COUPLING OF FINITE ELEMENT AND BOUNDARY ELEMENT METHODS FOR THE SCATTERING BY PERIODIC CHIRAL STRUCTURES *

Transcription

1 Journal of Computational Mathematics, Vol.26, No.3, 2008, COUPLING OF FINITE ELEMENT AND BOUNDARY ELEMENT METHODS FOR THE SCATTERING BY PERIODIC CHIRAL STRUCTURES * Habib Ammari Centre de Mathématiques Appliquées, CNRS UMR 764, Ecole Polytechnique 928 Palaiseau, France ammari@cmapx.polytechnique.fr Gang Bao Department of Mathematics Michigan State University, East Lansing, MI 48824, USA bao@math.msu.edu Dedicated to Professor Junzhi Cui on the occasion of his 70th birthday Abstract Consider a time-harmonic electromagnetic plane wave incident on a biperiodic structure in R 3. The periodic structure separates two homogeneous regions. The medium inside the structure is chiral and nonhomogeneous. In this paper, variational formulations coupling finite element methods in the chiral medium with a method of integral equations on the periodic interfaces are studied. The well-posedness of the continuous and discretized problems is established. Uniform convergence for the coupling variational approximations of the model problem is obtained. Mathematics subect classification: 65N30, 78A45, 35J20. Key words: Chiral media, Periodic structures, Finite element method, Boundary element method, Convergence.. Introduction Consider a time-harmonic electromagnetic plane wave incident on a biperiodic structure in R 3. By biperiodic structure or doubly periodic structure, we mean that the structure is periodic in two orthogonal directions. The periodic structure separates two homogeneous regions. The medium inside the structure is chiral and nonhomogeneous. The diffraction problem is to study the propagation of the reflected and transmitted waves away from the structure. Recently, there has been a considerable interest in the study of scattering and diffraction by chiral media. Such media are isotropic, reciprocal, and more importantly circularly birefringent, with potential applications in antennas, microwave devices, waveguides, and many other fields. In general, electromagnetic wave propagation in a chiral medium is governed by Maxwell s equations and a set of constitutive equations known as the Drude-Born-Fedorov constitutive equations, in which the electric and magnetic fields are coupled. The coupling is responsible for the chirality of the medium. It is measured by the magnitude of the chirality admittance β, which along with the dielectric coefficient ε and the magnetic permeability constant µ characterize completely the electromagnetic properties of the medium. On the other hand, periodic (gratings structures * Received December 2, 2007 / Revised version received March 8, 2008 / Accepted March 20, 2008 /

2 262 H. AMMARI AND G. BAO have received increasing attentions through the years because of important applications in integrated optics, optical lenses, anti-reflective structures, holography, lasers, communication, and computing. Chiral gratings provide an exciting combination of the medium and structure. The combination gives rise to new features and applications. For instance, chiral gratings are capable of converting a linearly polarized incident field into two nearly circularly polarized diffracted modes in different directions. For an interesting explanation and references of these equations and various physical and computational aspects of the electromagnetic wave propagation inside chiral media, we refer to Lakhtakia [39] and Lakhtakia, Varadan, and Varadan [40] (non-periodic chiral structures, and to Jaggar, et al. [38], Lakhtakia, Varadan, and Varadan [4], and Yueh and Kong [55] (periodic chiral structures. Results and additional references on closely related periodic achiral structures may be found in Petit [42] and Bao, Dobson, and Cox [6], Dobson and Friedman [33], Abboud [], Bao [3], Bao and Dobson [5], Bao and Zhou [8], Chen and Wu [26], Bao, Chen, and Wu [4], Arens, Chandler-Wilde, and DeSanto [2], and Rathsfeld, Schmidt, and Kleeman [5]. Other related recent results for Maxwell s equations in general media may be found in [7,27,35,36]. This paper is devoted to a new approach for solving the diffraction problem, which couples a finite element method (FEM in the nonhomogeneous chiral medium with a method of integral equations or boundary element method (BEM on the periodic interfaces. More precisely, the approach consists of two processes: First, a finite element method is used for solving the diffraction problem in the complicated structure of a nonhomogeneous and possibly chiral material. Second, a method of integral equations is developed to derive the exact boundary conditions. The fact that these exact boundary conditions are formulated on the surface of the structure implies that no mesh of the surrounding medium would be needed. In this work, the well-posedness of the continuous and discretized formulations is established. Uniform convergence for the coupling variational approximations of the model problem is obtained. We point out that the variational coupling formulations introduced here are extremely general in terms of material, grating geometry, as well as the incident angle. The material functions ε, µ, and β are only assumed to be bounded measurable. Also, a recent result of Torres [52] indicates that the boundary on which the integral equations are derived needs only be Lipschitz. Our present coupling approach is related to several other works in the literature. Levillain [43] implemented computationally several versions of a coupling procedure for Maxwell s equations in a three dimensional medium surrounding a bounded perfectly conducting body. de La Bourdonnaye [20] analyzed some coupling formulations for the Helmholtz equation as well as Maxwell s equations. Mathematical analysis of the coupling formulations in [43] has been carried out by Ammari and Nédélec [7,8]. The results of [7] and [8] are further extended in [9] to study coupling FEM/BEM formulations for Maxwell s equations with a Leontovich boundary condition. We also refer to Wendland [53] and Gatica and Wendland [34] for a survey of asymptotic error estimates for symmetric and nonsymmetric coupling of finite and boundary element methods and to Nédélec [49] for a recent survey of the integral equation methods for computational electromagnetics. Recently, in [2, 3], the authors have studied mathematical aspects of the diffraction problem by a periodic chiral structure. It is shown that for all but possibly a discrete set of parameters, the diffraction problem attains a unique quasi-periodic weak solution. Our proof is based on a Hodge decomposition lemma along with a new compact imbedding result. An important step of our approach is to reduce the diffraction problem into a bounded domain by using a pair of transparent boundary conditions. The approach in the present paper is different from

