Journal of Computational Physics

Developing a time-domain finite-element method for modeling of electromagnetic cylindrical cloaks

Jichun Li

^a,b,⇑,1

, Yunqing Huang

^a,2

, Wei Yang

^a,3

aHunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan 411105, China

bDepartment of Mathematical Sciences, University of Nevada Las Vegas, Las Vegas, NV 89154-4020, USA

a r t i c l e i n f o

Article history:

Received 9 September 2011

Received in revised form 21 November 2011 Accepted 25 December 2011

Available online 4 January 2012

Keywords:

Maxwell’s equations Invisibility cloak Finite element method

a b s t r a c t

In this paper we propose a time-domain finite element method for modeling of electro- magnetic cloaks. The permittivity and permeability of the cloak model are described by the Drude dispersion model. The model to be solved is quite challenging in that we have to solve a coupled problem with different partial differential equations given in different regions. Our method is based on a mixed finite element method using edge elements with different types of meshes in different regions. Numerical results demonstrate that our algo- rithm is quite effective for simulating cloaks in time-domain. To our knowledge, this is the first cloak simulation carried out by the time-domain finite element method.

Ó2012 Elsevier Inc. All rights reserved.

1. Introduction

The possibility of cloaking an object from detection by electromagnetic waves has recently become a very hot research topic. In 2006, Pendry et al.[25]and Leonhardt[14]independently showed that it is possible to create invisible cloaks for ray optics[14]and electromagnetic waves[25]by guiding light around a region as if nothing is there. In late 2006, a 2-D reduced cloak was successfully fabricated and demonstrated to work in the microwave frequency region[26]. This is the first practical realization of such a cloak, and the result matches with the computer simulation[6]performed using COMSOL mul- tiphysics finite element analysis software. These pioneering work inspired researchers in different disciplines around the world to pursue the human being’s invisibility dream.

Since 2006, many papers have been published on the study of using metamaterials[3,8,18]to construct invisibility cloaks of different shapes (e.g.[1,5,21,22,28]). Also cloaks operating from microwave frequencies to optical frequencies have been achieved, more details and references on cloaking can be found in recent reviews[4,9]. Some mathematical analysis has been carried out for cloaking in frequency domain[19,13].

Numerical simulation plays a very important role in modeling different cloaks and validating theoretical predictions. The time-domain finite difference (FDTD) method is a very popular technique used in this area, readers can find more details about the FDTD method and its applications in cloak simulation in a recent book[10]. Due to the major disadvantage of FDTD method in dealing with complex geometry[30], the finite element method (FEM) based commercial package COMSOL has

0021-9991/$ - see front matterÓ2012 Elsevier Inc. All rights reserved.

doi:10.1016/j.jcp.2011.12.026

⇑ Corresponding author at: Department of Mathematical Sciences, University of Nevada Las Vegas, Las Vegas, NV 89154-4020, USA. Tel.: +1 (702) 895 0355.

E-mail addresses:jichun@unlv.nevada.edu(J. Li),huangyq@xtu.edu.cn(Y. Huang),yangweixtu@126.com(W. Yang).

1 Supported by National Science Foundation Grant DMS-0810896.

2 This work was supported by National Science Foundation Grant DMS-0810896, and in part by the NSFC Key Project 11031006 and Hunan Provincial NSF project 10JJ7001.

3 Supported by Hunan Education Department Key Project 10A117 and Hunan Provincial Innovation Foundation for Postgraduate (CX2011B243).

Contents lists available atSciVerse ScienceDirect

Journal of Computational Physics

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / j c p

been extensively used in frequency-domain cloak simulation by engineers and physicists[6,4]. However, COMSOL cannot be used for time-domain cloak simulation due to its limitation on algorithmic development for time-domain cloak modeling.

On the other hand, the recently designed broadband cloaks[20]make the time-domain simulation more appealing and nec- essary. But little attention has been paid to the time-domain modeling of cloaks. To our best knowledge, the first time-do- main simulation of 2-D cloaking structures was carried out by Zhao et al.[31] in 2008 using FDTD method. Later, they generalized the same idea to 3-D cloak simulation in 2009[32]. It seems that there is no existing work on time-domain finite element (FETD) method developed for cloak simulation yet.

