Computer Physics Communications

Computer Physics Communications 261 (2021) 107769

Contents lists available atScienceDirect

Computer Physics Communications

journal homepage:www.elsevier.com/locate/cpc

The bi-Lebedev scheme for the Maxwell eigenvalue problem with 3D bi-anisotropic complex media

^✩

Xing-Long Lyu

^a

, Tiexiang Li

^a,b,∗

, Tsung-Ming Huang

^c,∗∗

, Wen-Wei Lin

^d

, Heng Tian

^d

aSchool of Mathematics, Southeast University, Nanjing 211189, People’s Republic of China

bNanjing Center for Applied Mathematics, Nanjing 211135, People’s Republic of China

cDepartment of Mathematics, National Taiwan Normal University, Taipei 116, Taiwan

dDepartment of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan

a r t i c l e i n f o

Article history:

Received 30 May 2020

Received in revised form 2 October 2020 Accepted 27 November 2020

Available online 10 December 2020

Keywords:

Maxwell eigenvalue problem Bi-Lebedev scheme

3D bi-anisotropic complex media Null-space free method Fast Fourier transform

a b s t r a c t

This paper focuses on studying the eigenstructure of generalized eigenvalue problems (GEPs) arising in the three-dimensional source-free Maxwell equations for bi-anisotropic complex media with a 3-by-3 permittivity tensorε >0, a permeability tensorµ >0, and scalar magnetoelectric coupling constants ξ = ¯ζ =ıγ. The bi-Lebedev scheme is appealing because it preserves the symmetry inherent to the Maxwell eigenvalue problem exactly and because full degrees of freedom of electromagnetic fields at each grid point are taken into account in the discretization. The resulting GEP has eigenvalues appearing in quadruples{±ω,± ¯ω}. We consider two main scenarios, whereγ < γ∗andγ > γ∗with γ∗as a critical value. In the former case, all the eigenvalues are real. In the latter case, the GEP has complex eigenvalues, and we particularly focus on the bifurcation of the eigenstructure of the GEPs.

Numerical results demonstrate that the newborn ground state has occurred afterγ = ˜γ > γ∗, and the associated eigenvector has an exotic phenomenon of localization. Moreover, the Poynting vectors of the newborn eigenvector not only are concentrated in the material but also display exciting patterns.

©2020 Elsevier B.V. All rights reserved.

1. Introduction

Mathematically, the propagations of electromagnetic fields in bi-isotropic and bi-anisotropic media are modeled by the three- dimensional (3D) source-free Maxwell equations in the frequency domain with the constitutive relations

∇ ×

E(r)

=

ı

ω

^B(r)

,

^(1a)

∇ ×

H(r)

= −

ı

ω

^D(r)

,

^(1b)

∇ ·

B(r)

=

0

, ∇ ·

D(r)

=

0

,

^(1c) where

ω

represents the frequency,ı

= √

−

1,E andH

∈

_C³are the electric and magnetic fields, respectively, andBandD

∈

_C³ are the magnetic induction and dielectric displacement, respec-

✩ The review of this paper was arranged by Prof. Hazel Andrew.

∗ Corresponding author at: School of Mathematics, Southeast University, Nanjing 211189, People’s Republic of China.

∗∗ Corresponding author.

E-mail addresses: [email protected](X.-L. Lyu),[email protected](T. Li), [email protected](T.-M. Huang),[email protected](W.-W. Lin), [email protected](H. Tian).

tively, at positionr

=

(x

,

^y

,

^z)

∈

_R³. For the linear nondispersive media,BandDsatisfy the following constitutive relations B(r)

= µ

^(r)H(r)

+ ζ

^(r)E(r)

,

^D(r)

= ε

^(r)E(r)

+ ξ

^(r)H(r)

,

⁽²⁾ where

ε

^{(r) and}

µ

(r) are the permittivity and permeability, re- spectively, and

ξ

^(r),

ζ

(r) are the magnetoelectric coupling pa- rameters. Here,

ε

^(r),

µ

^(r),

ξ

^{(r), and}

ζ

(r) are constants in the bi-isotropic media. They are generalized as 3

×

3 tensors with Hermitian positive definite (HPD) matrices

ε

^{(r) and}

µ

(r) for the bi-anisotropic media.

A null-space free method [1,2] is proposed to solve(1) with reciprocal bi-isotropic chiral media, where

ε

^(r)

>

^0,

µ

^(r)

=

1, and

ξ

^(r)

= ¯ ζ

^(r)

=

{ı

γ , γ ≥

0

,

^r

∈ /

air

,

0

,

^otherwise

.

⁽³⁾

For

γ > γ

^∗ (a critical value), a novel interesting physical phe- nomenon indicating that a new ground state is born and the corresponding electromagnetic field is localized in the chiral medium was first found in [2]. In this paper, the interest lies in developing a numerical method to simulate such a new physical phenomenon for the reciprocal bi-anisotropic chiral media with HPD

ε

^{(r) and}

µ

^{(r), and}

ξ

^(r),

ζ

^{(r) in(3).}

https://doi.org/10.1016/j.cpc.2020.107769

0010-4655/©2020 Elsevier B.V. All rights reserved.

