Numerical Study of the Plasma-Lorentz Model in Metamaterials

Transcription

1 J Sci Comput 3 54: 44 DOI.7/s Numerical Study of te Plasma-Lorentz Model in Metamaterials Jicun Li Yunqing Huang Wei Yang Received: 7 February / Revised: 3 May / Accepted: 3 May / Publised online: 6 May Springer Science+Business Media, LLC Abstract Since, te study of metamaterial as been a very ot topic due to its potential applications in many areas suc as design of invisibility cloa and sub-wavelengt imaging. Altoug several metamaterial models are often used by pysicists and engineers, te study of teir matematical properties as lagged beind. In tis paper, we initiate our investigation in te plasma-lorentz model. More specifically, we first discuss te well-posedness of tis model, ten develop two fully-discrete finite element metods for solving it. Detailed stability and error analysis are carried out, and 3-D numerical results justifying our teoretical analysis are presented. Keywords Maxwell s equations Metamaterial Plasma-Lorentz model Finite element metod Introduction In, Smit et al. [4] successfully constructed a composite medium wit simultaneously negative electric permittivity and magnetic permeability. An electromagnetic material wit tis property is usually called metamaterial. Later on, te first experimental demonstration of te negative refraction index was carried out for te metamaterial [4]. Since J. Li Department of Matematical Sciences, University of Nevada Las Vegas, Las Vegas, NV , USA jicun@unlv.nevada.edu Y. Huang W. Yang Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan 45, Cina Y. Huang uangyq@xtu.edu.cn W. Yang yangweixtu@6.com

2 J Sci Comput 3 54: 44, tere as been a tremendous growing interest in te study of metamaterial and its potential applications in areas ranging from electronics, telecommunications to sensing, radar tecnology, sub-wavelengt imaging, data storage, and design of invisibility cloa. More details on metamaterials can be found in some recent monograps suc as [9,, 7, 9] and references cited terein. Toug numerical simulations for metamaterials are often used by engineers and pysicists, tey mainly use te classic finite-difference time-domain FDTD metod [45]. However, it is nown tat te FDTD metod as a big disadvantage for solving problems wit complex geometries. Hence it would be quite interesting and useful to develop efficient and robust finite element metods FEMs for modeling metamaterials. Te metamaterial models can be described by te Maxwell s equations augmented wit some constitutive equations. Hence te study of metamaterial models is more complicated tan te standard Maxwell s equations in vacuum. Toug tere exist many excellent wor for Maxwell s equations in vacuum e.g., papers [, 4, 6 8, 4, 6, 9,,, 7, 34, 35], boos [, 8, 3] and references cited terein and in dispersive media e.g., [, 5,, 6, 8, 36 38, 44, 46], to our best nowledge, tere are not muc wor devoted to te development and analysis of FEMs for te Maxwell s equations wen metamaterials are involved [3]. In recent years, we made some initial effort [, 3 5] in developing and analyzing some FEMs for time-domain Maxwell s equations involving metamaterials. However, our previous papers were restricted to metamaterial models wose permittivity and permeability are given by te same type constitutive relations suc as te so-called Drude model or Lorentz model. In tis paper, we will investigate anoter type metamaterial model used by engineers and pysicists, in wic te permittivity is described by a plasma model, wile te permeability is described by te Lorentz model. To be specific, we denote tis model as te plasma- Lorentz model, wic as not been studied before from te matematics point view, toug tis model as been used by engineers and pysicists [3, 3, 33, 39, 4]. Here we first study its well-posedness, ten develop two fully-discrete finite element metods for solving tis model. Detailed stability analysis, error estimates and numerical results supporting te analysis are carried out. In tis paper, C> denotes a generic constant, wic is independent of te finite element mes size and time step size.leth α Ω 3 be te standard Sobolev space equipped wit te norm α and semi-norm α. Specifically, will mean te L Ω 3 -norm. We also introduce some common notation [3]: Hcurl; Ω = { v L Ω 3 ; v L Ω 3}, H curl; Ω = { v Hcurl; Ω; n v = on Ω }, H α curl; Ω = { v H α Ω 3 ; v H α Ω 3}, were α is a real number, and Ω is a bounded Lipscitz polyedral domain in R 3 wit connected boundary Ω.WeequipHcurl; Ω wit norm v,curl = v + curl v /, and H α curl; Ω wit norm v α,curl = v α + curl v α /. For clarity, we introduce te vector notation L Ω = L Ω 3, H α Ω = H α Ω 3, H α curl; Ω = H α curl; Ω 3. Te rest of te paper is organized as follows. In Sect., we present te governing equations for te plasma-lorentz metamaterial model. In Sect. 3, we develop two fully-discrete

