions [14,23]. The set is called the resolvent set of S(·). In the following we will assume that the resolvent set of S(·) is not empty. The complement of the resolvent set ρ(S(·)) in Λ is called the spectrum σ (S(·)). The spectrum σ (S(·)) has no accumulation points inside of Λ

Galerkin approximation
For the approximation of the eigenvalue problem (21) we consider a conforming Galerkin approximation. Let (V h ) h be a sequence of finite-dimensional subspaces of V and let P h : V → V h be the orthogonal projection of V onto V h . As usual we assume that The Galerkin approximation of the eigenvalue problem (21) reads as: find eigenpairs For the convergence analysis it is convenient to consider instead of the variational formulation (23) the equivalent operator formulation Further, we will also consider the eigenvalue problem for the operator-valued function J S(·), which is equivalent to the eigenvalue problem for S(·), i. e., (λ, u) is an eigenpair of the eigenvalue problem for J S(·) if and only if it is an eigenpair of the eigenvalue problem for S(·).

Regular convergence
In order to apply the convergence theory of [17,18] to the Galerkin eigenvalue problem (24) it is necessary to show that the sequence For the definition of the regular convergence we need the definition of a compact sequence first.
If the operator B is a compact perturbation of a coercive operator, then the sequence For the case that B satisfies only a generalized Gårding's inequality the regular convergence of (P h B P h ) h to B is in general not guaranteed. Sufficient conditions for that case are specified in the next lemma.
Sufficient conditions for the existence of a sequence of discrete operators (T h ) h such that (26) is satisfied will be provided next for the case of a stable splitting V = X ⊕ N and T = Θ, where Θ is defined by (19). First we need to define the gap δ V (U , W ) of two subspaces U and W of V : We say that the sequence (V h ) h satisfies a gap property with respect to the splitting V = X ⊕N if the following condition is satisfied: (GAP) There exist sequences (X h ) h and (N h ) h satisfying the following properties: This condition is a usual condition for Galerkin boundary element methods for Maxwell's equation, see e g. [4,5,[8][9][10]. We will discuss this in detail in Sect. 4 below.

Lemma 4
Let V = X ⊕ N be a stable splitting and suppose that (V h ) h satisfy (22) and the property (GAP). Then we have: (a) For any continuous projector Q : V → X which is onto in X , there exists a constant c > 0 such that for sufficiently small h we have Proposition 2 Let V = X ⊕ N be a stable splitting and Θ be defined by (19). Suppose that (V h ) h satisfy (22) and the property (GAP). Then there exists a sequence Proof Since V = X ⊕ N is a stable splitting there exists a continuous projection R : V → V with range X and kernel N .
From the decomposition V h = X h ⊕ N h and the fact that V h is finite-dimensional it follows that there exists a continuous projection R h : V h → V h with range X h and kernel N h . We will show that satisfies (27). The proof for that will be done in two steps.
(i) First we show that there exists a constant c > 0 such that for sufficiently small h By Lemma 4(b) the decomposition V h = X h ⊕ N h is uniformly stable for sufficiently small h and therefore inequality (29) follows.
(ii) Since Z = (I − R) and Z h = (I − R h ) we get from (29) also Hence, we have from which (27) follows.
Corollary 1 Let V = X ⊕ N be a stable splitting and λ ∈ Λ. Assume that S(λ) ∈ B(V , V ) fulfills a generalized Gårding's inequality of the form as in (20). Further, suppose that (V h ) h satisfy (22) and the property (GAP). Then (P h J S(λ)P h ) h converges regularly to J S(λ).

Asymptotic convergence results
In the next theorem we summarize main convergence results for the Galerkin approximation of the eigenvalue problem S(λ)u = 0 as given in (21). For additional convergence results we refer to [15][16][17][18].
be the weighted mean of all eigenvalues of the Galerkin eigenvalue problem (23) in D. Then it holds

