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On the approximation of dispersive electromagnetic eigenvalue problems in 2D

We consider time-harmonic electromagnetic wave equations in composites of a dispersive material surrounded by a classical material. In certain frequency ranges this leads to sign-changing permittivity and/or permeability. Previously meshing rules were reported, which guarantee the convergence of finite element approximations to the related scalar source problems. Here we generalize these results to the electromagnetic two dimensional vectorial equations and the related holomorphic eigenvalue problems. Different than for the analysis on the continuous level, we require an assumption on both contrasts of the permittivity and the permeability. We confirm our theoretical results with computational studies.


Introduction
For dispersive material laws [18,20,12,9] the magnetic permeability and/or the electric permittivity are negative in certain frequency ranges. In composites of classical and dispersive materials this leads to indefinite coefficients for the time-harmonic wave equations − div(µ(ω) −1 ∇u) − ω 2 (ω)u = 0, curl(µ(ω) −1 curl u) − ω 2 (ω)u = 0 and the well-posedness/Fredholmness of the associated boundary value problems/operators becomes a delicate question which depends on the contrasts κ µ −1 , κ at the materials interface. In [6] the Fredholmness for the scalar wave equation was thoroughly answered. For smooth interfaces the operator is Fredholm if κ µ −1 = 1. If the interface admits corners, then the operator is Fredholm if the contrast κ µ −1 is outside a critical interval I c −1, whose size depends on the angles of the corners. We refer to [3] for more details in the presence of corners and a technique to restore a Fredholm framework for contrasts inside of the critical interval. In [4,7] the analysis of [6] was extended to the two-and three-dimensional electromagnetic equations. Instead of the geometric techniques used in [6], the technique of [4,7] was to relate the Fredholmness to the well-posedness of the associated scalar equations (already discussed in [6]). The finite element approximation of scalar source problems in R 2 was discussed in [5] wherein meshing rules were reported which guarantee the convergence of approximations.
The applied technique of the former references to prove that an operator A is Fredholm, is to construct a bijective operator T such that A is weakly T -coercive. To explain this terminology recall that an operator A ∈ L(X) is called coercive, if inf u∈X, u X =1 | Au, u X | > 0 and weakly coercive if there exists a compact operator K ∈ L(X) such that A + K is coercive. For a bijective operator T ∈ L(X) the operator A ∈ L(X) is called (weakly) T -coercive, if T * A is (weakly) coercive. Here T * denotes the adjoint operator of T . Further we call Galerkin approximations with finite dimensional spaces X n ⊂ X to be T -compatible, if there exist operators T n ∈ L(X n ) with lim n→∞ T − T n L(Xn,X) = 0. This criterion does not only guarantee the convergence of approximations of source problems Au = f , but also of holomorphic eigenvalue problems A(ω)u = 0, see the framework [13,14] which is based on [16,17]. We also refer to the introduction of [14] for a historical recap of T-analysis.
In this article we consider the the two dimensional vectorial electromagnetic wave equation and the finite element approximation of the associated source and holomorphic eigenvalue problems. Different than for the analysis on the continuous level in 2D, we require an assumption on both contrasts of the permittivity and the permeability. We prove that the meshing rules of [5] guarantee the convergence of h-finite element approximations with Nédélec elements. To this end we generalize the approach of [5] and construct an apt T -operator by means of local pattern based reflection operators. In particular we employ a Helmholtz decomposition of H 0 (curl) = V ⊕ ⊥ ∇H 1 0 and introduce individual local reflection operators for each Helmholtz component.
The remainder of this article is structured as follows. In Section 2 we specify our setting and introduce the operator A to be investigated. In Section 3 we construct an apt T -operator and prove weak T -coercivity of A. In Section 4 we construct apt approximations T n of T and prove the convergence of approximations. In Section 5 we conduct computational studies to verify our theoretical results. In Section 6 we conclude and discuss the further outlook.