The combination of (1a), (2) and (1b)leads to the Maxwell eigenvalue problem (MEP)

[ 0

−∇×

∇×

0

] [E H ]

=

ı

ω

[

ε ξ

ζ µ

] [E

H ]

.

⁽⁴⁾

Bloch’s theorem [3] requires that the eigenfieldsEandHof(4)in 3D periodic media with lattice translation vectors

{

a_ℓ

}

³_ℓ=1satisfy the quasiperiodic condition

E(r

+

a_ℓ)

=

e^ı2πk·aℓE(r)

,

^H(r

+

a_ℓ)

=

e^ı2πk·aℓH(r)

, ℓ =

1

,

²

,

³

,

⁽⁵⁾ for a Bloch wave vectork.

In recent decades, several numerical methods, including plane- wave expansion methods [4,5], finite element methods [6–9], and finite difference (FD) methods [10–14], have been developed to solve the MEP (4); see the references therein for further details. The plane-wave method is widely used to find numerical solutions of Maxwell’s equations with periodic or quasiperiodic boundary conditions. The advantage is that the solution can be easily expanded as the superposition of a sequence of plane waves without any preprocessing. The finite element method is a popular choice for the simulation domain with an irregular shape and/or a complicated interior interface. To the best of our knowledge, in this case, the unstructured mesh is indispensable for this method to attain a desired accuracy, and the solution of an unstructured large sparse linear system is necessary and is usually beyond the scope of applications of the commonly used fast Fourier transform (FFT).

In 1966, a special FD method called Yee’s scheme [14] was developed that is attractive for simulating isotropic photonic crystals (i.e., without magnetoelectric coupling), owing to its sim- plicity and preservation of physical properties by which(4)can be conveniently discretized into a generalized eigenvalue problem (GEP). For 3D anisotropic photonic crystals, some researchers have proposed the Lebedev scheme [15–17] by which(1a)and (1b)are discretized on the Lebedev grid. A Lebedev grid can be seen as a superposition of the standard Yee grid and three shifted Yee grids, as illustrated inFig. 1.

This paper introduces the bi-Lebedev scheme [18,19] to solve the MEP (4) with 3D reciprocal bi-anisotropic complex media.

The so-called bi-Lebedev scheme artificially introduces another copy of the Lebedev grid that coincides with the one shown in Fig. 1(a) but with the color exchanged (red

→

blue, blue

→

red). Consequently, the same Lebedev scheme can be used to discretize (1a) and (1b) separately on these two replicas of the Lebedev grid with two different sets of variables,

{

E(red), H(blue),D(red), B(blue)} and

{

E(blue),H(red), D(blue), B(red)}, which are coupled with each other only when the bi-anisotropic constitutive relations(2)are evaluated [18,19].

In this paper, we make the following contributions to solving the MEP(4)with 3D bi-anisotropic complex media.

•

Using the bi-Lebedev scheme, we provide a detailed FD discretization of the MEP(4) together with the quasiperi- odic condition(5)with 3D bi-anisotropic complex media to produce a GEP. The matrices of the discrete permittivity

ε

and permeability

µ

preserve the HPD property if

ε

^and

µ

are HPD. With

ξ = ¯ ζ ∈

_Cin(3), eigenvalues of the resulting GEP appear as the pair

{ ω, − ω }

if

ω ∈

_R

∪

ıR, and they appear the quadruplet

{ ω, − ω, ω, ¯ − ¯ ω }

if

ω ∈

_C

\

(R

∪

ıR).

Moreover, by applying singular value decomposition (SVD) of the discrete single-curl operator, we develop a null-space free method to solve the resulting GEP.

•

We adopt a 3-by-3 permittivity HPD matrix

ε

and the per- meability

µ =

I₃ and take

ξ = ζ ¯

^{in (3) with}

γ >

γ

^∗

≈

1

.

204 to make the weight matrix [

ε ξ

ζ µ

]

indefinite,

which results in extremely complicated eigenstructures of the discrete MEP (4). Here, the weight matrix is HPD if

γ < γ

^∗ and singular if

γ = γ

^∗ (a critical value). With a similar derivation in Section3 of [2], the discrete MEP(4) has real eigenvalues

± ω

^for

γ < γ

^∗ and abundant 2

×

2 Jordan blocks at

ω = ∞

when

γ = γ

^{∗. For}

γ > γ

^{∗, a mass} of eigenvalue tetrads (

ω, − ω, ω, ¯ − ¯ ω

^{) with}

|

Re(

ω

⁾

| ≈

0 and

|

Im(

ω

⁾

| ≫

0 are created, some of which collide rapidly near the origin and then bifurcate into positive and negative real eigenvalues, respectively. The newborn positive eigenvalue pushes the original eigenvalues farther from zero, i.e., the newborn eigenstate possesses less energy (frequency) than that of the original ground state. Moreover, the associated eigenfields are highly confined in the bi-anisotropic chiral medium with negligible leakage to the outside.