3 J Sci Comput 3 54: 44 3 scemes, and prove te corresponding stability analysis and error estimates. Ten in Sect. 4, we present some numerical results justifying our teoretical analysis. Finally, we conclude our paper in Sect. 5. Te Governing Equations To simulate electromagnetic wave propagation, we ave to solve te general Maxwell s equations E = B t, D H = t,. were Ex,t and H x,t are te electric and magnetic fields, Dx,t and Bx,t are te corresponding electric and magnetic flux densities. In a general medium, D and B are related to E and H troug te constitutive relations D = ɛ E + P ɛe, B = μ H + M μh,. were ɛ and μ are te permittivity and permeability in free space, respectively, and P and M are polarization and magnetization, respectively. Moreover, ɛ and μ are te permittivity and permeability of te underlying medium, respectively. Te permittivity can be described by te cold electron plasma model [3, 33]: ɛω = ɛ ωp,.3 ωω + jν were ω is te excitation angular frequency, ω p > is te effective plasma frequency, ν is te loss parameter, and j = is te imaginary unit. On te oter and, te permeability can be described by te Lorentz model [39, 4]: Fω μω = μ,.4 ω + jγω ω were ω > is te resonant frequency, γ is te loss parameter, and F, is a parameter depending on te geometry of te unit cell of te metamaterial. Using a time-armonic variation of expjωt, and substituting.3 and.4 into., respectively, we obtain te time-domain equation for te polarization: P t and te equation for te magnetization: + ν P t = E,.5 M + γ M + ω t t M = μ Fω H..6 To facility te matematical study of te model, by introducing te induced electric current J = P and magnetic current K = M, we can write te time domain governing equations for te plasma-lorentz model as following: t t E ɛ = H J,.7 t

4 4 J Sci Comput 3 54: 44 H μ = E K, t.8 K μ ω F + γ t μ ω F K + μ F M = H,.9 M = K, μ F t μ F. J t + ν J = E.. To complete te model problem.7., we assume tat te boundary of Ω is perfect conducting so tat n E = on Ω,. were n is te unit outward normal to Ω. Furtermore, we assume tat te initial conditions are Ex, = E x, H x, = H x,.3 Kx, = K x, Mx, = M x, J x, = J x,.4 were E, H, K, M and J are some given functions. First, we ave te following stability for our model problem.7.. Lemma. For te solution E, H, K, M, J of te model problem.7. wit boundary condition. and initial conditions.3.4, te following stability olds true: ɛ Et + μ H t + μ ω F Kt + Mt μ F + J t ɛ E + μ H + μ ω F K + μ F M + J..5 Proof Note tat te problem.7. can be rewritten as werewedenoteut = E, H, K, M, J, operators A = diag ɛ I,μ I, C = diag,, γ Aut = B + Cut,.6 t μ ω F I, μ F I, μ ω F I,, ν I, I,

5 J Sci Comput 3 54: 44 5 and I I B = I μ F I μ F I, I ere I denotes a 3 3 identity matrix. Note tat te operator B is anti-symmetric. To prove te stability, multiplying.6 byu, ten integrating over Ω, and using te property u Bu =, we obtain d u Au = u Cu, dt integrating wic wit respect to t leads to te stability.5. Remar. Te stability.5 can be proved in a direct way. Multiplying.7. by E, H, K, M, J and integrating over Ω, ten adding te resultants togeter, we obtain [ d ɛ Et dt + μ H t + μ ω F Kt + Mt μ F + J t ] + ν J t ɛ ω + γ Kt p μ ω F =, integrating wic from to t leads to.5. Tis is te main reason wy we write te last tree governing equations.9. intisway. Now, let us prove te existence for our model problem.7.. Lemma. Te problem.7. as a unique solution E, H in Hcurl; Ω Hcurl; Ω. Proof From ordinary differential equation teory, we can prove tat te solutions of.5 and.6 wit zero initial conditions can be expressed as and P x,t= ɛ ω p ν t e νt s Ex,sds,.7 t Mx,t= μ Fω gt sh x,sds,.8 respectively. Here te ernel gt = γ α e t sinαt, wereα = ω γ. Using te definition J = P and K = M, ten substituting.7 and.8 into.7 t t and.8, respectively, we can rewrite.7 and.8 as follows: d AE + K E = LE + F,.9 dt