Galerkin approximation of the interior and exterior resonance problem
In this section we apply the abstract convergence results of Theorem 1 to the Galerkin approximation of the eigenvalue problem S(κ)u = 0.
Lemma 1 shows that S(κ) satisfies for all κ ∈ C \ {0} a generalized Gårding's inequality of the form (13), where R and Z are projections associated to the splitting V = X ⊕ N as given in (11). Further, (4) shows that ·, · τ = (J·, ·) V holds, where J := J V J × is an isometric isomorphism. Since S(·) is in C \ {0} in addition holomorphic, S(·) satisfies the assumption of Theorem 1. It remains to provide adequate ansatz spaces for the Galerkin approximation of the eigenvalue problem for S(·).
In what follows, we make the additional assumption that Ω i is a polyhedron, possibly curvilinear. Let (T h ) h be a sequence of regular triangulations of the boundary Γ with mesh size h. We denote by RT k (T h ) the space generated by Raviart-Thomas elements of order k on T h and by BMD k (T h ) the space generated by Brezzi-Douglas-Marini elements of order k on T h , where we refer to [5,8] for their definition. We adopt the convention of [5,8] that k = 0 means lowest order Raviart-Thomas or Brezzi-Douglas-Marini finite elements.
In the sequel V k h denotes either RT k (T h ) or BMD k (T h ). We consider the following Galerkin eigenvalue problem: In the next theorem we summarize the approximation properties of V k h which we need for the convergence analysis of the Galerkin eigenvalue problem (34).

Theorem 2
The following assertions hold true: where we refer to [ From the last theorem and the properties of S(·) it follows that we can apply the convergence results of Theorem 1 to the Galerkin eigenvalue problem (34). For the specification of the error estimates we consider the adjoint eigenvalue problem for S(·) with respect to the pairing ·, · τ . Let us first define the so-called adjoint function S * (κ) := (S(κ)) * . The adjoint eigenvalue problem for S(·) is then given by Note that the Fredholm alternative implies that κ is an eigenvalue of S(·) if and only if κ is an eigenvalue of S * (·).

Lemma 5
The following holds true: (i) It is sufficient to show that S(κ)u, r τ = u, −S(−κ)r τ (38) holds for all u, r ∈ γ i τ (C ∞ 0 (R 3 )). Let u, r ∈ γ i τ (C ∞ 0 (R 3 )), then we can use the integral representation for the pairings in (38) and get with (37): From the anti-symmetry of the pairing, the assertion follows.
(ii) The Fredholm alternative implies that κ is an eigenvalue of S(·) if and only if (S(κ)) * t = 0 for some t ∈ V \ {0}. The latter is by the result in (i) equivalent to S(−κ)t = 0.
In the next theorem we specify the convergence order of the eigenvalues and eigenfunctions of the Galerkin eigenvalue problem (34).
Proof As pointed out above the Galerkin eigenvalue problem satisfies the assumptions of Theorem 1. The error estimates follow then from the approximation property (35) of V k h and the fact that G (S(·), κ) = G(S * (·), κ), see Lemma 5(iv).

Scattering resonance problem for a penetrable scatterer
In this section we consider the scattering resonance problem for a penetrable scatterer. We now allow that the permittivity and the permeability are complex and frequency dependent. We assume that the frequency dependence is holomorphic in an open set Λ ⊂ C. Such configurations occur for example in the field of plasmonics or in the context of metamaterials.
The interior and exterior Calderón identities are used for deriving a boundary integral formulation of the scattering resonance problem (40). A function U 1 ∈ H(curl; Ω i ) is a solution of the Maxwell's equations in Ω i with wavenumber ω √ ε 1 μ 1 if and only if it satisfies the interior Calderón identity [8, Thm. 8] where We define for = 1, 2 the block operator We obtain the following boundary integral formulation of the scattering resonance problem (40) by using the interior and exterior Calderón identity and by setting (w, u) : Note that the eigenvalue problems (40) and (43) are not equivalent. If in the eigenvalue problem (40) ε 1 is interchanged with ε 2 and μ 1 with μ 2 , then one also obtains (43) as corresponding boundary integral formulation. However, the equivalence of (40) and (43) is guaranteed if for (43) a constraint is imposed, as shown next.