•

By virtue of the bi-Lebedev scheme, the eigenvectors of the discrete MEP(4)provide complete components of bothE(r) andH(r) at each grid point; therefore, the Poynting vector S

=

¹₂

ℜ

(E

× ¯

H) is instantly accessible at any grid point with- out any additional approximations once the eigenvector is given. In this work, we demonstrate that the Poynting vector associated with the newborn eigenvalue is also concentrated in the reciprocal bi-anisotropic chiral medium. More inter- estingly, the spatial distribution of Poynting vectors inside the chiral medium displays some peculiar patterns.

This paper is outlined as follows. In Section2, we define the notations of the discrete E(r), H(r),

ε

^(r),

µ

^(r),

ξ

^{(r) and}

ζ

^(r) on a Lebedev grid. In Section3, we provide the detailed matrix representation of bi-Lebedev scheme from which the MEP (4) is discretized into a sparse GEP of enormous dimension. The SVD of the discrete single-curl operator in this scheme and a null-space free method for the GEP are derived in Section 4.

Numerical results are provided in Section5to show the colliding eigenvalues and localization of eigenfields and Poynting vectors.

Finally, concluding remarks are given in Section6.

Notations. Bold letters denote vectors;I_n

= [

e₁

,

^e2

, . . . ,

^en

]

is the identity matrix of sizen. For matricesAandB,A^⊤andA^∗are the transpose and conjugate transpose, respectively; A

⊗

Band A

⊕

B

=

diag(A

,

B) are the Kronecker product and the direct sum ofAandB, respectively; vec(A) is the vectorization function of the matrixA.

2. Discretization ofE,^H, ε, µ, ξ, ζon the Lebedev grid First, it is worth noting that in this and the next two sections,

ξ

^{(r) and}

ζ

^(r)∗can be any 3-by-3 complex matrices.

Crystal structures can be classified as 14 Bravais lattices in 3D Euclidean space [20]. In fact, a primitive cell of a Bravais lattice, which is a parallelepiped formed by the lattice translation vectors

{

a_ℓ

}

³_ℓ=1, can be embedded into a minimally contained rectangular cuboid called the working cellΩc. Let the working cell be partitioned evenly by n₁, n₂ and n₃ grid points in the x-, y-, and z-direction, respectively, and

δ

x,

δ

y and

δ

z be the corresponding mesh lengths. For simplicity, we introduce the shorthand notations (r

,

^s

,

^t)

≡

(r

δ

x

,

^s

δ

y

,

^t

δ

z), wherer

,

^s

,

^t

∈

_R, and

ˆ

_i

≡

i

+

¹₂,

ˆ

_j

≡

j

+

¹₂,_k

ˆ ≡

k

+

¹₂ fori

=

0

,

¹

, . . . ,

ⁿ1

−

1

,

^j

=

0

,

¹

, . . . ,

ⁿ2

−

1

,

^andk

=

0

,

¹

, . . . ,

ⁿ3

−

1.

As mentioned in Section1, the components ofE(r) andH(r) attached to the Lebedev grid naturally form four groups shown in Fig. 1. Specifically, as shown inFig. 1(b), the components ofE(r) andH(r) on the standard Yee grid can be represented by E₁(

ˆ

_i

,

^j

,

^k)

,

^E2(i

, ˆ

_j

,

^k)

,

^E3(i

,

^j

,

_k)

ˆ ,

^(6a) H₁(i

, ˆ

_j

,

_k)

ˆ ,

^H2(

ˆ

_i

,

^j

,

_k)

ˆ ,

^H3(

ˆ

_i

, ˆ

_j

,

^k)

,

^(6b)

2

X.-L. Lyu, T. Li, T.-M. Huang et al. Computer Physics Communications 261 (2021) 107769

Fig. 1. (a) Illustration of the Lebedev grid and the collocatedEandHcomponents. (b) Standard Yee grid. (c) Yee grid shifted by (±^δx

2,±^δy

2,0). (d) Yee grid shifted by (±^δ₂^x,⁰,±^δ₂^z). (e) Yee grid shifted by (0,±^δ₂^y,±^δ₂^z). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

respectively. Next, as shown inFig. 1(c), we shift the components ofE(r) andH(r) in(6)by (

±

^δ₂^x

, ±

^δ₂^y

,

0), yielding

E₁(i

, ˆ

_j

,

^k)

,

^E2(

ˆ

_i

,

^j

,

^k)

,

^E3(

ˆ

_i

, ˆ

_j

,

_k)

ˆ ,

H₁(

ˆ

_i

,

^j

,

_k)

ˆ ,

^H2(i

, ˆ

_j

,

_k)

ˆ ,

^H3(i

,

^j

,

^k)

.

Then, as shown inFig. 1(d), we shift the components ofE(r) and H(r) in(6)by (

±

^δx

2

,

⁰

, ±

^δz

2), yielding E₁(i

,

^j

,

_k)

ˆ ,

^E2(

ˆ

_i

, ˆ

_j

,

_k)

ˆ ,

^E3(

ˆ

_i

,

^j

,

^k)

,

H1(

ˆ

_i

, ˆ

_j

,

^k)

,

^H2(i

,

^j

,

^k)

,

^H3(i

, ˆ

_j

, ˆ

_k)

.