6 6 J Sci Comput 3 54: 44 were we denote E = E, H, for te convolution product, F for source terms obtained by transforming a problem wit non-zero initial conditions to a problem wit zero initial conditions, and operators ɛ I A = 3 3 ɛ I, K = 3 3, L= 3 μ I 3 3 μ I In te above, I 3 is te 3 3 unit matrix, 3 is te zero matrix, ɛ = ɛ ωp ut e νt and ν μ = μ Fω gt. Hereut denotes te unit step function. Hence our problem.9 is a special case of Problem I of [5], wose existence and uniqueness is guaranteed by Teorem 3. of [5]. Finally, we can prove tat te Gauss s Law olds true for our model. Lemma.3 Assume tat te initial fields are divergence free: ɛ E =, μ H =, K =, M =, J =, ten for any time t>, te fields are still divergence free, i.e., we ave ɛ Et =, μ H t =, Kt =, Mt =, J t =. Proof Taing te divergence of.7, we ave ɛ E + J =.. t Similarly, taing te divergence of., we ave t J + ν J = ɛ ωp E.. Substituting. into., we ave t ɛ E + ν t ɛ E + ωp ɛ E =.. By te assumed initial conditions and., we ave initial conditions for ɛ E: ɛ E =, t ɛ E = J =..3 It is easy to see tat te ordinary differential equation. wit initial conditions.3 only as zero solution, i.e., ɛ Et. Oter divergence free conditions can be proved similarly.. 3 Fully-Discrete Scemes In tis section, we will develop two fully-discrete finite element metods for solving.7.. We assume tat te domain Ω is partitioned by a family of regular tetraedral or

7 J Sci Comput 3 54: 44 7 cubic meses T wit maximum mes size. Depending upon te regularity of te solution of.7., we can use a proper order Raviart-Tomas-Nédélec RTN mixed finite element space [3, 3]: For any l, on a tetraedral element, we can coose U = { u Hdiv; Ω : u K p l 3 p l x, K T }, V = { v Hcurl; Ω : v K p l 3 S l, K T },S l = { p p l 3, x p = }, wile on a cubic element we coose U = { u Hdiv; Ω : u K Q l,l,l Q l,l,l Q l,l,l, K T }, V = { v Hcurl; Ω : v K Q l,l,l Q l,l,l Q l,l,l, K T }. Here p denotes te space of omogeneous polynomials of degree, andq i,j, denotes te space of polynomials wose degrees are less tan or equal to i, j, in variables x,y,z, respectively. To impose te boundary condition., we denote V ={v V : v n = on Ω}. It is easy to see tat V U. 3. To carry out te error analysis below, we need to define two operators. Te first one is te standard L -projection operator P :ForanyH L Ω, P H U satisfies P H H, ψ =, ψ U. Anoter one is te standard Nédélec interpolation operator Π mapped from Hcurl; Ω to V. It is nown tat Π satisfies te following interpolation error estimate [3, 3]: E Π E + E Π E C l E l,curl, E H l curl; Ω, l, 3. and P as te projection error estimate: H P H C l H l, H H l Ω, l. 3.3 To define a fully discrete sceme, we divide te time interval,t into N uniform subintervals by points = t <t < <t N = T,weret =, = T/N, and denote te -t subinterval by I =[t,t ]. Moreover, we define u = u,for N,andte notation: 3. Te Cran-Nicolson Sceme δ u = u u /, u = u + u. We start wit a Cran-Nicolson type sceme for solving.7.: For =,,...,N, find E V, J V, H, K, M U suc tat ɛ δ E, φ H, φ + J, φ =, 3.4 μ δ H, ψ + E, ψ + K, ψ =, 3.5 μ ω F δ K, γ ψ + K μ ω F, ψ + M μ F, ψ = H, ψ, 3.6

8 8 J Sci Comput 3 54: 44 δ M μ F, ψ = K μ F, ψ, 3.7 δ J, ν φ + J, φ = E, φ, 3.8 old true for any φ V, ψ, ψ, ψ U, φ V, and are subject to te initial approximations E x = Π E x, H x = P H x, K x = P K x, M x = P M x, J x = Π J x. First, we can sow tat our sceme satisfies a discrete stability, wic as exactly te same form as te continuous case proved in Lemma.. Lemma 3. For any, we ave ɛ E + μ H + J + μ ω F K + μ F ɛ E + μ H + J + μ ω F K + μ F M M. 3.9 Proof Coosing φ = E, ψ = H, ψ = K, ψ = M, φ = J in , respectively, ten adding te resultants togeter, we ave E μ + H H + J + ν J + μ ω F K K γ + μ ω F K + M μ F M =, ɛ E J wic easily leads to 3.9. For te Cran-Nicolson sceme , we can prove te following optimal error estimate. Teorem 3. Let E m, H m, K m, M m, J m and E m, H m, Km, Mm, J m be te analytic and numerical solutions of.7. and at time t m, respectively. Under te following regularity assumptions: E t, J t L,T; H l curl; Ω, E tt, J tt, H tt, K tt, M tt, E tt, H tt L,T; L Ω, E, J, E L,T; H l curl; Ω, H, K, M L,T; H l Ω,