3 Coupling of Finite Element and Boundary Element Methods for the Scattering 263 the previous one in the following aspects: Here, the transparent boundary operators are not used and the exact radiation conditions are derived on the boundary of the chiral medium. No computation in the region surrounding the chiral medium is required. The well-posedness of the discretized problems excluding possibly a discrete set of singular frequencies is established and uniform convergence for the coupling FEM/BEM variational formulations is obtained. To the best of our knowledge, this paper presents the first coupling FEM/BEM variational approach for solving the diffraction problem. The approach has the potential for developing computationally attractive algorithms. It also gives a new proof of existence and uniqueness for solutions of the diffraction problem. The paper is outlined as follows. In Section 2, the Maxwell equations and the constitutive equations, the Drude-Born-Fedorov equations, are presented. Section 3 is devoted to both symmetric and nonsymmetric variational formulations of the diffraction problem, which couple BEM with FEM. The well-posedness of the coupling formulations is established. Results on existence and uniqueness of the weak quasi-periodic solutions are proved. In Section 4, uniform convergence for the coupling FEM/BEM variational approximations is obtained. The paper is concluded in Section 5 by some general remarks. 2. The Diffraction Problem Electromagnetic wave propagation in chiral media is governed by the time harmonic Maxwell equations and a set of constitutive equations, known as the Drude-Born-Fedorov equations, in which the electric and magnetic fields are coupled. The time harmonic Maxwell equations are (time dependence e iωt : E iωb = 0, (2. H + iωd = 0, (2.2 where E, H, D, and B denote the electric field, the magnetic field, the electric, and magnetic displacement vectors in R 3, respectively. The following Drude-Born-Fedorov equations hold: ( D = ε(x E + β(x E, (2.3 ( B = µ(x H + β(x H, (2.4 where ε is the electric permittivity, µ is the magnetic permeability, and β is the chirality admittance. The parameters β, ε, and µ characterize completely the electromagnetic properties of the medium. It is easily seen that the following equations are equivalent to the constitutive equations (2.3-(2.4: where k(x = ω ε(xµ(x. ( (k(xβ(x 2 D = ε(xe + iβ(x ω (k(x2 H, (2.5 ( (k(xβ(x 2 B = µ(xh iβ(x ω (k(x2 E, (2.6

4 264 H. AMMARI AND G. BAO Similarly, the Maxwell equations may be rewritten as In these equations, the parameter γ(x is defined as : E = (γ(x 2 β(xe + iωµ(x( γ(x k(x 2 H, (2.7 H = (γ(x 2 β(xh iωε(x( γ(x k(x 2 E. (2.8 (γ(x 2 = (k(x 2 (k(xβ(x 2. Throughout, we always assume that (k(xβ(x 2, x IR 3. Moreover, the above system may be shown to be equivalent in a weak sense to ( ω 2 β 2 εµ E ω 2 (εβe ω 2 εβ E ω 2 εe = 0, µ (2.9 E = (γ(x 2 β(xe + iωµ(x( γ(x k(x 2 H. (2.0 Standard ump conditions may be deduced from the above system. In fact, the tangential parts of the electric and magnetic fields are continuous across an interface. Let ν denote the unit normal to the interface. We then have [ν E] = 0, [ν β2 k 2 iωµ E γ2 β( k 2 β 2 ν E] = 0. iωµ We next specify the geometry of the problem. Let Λ and Λ 2 be two positive constants, such that the material functions ε, µ, and β satisfy, for any n, n 2 Z = {0, ±, ±2, }, ε(x + n Λ, x 2 + n 2 Λ 2, x 3 = ε(x, x 2, x 3, µ(x + n Λ, x 2 + n 2 Λ 2, x 3 = µ(x, x 2, x 3, β(x + n Λ, x 2 + n 2 Λ 2, x 3 = β(x, x 2, x 3. In addition, it is assumed that, for some fixed positive constant b, where ε, ε 2, µ, and µ 2 are positive constants. We make the following general assumptions: ε(x = ε, µ(x = µ, β(x = 0 for x 3 > b, ε(x = ε 2, µ(x = µ 2, β(x = 0 for x 3 < b, ε(x, µ(x, and β(x are all real valued L functions, ε(x ε 0, µ(x µ 0, and β 0, where ε 0 and µ 0 are positive constants; d(x = ( ω 2 β 2 εµ/µ d 0 > 0, for some positive constant d 0. Note that the second assumption is essential. Fortunately it appears to be common in the literature and ustifiable since β is generally small. The first assumption is a technical one. Analogous results may be possible for materials that absorb energy.

5 Coupling of Finite Element and Boundary Element Methods for the Scattering 265 Let be the domain where the material parameters ε, µ, and β are variable functions, be the domain above, and 2 be the domain below. Denote =, =, 2. Consider a plane wave in incident on. Here E in = se iq x, H in = pe iq x, (2. q = (α, α 2, β (0 = ω ε µ (cosθ cosθ 2, cosθ sin θ 2, sinθ is the incident wave vector whose direction is specified by θ and θ 2, with 0 < θ < π and 0 < θ 2 2π. The vectors s and p satisfy s = ωε (p q, q q = ω 2 ε µ, p q = 0. (2.2 We are interested in quasi-periodic solutions, i.e., solutions E and H such that the fields E α, H α defined by, for α = (α, α 2, 0, E α = e iα x E(x, x 2, x 3, (2.3 H α = e iα x H(x, x 2, x 3, (2.4 are periodic in the x direction of period Λ and in the x 2 direction of period Λ 2. Denote α = + iα = + i(α, α 2, 0. It is easy to see from (2.9 and (2.0 that E α and H α satisfy α (d α E α ω 2 α (εβe α ω 2 εβ α E α ω 2 εe α = 0, (2.5 α E α = (γ(x 2 β(xe α + iωµ(x( γ(x k(x 2 H α. (2.6 In order to solve (2.5-(2.6 we need to impose a radiation condition on the scattering problem. Due to the (infinite periodic structure, the usual Sommerfeld or Silver-Müller radiation condition is no longer valid [50]. The appropriate radiation condition may be derived as follows: Since E α is Λ periodic, we can expand E α in a Fourier series: E α (x = Eα in (x + U α (n (x 3e iαn x, (2.7 n Z where E in α = E in e iα x, α n = (2πn /Λ, 2πn 2 /Λ 2, 0, and U α (n (x 3 = Λ Λ Λ 2 0 Define for =, 2 the coefficients Λ2 0 (E α (x E in α (xe iαn x dx dx 2. β (n (α = { ω2 ε µ α n + α 2, ω 2 ε µ > α n + α 2, i α n + α 2 ω 2 ε µ, ω 2 ε µ < α n + α 2. (2.8 We assume that ω 2 ε α n +α 2 for all n Z 2, =, 2. This condition excludes resonances.