Last, as shown inFig. 1(e), we shift the components ofE(r) and H(r) in(6)by (0

, ±

^δy

2

, ±

^δz

2), yielding E₁(

ˆ

_i

, ˆ

_j

,

_k)

ˆ ,

^E2(i

,

^j

,

_k)

ˆ ,

^E3(i

, ˆ

_j

,

^k)

,

H₁(i

,

^j

,

^k)

,

^H2(

ˆ

_i

, ˆ

_j

,

^k)

,

^H3(

ˆ

_i

,

^j

, ˆ

_k)

.

Furthermore, we collect these four groups of the components of E(r) into four column vectors e_e1

,

^ee2

,

^ee3 and e_e4, and the four groups of components of H(r) into four column vectors h_f1

,

^hf2

,

^hf3 andh_f4. By defining _F(

ˇ : , : , :

)

≡

vec(F(

: , : , :

)), where F(

: , : , :

) is a three-way array, these vectors can be defined as follows:

e_e1

=

⎡

⎣

E

ˇ

₁(₀

ˆ : ˆ

n₁

−

1

,

⁰

:

n₂

−

1

,

⁰

:

n₃

−

1) E

ˇ

₂(0

:

n₁

−

1

,

₀

ˆ : ˆ

n₂

−

1

,

⁰

:

n₃

−

1) E

ˇ

3(0

:

n1

−

1

,

⁰

:

n2

−

1

,

₀

ˆ : ˆ

n3

−

1)

⎤

⎦

,

h_f1

=

⎡

⎣

H

ˇ

₁(0

:

n₁

−

1

,

₀

ˆ : ˆ

n₂

−

1

,

₀

ˆ : ˆ

n₃

−

1) H

ˇ

₂(₀

ˆ : ˆ

n₁

−

1

,

⁰

:

n₂

−

1

,

₀

ˆ : ˆ

n₃

−

1) H

ˇ

₃(₀

ˆ : ˆ

n₁

−

1

,

₀

ˆ : ˆ

n₂

−

1

,

⁰

:

n₃

−

1)

⎤

⎦

,

^(7a)

e_e2

=

⎡

⎣

E

ˇ

₁(0

:

n₁

−

1

,

₀

ˆ : ˆ

n₂

−

1

,

⁰

:

n₃

−

1) E

ˇ

₂(₀

ˆ : ˆ

n₁

−

1

,

⁰

:

n₂

−

1

,

⁰

:

n₃

−

1) E

ˇ

₃(₀

ˆ : ˆ

n₁

−

1

,

₀

ˆ : ˆ

n₂

−

1

,

₀

ˆ : ˆ

n₃

−

1)

⎤

⎦

,

h_f2

=

⎡

⎣

H

ˇ

₁(₀

ˆ : ˆ

n₁

−

1

,

⁰

:

n₂

−

1

,

₀

ˆ : ˆ

n₃

−

1) H

ˇ

₂(0

:

n₁

−

1

,

₀

ˆ : ˆ

n₂

−

1

,

₀

ˆ : ˆ

n₃

−

1) H

ˇ

₃(0

:

n₁

−

1

,

⁰

:

n₂

−

1

,

⁰

:

n₃

−

1)

⎤

⎦

,

^(7b)

e_e3

=

⎡

⎣

E

ˇ

₁(0

:

n₁

−

1

,

⁰

:

n₂

−

1

,

₀

ˆ : ˆ

n₃

−

1) E

ˇ

₂(₀

ˆ : ˆ

n₁

−

1

,

₀

ˆ : ˆ

n₂

−

1

,

₀

ˆ : ˆ

n₃

−

1) E

ˇ

₃(₀

ˆ : ˆ

n₁

−

1

,

⁰

:

n₂

−

1

,

⁰

:

n₃

−

1)

⎤

⎦

,

h_f3

=

⎡

⎣

H

ˇ

₁(₀

ˆ : ˆ

n₁

−

1

,

₀

ˆ : ˆ

n₂

−

1

,

⁰

:

n₃

−

1) H

ˇ

₂(0

:

n₁

−

1

,

⁰

:

n₂

−

1

,

⁰

:

n₃

−

1) H

ˇ

₃(0

:

n₁

−

1

,

₀

ˆ : ˆ

n₂

−

1

,

₀

ˆ : ˆ

n₃

−

1)

⎤

⎦

,

^(7c)

e_e4

=

⎡

⎣

E

ˇ

₁(₀

ˆ : ˆ

n₁

−

1

,

₀

ˆ : ˆ

n₂

−

1

,

₀

ˆ : ˆ

n₃

−

1) E

ˇ

₂(0

:

n₁

−

1

,

⁰

:

n₂

−

1

,

₀

ˆ : ˆ

n₃

−

1) E

ˇ

₃(0

:

n₁

−

1

,

₀

ˆ : ˆ

n₂

−

1

,

⁰

:

n₃

−

1)

⎤

⎦

,

h_f4

=

⎡

⎣

H

ˇ

₁(0

:

n₁

−

1

,

⁰

:

n₂

−

1

,

⁰

:

n₃

−

1) H

ˇ

₂(₀

ˆ : ˆ

n₁

−

1

,

₀

ˆ : ˆ

n₂

−

1

,

⁰

:

n₃

−

1) H

ˇ

₃(₀

ˆ : ˆ

n₁

−

1

,

⁰

:

n₂

−

1

,

₀

ˆ : ˆ

n₃

−

1)

⎤

⎦

.