Theorem 4
The following assertions hold true:  Conversely, suppose that (ω, w, u) is a solution of the eigenvalue problem (43) satisfying is a solution of the resonance problem (40) and (w, u) is a solution of (43). The interior Calderón identity (41) implies that 1 2 now that (ω, w, u), w, u ∈ V \ {0}, satisfies the eigenvalue problem (43) and that the equation 1 2 I − B 1 (ω)) w, u) = 0 holds. We first show that Subtracting the second equation from the first equation in (44) and using that (w, u) satisfies the eigenvalue problem (43) gives Hence, E 1 and E 2 satisfy the transmission conditions of the eigenvalue problem (43). From the assumption 1 2 I − B 1 (ω)) w, u) = 0 and the first equation in (44), we finally get For the analysis of the eigenvalue problem (43) we consider the following antisymmetric pairing on the product space V 2 × V 2 → C: In the next lemma we collect results on boundary integral operators which we need in order to prove that B 1 (ω) + B 2 (ω) satisfies a generalized Gårding's inequality.
Consider the scaling Since for any K : V 2 → V 2 and for any u ∈ V 2 it holds where R := diag(U, U), D := diag(μκ −1 I, I), C := D −1 KD and T := DRD, we will first determine an adequate compact perturbation K of A. Let us define the operatorŜ(κ) : V → V by where we refer to (8) for the definition of A(κ) and V (κ). Since M(κ)−M(0) and κS(κ)−Ŝ(κ) are compact operators in V [9, p. 472] the operator is a compact perturbation of A in V 2 . Let u ∈ V 2 and Du=:v=: according to the splitting V = X ⊕ N as given in (11), Inequality (49) follows now with the same arguments as for Re(κ) = 0.
The assumptions on the material parameters ε and μ in (55) and (56) of Theorem 5 are satisfied for example if the scatterer and background medium are dielectrics but also for typical configurations of scattering problems in plasmonics.

Galerkin approximation
For the Galerkin approximation of the eigenvalue problem (43) we use is an open set such that μ (ω)ε j (ω), ∈ {1, 2}, are holomorphic on Λ and that the assumptions in (55) and (56) of Theorem 5 are satisfied. The Galerkin eigenvalue problem reads as follows: We use the abstract results of Sect. 3 in order to show the convergence of the eigenvalues and eigenfunctions of the Galerkin eigenvalue problem (59). Define Further, let P h : V → V k h be the orthogonal projection of V onto V h , and define P h := diag(P h , P h ) and J := 0 J J 0 .
Then, from Corollary 1 it follows that the sequence (P h J(B 1 (ω) + B 2 (ω))P h ) h converges regularly to J(B 1 (ω) + B 2 (ω)) for all ω ∈ Λ. As a consequence, we can apply the convergence results of Theorem 1 to the eigenvalues and eigenfunctions of the Galerkin eigenvalue problem (59).

Numerical examples
In this section we report on results from some numerical experiments for the approximation of the eigenvalues of the boundary integral formulations of the interior and exterior resonance problem (1) and (2), and of the scattering resonance problem for the penetrable scatterer (40). In all experiments Raviart-Thomas elements of lowest order k = 0 are used. For the computations of the boundary element matrices the open-source library BEM++ [31] is employed. The Galerkin approximations of the eigenvalue problems for S(·) and for B 1 (·) + B 2 (·) result in holomorphic matrix eigenvalue problems in C \ {0} and in Λ, respectively. The related matrix-valued functions are denoted by S h (·) and B 1,h (·) + B 2,h (·). For the numerical solution of the matrix eigenvalue problems we use the contour integral method as given in [2]. For other variants of the contour integral method we refer to [1,38]. The contour integral method is a reliable method for the approximation of all eigenvalues which lie inside of a given contour in the complex plane, and for the approximation of the corresponding eigenvectors. The method is based on the contour integration of the resolvent, (S h (·)) −1 and (B 1,h (·) + B 2,h (·)) −1 in our case, and utilizes that the eigenvalues of eigenvalue problems for holomorphic matrix-valued functions are poles of the resolvent. By contour integration of the resolvent applied to some randomly chosen set of test vectors a reduction of the holomorphic eigenvalue problem to an equivalent linear eigenvalue problem is possible such that the eigenvalues of the linear eigenvalue problem coincide with the eigenvalues of the nonlinear eigenvalue problem inside the contour.
The main computational cost of the numerical implementation of the contour integral method consists in the approximation of the contour integral of the resolvent. In our numerical experiments we use for that the composite trapezoidal rule as suggested in [2]. In general, with such an approximation of the contour integral an exponential convergence order for the approximation of the eigenvalues with respect to number of quadrature nodes is achieved [2]. The composite trapezoidal rule requires in each quadrature node ξ j , j = 1, . . . , N , the application of (S h (ξ j )) −1 and (B 1,h (ξ j ) + B 2,h (ξ j )) −1 , respectively, to some randomly chosen set of test vectors, for which in our numerical experiments an LU decomposition is utilized.