6 266 H. AMMARI AND G. BAO For convenience, we also introduce the following notation: Λ + = {n Z2 : Im(β (n = 0}, Λ = {n Z2 : Im(β (n 0}. Observe that inside ( =, 2, ε = ε, µ = µ, and β = 0, Maxwell s equations then become ( α + ω 2 ε µ E α = 0, (2.9 where α = + 2iα α 2. Since the medium in ( =, 2 is homogeneous, the method of separation of variables implies that E α can be expressed as a sum of plane waves: where the A (n E α = E in α,(x + n Z H α = H in α, (x + n Z and B (n H in α (x in x 3 > b and E in α,2 A (n e ±iβ(n x 3+iα n x, =, 2, in ( + x 3 > b, (2.20 B (n e ±iβ(n x 3+iα n x, =, 2, in ( + x 3 > b, (2.2 are constant (complex vectors and Eα,(x in = Eα in (x, Hα,(x in = (x = Hin α,2 (x = 0 in x 3 < b. The following radiation condition is employed: Since β n is real for at most finitely many n, there are only a finite number of propagating plane waves in the sum (2.20, the remaining waves are exponentially decaying (or radiated as x 3. We will insist that E α is composed of bounded outgoing plane waves in and 2, plus the incident (incoming wave in. From (2.7 and (2.8 we deduce Define E (n α (x 3 = { U α (n (be iβ(n (x 3 b, in x 3 > b, U α (n ( be iβ(n 2 (x 3+b, in x 3 < b. Λ = Λ Z Λ 2 Z {0} R 3. (2.22 Since the fields E α are Λ-periodic, we can move the problem from R 3 to the quotient space R 3 /Λ. For the remainder of the paper, we shall identify with the cylinder /Λ, and similarly for the boundaries /Λ. Thus from now on, all functions defined on,, and are implicitly Λ-periodic. Define div α by div α u = α u = ( x + iα u + ( x2 + iα 2 u 2. In the future, for simplicity, we shall drop the subscript α. Denote by div Γ, Γ, Γ, and curl Γ, the surface divergence, the surface gradient, the surface vector rotational, and the surface scalar rotational, respectively. Their meanings should be clear from the contexts. Let H m be the mth order L 2 -based Sobolev spaces of complex valued functions and H m p ( be the subset of all functions in H m ( which are the restrictions to of the Λ-periodic functions in H m loc (R2 ( b, b. The spaces H m p ( and H m p ( may be defined similarly. Consider further the notation: { H(curl, = v L 2 ( 3, v L 2 ( 3}, TH s ( = { u H s ( 3, u. n = 0 }, { } TH s (div, = u TH s (, div Γ u H s (, { } TH s (curl, = u TH s (, curl Γ u H s (.

7 Coupling of Finite Element and Boundary Element Methods for the Scattering 267 Next, introduce the periodic Green kernel G, for =, 2, which satisfies the radiation condition (2.22 and the Helmholtz equation: From [50], consider formally the sum We have u + ω 2 ε µ u = 0 in IR 3. G = i 2Λ Λ 2 n Z β (n 2 e iαn. x e iβ(n x 3. (2.23 Lemma 2.. The sum (2.23 defines an L 2 loc (IR3 periodic function which satisfies (i G + ω 2 ε µ G = n Z 2 δ(λ n in (C 0 (IR3, (ii G is a C function in IR 3 \ n Z 2{Λ n }, (iii G satisfies the radiation condition (2.22. Here Λ n = (n Λ, n 2 Λ 2, 0, n = (n, n 2 Z 2 and δ(λ n is the Dirac measure at Λ n. Note that if β (n 0, n Z 2, for x Λ n, the series in (2.23 converges uniformly in compact sets but cannot be component-wise differentiated with respect to x 3 at x 3 = 0. The following identity holds. Lemma 2.2. where x n = x + Λ n. G = n Z 2 eiω εµ x n iα. xn e 4π x n, (2.24 See Morelot [46] for a proof of (2.24 by the Poisson summation formula. An analogous representation of the periodic Green kernel in the case of a single periodic surface was given by Chen and Friedman [25] and Bruno and Reitich [24]. From now on, we denote by G (x, y = G (x y for x = (x, x 2, x 3, y = (y, y 2, y 3 IR 3. We have the following additional result about the singularity of the kernel G. Lemma 2.3. The function G (x, y 4π x y + i ( α log x y + α 2 log x 2 y 2 2π is a continuous function as x y 0. Proof. Recalling that = α. α, it is easy to see that εµ ( + ω 2 ε µ (G eiω x y = iα ( e iω ε µ 4π x y 2π. x y εµ x y + α 2 eiω, 4π x y 4π x y for any x = (x, x 2, x 3, y = (y, y 2, y 3 (0, Λ (0, Λ 2 IR. By the standard elliptic theory, the kernel G e iω ε µ / x y has the same singularity when x y 0 as R with R = iα 2π. ( x y. Moreover, R behaves like i(2π (α log x y + α 2 log x 2 y 2 + O( x y log x y. The conclusion follows from the continuity of the function x y log x y as x y 0.

8 268 H. AMMARI AND G. BAO Lemma 2.4. There exist three positive constants C, C, and C, such that C θ 2 H /2 (Γ G (x, yθ(xθ(ydγ(ydγ(x for any θ H /2 (. C θ 2 H /2 ( C θ 2 H 3/2 (, (2.25 Proof. By Lemma 2.3, it suffices to prove that there exist positive constants C and C such that C θ 2 H /2 (Γ x y θ(xθ(ydγ(ydγ(x C θ 2 H /2 (Γ, (2.26 for any θ H /2 (. The coercivity estimate is classical if the boundary Γ is closed. Let be a bounded and closed boundary such that 2, ϕ on, and ϕ 0 outside of 3 /2 (component-wise. Denote θ = ϕθ for any θ H /2 (. According to [49] it is clear that x y θ(x θ(ydγ(ydγ(x C θ 2 H /2 ( Γ, which along with x y θ(x θ(ydγ(ydγ(x 2 and yield the estimate ( θ 2 H /2 ( θ 2 H /2 ( θ 2 H /2 ( x y θ(xθ(ydγ(ydγ(x 3. Continuous Coupling FEM/BEM Formulations In this section, we derive coupling FEM/BEM formulations for solving the diffraction problem. The well-posedness of the continuous coupling formulations is established. We also prove that the derived coupling formulations are of Fredholm type and they do not generate spurious eigenfrequencies at the continuous level since they are completely equivalent to Equations (2.5-(2.6 which along with the radiation condition (2.22 govern the diffraction from periodic chiral structures. In the following, we first state a useful Hodge decomposition lemma, a classical compactness result, and a trace regularity result. We then derive coupling FEM/BEM formulations. We also study questions on existence and uniqueness of the solutions. 3.. Hodge decomposition and compactness Assume that is connected and is simply connected. Lemma 3.. (a Let { IM( = u H(curl,, The following Hodge decomposition holds: } ε u. q = 0, q H (. H(curl, = IM( H (,