9 J Sci Comput 3 54: 44 9 tere exists a constant C>independent of te mes size and time step, suc tat max ɛ E n E n + μ n H n H n + J n J n + μ ω F K n K n + M n M n C l + 4, μ F were l is te order of te basis functions in spaces U and V. Proof Integrating.7. from t to t, ten multiplying test functions φ V, ψ, ψ, ψ U, φ V and integrating te resultant equations over domain Ω, we obtain ɛ δ E, φ H, φ + J, φ I =, 3. I μ δ H, ψ + E, ψ + K, ψ I =, 3. I μ ω F δ K γ, ψ + μ ω F K, ψ + M, ψ I μ F I = H, ψ, 3. I δ M, ψ μ F = K, ψ μ F, I δ J ν, φ + J, φ = E, φ I. 3.3 I Let us denote ξ = Π E E, η = P H H, ξ = P K K, ξ = P M M, ζ = Π J J. Subtracting from gives us te error equations ɛ δ ξ, φ η, φ + ζ, φ = ɛ δ Π E E, φ P H H, φ I + Π J J, φ, 3.4 I μ δ η, ψ + ξ, ψ + ξ, ψ = μ δ P H H, ψ + Π E I E, ψ + P K K, ψ, 3.5 I

10 3 J Sci Comput 3 54: 44 μ ω F δ ξ, γ ψ + ξ μ ω F, ψ + μ F = μ ω F δ P K K γ, ψ + μ ω F + P M M, ψ μ F I δ ξ μ F, ψ ξ μ F, ψ = δ P M M, ψ μ F μ F ξ, ψ η, ψ P K P H K, ψ I H, ψ, 3.6 I P K K, ψ, 3.7 I δ ζ, ν φ + ζ, φ ξ, φ = δ Π J J ν, φ + Π J J, φ I Π E E, φ. 3.8 I Coosing φ = ξ, ψ = η, ψ = ξ, ψ = ξ, φ = ζ in , respectively, ten adding te resultants togeter and using te projection property, we ave ɛ ξ ξ μ + η η + ζ ɛ ω p ζ ν + ζ + ξ μ ω F ξ γ + ξ μ ω F + ξ μ F ξ = ɛ δ Π E E, ξ H H, ξ + Π J J, ξ I I + Π E E, η + K K, η I I γ + μ ω F K K, ξ + M M, ξ I μ F I H H, ξ K K, ξ I μ F I + δ Π J J, ζ ν ɛ ω + Π p J J, ζ I Π E E, ζ. 3.9 I Te rest proof is standard see our early wor [4] by estimating eac term on te rigt and side of 3.9, and using te triangle inequality and te estimates

11 J Sci Comput 3 54: Te Leap-Frog Sceme We construct a leap-frog type sceme for solving.7.: For =,,...,finde V, J + V, H +, K, M+ U suc tat E ɛ E, φ H, φ + J, φ =, 3. + H H μ μ ω F, ψ + E, ψ + K, ψ =, 3. K K, ψ + γ μ ω F = H, ψ, + M M μ F, ψ = K μ F, ψ, + J J, φ + ν + J + J K + K, ψ + M, ψ μ F 3., φ = E, φ, 3.3 old true for any φ V, ψ, ψ, ψ U, φ V, and are subject to te initial approximations E x = Π E x, [ H x = P H x μ E x + K x ], K x = P K x, M x = P [M x + ] K x, J x = Π [ J x + ɛ ω p E x νj x ] First, let us prove te following discrete stability for te leap-frog sceme Lemma 3.3 Denote c v = μ ɛ for te speed of ligt in vacuum, and a constant c inv > in te standard inverse estimate u c inv u, u V. 3.6 Under te time step constraint { = min ω F, ω, ω p, c v c inv }, 3.7

12 3 J Sci Comput 3 54: 44 for any we ave ɛ E + μ + H + + J [ C ɛ E + μ H + were C> is independent of and. + K μ ω F J + + M μ F + K μ ω F + μ F M ], 3.8 Proof Coosing φ = E + E, ψ = H + ψ = M + + M, φ = J + + J te resultants togeter, we ave ɛ E E + μ H + H + μ ω F K K + μ F H J + H = [ H + [ H + H, ψ = K + K, in , respectively, ten adding + M + M, E + E E, H + + H J + J, E + E + E, J + + J K, H + + H +, K + K, E, K μ F M + H, E ] + + H, K, K + K ] + [ E, J + μ F Summing up 3.9for from to n,weobtain ɛ E n E + μ H n+ H + K n μ ω F K + μ F μ F K, M + + M E, J ] [ K, M + K, M ] M n+ ɛ ω p J n+ J M [ H, E n+ H ] [, E n + E n, J n+ E, J ] + [ H, K n+ H ], K n [ + K n μ F, M n+ K, M ]. 3.3 Using te inverse estimate 3.6 and te basic aritmetic-geometric mean inequality we ave ab δa + 4δ b,