^(7d)

For convenience, we define e_e

= [

e^⊤_e1

,

^e⊤_e2

,

^e⊤_e3

,

^e⊤_e4

]

^⊤

forE(r) sampled at midpoints of edges and centroids, and h_f

= [

h^⊤_f1

,

^h⊤f2

,

^h⊤f3

,

^h⊤f4

]

^⊤

forH(r) sampled at face centers and vertices.

In the bi-Lebedev scheme, by just interchanging the symbols eandhin(7), we can define

e_f

= [

e^⊤_f1

,

^e⊤f2

,

^e⊤f3

,

^e⊤f4

]

^⊤

forE(r) sampled at face centers and vertices, and h_e

= [

h^⊤_e1

,

^h⊤_e2

,

^h⊤_e3

,

^h⊤_e4

]

^⊤

forH(r) sampled at midpoints of edges and centroids.

Next, we consider the anisotropic permittivity

ε

(r), permeabil- ity

µ

(r) and magnetoelectric coupling tensors

ξ

^{(r) and}

ζ

^{(r)∗. For} convenience, we lets

= ε, µ, ξ, ζ

^{and writes(r)}

= [

s_pq(r)

]

³_p,q=1. In addition, we defineS⁽c)(s) for c

=

e or f as the corresponding assembled weight matrix. The detailed definition can be found in Appendix A.

Theorem 1. If s

= [

s_pq(r)

] ∈

_C^3×3 is HPD, for s

= ε

^or

µ

^{, then}

S⁽c)(s)for c

=

e or f is also HPD.

Proof. SeeAppendix C.1. □

3. Discretization of MEP with 3D Bi-anisotropic complex me- dia

In this section, we present the detailed discretization of the MEP (4) using the bi-Lebedev scheme. To simplify the nota- tion of the discrete single-curl operator, we define a function C(X₁

,

^X2

,

^X3) as

C(X₁

,

^X2

,

^X3)

=

[ ₀

−

X₃ X₂ X₃ 0

−

X₁

−

X₂ X₁ 0 ]

,

whereX₁,X₂andX₃are three square matrices of the same size.

Without loss of generality, hereafter, we take the face-centered cubic (FCC) lattice as an example. The discrete single-curl opera- tor with the quasiperiodic condition(5)using Yee’s scheme can be represented by the matrixC(C₁

,

^C2

,

^C3) [1,12], where

C₁

=

¹

δ

x

I_n2n3

⊗

K_1,n1(e^ıkˆ·a1)

,

^Km1,m2(X)

=

[0 I_m1(m2−₁₎

X 0

]

−

I_m1m2

,

(8a) C₂

=

¹

δ

y

I_n3

⊗

K_n1,n2(e^ıkˆ·a2J₁)

,

^J1

=

[ 0 e^−ıkˆ·a1In₁/²

I_n1/2 0 ]

,

^(8b)

C₃

=

¹

δ

z

K_n1n2,n3(e^ıkˆ·a3J₂)

,

^J2

=

[ 0 e^−ıkˆ·a2I_n2/3

⊗

I_n1 I_2n2/3

⊗

J₁ 0

]

,

(8c) with_k

ˆ =

2

π

k. In passing, the most general expressions ofC₁

,

^C2

andC₃for any Bravais lattice corresponding to(8)can be found in [21].

3.1. Matrix representation of

∇ ×

E

=

ı

ω

^Bat face centers

In this subsection, we derive the matrix representations of the FD discretization of (1a)on subgrids 1

,

²

,

3 and 4 illustrated in Figs. 1(b),1(c),1(d), and 1(e), respectively. Specifically, we first write down the central FD approximation of(1a)in the compo- nentwise form at each grid point and then recast all formulas for each subgrid into matrix–vector form. The detailed explanation can be found in Appendix B. Moreover, the details of how to incorporate the quasiperiodic condition (5)into the FD formula, which have been thoroughly discussed in [21] for the Yee grid (i.e., subgrid 1), hold for subgrids 2,3 and 4.