9 Coupling of Finite Element and Boundary Element Methods for the Scattering 269 where the orthogonality is with respect to the product ((, defined by ((u, v = u. v + ε u. v. (b The Hodge decomposition holds: TH /2 (div, = Γ H 3/2 ( /C Γ H /2 ( /C, where the orthogonality is with respect to the duality product between TH /2 ( and TH /2 (. Proof. Let E H(curl,. Since is connected, i.e., the space of Neumann fields in is trivial, there exists a unique u satisfying u = E in, div εu = 0 in, u. n = 0 on. Further, since (div Γ (E Γ u Γ H /2 (, there exists a unique solution p H ( to the boundary value problem divε p = 0 in, p = (div Γ (E Γ u Γ on. It is clear that E = u + p and the uniqueness of the decomposition (a is obvious. Since is simply connected, the decomposition (b follows immediately from [20]. Note that in the case where is non-simply connected, the finite dimensional vector space { } { } N( = θ, Γ θ = 0 = θ, div Γ θ = 0, curl Γ θ = 0, is nontrivial and (b should be replaced with, see for instance ([4], section 4, TH /2 (div, = Γ H 3/2 ( /C Γ H /2 ( /C N(. This completes the proof of the lemma. Remark 3.. A decomposition similar to (a was originally introduced by Birman and Solomyak [2] to regularize Maxwell s equations in a bounded domain. A decomposition similar to (b was used in [5]. Lemma 3.2. The imbedding from IM( to L 2 ( 3 is compact. Proof. For any sequence u m IM( let ũ m be its periodic extension. Denote = (0, Λ (0, Λ 2 O. Let = ( Λ, 2Λ ( Λ 2, 2Λ 2 O and ϕ be a smooth function with ϕ(x, x 2 (x, x 2 [0, Λ ] [0, Λ 2 ], ϕ(x, x 2 0 (x, x 2 ([ Λ, Λ /2] [3Λ /2, 2Λ ] ([ Λ 2, Λ 2 /2] [3Λ 2 /2, 2Λ 2 ]. Then ϕũ m. n = 0 on, ϕũ m L 2 ( 3, ϕũ m L 2 ( 3, divϕũ m L 2 (.

10 270 H. AMMARI AND G. BAO Further, ϕũ m = u m in and there exists a constant C independent of m, such that ϕũ m H(curl, C u m IM(. (3. If the sequence {ũ m } is bounded in H(curl, then we can extract by Weber [54] a subsequence that converges strongly in L 2 ( 3. The compactness of the imbedding follows from (3.. Lemma 3.3. ([3] For any η > 0, the following estimate holds: u n TH /2 ( η u L 2 ( 3 + η u L 2 ( Periodic integral representations Denote E in, E = E i in, E 2 in 2, H in, H = H i in, H 2 in 2. We now derive periodic integral representations for E and H inside. Lemma 3.4. The following periodic integral representation formulas hold: E = E in G M i G div Γ J iωµ G J, x, (3.2 Γ ωε Γ H = H in G J + i G div Γ M + iωε G M, x, (3.3 ωµ where M = E n Γ = E i n Γ and J = H n Γ = H i n Γ. Proof. Without loss of generalities, it suffices to establish the periodic integral representation (3.2 for E in. Let b = {x 3 < b}. Multiplying both sides of the Maxwell equations satisfied by E and integrating by parts over b, we get by some standard manipulations [28] that E (x = + +iωµ Γ G M i ωε G E e 3 + i ωε G E e 3, x b. G div Γ J iωµ G J Γ Γ G div x3=b(e e 3 Rewrite the terms on x 3 = b G E e 3 + i G div x3=b(e e 3 + iωµ ωε = G E in e 3 + i G div x3=b(e in e 3 + iωµ x 3=b ωε + G (E E in e 3 + i ωε +iωµ G (E E in e 3. G div x3=b((e E in e 3 G E e 3 G E in e 3

11 Coupling of Finite Element and Boundary Element Methods for the Scattering 27 It is easily seen that E in (x = G E in e 3 + i G div x3=b(e in e 3 ωε x 3=b + iωµ G E in e 3, for x b. To prove the periodic integral representation (3.2, it is sufficient to show that the quantity G (E E in e 3 + i G div x3=b((e E in e 3 ωε x 3=b +iωµ G (E E in e 3 goes to 0 as b + uniformly in x. Write G = G + + G, where G ± = i 2Λ Λ 2 n Λ ± β (n e iαn. x e iβ(n x 3. From B (n = i ωµ A (n (ib (n e 3 + iα n, n Z 2, where A (n and B (n are defined by (2.20-(2.2, we obtain after some simple calculations that +iωµ G + (E E in e 3 + i ωε G + div ((E E in e 3 G + (E E in e 3 = 0. (3.4 On the other hand, the quantity G (E E in e 3 + i G ωε div ((E E in e 3 +iωµ G (E E in e 3 is exponentially decaying as b + hence the conclusion follows from (3.4. Next, we determine the unknowns M and J in TH /2 (div, as well as the fields E i and H i in H(curl,. To derive periodic integral equations on from the periodic integral representations (3.2-(3.3, the following lemma is needed. Lemma 3.5. For any v TH /2 (div,, lim n (x 0 G (x, yv(y dγ(y x x 0 = v(x 0 + n (x 0 x G (x, y v(y dγ(y. 2

12 272 H. AMMARI AND G. BAO Proof. The key of the proof is to observe by Lemma 2.3 that G (x, y 4π x y + i ( α log x y + α 2 log x 2 y 2 2π is a continuous function even as x y 0. Hence ( lim n (x 0 G (x, y x x 0 4π x y ( = n (x 0 G (x 0, y 4π x 0 y v(y dγ(y. By Müller [47], we have lim n (x 0 v(y dγ(y x x 0 x y = v(x 0 + n (x 0 x v(y dγ(y, 2 x y lim n (x 0 x x 0 x y v(y dγ(y = n (x 0 v(y dγ(y v(y dγ(y. x 0 y The proof is now complete. Taking the limit of (3.2-(3.3 tangentially on, we obtain by Lemma 3.5 the periodic integral equations on : 2 n (x E i (x = n (x E in (x i n (x ( Γ G (x, ydiv Γ J (y dγ(y dγ(x ωε iωµ n (x ( G (x, yj (y dγ(y dγ(x n (x ( x G (x, y M (y dγ(y dγ(x (3.5 and 2 n (x H i (x = n (x H in (x + i n (x ( Γ G (x, ydiv Γ M (y dγ(y dγ(x ωµ +iωε n (x ( G (x, ym (y dγ(y dγ(x n (x ( x G (x, y J (y dγ(y dγ(x. ( Derivations of the coupling FEM/BEM formulations Since the singularity of the kernel G (x, y behaves like (4π x y, classical results from potential theory [47] yield that each term in (3.5-(3.6 belongs to TH /2 (curl,. By the classical duality result: (TH /2 (curl, = TH /2 (div,, we can make sense of the duality