In this paper, we propose a FETD method to simulate a 2-D cylindrical cloak. This problem is quite challenging, since we have to solve a coupled problem with different partial differential equations in different regions, and the material parameters are highly anisotropic and nonhomogeneous. The rest of the paper is organized as follows. We first describe the 2-D cylin- drical cloak modeling equations in Section2. In Section3, we develop a FETD method for solving the model problem. Then in Section4, we present a detailed stability analysis of our scheme. Numerical results showing the cloaking phenomena ob- tained by our method are illustrated in Section5. We conclude the paper in Section6.

2. The modeling equations

The cloak modeling is based on the Faraday’s law and Ampere’s law written as follows:

@B

@t ¼ rE; ð1Þ

@D

@t ¼rH ð2Þ

and the constitutive relations

D¼

e

E; ð3Þ

B¼

l

H; ð4Þ

whereEandHare the electric and magnetic fields respectively,DandBare the electric displacement and magnetic induc- tion respectively,

e

and

l

are cloak permittivity and permeability, respectively. For the cylindrical cloak, the ideal material parameters in the polar coordinate system are given by[25]:

e

r¼

l

_r¼rR1

r ;

e

/¼

l

_/¼ r rR1

;

e

z¼

l

_z¼ R2

R2R1

2

rR1

r ; ð5Þ

whereR1andR2are the inner and outer radius of the cloak. In this case,Ebecomes a 2-D vector, andHis a scalar, i.e., we can writeE= (Ex,Ey)⁰andH=Hz, where the subindexx,yorzdenotes the component in each direction.

Transforming the polar coordinate system to the Cartesian coordinate system, we can obtain[31]:

e

0

e

r

e

/E¼

e

rsin²/þ

e

/cos²/ ð

e

/

e

rÞsin/cos/

ð

e

/

e

rÞsin/cos/

e

rcos²/þ

e

/sin²/

" #

D: ð6Þ

Because the material parameters given in(5)can not be used directly to simulate the time-domain cloak, we have to map the parameters using the dispersive medium models. Here we consider the Drude model for the permittivity:

e

rð

x

Þ ¼1

x

²p

x

²j

xc

^{; ð7Þ}

where

c

P0 andxp> 0 are the collision and plasma frequencies, respectively. Substituting(7)into(6)and using the follow- ing rules

j

x

!@

@t;

x

²! @²

@t²; ð8Þ

we have (detailed derivation see[31]):

e

0

e

_/ ^@

2

@t²þ

c

^@

@tþw²_p

! E¼ @²

@t²þ

c

^@

@tþw²_p

!

MADþ

e

_/ ^@

2

@t²þ

c

^@

@t

!

MBD; ð9Þ

where we denoteD= (Dx,Dy)⁰and MA¼ sin²/ sin/cos/

sin/cos/ cos²/

" #

; MB¼ cos²/ sin/cos/

sin/cos/ sin²/

" #

: Similarly, we map the permeability using the Drude model[31]:

l

_zð

x

Þ ¼A 1

x

²_pm

x

²j

xc

m

!

; ð10Þ

whereA¼_R^R2

2R1, andxpm> 0 and

c

mP0 are the magnetic plasma and collision frequencies, respectively. Substituting(10) into(4), we obtain

Bz¼

l

_o

l

_zHz¼

l

₀A 1

x

²_pm

x

²j

xc

m

! Hz:

Then using rules(8), we have

@²

@t²þ

c

_m^@

@t

!

Bz¼

l

₀A @²

@t²þ

c

_m^@

@tþ

x

²_pm

!

Hz: ð11Þ

To carry out the cloak simulation, we have to reduce the infinite domain problem to a bounded domain by using Bereng- er’s perfectly matched layer (PML)[2]to absorb waves leaving the computational domain without introducing reflections.