13 J Sci Comput 3 54: H n+, E n = μ H n+,c v ɛ E n δ μ H n+ + c vc inv ɛ E n 4δ. Similarly, we ave E n, J n+ = ω p ɛ E n, J n+ δ J n+ + ω p ɛ E n 4δ, H n+, K n μ = H n+, ω F K μ n ω F δ 3 μ H n+ + ω F K n 4δ 3 μ ω F, and μ F K n, M n+ = μ ω F Kn, ω μ F Mn+ δ 4 μ ω F K n + ω 4δ 4 μ F M n+. Substituting te above inequalities into 3.3, we ave c vc inv ω p ɛ E n 4δ 4δ + δ δ 3 μ H n+ + δ J n+ + ω F δ 4 K n 4δ 3 μ ω F + ω n+ M 4δ 4 μ F ɛ E + μ H + J ɛ ω + p μ ω F K + M μ F + H, E E, J + H, K K, M. 3.3 Under te coice δ = δ = δ 3 = δ 4 = 4,and { = min ω F, ω, ω p, c v c inv we can easily see tat te coefficients on te left and side of 3.3 are all larger tan, wic easily leads to te stability 3.8. },

14 34 J Sci Comput 3 54: 44 For te leap-frog sceme , we can prove te following optimal error estimate. Teorem 3.4 Let E m, H m+, K m, M m+, J m+ and E m, H m+, K m, Mm+, J m+ be te analytic and numerical solutions of.7. and , respectively. Under te same regularity assumptions as Teorem 3., tere exists a constant C>indepen- dent of and suc tat max n ɛ E n E n + μ H n+ n+ H + + μ ω F K n K n + M n+ n+ M μ F were l is te order of te basis functions in spaces U and V. J n+ J n+ C l + 4, Proof Integrating.7and.9 from t to t,.8,.. from t to t +,ten multiplying proper test functions φ V, ψ, ψ, ψ U, φ V and integrating te resultants over domain Ω,weobtain E E t t ɛ, φ H, φ + J, φ t =, 3.3 t H + H t+ μ, ψ + E, ψ t t+ + K, ψ t =, 3.33 K K μ ω F, ψ + γ t μ ω F K, ψ + t M, ψ t μ F t t = H, ψ, 3.34 t M + M, ψ μ F = t+ K, ψ μ F, t ɛ ω p Let us denote J + J, φ + ν t+ t+ J, φ = E, φ t t ξ = Π E E, η = P H H, ξ = P K K, ξ = P M M, ζ = Π J J. Subtracting from gives us te error equations: ɛ δ ξ, φ η, φ + ζ, φ = ɛ δ Π E E, φ P H t H, φ + Π J t J, φ t, 3.36 t

15 J Sci Comput 3 54: μ δ η +, ψ + ξ, ψ + ξ, ψ, ψ = μ δ P H + H + + Π E μ ω F δ ξ, ψ + t+ t E, ψ + P K t+ K, ψ, 3.37 t γ ξ μ ω F, ψ + μ F = μ ω F δ P K K γ, ψ + μ ω F + P M t M, ψ μ F t δ ξ +, ψ μ F ξ μ F, ψ = δ P M + M +, ψ μ F μ F δ ζ +, φ + ν + ν ɛ ω p Π J ξ, ψ η, ψ P K P H t t K, ψ t H, ψ t, 3.38 P K t+ K, ψ, 3.39 t ζ, φ ξ, φ = δ Π J + J +, φ t+ J, φ Π E t+ E, φ. 3.4 t t Coosing φ = ξ, ψ = η, ψ = ξ, ψ = ξ, φ = ζ in , respectively, ten adding te resultants togeter and using te projection property of P,we ave ɛ ξ ξ μ + η + η + ξ μ ω F ξ + μ F ɛ δ Π E E, ξ H + Π J J, ξ t t + Π E γ + μ ω F t+ t K t + ζ ζ ξ + ξ t H, ξ t E, η + K t+ K, η t + M t M, ξ μ F t t K, ξ