DenoteZ⁽f)

pq(

·

)

=

S⁽f)

pq(

· , ζ

^{) andM(}pq^f)(

·

)

=

S⁽f)

pq(

· , µ

), which are defined in(A.1)inAppendix A. Using the FD discretization of(1a) at

•

(i

, ˆ

_j

, ˆ

_{k), (}

ˆ

_i

,

^j

,

_{k) and (}

ˆ ˆ

_i

, ˆ

_j

,

k), respectively, in Subgrid 1 (Fig. 1(b));

•

(

ˆ

_i

,

^j

, ˆ

_{k), (i}

, ˆ

_j

,

_{k) and (i}

ˆ ,

^j

,

k), respectively, in Subgrid 2 (Fig. 1(c));

•

(

ˆ

_i

, ˆ

_j

,

^{k), (i}

,

^j

,

^{k) and (i}

, ˆ

_j

,

k), respectively, in Subgrid 3

ˆ

(Fig. 1(d));

•

(i

,

^j

,

^{k), (}

ˆ

_i

, ˆ

_j

,

^{k) and (}

ˆ

_i

,

^j

,

k), respectively, in Subgrid 4

ˆ

(Fig. 1(e)),

the resulting matrix representations are C(C₁

,

^C2

,

^C3)e_e1

=

ı

ω

⁽[

Z⁽f) 11 Z⁽f)

12 Z⁽f) 13 Z⁽f)

14

] e_f

+

[

M⁽f) 11 M⁽f)

12 M⁽f) 13 M⁽f)

14

]

h_f)

,

^(9a)

C(

−

C₁^∗

, −

C₂^∗

,

^C3)e_e2

=

ı

ω

⁽[

Z⁽f) 21 Z⁽f)

22 Z⁽f) 23 Z⁽f)

24

] e_f

+

[

M⁽f) 21 M⁽f)

22 M⁽f) 23 M⁽f)

24

]

h_f)

,

^(9b)

C(

−

C₁^∗

,

^C2

, −

C₃^∗)e_e3

=

ı

ω

⁽[

Z⁽f) 31 Z⁽f)

32 Z⁽f) 33 Z⁽f)

34

] e_f

+

[

M⁽f) 31 M⁽f)

32 M⁽f) 33 M⁽f)

34

]

h_f)

,

^(9c)

C(C₁

, −

C₂^∗

, −

C₃^∗)e_e4

=

ı

ω

⁽[

Z⁽f) 41 Z⁽f)

42 Z⁽f) 43 Z⁽f)

44

] e_f

+

[

M⁽f) 41 M⁽f)

42 M⁽f) 43 M⁽f)

44

]

h_f)

,

^(9d)

respectively. The detailed derivation of (9a) is introduced in Appendix B.1;(9b),(9c)and(9d)can be obtained similarly. From (9),(1a)is discretized at face centers and vertices into

C

˜

e_e

=

ı

ω

(

Z₍f)e_f

+

M₍f)h_f

)

,

^(10a)

where C

˜ =

diag(

C(C₁

,

^C2

,

^C3)

,

^C(

−

C₁^∗

, −

C₂^∗

,

^C3)

,

^C(

−

C₁^∗

,

^C2

, −

C₃^∗)

,

C(C₁

, −

C₂^∗

, −

C₃^∗))

.

^(10b)

3.2. Matrix representation of

∇ ×

E

=

ı

ω

^Bat midpoints of edges In this subsection, we discretize(1b)on subgrids 1

,

²

,

^{3 and 4} illustrated inFigs. 1(b),1(c),1(d), and1(e), respectively. Similarly, we recast all formulas for each subgrid into matrix–vector form.

Using the FD discretization for(1a)at

•

(

ˆ

_i

,

^j

,

^{k), (i}

, ˆ

_j

,

^{k), and (i}

,

^j

,

k), respectively, in Subgrid 1

ˆ

(Fig. 1(b));

•

(i

, ˆ

_j

,

^{k), (}

ˆ

_i

,

^j

,

^{k) and (}

ˆ

_i

, ˆ

_j

,

k), respectively, in Subgrid 2

ˆ

(Fig. 1(c));

•

(i

,

^j

,

_{k), (}

ˆ ˆ

_i

, ˆ

_j

,

_{k) and (}

ˆ ˆ

_i

,

^j

,

k), respectively, in Subgrid 3 (Fig. 1(d));

•

(

ˆ

_i

, ˆ

_j

,

_{k), (i}

ˆ ,

^j

,

_{k) and (i}

ˆ , ˆ

_j

,

k), respectively, in Subgrid 4 (Fig. 1(e)),

the resulting matrix representations are C(C₁

,

^C2

,

^C3)^∗e_f1

=

ı

ω

([

Z⁽e) 11 Z⁽e)

12 Z⁽e) 13 Z⁽e)

14

]e_e

+

[

M⁽e) 11 M⁽e)

12 M⁽e) 13 M⁽e)

14

]h_e)

,

^(11a)

C(

−

C₁^∗

, −

C₂^∗

,

^C3)^∗e_f2

=

ı

ω

([

Z⁽e) 21 Z⁽e)

22 Z⁽e) 23 Z⁽e)

24

]e_e

+

[

M⁽e) 21 M⁽e)

22 M⁽e) 23 M⁽e)

24

]h_e)

,

^(11b)

C(

−

C₁^∗

,

^C2

, −

C₃^∗)^∗e_f3

=

ı

ω

([

Z⁽e) 31 Z⁽e)