Specification of the problem
Let Ω ⊂ R 2 be a bounded Lipschitz domain (open and connected) with unit outward normal vector ν and unit tangential vector τ . Let Ω − ⊂ Ω be a Lipschitz polyhedron with ∂Ω − ∩ ∂Ω = ∅ and let Ω + := Ω \ Ω − , Σ := ∂Ω − . Let µ and be the magnetic permeability and the electric permittivity. We consider the case of a composite material whereby Ω + is a classical material and Ω − is a dispersive material, i.e. | Ω− and µ| Ω− depend on the frequency ω. We formulate all our theory in terms of the spectral parameter λ := ω 2 . To simplify the presentation we assume that µ, are scalar and constant in Ω ± and that in Ω − the material is modeled by a Drude law. To be precise let µ + , µ − , + , − be positive constants and ω µ , ω be real non-negative constants and whereby ω µ and ω are the resonant frequencies. We introduce the contrasts Note that our forthcoming theory in Sections 3 and 4 can easily be adapted to non-homogeneous coefficients µ ± , ± , although in this case the definition of the contrasts is more technical (see, e.g. [6,Theorem 4.3]). Let ∂ xn u be the partial derivative of a function u with respect to the variable x n . For a two-vector function u = (u 1 , u 2 ) let curl u := ∂ x1 u 2 − ∂ x2 u 1 . For a scalar function u let Curl u := (∂ x2 u, −∂ x1 u) . For s ≥ 0 and a Lipschitz domain , u, u ∈ H s (curl; D) and H(curl; D) := H 0 (curl; D). We consider the electromagnetic time-harmonic wave equation Here we consider either a source problem for which a solution u ∈ H(curl; Ω) is sought for given λ and right hand side f . Or we consider the eigenvalue problem for which f = 0 and a pair (λ, u) is sought (with non-trivial u). To formulate these in a functional setting we introduce the following. For two Banach spaces X, Y denote L(X, Y ) the space of bounded linear operators from X to Y with norm A L(X,Y ) := sup u∈X, u X =1 Au Y and set L(X) := L(X, X). Further let X := H 0 (curl; Ω) := {u ∈ H(curl; Ω) : τ · u = 0 on ∂Ω} with scalar product u, u X := u, u H(curl;Ω) . We consider the operator A(λ) ∈ L(X) defined by for all u, u ∈ X and λ ∈ C \ {0, ω 2 µ }. Then the operator formulation of the source problem is to find for given λ ∈ C \ {0, ω 2 µ } andf ∈ X a solution u ∈ X such that A(λ)u =f . The operator formulation of the eigenvalue problem is to find (λ, u) ∈ (C \ {0, ω 2 µ }) × (X \ {0}) such that A(λ)u = 0. In this article we are concerned with two goals. The first is to prove that A(λ) is Fredholm with index zero for λ ∈ Λ, whereby Λ ⊂ C is an open connected set which will be specified later on. The second one is two prove that A(λ) can be sufficiently approximated by convenient H(curl)-finite element methods with the meshing rules introduced in [5]. The means to establish the former goals are to construct for each λ ∈ Λ a bijective operator T (λ) ∈ L(X) such that A(λ) is weakly T (λ)-coercive. In this situation we say that A(·) is weakly T (·)coercive, from which it readily follows that A(λ) is Fredholm with index zero for all λ ∈ Λ. For the analysis of approximations of source and eigenvalue problems by Galerkin methods with discrete spaces X n ⊂ X we employ the framework of T (·)-compatible approximations [14,13]. To this end we need to construct operators T n (λ) ∈ L(X n ) such that lim n→∞ T (λ)−T n (λ) L(Xn,X) = 0 for each λ ∈ Λ.
Let us examine the properties of A(λ) for λ in different regions of C. At first we note the singular behaviour at λ = 0, ω 2 µ and the degenerate behaviour at λ = ω 2 . On the other hand A(λ) is clearly coercive for λ < 0. Also for λ ∈ C \ R one can check that the coefficients 1/µ(λ)| Ω+ , −λ (λ)| Ω+ , 1/µ(λ)| Ω− , −λ (λ)| Ω− are contained in a closed salient sector. Hence for λ ∈ C \ R the operator A(λ) is coercive too. Thus with R + := {x ∈ R : x > 0} we set For the analysis of A(λ) for λ > 0 we introduce the following Helmholtz decomposition. Let and P W , P V be the respective orthogonal projections and P W0 ∈ L(X, W 0 ) be such that P W u = ∇P W0 u for all u ∈ X. We recall the embeddings whereby the first embedding can be seen as in [8]. Thence for λ > max(ω 2 µ , ω 2 ) the operator A(λ) is weakly T (λ)-coercive with On the contrary if λ ∈ (0, max(ω 2 µ , ω 2 )) the study of A(ω) requires much more sophisticated techniques and will be dealt with in Section 3.

Weak T-coercivity
In this section we consider the case λ ∈ (0, max(ω 2 µ , ω 2 )). For contrasts κ µ −1 (λ), κ (λ) outside a critical interval (which will be specified) we construct an apt operator T (λ), such that A(λ) is weakly T (λ)-coercive. The definition of T (λ) involves cut-off functions χ n and local reflection operators R V,± n , R W0,± n . In Subsection 3.1 we formulate assumptions on those, introduce T (λ) under this pretext and prove the weak T (λ)-coercivity of A(λ). In Subsection 3.2 we explicitly construct cut-off functions and local reflection operators which satisfy the former requirements. In Subsection 3.3 we conclude and formulate our results.