16 36 J Sci Comput 3 54: 44 H t H, ξ t + ɛ ω p ν + δ Π J + J +, ζ K t+ K, ξ μ F t Π J t+ J, ζ Π E t+ E, ζ. 3.4 t t Te rest proof follows similarly to our early wor [3] by estimating eac term on te rigt and side of 3.4, and using te triangle inequality and te estimates Remar 3. We want to remar tat te Cran-Nicolson sceme as an nonsymmetric linear system of as many as 5 unnown functions five unnown 3D variables, wic results a very large-scale system even for linear edge elements. Hence direct solving te coupled system is quite callenging. In tis aspect te leap-frog sceme is more practical, since eac time we only need solve one unnown variable. Of course, we ave te CFL time step constraint. More efficient algoritms will be explorered in te future. 4 Numerical Results In tis section, we implemented te leap-frog sceme for te lowest-order Raviart-Tomas-Nedelec cubic element i.e., l = inu and V to confirm our teoretical analysis and effectiveness of our algoritm. For simplicity, we assume tat Ω is te unit cube [, ] 3 and te time interval is I =[, ]. To rigorously cec te convergence rate, we construct an analytical solution of.7. wit all pysical parameters in.7. being one except tat γ = ω =,F =, and wit a source term f x,t and gx,t added to te rigt and sides of.7 and.8, respectively. More specifically, te analytical solution is as follows: E x A cos πxsin πy sin πz Ex,t= E y = B sin πxcos πy sin πz e t cos t, E z C sin πxsin πy cos πz Kx,t= H x πc Bsin πxcos πy cos πz H x,t= H y = πa Ccos πxsin πy cos πz e t cos t, H z πb A cos πxcos πy sin πz K x K y K z πc Bsin πxcos πy cos πz [ = πa Ccos πxsin πy cos πz e t πb A cos πxcos πy sin πz t sin t + sin t + ], t cos t M x πc Bsin πxcos πy cos πz Mx,t= M y = πa Ccos πxsin πy cos πz e t t sin t, M z πb A cos πxcos πy sin πz

17 J Sci Comput 3 54: J x A cos πxsin πy sin πz J x,t= J y = B sin πxcos πy sin πz e t sin t, J z C sin πxsin πy cos πz were constants A =,B =,C = 3. It is easy to cec tat our constructed solutions satisfy te boundary conditions n E = on Ω, and te Gauss s Law Ex,t=, H x,t=, x,t Ω [, ]. Furtermore, te source terms f and g are and f x,t= E t A 3Aπ cos πxsin πy sin πz H + J = B 3Bπ sin πxcos πy sin πz e t cos t, C 3Cπ sin πxsin πy cos πz gx,t= H + E + K t πc Bsin πxcos πy cos πz [ = πa Ccos πxsin πy cos πz e t πb A cos πxcos πy sin πz t sin t sin t + ]. t cos t In practical implementation of wit added source terms f, g and our assumed parameter values, te sceme can be implemented as follows: at eac time step, we first solve E and K can be done in parallel from [ E, φ = E, φ + H, φ J, φ + f ], φ, 4. K, ψ = K [, ψ + + H, ψ + M ], ψ, 4. respectively; ten solve H +, M +, J + can be done in parallel from respectively. + H, ψ = H [ ], ψ + g, ψ E, ψ K, ψ, M, ψ = M, ψ + K, ψ, J, φ = J +, φ + E +, φ, 4.5

18 38 J Sci Comput 3 54: 44 Table Te L errors obtained wit fixed =. on uniform cubic meses Mes = /4 = /8 Rate = /6 Rate = /3 Rate = /64 Rate E H K M J

19 J Sci Comput 3 54: Table Total degrees of freedom DOFs on uniform cubic meses and CPU time Mes Total edge DOFs Total face DOFs CPU in seconds = / = / = / = / = / Fig. Electric fields at T = obtained on tetraedral meses: on 4 4 4mesleftand8 8 8mes rigt We tested te algoritm on bot uniformly refined cubic and tetraedral meses wit various time step size. Exemplary convergence results obtained wit =. are presented in Tables and 3 for cubic and tetraedral meses, respectively. Te results in bot tables clearly sow O convergence in te L norm as our teoretical analysis proved in Teorem 3.4. Te corresponding total degrees of freedom and CPU time are sown in Tables and 4, from wic we can see tat our algoritm is quite efficient. Note tat all our tests are carried out under MATLAB 7. running on a Dell destop wit GB memory and.93 GHz CPU. In Fig. we presented te electric fields obtained wit =. on two different tetraedral meses. Since suc 3D figures are not easy to see clearly, ten we presented some slice cuts in Figs. and 3 on te plane z =.4, i.e., we plotted te electric field E x,e y and H x,h y on z =.4. 5 Concluding Remars In tis paper, we carried out te first matematical study of anoter popular metamaterial model we named it as te plasma-lorentz model used by pysicists and engineers. Here we discussed te well-posedness of tis model, developed two fully-discrete finite element metods for solving it, and proved te corresponding stability analysis and error estimates. Numerical results supporting our analysis are presented. More recently developed numerical scemessucaspfems[, 43] and DG metods [8], and practical applications of tis model will be investigated in our future wor.