32 Z⁽e) 33 Z⁽e)

34

]e_e

+

[

M⁽e) 31 M⁽e)

32 M⁽e) 33 M⁽e)

34

]h_e)

,

^(11c)

C(C₁

, −

C₂^∗

, −

C₃^∗)^∗e_f4

4

X.-L. Lyu, T. Li, T.-M. Huang et al. Computer Physics Communications 261 (2021) 107769

=

ı

ω

([

Z⁽e) 41 Z⁽e)

42 Z⁽e) 43 Z⁽e)

44

]e_e

+

[

M⁽e) 41 M⁽e)

42 M⁽e) 43 M⁽e)

44

]h_e)

,

^(11d)

respectively. In summary, from(11),(1a)is discretized at mid- points of edges into

C

˜

^∗e_f

=

ı

ω

(

Z₍e)e_e

+

M₍e)h_e)

,

⁽¹²⁾

where_C

˜

is defined in(10b).

3.3. Matrix representation of

∇ ×

H

= −

ı

ω

^Dat midpoints of edges Discretization of(1b)at the midpoints of edges and face cen- ters is a verbatim repetition of the derivations in Sections3.1and 3.2, except thatE(r) andB(r) are replaced withH(r) andD(r), respectively, and

ω

is replaced with

− ω

. The detailed derivations of Section3.3are located inAppendix B.2and those of Section3.4 inAppendix B.3.

From (B.3a), (B.3b), (B.3c) and (B.3d) in Appendix B.2, the discretization of(1b)at midpoints of edges can be represented as

C

˜

^∗h_f

= −

ı

ω

(

E₍e)e_e

+

X₍e)h_e)

.

⁽¹³⁾

3.4. Matrix representation of

∇ ×

H

= −

ı

ω

^Dat face centers From (B.4a), (B.4b), (B.4c) and (B.4d) in Appendix B.3, the discretization of(1b)at face centers can be represented as C

˜

h_e

= −

ı

ω

(

E₍f)e_f

+

X₍f)h_f

)

.

⁽¹⁴⁾

Finally, from (10a), (12), (13) and (14), the MEP (4) is dis- cretized into the following GEP

⎡

⎢

⎢

⎣

0 0

− ˜

C^∗ 0

0 0 0

− ˜

C

C

˜

0 0 0

0 _C

˜

^∗ _{0 0}

⎤

⎥

⎥

⎦

⎡

⎢

⎣ e_e e_f h_f h_e

⎤

⎥

⎦

=

ı

ω

⎡

⎢

⎣

E₍e) 0 0 X₍e)

0 E₍f) X₍f) 0 0 Z₍f) M₍f) 0 Z₍e) 0 0 M₍e)

⎤

⎥

⎦

⎡

⎢

⎣ e_e e_f h_f h_e

⎤

⎥

⎦

≡

ı

ω

[E X

Z M

]

⎡

⎢

⎣ e_e e_f h_f h_e

⎤

⎥

⎦

.

⁽¹⁵⁾

Assume

ξ

^(r)

= ζ

^{(r)∗. In GEP (15), E and M are Hermi-} tian and X

=

Z^∗. Thus, it is easily seen that(15) is a skew- Hermitian/skew-Hermitian pencil; then, it becomes self-evident that

ω

^and

ω ¯

are both eigenvalues of the GEP(15). Furthermore, if (

ω, [

e^⊤_e

,

^e⊤_f

,

^h⊤_f

,

^h⊤e

]

^⊤) is an eigenpair of(15), then it is easily seen that (

− ω, [−

e^⊤_e

,

^e⊤f

,

^h⊤f

, −

h^⊤_e

]

^⊤) is also an eigenpair of(15).

In light of these properties, we have the following theorem.

Theorem 2. Eigenvalues of the GEP(15)appear as the pair

{ ω, − ω }

if

ω ∈

_R

∪

ıR, and they appear as the quadruplet

{ ω, − ω, ω, ¯ − ¯ ω }

if

ω ∈

_C

\

(R

∪

ıR).

4. SVD and the fast Eigensolver

Based on the exquisite skills developed in [1,2], we derive the SVD of_C

˜

_in(10b)and propose a fast eigensolver for the GEP(15).

Theorem 3([1,11]). Let C_ℓ,

ℓ =

1

,

²

,

³be defined in(8). Then they are simultaneously diagonalized by the unitary matrix T

∈

_C^n×n, i.e.,

C_ℓT

=

TΛℓ

, ℓ =

1

,

²

,

³

,

⁽¹⁶⁾

whereΛℓis the eigenvalue matrix for C_ℓ. The detailed expressions of T andΛℓcan be found in [11,21].

From(16),C(C₁

,

^C2

,

^C3),C(

−

C₁^∗

, −

C₂^∗

,

^C3),C(

−

C₁^∗

,

^C2

, −

C₃^∗) and C(C₁

, −

C₂^∗

, −

C₃^∗) in (B.2), (9b), (9c) and (9d), respectively, are equal to

(I₃

⊗

T)˜_Λ(I₃

⊗

T^∗)

,

⁽¹⁷⁾

where

˜Λ

=

⎡

⎣

0

−˜

_Λ3 ˜_Λ2

˜Λ3 0

−

_Λ˜1

−˜

_Λ2 ˜_Λ1 0

⎤

⎦

,

⁽¹⁸⁾

with (_Λ˜1

,

˜_Λ2

,

˜_Λ3)

=

(Λ1

,

Λ2

,

Λ3), (

− ¯

_Λ1

, − ¯

_Λ2

,

Λ3), (

− ¯

_Λ1

,

Λ2

,

− ¯

_Λ3) and (Λ1

, − ¯

_Λ2

, − ¯

_Λ3), respectively. Let˜_Λℓ

=

diag

{ ˜ λ

1ℓ

, λ ˜

2ℓ

, . . . , λ ˜

nℓ

} , ℓ =

1

,

²

,

3. Doing a perfect shuffle of this ˜_Λ, i.e., multiplying

P

= [

e₁

,

^en+1

,

^e2n+1

,

^e2

,

^en+2

,

^e2n+2

, . . . ,

^en

,

^e2n

,

^e3n

] ∈

_R^3n×3n andP^⊤to˜_Λfrom the right and left, respectively, we can trans- form˜_Λto a block diagonal matrix

P^⊤˜_ΛP

=

L₁

⊕

L₂

⊕ · · · ⊕

L_n

,

with

L_m

=

⎡

⎣

0

− ˜ λ

m,3

λ ˜

m,2

λ ˜

m,3 0

− ˜ λ

m,1

− ˜ λ

m,²

λ ˜

m,¹ 0

⎤

⎦

,

^m

=

1

,

²

, . . . ,

ⁿ

.

Theorem 4. Letv₀

≡

[

ℓ

1

ℓ

2

ℓ

3

]⊤

be a nonzero vector and

L

=

[ ₀

− ℓ

3

ℓ

2

ℓ

3 0

− ℓ

1

− ℓ

2

ℓ

1 0 ]

.

⁽¹⁹⁾

Choose

α, β ∈

_Rsuch thatv₁

≡

[

β ℓ ¯

3

− ¯ ℓ

2

ℓ ¯

1

− α ℓ ¯

3

α ℓ ¯

2

− β ℓ ¯

1

]⊤

is a nonzero vector. Definev₂

= ¯

v₀

× ¯

v₁. Then, V

≡

[ _v0

√

ℓq v₁

∥_v1∥₂ v₂

√

ℓq∥_v1∥₂

]

,

^U

=

[ _v¯₀

√

ℓq v¯₂

√

ℓq∥_v1∥₂

−

_∥^¯v1

v1∥₂

]

are unitary matrices with

ℓ

q

= | ℓ

1

|

²

+ | ℓ

2

|

²

+ | ℓ

3

|

², and L^∗L

=

V diag(

0

, ℓ

q

, ℓ

q

)V^∗

,

^LL∗

=

U diag( 0

, ℓ

q

, ℓ

q

)U^∗

.

⁽²⁰⁾ Furthermore, L has the following SVD

L

=

U diag (

0

,

√

ℓ

q

,

√

ℓ

q

)

V^∗

.

⁽²¹⁾

Proof. SeeAppendix C.2. □ Define

[Ψ0 Ψ1 Ψ2

] (22)

≡

⎡

⎣

˜Λ1 Λq

−

˜_Λ1˜_Λ

∗ s _Λ˜

∗ 3

−

˜_Λ

∗ 2

˜Λ2 Λq

−

˜_Λ2˜_Λ

∗ s _Λ˜

∗ 1

−

˜_Λ

∗ 3

˜Λ3 Λq

−

˜_Λ3˜_Λ

∗ s _Λ˜

∗ 2

−

˜_Λ

∗ 1

⎤

⎦diag

×

(

Λ⁻q¹²

,

(

3Λ²q

−

_Λq˜_Λp )⁻¹₂

,

(

3Λq

−

˜_Λp )⁻¹₂

)

with

˜Λp

=

˜_Λs˜_Λ

∗

s

≡

(˜_Λ1

+

˜_Λ2

+

˜_Λ3)(˜_Λ1

+

˜_Λ2

+

˜_Λ3)^∗

,

Λq

=

_Λ^∗_1Λ1

+

_Λ^∗_2Λ2

+

_Λ^∗_3Λ3

.

FromTheorem 4, we have the SVD of_Λ˜as

˜Λ

=

[_Ψ

¯

0 _Ψ

¯

1

− ¯

_Ψ2]

diag(0

,

Λ¹q^/2

,

Λ¹q^/2)[

Ψ0 Ψ2 Ψ1

]^∗

.

⁽²³⁾ Denote

{

_Ψ0

,

Ψ1

,

Ψ2

}

in (22)by

{

_Ψm0

,

Ψm1

,

Ψm2

}

⁴_m=1 for the four cases of

{˜

_Λℓ

}

³_ℓ=1as in(18). Substituting(23)into(17), we obtain the SVD ofC, as shown inTheorem 5below.