The two dimensional Berenger PML governing equations can be written as:

e

0

@Ex

@t þ

r

yEx¼@ðHzxþHzyÞ

@y ; ð12Þ

e

0

@Ey

@t þ

r

xEy¼ @ðHzxþHzyÞ

@x ; ð13Þ

l

₀^@Hzx

@t þ

r

mxHzx¼ @Ey

@x; ð14Þ

l

₀^@Hzy

@t þ

r

myHzy¼@Ex

@y; ð15Þ

where the parameters

r

i,

r

mi,i=x,y, are the homogeneous electric and magnetic conductivities in thexandydirections, respectively.

For easy implementation of our algorithm given below, we rewrite(12) and (13)in the vector form:

e

0

@E

@tþ

r

y 0

0

r

x

E¼rH; ð16Þ

where we used the 2-D vector curl operator rH¼

@H

@y

^@H_@x

!

; forH¼HzxþHzy:

3. A fully discrete explicit finite element scheme

To design our mixed finite element method, we partitionXby a family of regular meshesT^hwith maximum mesh sizeh.

To accommodate our problem easily, we use a mesh with mixed types of elements: triangles in the cloak and free space re- gion; rectangles in the PML region, cf.Fig. 1(b) below. For simplicity, currently we implement our algorithm using the low- est-order Raviart–Thomas–Nédélec’s mixed spacesUhandVhgiven as follows[23,24]: for any rectangular elementK2T^h, we choose

Uh¼nwh2L²ðXÞ:whj_K2Q0;0; 8K2T^ho

;

Vh¼n/_h2Hðcurl;XÞ:/_hjK2Q0;1Q1;0; 8K2T^ho

;

whereQi,jdenotes the space of polynomials whose degrees are less than or equal toiandjin variablesxandy, respectively.

Here we denote the spaceH(curl;X) = {v2(L²(X))²;rv2(L²(X))²}. While on a triangular element, we choose Uh¼nwh2L²ðXÞ:whj_K is a piecewise constant;8K2T^ho

;

Vh¼n/_h2Hðcurl;XÞ:/_hjK¼spanfkirkjkjrkig; i;j¼1;2;3; 8K2T^ho

; wherekidenotes the standard linear basis function at vertexiof elementK.

To impose the perfect conducting boundary conditionnE=0, we introduce the space V⁰_h¼ f/h2Vh;n/h¼0on@Xg:

To define a fully-discrete scheme, we divide the time intervalI= [0,T] intoNuniform subintervalsIi= [ti1,ti] by pointsti=i

s

, i= 0, 1,. . .,N, where

s

=T/N. Furthermore, we denoteuⁿ=u(,t_n) andu^nþ12¼u;ðnþ¹₂Þ

s

, and introduce some operators:

dsuⁿ¼uⁿuⁿ¹

s

^{; d}

2

suⁿ¼uⁿ2uⁿ¹þuⁿ²

s

^{2 ;}

d2suⁿ¼uⁿuⁿ² 2

s

^{; u}

n1¼uⁿþ2uⁿ¹þuⁿ²

4 ; buⁿ¼uⁿþuⁿ¹

2 :

With the above preparation, we now construct a leap-frog type scheme for solving the modeling equations in the cloak re- gion: forn= 1, 2,. . ., findD^nþ_h ¹²;E^nþ_h ¹²2V⁰_h; B^nþ1_h ;H^nþ1_h 2Uhsuch that

dsD^nþ_h ¹²;/_h

Hⁿ_h;r/_h

¼0; ð17Þ

e

0

e

/d²_sE^nþ_h ¹²;/~h

þ

ce

0

e

/d2sE^nþ_h ¹²;/~h

þ

x

²_p

e

0

e

/Eⁿ_h ¹²;/~h

¼ MAþ

e

/MB

d²_sD^nþ_h ¹²;/~h

þ

x

²_pMADⁿ_h ¹²;/~h

þ

c

ðMAþ

e

/MBÞd2sD^nþ_h ¹²;/~h

; ð18Þ

dsB^nþ1_h ;w_h

þrE^nþ_h ¹²;w_h

¼0; ð19Þ

l

₀Ad²_sH^nþ1_h ;w~h

þ

l

₀A

c

md2sH^nþ1_h ;w~h

þ

l

₀A

x

²pmHⁿ_h;w~h

¼ d²_sB^nþ1_h ;w~h

þ

c

md2sB^nþ1_h ;w~h

ð20Þ hold true for any/_h;/~h2V⁰_h; w_h;w~h2Uh.