Remark 6.1
For the analysis of the contour integral method it is usually assumed that the underlying matrix-valued function of the eigenvalue problem is holomorphic inside the contour. However, the contour integral method is also suitable for eigenvalue problems where the underlying function has isolated singularities in the case of that the resolvent has a holomorphic continuation in the neighborhood of the singularities. The reason for that is that the contour integral method operates on the resolvent of the eigenvalue problem and approximates the poles of the resolvent. Our numerical experiments indicates that there is a holomorphic continuation of (S h (·)) −1 as well as of (B 1,h (·) + B 2,h (·)) −1 in the neighborhood of 0, see Figs. 1, 2 and 3, respectively. The conjecture is that this property holds in general if the material parameters are holomorphic in the neighborhood of 0.

Interior and exterior resonance problem
In the first numerical examples we consider the Galerkin approximation (34) of the boundary integral formulation (10) of the interior and exterior resonance problem (1) and (2). The unit cube and the unit ball are chosen for Ω i . In all examples in this subsection we take ε = μ = 1.0.

Interior resonance problem for the unit cube
For the unit cube only the interior resonances are known analytically. Therefore we restrict ourselves to the approximation of the resonances of the interior problem. The interior resonances have the form κ = πk, where k = k 2 1 + k 2 2 + k 2 3 with k 1 , k 2 , k 3 ∈ N 0 and k 1 k 2 + k 2 k 3 + k 3 k 1 > 0 [3,Sect. 6]. For the contour integral method we choose as contour the ellipse ϕ(t) = c + a cos(t) + ib sin(t), t ∈ [0, 2π], with c = 5.0, a = 5.1 and b = 0.5. There are seven distinct eigenvalues within this ellipse which have a total algebraic multiplicity of 32. The number of quadrature nodes for the approximation of the contour integral is 25. In Table 1 the errors of the approximations of the two smallest resonances and of the largest resonance for discretizations with mesh sizes h = √ 22 − j , j = 2, . . . , 5, are given. We observe that the experimental orders of convergence (eoc) match the theoretical predicted cubic asymptotic convergence order, see (39). Further, for the specified mesh sizes all exact resonances are approximated with the right multiplicities and no spurious resonances occur within the ellipse. A plot of the computed resonances for the mesh with mesh size h = √ 22 −4 is given in Fig. 1. The experiments also confirm the mentioned conjecture that the resolvent S h (·) −1 can be holomorphically extended to κ = 0. We want to mention that for coarser discretizations with 18 and 72 degrees of freedom, which corresponds to the mesh sizes h = √ 2 and h = √ 22 −1 , only the two smallest resonances are approximated well and that for these discretizations the cubic convergence order is no longer observable.

Interior and exterior resonance problem for the unit ball
The interior and exterior resonances of the unit ball can be represented as zeros using the spherical Bessel and Hankel functions [29]. We denote by j n the spherical Bessel functions of the first kind and as before by h (1) n the spherical Hankel functions of the first kind. The set of the interior resonances is given by {κ ∈ R \ {0} : j n (κ) = 0 or j n (κ) + κ j n (κ) = 0, n ∈ N}, and that of the exterior resonances by As contour for the contour integral method the ellipse ϕ(t) = c + a cos(t) + ib sin(t), t ∈ [0, 2π], is chosen with c = 5.0, a = 5.3 and b = 1.0. There are 23 distinct eigenvalues within this ellipse having a total algebraic multiplicity of 157. For the approximation of the contour integral 100 quadrature nodes are taken. In Table 2 the errors of the smallest and the largest interior and exterior resonance in modulus inside the ellipse are given. The experimental convergence order is in contrast to the cube of one order reduced since the sphere is approximated by flat triangles. Again, all exact resonances are approximated with the right multiplicities and no spurious resonances occur. In Fig. 2 the computed resonances by the contour integral method for the mesh with mesh size h = 0.082 are plotted.