20 4 J Sci Comput 3 54: 44 Table 3 Te L errors obtained wit fixed =. on uniform tetraedral meses Mes = /4 = /8 Rate = /6 Rate = /3 Rate E H K M J

21 J Sci Comput 3 54: 44 4 Table 4 Total degrees of freedom DOFs on uniform tetraedral meses and CPU time Mes Total edge DOFs Total face DOFs CPU in seconds = / = / = / = / Fig. Slice cuts at T = obtained on cubic meses: electric field E x,e y on 8 8mestop left and 6 6 mes top rigt; magnetic field H x,h y on 8 8mesbottom left and 6 6 mes bottom rigt Acnowledgements Tis wor was supported by National Science Foundation grant DMS-8896, and in part by te NSFC Key Project 36 and Hunan Provincial NSF project JJ7.

22 4 J Sci Comput 3 54: 44 Fig. 3 Slice cuts at T = obtained on tetraedral meses: electric field E x,e y on 8 8mestop left and 6 6 mes top rigt; magnetic field H x,h y on 8 8mesbottom left and 6 6 mes bottom rigt References. Bans, H.T., Boil, V.A., Gibson, N.L.: Analysis of stability and dispersion in a finite element metod for Debye and Lorentz media. Numer. Metods Partial Differ. Equ. 5, Bec, R., Hiptmair, R., Hoppe, R.H.W., Wolmut, B.: Residual based a posteriori error estimators for eddy current computation. Modél. Mat. Anal. Numér. 34, Boardman, A.D., Marinov, K.: Electromagnetic energy in a dispersive metamaterial. Pys. Rev. B 73, Bocev, P.B., Garasi, C.J., Hu, J.J., Robinson, A.C., Tuminaro, R.S.: An improved algebraic multigrid metod for solving Maxwell s equations. SIAM J. Sci. Comput. 5, Bonnet-Ben Dia, A.S., Ciarlet, P. Jr., Zwölf, C.M.: Time armonic wave diffraction problems in materials wit sign-sifting coefficients. J. Comput. Appl. Mat. 34, Cen, Z., Du, Q., Zou, J.: Finite element metods wit matcing and nonmatcing meses for Maxwell equations wit discontinuous coefficients. SIAM J. Numer. Anal. 37, Ciarlet, P. Jr., Zou, J.: Fully discrete finite element approaces for time-dependent Maxwell s equations. Numer. Mat. 8,