In order to couple(20)well with the PML Eqs.(14) and (15), we split(20)into

l

₀Ad²_sH^nþ1_zx;h;w~h

þ

l

₀A

c

_md2sH^nþ1_zx;h;w~h

þ

l

₀A

x

²_pmHⁿ_zx;h;w~h

¼1

2 d²_sB^nþ1_h ;w~h

þ1

2

c

_md2sB^nþ1_h ;w~h

; ð21Þ

l

₀Ad²_sH^nþ1_zy;h;w~h

þ

l

₀A

c

_md2sH^nþ1_zy;h;w~h

þ

l

₀A

x

²_pmHⁿ_zy;h;w~h

¼1

2 d²_sB^nþ1_h ;w~h

þ1

2

c

_md2sB^nþ1_h ;w~h

: ð22Þ

Similarly, we can construct a leap-frog type scheme for solving the Eqs. (16), (14) and (15) in the PML region: find E^nþ_h ¹²2V⁰_h; H^nþ1_zx;h;H^nþ1_zy;h 2Uh such that

e

0 dsE^nþ_h ¹²;/~h

þ

r

y 0 0

r

x

bE^nþ_h ¹²;/~h

¼ Hⁿ_zx;hþHⁿ_zy;h;r/~h

; ð23Þ

l

₀ dsH^nþ1_zx;h;w1;h

þ

r

mxHb^nþ1_zx;h;w1;h

¼ @

@xE^nþ_y;h¹²;w1;h

; ð24Þ

PML PML

wave source

PML PML

PEC

cloak air

Fig. 1.(a) The cloak modeling setup; and (b) a coarse mesh.

l

₀dsH^nþ1_zy;h;w_2;h

þ

r

myHb^nþ1_zy;h;w_2;h

¼ @

@yE^nþ_x;h¹²;w_2;h

ð25Þ hold true for any/~h2V⁰_h; w_1;h;w_2;h2Uh.

In summary, our mixed finite element time-domain algorithm for modeling the invisible cloak can be performed in the following steps: first, construct a proper meshThofX, choose a proper time step size

s

and proper initial conditions; then at each time stepn, perform theFETD Algorithm:

1. Solve(17)forD^nþ_h ¹²onTh.

2. Solve(18) and (23)forE^nþ_h ¹²onTh. 3. Solve(19)forB^nþ1_h onTh.

4. Solve(21) and (24)forH^nþ1_zx;h onTh. 5. Solve(22) and (25)forH^nþ1_zy;h onTh.

6. CalculateH^nþ1_h ¼H^nþ1_zx;hþH^nþ1_zy;h, then go back to step 1 and repeat the above process. Note that in the free space region,E^nþ_h ¹² andHⁿ⁺¹are updated using(23)–(25)with

r

x=

r

y=

r

mx=

r

my= 0.

4. Stability analysis

In this section, we present the stability analysis for our proposed scheme. Letcv¼1=^pffiffiffiffiffiffiffiffiffiffi

0

l

₀be the wave propagation speed in vacuum,cinv> 0 is a constant from the inverse estimate

krw_hk0;X6cinvh¹kwhk0;X; 8w_h2Vh; ð26Þ wherekk0,Xdenotes theL²norm over domainX.

We start with the easy stability analysis in the PML and free space regionXr: Theorem 4.1.Assume that the time step

s

^{6 h}

c_vcinv; ð27Þ

then for any nP2 and the solutionEⁿ_h ¹²;Hⁿ_h¼Hⁿ_zx;h;Hⁿ_zy;h

of(23)–(25), we have

0 Eⁿ_h ¹²

2 0;Xr

þ

l

₀ Hⁿ_h ²_0;X

r 63

0 E¹²_h

2 0;Xr

þ

l

₀ H¹_h ²

0;Xr

;

where we denote Hⁿ_h ²_0;X

r¼ Hⁿ_zx;h ²

0;Xr

þ Hⁿ_zy;h ²

0;Xr

.