Scattering resonance problem for a penetrable scatterer
In this subsection we consider the Galerkin approximation of the boundary integral formulation (43) of the scattering resonance problem for the penetrable scatterer (40). The domains Ω i for the numerical examples are again the unit cube and the unit ball. As approximation

Unit cube
For this example the material parameters are set to ε 1 = 4.0 and ε 2 = μ 1 = μ 2 = 1.0. The exact resonances of the resonance problem for a penetrable cube are not known. As reference resonances the computed resonances of a very fine mesh with mesh size h = 0.03125 are taken. As contour for the contour integral method we chose the ellipse ϕ(t) = c + a cos(t) + ib sin(t), t ∈ [0, 2π], with c = 2.5, a = 2.7 and b = 0.5. For the approximation of the contour integral 20 quadrature nodes are used. The numerical experiments suggest a cubic asymptotic convergence order, see Table 3, which is in accordance with the theoretical results.
The computed approximations of the resonances by the contour integral method inside the ellipse for h = 0.177 are plotted in Fig. 3. The numerical experiments indicate that the resolvent (B 1,h (·) + B 2,h (·)) −1 can be holomorphically extended to ω = 0.

Unit ball
In the last example we consider a plasmonic resonance problem for a golden nano-sphere with diameter d = 100nm embedded in a host medium with refractive index of 1.5 [35]. A Drude model for the permittivity of gold is taken of the form [30] with ε 0 being the vacuum permittivity, ω p = 1.26 × 10 16 s −1 and γ = 1.41 × 10 14 s −1 . The other material parameter are chosen as ε 2 = 1.5ε 0 , μ 1 = μ 2 = μ 0 , where μ 0 is the vacuum permeability.
The exact resonances of the scattering resonance problem for a penetrable scatterer for a ball with radius R can be determined by the Mie series method [35,Supplemental material,Sect. 3]. A number ω ∈ C is a resonance if it satisfies either  or for some n ∈ N, where κ = ω √ ε (ω)μ (ω), = 1, 2. In plasmonics those scattering resonances are of interest which are close to the frequency range of light which corresponds to [1.7, 3.1] −1 , where is the reduced Planck constant in eV s given by = 6.58211957e−16. For the contour integral method as contour an ellipse is chosen with ϕ(t) = −1 [c + a cos(t) + ib sin(t)], t ∈ [0, 2π], with c = 2.5, a = 1.4 and b = 0.75. The number of quadrature nodes for the approximation of the contour integral is 20. Three distinct resonances with total algebraic multiplicity of 15 lie inside the ellipse, which are solutions of Eq. (61) for n = 1, 2, 3.
The principal square root, with the branch cut along the non-positive real axis, applied to the permittivity ε 1 (ω) defined in (60) is non-continuous along the contour. Instead we take the square root with the branch cut {z = r (cos(φ 0 ) + i sin(φ 0 )) : r ≥ 0}, φ 0 = 1.9π, which guarantees that √ ε 1 (ω)μ 1 is continuous along the contour and inside of it. In Table 4   the relative error and the experimental order of convergence for the approximation of the resonances inside the contour are given. The convergence order is compared to the cube of one order reduced as expected since the sphere is approximated by flat triangles. Again, all eigenvalues inside the ellipse are approximated with the right multiplicity and no spurious eigenvalues occur. In Fig. 4 the computed resonances by the contour integral method for the mesh with mesh size h = 7.67 are plotted.

Conclusions
In this paper we have analyzed the Galerkin approximation of boundary integral formulations for three different types of electromagnetic resonance problems. We have considered the single layer boundary integral formulation for the interior and exterior resonance problem with perfect conducting boundary conditions and a first kind boundary integral formulation of the resonance problem for the penetrable scatterer. These boundary integral formulations are eigenvalue problems for holomorphic Fredholm operator-valued functions. For the numerical approximation of these eigenvalue problems a Galerkin approximation with Raviart-Thomas and Brezzi-Douglas-Marini type elements are considered. The extension of recent results [15,16] on the regular approximations of operators satisfying a generalized Gårding's inequality enables us to apply the classical convergence theory on the approximation of eigenvalue problems for holomorphic Fredholm operator-valued functions to the proposed boundary element approximations. Numerical examples confirm the theoretical convergence results and show that the asymptotic convergence behavior appears on rather coarse meshes.
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