23 J Sci Comput 3 54: Cocburn, B., Li, F., Su, C.-W.: Locally divergence-free discontinuous Galerin metods for te Maxwell equations. J. Comput. Pys. 94, Cui, T.J., Smit, D.R., Liu, R. eds.: Metamaterials: Teory, Design, and Applications. Springer, Berlin 9. Demowicz, L., Kurtz, J., Pardo, D., Paszynsi, M., Racowicz, W., Zdune, A.: Computing wit p- Adaptive Finite Elements. Vol.. Frontiers: Tree Dimensional Elliptic and Maxwell Problems wit Applications. CRC Press/Taylor & Francis, Boca Raton 6. Donderici, B., Teixeira, F.L.: Mixed finite-element time-domain metod for Maxwell equations in doubly-dispersive media. IEEE Trans. Microw. Teory Tec. 56, 3 8. Engeta, N., Ziolowsi, R.W.: Electromagnetic Metamaterials: Pysics and Engineering Explorations. Wiley-IEEE Press, New Yor 6 3. Fernandes, P., Raffetto, M.: Well posedness and finite element approximability of time-armonic electromagnetic boundary value problems involving bianisotropic materials and metamaterials. Mat. Models Metods Appl. Sci. 9, Fezoui, L., Lanteri, S., Lorengel, S., Piperno, S.: Convergence and stability of a discontinuous Galerin time-domain metod for te eterogeneous Maxwell equations on unstructured meses. Mat. Model. Numer. Anal. 39, 49C Frantzesais, D.J., Ioannidis, A., Roac, G.F., Stratis, I.G., Yannacopoulos, A.N.: On te error of te optical response approximation in ciral media. Appl. Anal. 8, Gopalarisnan, J., Pascia, J.E., Demowicz, L.F.: Analysis of a multigrid algoritm for time armonic Maxwell equations. SIAM J. Numer. Anal. 4, Hao, Y., Mittra, R.: FDTD Modeling of Metamaterials: Teory and Applications. Artec House, Boston 8 8. Hestaven, J.S., Warburton, T.: Nodal Discontinuous Galerin Metods: Algoritms, Analysis, and Applications. Springer, New Yor 8 9. Hiptmair, R.: Finite elements in computational electromagnetism. Acta Numer., Houston, P., Perugia, I., Scötzau, D.: Energy norm a posteriori error estimation for mixed discontinuous Galerin approximations of te Maxwell operator. Comput. Metods Appl. Mec. Eng. 94, Huang, Y., Li, J.: Recent advances in time-domain Maxwell s equations in metamaterials. In: Zang, W., et al. eds. Lecture Notes in Computer Sciences, vol. 5938, pp Hyman, J.M., Sasov, M.: Mimetic discretizations for Maxwell s equations. J. Comput. Pys. 5, Li, J.: Numerical convergence and pysical fidelity analysis for Maxwell s equations in metamaterials. Comput. Metods Appl. Mec. Eng. 98, Li, J.: Finite element study of te Lorentz model in metamaterials. Comput. Metods Appl. Mec. Eng., Li, J., Wood, A.: Finite element analysis for wave propagation in double negative metamaterials. J. Sci. Comput. 3, Lu, T., Zang, P., Cai, W.: Discontinuous Galerin metods for dispersive and lossy Maxwell s equations and PML boundary conditions. J. Comput. Pys., Lin, Q., Yan, N.: Global superconvergence for Maxwell s equations. Mat. Comput. 69, Maradei, F., Ke, H., Hubing, T.H.: Full-wave model of frequency-dispersive media wit Debye dispersion relation by circuit-oriented FEM. IEEE Trans. Electromagn. Compat. 5, Maros, P., Sououlis, C.M.: Wave Propagation: From Electrons to Potonic Crystals and Left-Handed Materials. Princeton University Press, Princeton 8 3. Mon, P.: Finite Element Metods for Maxwell s Equations. Oxford University Press, Oxford 3 3. Nédélec, J.-C.: Mixed finite elements in R 3. Numer. Mat. 35, Pendry, J.B.: Negative refraction maes a perfect lens. Pys. Rev. Lett. 85, Pendry, J.B., Holden, A.J., Robbins, D.J., Stewart, W.J.: Magnetism from conductors and enanced nonlinear penomena. IEEE Trans. Microw. Teory Tec. 47, Qiao, Z., Yao, C., Jia, S.: Superconvergence and extrapolation analysis of a nonconforming mixed finite element approximation for time-armonic Maxwell s equations. J. Sci. Comput. 46, Rieben, R.N., Rodrigue, G.H., Wite, D.A.: A ig order mixed vector finite element metod for solving te time dependent Maxwell equations on unstructured grids. J. Comput. Pys. 4, Rodriguez-Esquerre, V.F., Kosiba, M., Hernandez-Figueroa, H.E.: Frequency-dependent envelope finite-element time-domain analysis of dispersion materials. Microw. Opt. Tecnol. Lett. 44, Sceid, C., Lanteri, S.: Convergence of a discontinuous Galerin sceme for te mixed time domain Maxwell s equations in dispersive media. INRIA preprint, No. 7364, May

24 44 J Sci Comput 3 54: Saw, S.: Finite element approximation of Maxwell s equations wit Debye memory, Advances in Numerical Analysis, Vol., Article ID 9383, 8 pp. 39. Selby, R.A., Smit, D.R., Nemat-Nasser, S.C., Scultz, S.: Microwave transmission troug a twodimensional, isotropic, left-anded metamaterial. Appl. Pys. Lett. 784, Selby, R.A., Smit, D.R., Scultz, S.: Experimental verification of a negative index of refraction. Science 9, Smit, D.R., Kroll, N.: Negative refractive index in left-anded materials. Pys. Rev. Lett. 85, Smit, D.R., Padilla, W.J., Vier, D.C., Nemat-Nasser, S.C., Scultz, S.: Composite medium wit simultaneously negative permeability and permittivity. Pys. Rev. Lett. 84, Solin, P., Dubcova, L., Cerveny, J., Dolezel, I.: Adaptive p-fem wit arbitrary-level anging nodes for Maxwell s equations. Adv. Appl. Mat. Mec., Stoyov, N.S., Kuien, T.A., Lowery, M.M., Taflove, A.: Finite-element time-domain algoritms for modeling linear Debye and Lorentz dielectric dispersions at low frequencies. IEEE Trans. Biomed. Eng. 5, Taflove, A., Hagness, S.C.: Computational Electrodynamics: Te Finite-Difference Time-Domain Metod. Artec House, Boston 46. Wang, B., Xie, Z., Zang, Z.: Error analysis of a discontinuous Galerin metod for Maxwell equations in dispersive media. J. Comput. Pys. 9,