Proof.Choosing/~_h¼2

s

^bE^nþ_h ¹²; w_1;h¼2

s

H^bnþ1_zx;h andw_2;h¼2

s

H^bnþ1_zy;h in(23)–(25), respectively, and using the 2-D curl notation, we obtain

0 E^nþ_h ¹²

2 0;Xr

Eⁿ_h ¹²

2 0;Xr

þ

l

₀ H^nþ1_h ²

0;Xr

Hⁿ_h ²_0;X

r

62

s

Hⁿ_h;rEb^nþ_h¹²

rE^nþ_h ¹²;Hb^nþ1_h

¼

s

Hⁿ_h;rEⁿ_h ¹²

H^nþ1_h ;rE^nþ_h ¹²

; ð28Þ

where in the last step we used the identity Hⁿ_h;rE^nþ_h¹²þEⁿ_h ¹²

rE^nþ_h ¹²;H^nþ1_h þHⁿ_h

¼Hⁿ_h;rEⁿ_h ¹²

H^nþ1_h ;rE^nþ_h ¹² :

Summing up(28)fromn= 1 toN1, we obtain

0 E^N_h ^{12 2}

0;Xr

E¹²_h ²

0;Xr

þ

l

₀ H^N_h ²

0;Xr

H¹_h ²

0;Xr

6

s

H¹_h;rE¹²_h

H^N_h;rE^N_h ¹²

: ð29Þ

Using the definitioncv¼1=^pffiffiffiffiffiffiffiffiffiffi

0

l

₀, the inverse estimate(26), the Cauchy–Schwarz inequality, and the time constraint (27), we have

s

H^N_h;rE^N_h ¹²

¼ ffiffiffiffiffiffi

l

₀

p H^N_h;

s

c_v ffiffiffiffiffi

0

p rE^N_h ¹²

61

2

l

₀ H^N_h ²

0;Xr

þ ð

s

c_vÞ²

0 rE^N_h ¹²

2 0;Xr

61

2

l

₀ H^N_h ²

0;Xr

þ

s

c_vcinvh¹2

0 E^N_h ^{12 2}

0;Xr

61

2

l

₀ H^N_h ²

0;Xr

þ1

2

0 E^N_h ^{12 2}

0;Xr

: ð30Þ

By the same technique, we obtain

s

H¹_h;rE¹²_h 61

2

l

₀ H¹_h ²

0;Xr

þ1 2

0 E¹²_h

2 0;Xr

: ð31Þ

Substituting the estimates(30) and (31)into(29)completes the proof. h

Remark 4.1. To see how largecinv in(26)can be, let us consider an arbitrary rectangular elementK= [xchx,xc+hx] [ychy,yc+hy], on which the lowest edge element basis functions are:

w^h₁¼

ðycþhyÞy 4hxhy

0

!

; w^h₂¼ 0

xðxchxÞ 4hxhy

!

; w^h₃¼

ðychyÞy 4hxhy

0

!

; w^h₄¼ 0

xðxcþhxÞ 4hxhy

! : Herew^h_j; j¼1;2;3;4;start from the bottom edge and orient counterclockwisely.

For simplicity, we assume that the meshT^hofXis formed byNrectanglesK. In this case, it is easy to check that the 2D curl ofw^h_j satisfies

Z

X

rw^h_j

²dx dy¼X

K2T^h

Z

K

1 4hxhy

2

dx dy¼ N 4hxhy

andw^h_j satisfies Z

X

w^h₁

2

dx dy¼X

K2T^h

Z

K

y_cþhyy 4hxhy

2

dx dy¼X

K2T^h

2hx

ð4hxhyÞ²1

3 ðycþhyyÞ³

ycþhy

y¼y_chy

¼Nhy

3hx

;

Z

X

w^h₃

2

dx dy¼Nhy

3hx

; Z

X

w^h₂

2

dx dy¼ Z

X

w^h₄

2

dx dy¼Nhx

3hy

;

from which we can see that rw^h_j

2 0;X

w^h_j

2 0;X

¼ 3

4h²_y; j¼1;3:

Similarly, we have rw^h_j

2 0;X

w^h_j

2 0;X

¼ 3

4h²_x; j¼2;4:

Denoteh= max{hx,hy}. Hence we have cinvP

ffiffiffi3 4 r h

hx

or cinvP ffiffiffi3 4 r h

hy

; ð32Þ

which means thatcinvcan be very large for anisotropic meshes. But for the often used shape regular mesh,cinvshould not be that large. Of course, exact estimate ofcinvreally depends on the mesh and the order of the basis functions. To our knowledge, there is no general formular forcinv.