13 Coupling of Finite Element and Boundary Element Methods for the Scattering 273 products of (3.5-(3.6 with test functions in TH /2 (div,. Multiplying both sides of the equation (3.6 by J t TH /2 (div, and integrating it over, we obtain i ωε G div Γ J div Γ J t iωµ G J. J t E i. J t 2 Γ x G M. J t = E in. J t, Jt TH /2 (div,. (3.7 Using the Hodge decomposition Lemma 3., we decompose (3.7 into the following two variational formulations: ϕ t H3/2 ( and ψ t H/2 (, i ωε G Γ ϕ Γ ϕ t iωµ G Γ ϕ. Γ ϕ t iωµ G Γ ψ. Γ ϕ t Γ x G M. Γ ϕ t E i. Γ ϕ t 2 = E in. Γ ϕ t (3.8 and iωµ G Γ ψ. Γ ψ t iωµ G Γ ϕ. Γ ψ t x G M. Γ ψ t E i. Γ ψ t = 2 E in. Γ ψ t, (3.9 where the unknowns are ϕ H 3/2 ( and ψ H /2 (. The unknown function J = Γ ϕ + Γ ψ is in TH /2 (div,. In the chiral medium, we solve the following problem for E i H(curl, in a weak sense: d E i ω 2 εβ E i ω 2 (εβe i ω 2 εe i = 0 in, (3.0 E i n = M on. (3. Multiplying both sides of (3.0 by E t H(curl, and integrating by parts over yield d E i. E t ω 2 εe i. E t ω 2 εβe i. E t ω 2 εβ E i. E t iω J. E t iω Γ J 2. E t = 0, Γ 2 (3.2 for any E t H(curl,. By the Hodge decomposition Lemma 3., we write E i = u i + p i, where u i IM( and p i H (. Replacing E i with u i + p i in the variational equation (3.2, we obtain d u i. u t ω 2 εu i. u t ω 2 εβu i. u t ω 2 εβ p i. u t ω 2 εβ u i. u t iω J. u t iω J 2. u t = 0, u t IM(, (3.3 Γ Γ 2

14 274 H. AMMARI AND G. BAO and ω 2 ε p i. p t ω 2 εβ u i. p t iω J. p t iω Γ J 2. p t = 0, Γ 2 p t H (. (3.4 From now on, we denote u Γ = u i,γ = n (n u i on. Theorem 3.. If E and H are solutions of the Maxwell equations (2.5-(2.6 together with the radiation condition (2.22, then u i, p i, J, ϕ, and ψ defined by E i = u i + p i, J = H i n Γ = Γ ϕ + Γ ψ are solutions of the variational formulation (3.3-(3.4-(3.8-(3.9. The converse is also true. If u i, p i, ϕ, ψ, and J = Γ ϕ + Γ ψ are solutions of (3.3-(3.4-(3.8-(3.9 and satisfy that then 2 J = H in n ( + i ωµ Γ G J n G div Γ ((u Γ + Γ p i n n +iωε G ((u Γ + Γ p i n n, (3.5 E in, E = E i in i, E 2 in 2, H in, H = H i in i, H 2 in 2, (3.6 where E and H are determined from the periodic integral representations (3.2-(3.3 and H i = i(ωµ u i, are solutions of the Maxwell equations (2.5-(2.6 along with the radiation condition (2.22. Proof. By the Hodge decomposition Lemma 3., the periodic integral representations (3.2- (3.3 and some integrations by parts, we easily establish that if E and H are solutions of the Maxwell equations (2.5-(2.6 along with the radiation condition (2.22. Thus u i, p i, J, ϕ, and ψ defined by E i = u i + p i, J = H i n Γ = Γ ϕ + Γ ψ, are solutions of the variational formulation (3.3-(3.4-(3.8-(3.9. Now, assume that u i, p i, ϕ, ψ, and J = Γ ϕ + Γ ψ are solutions of (3.3-(3.4- (3.8-(3.9 and also satisfy the periodic integral equation (3.5. Adding equations (3.3 and (3.4, we get once again by the Hodge decomposition Lemma 3. that d E i. E t ω 2 εe i. E t ω 2 εβe i. E t ω 2 εβ E i. E t iω J. E t iω J 2. E t = 0, Γ Γ 2 for any E t H(curl, with E i = u i + p i in. Consequently, d E i ω 2 εβ E i ω 2 (εβe i ω 2 εe i = 0 in, d E i n ω 2 εβe i n = iωj on. From (3.8-(3.9, we obtain that the fields E and E 2 given by the periodic integral representations (3.2, where M = (u i + p i n on,

15 Coupling of Finite Element and Boundary Element Methods for the Scattering 275 are solutions of the Maxwell equations E ω 2 ε µ E = 0 in, along with the radiation condition (2.22. In addition, if (3.5 holds, then the fields E satisfy ( E n = iωµ J on. But E n = E i n on. Thus, from the ump conditions (2., it follows that E and H of the form (3.6 with H i = i(ωµ u i are solutions of the Maxwell equations (2.5-(2.6 together with the radiation condition (2.22. The proof is now complete. To derive coupling FEM/BEM variational formulations for solving the diffraction problem, we also need the following two lemmas. These lemmas are known in case of a closed boundary Γ and the usual Green kernel of the Helmholtz equation in IR 3 [7]. Lemma 3.6. If J, u Γ, and p i satisfy the periodic integral equation (3.5, then J. u t Γ 2 = i ωµ iωε 2 G curl Γ u Γ curl Γ u t G Γ p i. u t n iωε G u Γ n. u t n ( n(xg J x G (n (x n (y.j.u t + (H in n.u t, u t TH /2 (curl,, (3.7 J. Γ p t = iωε G Γ p i. Γ p t ( n(xg J x G (n (x n (y.j. Γ p t Γ iωε G u Γ n. Γ p t + (H in n. Γ p t, p t H /2 (. (3.8 Proof. It suffices to establish (3.7. Multiplying Γ G div Γ (u Γ n by u t n for any u t TH /2 (curl, and integrating by parts over gives = Γ G div Γ (u Γ n u t n G div Γ (u Γ n div Γ (u t n = G curl Γ u Γ curl Γ u t. (3.9 The formula (3.7 follows from an integration by parts of the periodic integral equation (3.6 along with (3.9. We also need the following technical lemma.

16 276 H. AMMARI AND G. BAO Lemma 3.7. For any ϕ, ψ H /2 (, and v Γ TH /2 (, the following identities hold: G Γ ϕ. Γ ψ = (n (x n (y y G. Γ ψ ϕ, (3.20 G Γ ϕ.v Γ = (n (x n (y y G. v Γ ϕ, (3.2 Γ G. (( Γ ϕ ( Γ ψ = ω 2 ε µ G ψ ((n (x n (y. Γ ϕ, G. ((v Γ n Γ ψ = ω 2 ε µ We are now ready to state and prove the following theorem. (3.22 G ψ (n (x n (y.v Γ. (3.23 Theorem 3.2. The variational formulation (3.3-(3.4-(3.8-(3.9 together with the periodic integral equations (3.5 yield the following coupling FEM/BEM formulations: (A: obtained by replacing the terms J. u t and J. p t with their expressions from (3.7-(3.8; (B: obtained by dividing each of the terms J. u t and J. p t into two halves and then replacing 2 J. u t and 2 J. p t with their expressions from (3.7-(3.8. Note that an important advantage of (B is that the coupling formulation is symmetric. Furthermore, we may derive the third coupling FEM/BEM variational formulation. Similar to (3.2, the following variational formulation may be obtained for the magnetic field H: d H i. H t ω 2 µh i. H t ω 2 µβh i. H t ω 2 for any H t H(curl,, where µβ H i. H t iω M. H t iω Γ M 2. H t = 0, Γ 2 (3.24 d = ω2 β 2 εµ, H = ε H in, H i in i, H 2 in 2. Represent by the Hodge decomposition Lemma 3., H i = v i + q i, where v i IM( and q i H (. It follows from the periodic integral equation (3.5 and the identities (3.7-(3.8 that 2 M. vγ t = i G curl Γ v Γ curl Γ vγ t ωε iωµ G Γ q i. vγ t n Γ ( n(xg M x G (n (x n (y.m.u t iωµ G v Γ n. vγ t n + (E in n.vγ t, vγ t TH /2 (curl,, (3.25

17 Coupling of Finite Element and Boundary Element Methods for the Scattering 277 M. Γ q t = iωµ G Γ q i. Γ q t iωµ G v Γ n. Γ q t 2 Γ ( n(xg M x G (n (x n (y.m. Γ q t + (E in n. Γ q t, q t H /2 (. (3.26 Therefore, we obtain the third coupling FEM/BEM variational formulation. Theorem 3.3. The variational formulation (3.3-(3.4-(3.8-(3.9 together with the periodic integral equations (3.5 yield the following coupling FEM/BEM formulation: (C: obtained by replacing H i by v i + q i in (3.24, E i by u i + p i in (3.2, and the terms J. E t and M. H t with their expressions from (3.7-(3.8 and (3.25-( Existence and uniqueness results In this subsection, we show that the coupling FEM/BEM variational formulations (A, (B, and (C are of Fredholm type. Hence, for all but possibly a discrete sequence of parameters, there exist unique solutions to (A, (B, and (C. The results also yield a new proof of the uniqueness theorem for the diffraction problem. Theorem 3.4. Each one of the variational formulations (A, (B, and (C is of Fredholm type. Proof. It is sufficient to prove the theorem for the variational formulation (B. The same arguments will prove the theorem for the other two formulations (A and (C. Denote the left hand side terms of (B by a (u, u t, a 2 (p, p t, a 3 (ϕ, ϕ t, and a 4(ψ, ψ t, respectively. We take u t = u, p t = p, ϕ t = ϕ, and ψ t = ψ and consider the quantity Note that a (u, u a 2 (p, p iωa 3 (ϕ, ϕ + iωa 4 (ψ, ψ. p. Γ ψ = Γ ψ. p = 0. Next, Lemma 3.7 (3.20-(3.2 and the fact that the kernel (n (x n (y x G is of order one yield G Γ p. u Γ, G u Γ n. Γ p, G Γ p. Γ p, G Γ ψ. Γ ϕ, and G Γ ϕ. Γ ψ are compact. Next, by the Cauchy-Schwartz inequality, for any η εβ(u. u u. u η u 2 L 2 ( + 3 C(η u 2 L 2 ( 3. Now, we estimate the term G Γ ϕ. Γ ϕ. According to Lemma 2.4 G Γ ϕ. Γ ϕ C ϕ 2 H /2 (Γ.

18 278 H. AMMARI AND G. BAO By the trace regularity result stated in Lemma 3.3, we have G u Γ n. u Γ n C u Γ 2 TH /2 ( η u 2 L 2 ( 3 + C(η u 2 L 2 ( 3, and u Γ. ϕ u Γ TH /2 (Γ ϕ H 3/2 (, (η u L 2 ( 3 + η u L 2 ( 3 ϕ H 3/2 (, η (η u L 2 ( 32 + η ϕ 2 H 3/2 ( + η η u 2 L 2 ( 3 + 2η ϕ 2 H 3/2 ( + η 3 u 2 L 2 ( 3. ( η u L 2 ( η ϕ 2 H 3/2 ( Combining the above estimates and observing that { } Re εβ p. u εβ u. p = 0, { [ ]} Re i Γ ψ. u + u Γ. Γ ψ = 0, { [ ]} Re i Γ ϕ. p + p. Γ ϕ = 0, and the fact that the term µ G curl Γ u Γ 2 has the favorable sign, we obtain from Lemma 2.4 that for any u IM(, p H (, ϕ H 3/2 (, and ψ H /2 (, { } Re a (u, u a 2 (p, p + iωa 3 (ϕ, ϕ iωa 4 (ψ, ψ } C { u 2 IM( + p 2 H ( + ϕ 2 H 3/2 (Γ + ψ 2 H /2 ( } C 2 { u 2 L 2 ( + 3 p 2 L 2 ( + ϕ 2 H /2 (Γ + ψ 2 L 2 (, where C and C 2 are two positive constants. Since the imbedding from IM( to L 2 ( 3 is compact (Lemma 3.2, the Fredholm alternative holds for (B. The proof is now complete. 4. Discrete Problems This section is devoted to the study of discretization of the coupling FEM/BEM formulations. Our main result is a uniform convergence theorem for the coupling approach. 4.. Convergence analysis We discretize the coupling FEM/BEM variational formulations by using a family of finite element subspaces V h in H(curl,, where the parameter h is the maximum diameter of the

19 Coupling of Finite Element and Boundary Element Methods for the Scattering 279 elements for a given finite element mesh. The domain can be meshed by using curvilinear tetrahedra. Assume that the subspace V h satisfies the Hodge decomposition given in Lemma 3.. Then any vector E h V h can be represented as E h = u + p, where u IM( and p H (. We also require that R h u u L2 ( 3 0, h 0, E h H(curl, where R h : H 2 ( 3 V h is an interpolation operator. The family of finite element subspaces used to discretize the vector unknown u is the proection of V h on IM(. An example which satisfies the assumptions is the family of Nédélec s finite elements (tetrahedra in H(curl [48]. The Nédélec element has the property that S h V h, where S h is the usual P -Lagrange finite element approximation. Natural approximations for ϕ H 3/2 ( and ψ H /2 ( are: (i P -Lagrange finite element approximation for ψ and then Γ ψ is the space of Raviart-Thomas; and (ii any C finite element for ϕ. Now, let X h X = IM( H ( H 3/2 ( H /2 ( be the discretized subspace. By essentially identical arguments as in [7], the Babuska-Brezzi condition may be verified. Lemma 4.. There is a positive constant C independent of h such that A((u h, p h, ϕ h, ψ h, (u t h sup, pt h, ϕt h, ψt h (u t h,pt h,ϕt h,ψt h X h\{0} (u t h, pt h, ϕt h, ψt h C (u h, p h, ϕ h, ψ h, (u h, p h, ϕ h, ψ h X h. From the above lemma, the convergence result below follows. Theorem 4.. There exists h 0 > 0, such that, for 0 < h < h 0, the discrete solution E h = u h + p h is well defined with the following error estimate: E i E h H(curl, C where C is a positive constant independent of h. inf E i F h H(curl,, (4. F h X h Remark 4.. Note that in general the estimate (4. may not be improved. This is essentially due to the fact that the solution is only in H(curl, for bounded measurable material parameters ε, µ and β. Better convergence results would be possible with additional smoothness assumptions on the solutions Approximation of the periodic integral operators In practice, one cannot compute the kernels (G =,2 from the full infinite series expansions (2.23. It is thus necessary to obtain appropriate error estimates when truncations of the series expansions take place. In the following, we show that by extracting the principle singularity (4π x y from G (x, y, the uniform error estimates remain valid, provided that sufficiently many terms in the expansions of the operators G (x, y 4π x y

20 280 H. AMMARI AND G. BAO are taken. Write G (x, y x3=y 3 = + i 2Λ Λ 2 i 2Λ Λ 2. (x y e iαn n Z n n 2 2 n Z 2 ( β (n e iαn. (x y. n 2 + n 2 2 Then where G (x, y x3=y 3 = R (x, y = i 2Λ Λ 2 K (x, y = i 2Λ Λ 2. (x y e iαn n Z n n 2 2 n Z 2 ( β (n 4π x y + R (x, y + K (x, y, n 2 + n 2 2 4π x y = e iαn. (x y. n Z 2 γ (n e iαn. (x y, The kernel R (x, y has a singularity like iαπ log x y as x y 0 and the kernel K (x, y is continuous as x y 0. Lemma 4.2. There exists a positive constant M 0, such that for M > M 0 ( e iαn. (x y θ(xθ(y dγ(x dγ(y n M+, n 2 M+ γ (n CM θ 2 H /2 ( (4.2 for any θ H /2 (, where C is a positive constant independent of M. We recall that there are two positive constants C and C 2 with C θ 2 H /2 (Γ x y θ(xθ(y dγ(x dγ(y C2 θ 2 H /2 (Γ. Let the truncated kernel G M be defined by G M (x, y = 4π x y + + i 2Λ Λ 2 n M, n 2 M n M, n 2 M ( β (n γ (n iαn. (x y e e iαn. (x y. n 2 + n 2 2 By replacing G with G M in the coupling FEM/BEM variational formulations (A, (B, and (C, the following error estimate holds. Theorem 4.2. There exist h 0 and M 0, such that for 0 < h < h 0 and M > M 0 { } E i E h H(curl, C inf E i F h H(curl, + M E i H(curl,, F h X h where C is a constant independent of h and M.

21 Coupling of Finite Element and Boundary Element Methods for the Scattering 28 Remark 4.2. By assuming a complex frequency: ω = ω + iω (ω > 0, Morelot [46] proved that the estimate n M+, n 2 M+ β (n e iαn. x e iβ(n x 3 M min(λ,λ 2 C e ω ω, holds uniformly in compact sets, where the constant C is independent of M. Note that Morelot s estimate breaks down when ω = 0. We also refer to [46] for a computation of the kernel G by using the Ewald transform. 5. Concluding Remarks We have presented and analyzed several coupling FEM/BEM formulations for solving the diffraction from periodic chiral structures. It has been shown that the proposed numerical approximations attain unique solutions with uniform convergence properties. An interesting future proect is to decouple the finite element and integral equations solutions by using an iterative procedure. The integral equations could then be solved by a fast algorithm, for example a multipole method, independently of the finite element solutions. The fast algorithm would significantly reduce the CPU time for the integral equation portion of the code. Acknowledgments. The research of G. Bao was supported in part by the NSF grants DMS , CCF , EAR , the ONR grant N , the National Science Foundation of China grant References [] T. Abboud, Formulation variationnelle des équations de Maxwell dans un réseau bipériodique de R 3, C. R. Acad. Sci. Paris, Série I, Math., 37 (993, [2] H. Ammari and G. Bao, Analysis of the diffraction from periodic chiral structures, C. R. Acad. Sci., Paris, Série I, Math., 326 (998, [3] H. Ammari and G. Bao, Electromagnetic diffraction by periodic chiral structures, Math. Nachr., 25 (2003, 3-8. [4] H. Ammari, M. Laouadi, and J.C. Nédélec, Low frequency behavior of solutions to electromagnetic scattering problems in chiral media, SIAM J. Appl. Math., 58 (998, [5] H. Ammari and J.C. Nédélec, Time-harmonic electromagnetic fields in chiral media, in Modern Mathematical Methods in Diffraction Theory and its Applications in Engineering, Verf. Math. Phys., 42 (997, [6] H. Ammari and J.C. Nédélec, Time-harmonic electromagnetic fields in thin chiral curved layers, SIAM J. Math. Anal., 29 (998, [7] H. Ammari and J.-C. Nédélec, Couplage éléments finis équations intégrales pour la résolution des équations de Maxwell en milieu hétérogène, in Equations aux Dérivées Partielles et Applications. Articles dédiés à Jacques-Louis Lions, Elsevier, Paris, 998, [8] H. Ammari and J.-C. Nédélec, Coupling of finite and boundary element methods for the timeharmonic Maxwell equations. Part II: a symmetric formulation, Operator Theory: Advances and Applications 08. The Maz ya Anniversary Collection, Rossmann et al. (Eds, 0 (999, [9] H. Ammari and J.-C. Nédélec, Coupling integral equations method and finite volume elements for the resolution of the Leontovich boundary value problem for the time-harmonic Maxwell equations in three dimensional nonhomogeneous media, Mathematical Aspects of Boundary Element Methods, Research Notes in Mathematics, 44 (999, -22.