Abstract framework
We follow [5, p. 813-814] and introduce some geometry based tools which will be necessary to construct T (λ). Recall that Σ is a polygonal line without endpoints. Let N be the number of its corners, (c n ) n=1,...,N be its corners, (α n ) n=1,...,N be the corner angles measured in Ω + , and (e n ) n=1,...,N be its edges. Assumption 3.1 (cut-off functions). Let χ n ∈ C ∞ (Ω, [0, 1]), n = 1, . . . , 2N be smooth cut-off functions. We associate the first N functions to the corners of Σ and the second N functions to the edges of Σ. In particular for each n = 1, . . . , N let χ n have a support localized in a neighborhood of the interface, such that χ n = 1 in a neighborhood of the corner c n . Further let χ n , n = 1, . . . , N have mutually disjoint supports. For each n = 1, . . . , N let χ N +n have a support localized in a neighborhood of the edge e n , such that χ n = 0 on Σ \ e n . Additionally let 2N n=1 χ n = 1 in a neighborhood of Σ. For χ n satisfying Assumption 3.1 we further define S n := supp χ n as the support of the cut-off function χ n . Next we introduce our assumptions on the local reflection operators. We introduce separate reflection operators for each Helmholtz component. To this end let If necessary we shorten the notation and write for a function u defined on Ω and an operator R acting from X ± or W 0,± the short form Ru instead of Ru| Ω± . We note that in the following assumption the reflection operators R V,± n act from X ± (and not only from restrictions of V , i.e. V ± ).
For χ n and R V,± n , R W0,± n satisfying Assumptions 3.1 and 3.2 respectively let Subsequently we define In addition we set T V,0 := I X and T W0, . It is then natural to consider the composed operator and observe T (λ) ∈ L(X). Now we are able to proof a similar result as in Lemma 2 of [5].
Proof. We present only the case κ µ −1 (λ), κ (λ) ∈ (−1, 0). The other cases can be treated analogously. We abbreviate v := P V u, w 0 := P W0 u, v := P V u and w 0 := P W0 u and define A 1 , A 2 , A 3 ∈ L(X) by Then we can follow the proof of [5, Lemma 2] line by line to obtain by means of a weighted Youngs inequality that Since V and W are orthogonal and due to curl Thus A 1 + A 2 is coercive. The operator A 3 is compact due to (4) and Assumption 3.2.
We continue with a discussion on the bijectivity of T (λ). It can easily be seen that ( . For this reason we are not able to prove the bijectivity of T (λ) via the convenient way (T (λ)T (λ) = I X ). However we can obtain a weaker result.
is bijective.
Proof. We abbreviate T := T (λ) and compute Thus if we can prove that be the embedding operator. Due to (4) and Assumption 3.

Summary of results
We combine the results from Subsections 3.1 and 3.2.
Proof. Follows from Proposition 3.5 and Subsection 3.2.
Corollary 3.7. Let the assumptions of Theorem 3.6 be satisfied. Then the spectrum of A(·) : Λ α → L(X) is discrete, has no accumulation points in Λ α and every point in the spectrum is an eigenvalue with finite algebraic multiplicity.
Proof. Follows from Theorem 3.6 and the analytic Fredholm theorem.
Remark 3.8. The studies in the preceding articles [6,4,7,5] considered the case of dispersionless µ, and were not able to obtain that the resolvent set of the corresponding operator function is non-empty. Though for dispersive µ, it is no problem to obtain coercivity for λ ∈ C \ R + .

T-compatible approximation
In this section we analyze the approximation of A(·) by Galerkin methods. To this end we employ the framework of T (·)-compatible approximations [14]. In
For the proofs of Lemmata 4.8 and 4.9 we adapt the discrete commutator technique [2].
The above equality holds, because Π X∓,n χ m (x)R V,± m π Xn P V u | B hn (x)∩Ω∓ depends only on and because Π X∓,n is a projection. We continue to estimate h t n u X . Altogether we obtain (T V,± − T V,± n )π Xn P V L(Xn,X) h t n + (I X − π Xn )P V L(Xn,X) which proves the claim.  Let T (·) and T n (·) be as defined in (3), (5), (7) and (13) respectively. Then for each λ ∈ Λ R V ,R W 0 it holds lim n→∞ T (λ) − T n (λ) L(Xn,X) = 0.
Remark 4.11. Note that [5] considers only lowest order finite elements and estimates the term similar to (1 − Π X∓,n )χ m R V,± m π Xn P V u in a direct fashion. Instead, one can apply the discrete commutator technique [2] as in Lemma 4.8 to generalize the results of [5] to high order methods.
For the construction of such meshes we refer to [5] and Figures 1a and 1b. We note that for a technical reason our Definition 4.12 is slightly different to [5,Definition 3]. We formulate our definition in terms of the slightly lager patches Ω n instead of S n . Lemma 4.13. Assume that Ω is a Lipschitz polyhedron and that all angles (α n ) n=1,...,N of the interface Σ are in 2πQ. Let χ n and R V,± n , R W0,± n be as defined in Subsection 4.2. Let (T n ) n∈N be a sequence of shape-regular and locally R-conform triangular meshes of Ω with maximal element diameter h n → 0 as n → ∞. Let X n be a Nédélec finite element space on the mesh T n such that the polynomial degree k ∈ N is uniform on all triangles ∆ ∈ Proof. Let (W 0,n , X n , J n ) be the subcomplex of (W 0 , X, J) associated to X n , see e.g. [19]. Then the existence of L 2 -uniformly bounded cochain projections π W0,n , π Xn , π Jn follows as in [1,Theorem 5.9]. Hence Assumption 4.1 is satisfied. We can use the Scott-Zhang-interpolant as local projection Π W0,n , see e.g. [10,Lemma 1.130]. We can use the standard H(curl)-interpolant as local projection Π Xn , which is defined in terms of the degrees of freedom and basis functions [19, p. 62, 80]. Here we exploit that in the special 2D setting the H(curl)-interpolant is bounded on H(curl). Indeed we recall that the degrees of freedom are q l , τ · u H 1/2 (e)×H −1/2 (e) , (q l ) l=0,...,k basis of P k (e), for each edge e in the skeleton of T n , and s l , curl u L 2 (∆) , s l basis for P k−1 (∆)/R, ∇p l , u L 2 (∆) , p l basis for λ 1 λ 2 λ 3 P k−2 (∆), for each triangle ∆ ∈ T n . Hereby P k denotes the space of polynomials of degree lower equal than k and λ j , j = 1, 2, 3 denote the barycentric coordinates. Thus Assumption 4.2 is satisfied. It can easily be seen that Assumption 4.3 is satisfied by finite element spaces. Since the meshes T n are locally R-conform and the polynomial degree k is locally uniform too, Assumption 4.4 is satisfied as well.

Summary of results
Theorem 4.14. Assume that Ω is a Lipschitz polyhedron and that all angles (α n ) n=1,...,N of the interface Σ are in 2πQ. Let λ ∈ Λ α be in the resolvent of A(·), f ∈ X and u ∈ X be the solution to A(λ)u = f .
Let (T n ) n∈N be a sequence of shape-regular and locally R-conform triangular meshes of Ω with maximal element diameter h n → 0 as n → ∞. Further let k 0 ∈ N and X n be a Nédélec finite element space on the mesh T n with maximal local polynomial degrees k ≤ k 0 and such that the local polynomial degree k ∈ N is uniform on all triangles ∆ ∩ 2N n=1 Ω n = ∅. Then there exist constants C, n 0 > 0 such that for all n > n 0 there exists a unique solution u n ∈ X n to the approximated equation P Xn A(λ)u n = P Xn f which satisfies u − u n H(curl;Ω) ≤ C inf u n ∈Xn u − u n H(curl;Ω) .
Let (T n ) n∈N be a sequence of shape-regular and locally R-conform triangular meshes of Ω with maximal element diameter h n → 0 as n → ∞. Further let k 0 ∈ N and X n be a Nédélec finite element space on the mesh T n with maximal local polynomial degrees k ≤ k 0 and such that the local polynomial degree k ∈ N is uniform on all triangles ∆ ∩ 2N n=1 Ω n = ∅. Consider the discrete eigenvalue problem to find (λ n , u n ) ∈ Λ α × X n \ {0} such that P Xn A(λ n )u n = 0.
Then the eigenvalues λ n and eigenfunctions u n converge to λ and u respective in the sense of [14, i)-vii)]. For example, ifΛ ⊂ Λ α is a compact set with rectifiable boundary ∂Λ ⊂ ρ A(·) and a simple eigenvalue {λ} =Λ ∩ σ A(·) with normalized eigenfunction u ∈ X, then there exist n 0 ∈ N and c > 0 such that for all n > n 0 the setΛ ∩ σ P Xn A(·)| Xn consists of a simple eigenvalue λ n with normalized eigenfunction u n ∈ X n and |λ − λ n | ≤ c inf To compute the eigenvalues of the rational matrix function A n (·) it is most convenient to linearize it and subsequently to apply a linear eigenvalue solver. To this end let Y n := {(curl u)| Ω− : u ∈ X n } ⊂ L 2 (Ω − ) with associated orthogonal projection p Yn ∈ L(L 2 (Ω − ), Y n ) and Then for λ = ω 2 µ the function u ∈ X n , solves A n (λ)u = 0 if and only if (u, v) ∈ X n × Y n solves for all (u , v ) ∈ X n × Y n . However, if µ − is not constant then the above correspondence is not true. In this case one can choose e.g. v := 1 λ−ω 2 µ curl u| Ω− as auxiliary variable. Then the correspondence still holds, but the obtained matrix stencil is not selfadjoint. Alternatively, we can analyze the selfadjoint matrix stencil in (15) as follows. If (u, v) ∈ X n × Y n solves (15), then u ∈ X n solves (A n (λ) + K n (λ))u = 0 with K n (λ) ∈ L(X n ) defined by (a) Geometrical setup to build locally Rconform meshes.

Source problem
We consider the source problem (1) with right hand side f = (1, 1) , λ = 1 and ω µ = ω = √ 2, µ + = 1, µ − = 1/10, + = 1, − = 10. Thus we obtain the contrasts κ µ −1 (λ) = κ (λ) = −10. We consider a sequence of locally R-conform meshes (T n ) nN obtained from uniform refinements of an initial mesh T 0 with maximal element diameter h max = 0.2. Subsequently we build X n as lowest order Nédélec elements on T n . As we do not have an analytical solution at our disposal, we take the finite element solution on the finest mesh as reference. In Figure 2 we present the relative errors in the H(curl) and L 2 norms. To gain more confidence in our approximations we compute a second reference solution in an alternative way. By means of a Helmholtz decomposition we can represent u = (λ) −1 Curl v with v ∈ H 1 (Ω) being the solution to ν · (λ) −1 ∇v = 0 on ∂Ω, with potential f 0 (x) = x 1 − x 2 , Curl f 0 = f . Recall that the approximation of (16) was already analyzed in [5]. We compute an approximate solution v n ∈ H 1 (Ω) to (16) with scalar finite elements on the finest mesh T n and set u n := (λ) −1 Curl v n ∈ (L 2 (Ω)) 2 . Consequently we can compare u n to u n in the L 2 norm, see the last column in Figure 2. In accordance with our theory we observe decreasing errors in Figure 2. In Figure 3 we plot the computed solution.

Eigenvalue problem
Let ω µ = 2, ω = √ 2, µ + = 1, µ − = 10, + = 1, − = 10. Then κ µ −1 (λ) ∈ I α if and only if λ ∈ [8/3, 200/51] and κ (λ) ∈ I α if and only if λ ∈ [4/3, 100/51]. We consider the eigenvalue problem (1), i.e. we search for (λ, u) ∈ C × X \ {0} which solve (1) with f = 0. Since this is a rational eigenvalue problem, we linearize it with the auxiliary variable v := ωµ √ µ− 1 λ−ω 2 µ curl u| Ω− as in Section 4.3. Subsequently we apply the Arnoldi algorithm of NG-Solve to solve the linear matrix eigenvalue problem. As for the source problem we use the associated equation for the Curl-potential to compute a second reference solution. In Figure 4 we present the computed spectrum. Outside the interval [4/3, 100/51] we note a good accordance of the eigenvalues between the two computations. Inside the interval [4/3, 100/51] (between the green boxes) we note an accumulation of eigenvalues, which is to be expected since the operator function is not Fredholm in this interval. In addition we also observe a good accordance of the eigenvalues inside [8/3, 200/51], which is not covered by our theory. To compute errors we choose the biggest eigenvalue below 4/3 (marked with a magenta box) and list in Figure 5 the associated errors. In accordance with our theory we observe decreasing errors in Figure 5. In Figure 6 we plot the corresponding eigenfunction.

Conclusion
In this article we extended the analysis of [5] to dispersive time-harmonic twodimensional vectorial electromagnetic wave equations. For the definition of our T -operator we introduced reflection operators for each Helmholtz component, whereby both individual reflection operators are based on the geometric construction in [5]. We showed that h-finite element methods with Nédélec elements and locally R-conform meshes are T -compatible. Consequently the Tcompatibility ensures [14] that the holomorphic eigenvalue problems are approximated reliably. We underlayed our results with computational experiments. However, our analysis requires an additional assumption on the contrast κ µ −1 , which is not necessary for the results on the continuous level in 2D [4]. Indeed, our computational experiments suggest that the finite element method yields