The rest of the section is devoted to the stability analysis on the cloak regionXc=XnXr. For simplicity, in the rest of this section, we usekk0to denotek k_0;Xc.

Lemma 4.1

0

2 ffiffiffiffiffi

/

p dsE^nþ_h ¹²

2 0 ffiffiffiffiffi

/

p dsEⁿ_h ¹²

2 0

þ

0

x

²_p

2

ffiffiffiffiffi

/

p bE^nþ_h ¹²

2

0 p bffiffiffiffiffi

/Eⁿ_h ¹²

2 0

þ

sc

0 ffiffiffiffiffi

/

p dsbE^nþ_h ¹²

2 0

6

s

ðMAþ

/MBÞd²sD^nþ_h ¹²;dsbE^nþ_h ¹²

þ

sc

d2ðMAþ

/MBÞdsDb^nþ_h ¹²;dsDb^nþ_h ¹² þ 1

4d2

ðMAþ

/MBÞdsbE^nþ_h ¹²;dsbE^nþ_h ¹²

þ

sx

²_p

2 d3

0 ffiffiffiffiffi

/

p dsbE^nþ_h ¹²

2

0þ 1

2d3

0

Db^nþ_h ¹²

2 0þ Dbⁿ_h ¹²

2 0

: ð33Þ

Proof. Choosing/~h¼

s

dsbE^nþ_h ¹²in(18), we obtain

0

/d²_sE^nþ_h ¹²;

s

dsEb^nþ_h¹²

¼

0

/dsE^nþ_h ¹²dsEⁿ_h ¹²

;dsbE^nþ_h ¹²

¼

0

2 ffiffiffiffiffi

/

p dsE^nþ_h ¹²

2 0 ffiffiffiffiffi

/

p dsEⁿ_h ¹²

2 0

;

c

0

/

E^nþ_h ¹²Eⁿ_h ³²

2

s

^;

s

dsbE^nþ_h ¹² 0

B@

1 CA

¼

sc

0

2

/dsE^nþ_h ¹²þdsEⁿ_h ¹²

;dsEb^nþ_h¹²

¼

sc

0 ffiffiffiffiffi

/

p dsbE^nþ_h ¹²

2 0;

x

²_p

0

/

E^nþ_h ¹²þ2Eⁿ_h ¹²þEⁿ_h ³²

4 ;bE^nþ_h ¹²Ebⁿ_h ¹² 0

B@

1 CA

¼

0

x

²p

2

ffiffiffiffiffi

/

p bE^nþ_h ¹²

2

0 p bffiffiffiffiffi

/Eⁿ_h ¹²

2 0

: Using the facts thatjM_Aj6I,jM_Bj6Iand 1<_R^R2

2R1

6

e

/, and the arithmetic–geometric mean inequalityjabj6da²þ_4d¹b², we have

c

ðMAþ

e

/MBÞd2sD^nþ_h ¹²;

s

dsbE^nþ_h ¹²

¼

sc

ðMAþ

e

/MBÞdsDb^nþ_h ¹²;dsbE^nþ_h ¹² 6

sc

d2ðMAþ

e

/MBÞdsDb^nþ_h ¹²;dsDb^nþ_h ¹²

þ 1 4d2

ðMAþ

e

/MBÞdsEb^nþ_h ¹²;dsEb^nþ_h ¹²

and

x

²_pMA

D^nþ_h ¹²þ2Dⁿ_h ¹²þDⁿ_h ³² 4 ;