22 282 H. AMMARI AND G. BAO [0] H. Ammari and J.-C. Nédélec, Analysis of the diffraction rom chiral gratings, Mathematical Modelling in Optical Science, ed. by Bao, Cowsar, and Masters, Chapter 6, 79-06, SIAM Frontiers in Applied Mathematics, 200. [] H. Ammari, K. Hamdache and J.-C. Nédélec, Chirality in the Maxwell equations by the dipole approximation method, SIAM J. Appl. Math., 59 (999, [2] T. Arens, S. N. Chandler-Wilde, and J. A. DeSanto, On integral equation and least squares methods for scattering by diffraction gratings, Comm. Comput. Phys., (2006, [3] G. Bao, Variational approximation of Maxwell s equations in biperiodic structures, SIAM J. Appl. Math., 57 (997, [4] G. Bao, Z. Chen, and H. Wu, Adaptive finite element method for diffraction gratings, J. Opt. Soc. Am. A, 22 (2005, [5] G. Bao and D. Dobson, On the scattering by a periodic structure, Proc. Amer. Math. Soc., 28 (2000, [6] G. Bao, D. Dobson, and J. A. Cox, Mathematical studies of rigorous grating theory, J. Opt. Soc. Am. A, 2 (995, [7] G. Bao and P. Li, Inverse medium scattering problems in near-field optics, J. Comput. Math., 25:3 (2007, [8] G. Bao and A. Zhou, Analysis of finite dimensional approximations to a class of partial differential equations, Math. Method. Appl. Sci., 27 (2004, [9] A. Bendali, Numerical analysis of the exterior boundary value problem for the time-harmonic Maxwell equations by a boundary finite element, Part I: the continuous problem, Math. Comput., 43 (984, [20] A. de La Bourdonnaye, Some formulations coupling finite element and integral equation method for Helmholtz equation and electromagnetism, Numer. Math., 69 (995, [2] M. Birman and M. Z. Solomyak, L 2-Theory of the Maxwell operator in arbitrary domains, Russ. Math. Surv+, 42 (987, [22] F. Brezzi and C. Johnson, On the coupling of boundary integral and finite element methods, Calcolo, 6 (979, [23] U. Brink and E. P. Stephan, Convergence rates for the coupling of FEM and BEM, IMA J. Numer. Anal., 6 (996, [24] O. P. Bruno and F. Reitich, Solution of a boundary value problem for the Helmholtz equation via variation of the boundary into the complex domain, Proc. Royal Soc. Edinburgh, 22 A (992, [25] X. Chen and A. Friedman, Maxwell s equations in a periodic structure, Trans. Amer. Math. Soc., 323 (99, [26] Z. Chen and H. Wu, An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures, SIAM J. Numer. Anal., 4 (2003, [27] G. Cohen and M. Durufle, Non spurious spectral-like element methods for Maxwell s equations, J. Comput. Math., 25:3 (2007, [28] D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory, Springer-Verlag, 992. [29] M. Costabel, Symmetric methods for coupling of finite elements and boundary elements, in Boundary Elements IV, Vol., Compt. Mech. Southampton, 987, [30] M. Costabel and E. P. Stephan, Strongly elliptic boundary integral equations for electromagnetic transmission problems, Proc. Royal Soc. Edinburgh, 09 A (988, [3] M. Costabel and E. P. Stephan, Coupling of finite elements and boundary elements for inhomogeneous transmission problems in IR 3, in Mathematics of Finite Elements and Applications VI, J. R. Whiteman, ed., Academic Press, London, , 988. [32] M. Costabel and E. P. Stephan, Coupling of finite and boundary element methods for an elastoplastic interface problem, SIAM J. Numer. Anal., 27 (990,

23 Coupling of Finite Element and Boundary Element Methods for the Scattering 283 [33] D. Dobson and A. Friedman, The time-harmonic Maxwell equations in a doubly periodic structure, J. Math. Anal. Appl., 66 (992, [34] G. N. Gatica and W. L. Wendland, Coupling of mixed finite elements and boundary elements for linear and nonlinear elliptic problems, Appl. Anal., 63 (996, [35] T. Hagstrom and S. Lau, Radiation boundary conditions for Maxwell s equations: a review of accurate time-domain formulations, J. Comput. Math., 25:3 (2007, [36] T. Huttunen and P. Monk, The use of plane waves to approximate wave propagation in anisotropic media, J. Comput. Math., 25:3 (2007, [37] C. Johnson and J.-C. Nédélec, On the coupling of boundary integral and finite element methods, Math. Comp., 35 (980, [38] D.L. Jaggar, N. Engheta, M. Kowarz, P. Pelet, J. Liu, and Y. Kim, Periodic chiral structures, IEEE T. Antenn Propag, 37 (989, [39] A. Lakhtakia, Beltrami fields in chiral media, World Scientific Series in Contemporary Chemical Physics-Vol. 2, 994. [40] A. Lakhtakia, V.K. Varadan, and V.V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media, Lecture Notes in Physics, 355 (989. [4] A. Lakhtakia, V.K. Varadan, and V.V. Varadan, Scattering by periodic achiral-chiral interfaces, J. Opt. Soc. Am. A, 6 (989, [42] Electromagnetic Theory of Gratings, Topics in Current Physics, Vol. 22, edited by R. Petit, Springer-Verlag, Heidelberg, 980. [43] V. Levillain, Couplage éléments finis-équations intégrales pour la résolution des équations de Maxwell en milieu hétérogène, Ph.D. Thesis, Ecole Polytechnique, 99. [44] S. Meddahi, An optimal iterative process for the Johnson-Nedelec method of coupling boundary and finite elements, SIAM J. Numer. Anal., 35 (998, [45] S. Meddahi, J. Valdés, O. Menéndez, and P. Pérez, On the coupling of boundary integral and mixed finite element methods, J. Compt. Appl. Math., 69 (996, [46] A. Morelot, Etude d une méthode numérique de simulation de la diffraction d une onde électromagnétique par un réseau bi-périodique, Ph.D. Thesis, Ecole Polytechnique, 992. [47] C. Müller, Foundations of the mathematical theory of electromagnetic waves, Springer-Verlag, New York, 969. [48] J.-C. Nédélec, A new family of mixed finite elements in IR 3, Numer. Math., 50 (986, [49] J.-C. Nédélec, New trends in the use and analysis of integral equations, Proc. Symp. Appl. Math., 48 (994, [50] J.-C. Nédélec and F. Starling, Integral equations methods in a quasi-periodic diffraction problem for the time-harmonic Maxwell s equations, SIAM J. Math. Anal., 22 (99, [5] A. Rathsfeld, G. Schmidt, and B. H. Kleemann, On a fast integral equation method for diffraction gratings, Comm. Comput. Phys., (2006, [52] R. H. Torres, Maxwell s equations and dielectric obstacles with Lipschitz boundaries, J. Lond. Math. Soc., 57 (998, [53] W. L. Wendland, On asymptotic error estimates for combined BEM and FEM, in Finite Element and Boundary Element Techniques from Mathematical and Engineering Point of View, CISM Courses and Lectures, 30, Springer, Viennna, 988, [54] Ch. Weber, A local compactness theorem for Maxwell s equations, Math. Method. Appl. Sci., 2 (980, [55] S. Y. Yueh and J. A. Kong, Analysis of diffraction from chiral gratings, J. Elctro. Waves Appl., 5 (99,