s

dsEb^nþ_h ¹²

! 6

sx

²_p

2 dsEb^nþ_h ¹²

0

Db^nþ_h ¹²þDbⁿ_h ¹²

0

6

sx

²_p

2 d3

e

0 pffiffiffiffiffi

e

/

dsEb^nþ_h ^{12 2}

0þ 1 2d3

e

0

Db^nþ_h ^{12 2}

0þ Dbⁿ_h ^{12 2}

0

; where in the last step, we used the fact that d_sbE^nþ_h ¹² 6 ffiffiffiffiffi

e

/

p d_sEb^nþ_h ¹²

0. The proof completes by adding the above inequalities together. h

Lemma 4.2 1

2

l

₀AðMAþ

e

/MBÞdsH^nþ1_h ;dsH^nþ1_h

ðM Aþ

e

/MBÞdsHⁿ_h;dsHⁿ_h

h i

þ

sl

₀A

c

mðMAþ

e

/MBÞdsHb^nþ1_h ;dsHb^nþ1_h þ

l

₀A

x

²_pm

2 ðMAþ

e

/MBÞHb^nþ1_h ;Hb^nþ1_h

ðM Aþ

e

/MBÞHbⁿ_h;Hbⁿ_h

h i

6

sc

_m d8ðMAþ

e

/MBÞdsBb^nþ1_h ;dsbB^nþ1_h þ 1

4d8

ðMAþ

e

/MBÞdsHb^nþ1_h ;dsHb^nþ1_h

þ

s

ðMAþ

e

/MBÞd²_sB^nþ1_h ;dsHb^nþ1_h :

ð34Þ

Proof.An equivalent variable coefficient form of(20)can be written as ðMAþ

/MBÞ

l

₀Ad²_sH^nþ1_h þ

l

₀A

c

_md2sH^nþ1_h þ

l

₀A

x

²_pmHⁿ_h

;w~h

¼ ðMAþ

/MBÞd²_sB^nþ1_h ;w~h

h

þ ðM Aþ

/MBÞ

c

_md2sB^nþ1_h ;w~hÞi

: ð35Þ

Choosingw~h¼

s

dsHb^nþ1_h in(35), we obtain

l

₀A ðMAþ

/MBÞdsH^nþ1_h dsHⁿ_h

;dsHb^nþ1_h

¼1

2

l

₀A ðMAþ

/MBÞdsH^nþ1_h ;dsH^nþ1_h

ðM Aþ

/MBÞdsHⁿ_h;dsHⁿ_h

;

l

₀A

c

mðM_Aþ

e

/M_BÞd2sH^nþ1_h ;

s

d_sHb^nþ1_h

¼

sl

₀A

c

mðM_Aþ

e

/M_BÞdsHb^nþ1_h ;d_sHb^nþ1_h

;

l

₀A

x

²pm

4 ðM_Aþ

e

/M_BÞH^nþ1_h þHⁿ_hþHⁿ_hþHⁿ¹_h

;Hb^nþ1_h Hbⁿ_h

¼

l

₀A

x

²pm

2 ðM_Aþ

e

/M_BÞHb^nþ1_h ;Hb^nþ1_h

ðM _Aþ

e

/M_BÞHbⁿ_h;Hbⁿ_h

h i

;

c

_mðMAþ

e

/MBÞd2sB^nþ1_h ;

s

dsHb^nþ1

¼

s c

_mðMAþ

e

/MBÞdsbB^nþ1_h ;dsHb^nþ1 6

s c

m d₈ðM_Aþ

e

/M_BÞdsbB^nþ1_h ;d_sbB^nþ1_h

þ 1 4d8

ðM_Aþ

e

/M_BÞdsHb^nþ1_h ;d_sHb^nþ1_h

: Summing up the above estimates concludes the proof. h

Lemma 4.3

s

ðMAþ

/MBÞd²sD^nþ_h ¹²;dsbE^nþ_h ¹²

þ

s

ðMAþ

/MBÞd²sB^nþ1_h ;dsHb^nþ1_h

¼

s

2 ðMAþ

/MBÞdsHⁿ_h;rdsEⁿ_h¹²

ðM Aþ

/MBÞdsH^nþ1_h ;rdsE^nþ_h ¹²

h i

þ

s

dsHⁿ_h;rðMAþ

/MBÞ dsbE^nþ_h ¹²

: ð36Þ

Proof. For an arbitrary functionf(x), multiplying(2)byf(x)/and integrating overX, we can obtain fðxÞ@D

@t;/

¼ ðrH;fðxÞ/Þ ¼ ðH;r ðfðxÞ/ÞÞ; 8/2H0ðcurl;XÞ; ð37Þ

from which we can obtain an equivalent form of(17):

fðxÞdsD^nþ_h ¹²;/_h

Hⁿ_h;r ðfðxÞ/hÞ

¼0; 8/_h2V⁰_h: ð38Þ

Subtracting(38)from itself withnreduced by 1, then dividing by

s

, we obtain fðxÞd²_sD^nþ_h ¹²;/_h

dsHⁿ_h;r ðfðxÞ/_hÞ

¼0; 8/_h2V⁰_h: ð39Þ

On the other hand, it is easy to see that an equivalent form of(19)is:

fðxÞdsB^nþ1_h ;w_h

þfðxÞrE^nþ_h ¹²;w_h

¼0; 8w_h2Uh; ð40Þ

from which we obtain fðxÞd²_sB^nþ1_h ;w_h

þfðxÞrdsE^nþ_h ¹²;w_h

¼0; 8w_h2Uh: ð41Þ

Choosing/_h¼

s

d_sEb^nþ_h ¹²; w_h¼

s

d_sHb^nþ1_h in(39) and (41), respectively, we have

s

fðxÞd²_sD^nþ_h ¹²;dsbE^nþ_h ¹²

þ

s

fðxÞd²_sB^nþ1_h ;dsHb^nþ1_h

¼

s

dsHⁿ_h

;rfðxÞdsE^nþ_h ¹²

fðxÞrdsbE^nþ_h ¹²;dsHb^nþ1_h

h i

: ð42Þ

Using the identity

dsHⁿ_h;fðxÞrdsEb^nþ_h ¹²

fðxÞrdsE^nþ_h ¹²;dsHb^nþ1_h

¼1

2 dsHⁿ_h;fðxÞrdsEⁿ_h ¹²

dsH^nþ1_h ;fðxÞrdsE^nþ_h ¹²

h i

; ð43Þ

the formula

r ðfðxÞuÞ ¼fðxÞruþrfðxÞ u

and choosingf(x) =MA+

/MBin(42), we conclude the proof. h

Lemma 4.4. For any vector (u, v)⁰, we have ðu;

v

^ÞðMAþ

/MBÞ u

v

^>u

2þ

v

^2:

Proof. Using the definitions ofM_AandM_B, and the fact that

/> 1, we have

ðu;

v

^ÞðMAþ

/MBÞ u

v

^{¼ ðu}

2sin²/2u

v

^sin/cos/þ

v

^{2cos2/Þ þ}

/u²cos²/þ2u

v

^sin/cos/þ

v

^2sin2/

¼ ðusin/

v

^cos/Þ2þ

/ðucos/þ

v

^{sin/Þ2>ðusin/}

v

^{cos/Þ2þ ðucos/þ}

v

^sin/Þ2

¼u²þ

v

^2:

Combining the above estimates, we finally have the following stability result on the cloaking region:

Theorem 4.2. Under the time step constraint(27),for any NP1, we have

0

ffiffiffiffiffi

/

p dsE^Nþ_h ¹²

2

0þ

x

²_p p b^{ffiffiffiffiffi}

/E^Nþ_h ¹²

2 0

þ

l

₀A dsH^Nþ1_h ²

0þ

x

²_pm Hb^Nþ1_h ²

0

6C

0 ffiffiffiffiffi

/

p dsE¹²_h

2

0þ

0

x

²_p p b^{ffiffiffiffiffi}

/E¹²_h

2

0þ

l

₀A ffiffiffiffiffi

/

p dsH¹_h ²

0þ

l

₀A

x

²_pm p b^{ffiffiffiffiffi}

/H¹_h ²

0

: