Numerical Study of the Plasma-Lorentz Model in Metamaterials

DOI 10.1007/s10915-012-9608-5

Numerical Study of the Plasma-Lorentz Model in Metamaterials

Jichun Li·Yunqing Huang·Wei Yang

Received: 17 February 2012 / Revised: 3 May 2012 / Accepted: 13 May 2012 / Published online: 26 May 2012

© Springer Science+Business Media, LLC 2012

Abstract Since 2000, the study of metamaterial has been a very hot topic due to its potential applications in many areas such as design of invisibility cloak and sub-wavelength imaging.

Although several metamaterial models are often used by physicists and engineers, the study of their mathematical properties has lagged behind. In this paper, we initiate our investiga- tion in the plasma-Lorentz model. More specifically, we first discuss the well-posedness of this model, then develop two fully-discrete finite element methods for solving it. Detailed stability and error analysis are carried out, and 3-D numerical results justifying our theoret- ical analysis are presented.

Keywords Maxwell’s equations·Metamaterial·Plasma-Lorentz model·Finite element method

1 Introduction

In 2000, Smith et al. [42] successfully constructed a composite medium with simultane- ously negative electric permittivity and magnetic permeability. An electromagnetic material with this property is usually called metamaterial. Later on, the first experimental demon- stration of the negative refraction index was carried out for the metamaterial [40]. Since

J. Li (

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

e-mail:[email protected]

Y. Huang·W. Yang

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

Y. Huang

e-mail:[email protected] W. Yang

e-mail:[email protected]

2000, there has been a tremendous growing interest in the study of metamaterial and its po- tential applications in areas ranging from electronics, telecommunications to sensing, radar technology, sub-wavelength imaging, data storage, and design of invisibility cloak. More details on metamaterials can be found in some recent monographs such as [9,12,17,29]

and references cited therein.

Though numerical simulations for metamaterials are often used by engineers and physi- cists, they mainly use the classic finite-difference time-domain (FDTD) method [45]. How- ever, it is known that the FDTD method has a big disadvantage for solving problems with complex geometries. Hence it would be quite interesting and useful to develop efficient and robust finite element methods (FEMs) for modeling metamaterials.

The metamaterial models can be described by the Maxwell’s equations augmented with some constitutive equations. Hence the study of metamaterial models is more complicated than the standard Maxwell’s equations in vacuum. Though there exist many excellent work for Maxwell’s equations in vacuum (e.g., papers [2,4,6–8,14,16,19,20,22,27,34,35], books [10,18,30] and references cited therein) and in dispersive media (e.g., [1,5,11, 26,28,36–38,44,46]), to our best knowledge, there are not much work devoted to the development and analysis of FEMs for the Maxwell’s equations when metamaterials are involved [13]. In recent years, we made some initial effort [21,23–25] in developing and analyzing some FEMs for time-domain Maxwell’s equations involving metamaterials. How- ever, our previous papers were restricted to metamaterial models whose permittivity and permeability are given by the same type constitutive relations such as the so-called Drude model or Lorentz model.

In this paper, we will investigate another type metamaterial model used by engineers and physicists, in which the permittivity is described by a plasma model, while the permeability is described by the Lorentz model. To be specific, we denote this model as the plasma- Lorentz model, which has not been studied before from the mathematics point view, though this model has been used by engineers and physicists [3,32,33,39, 41]. Here we first study its well-posedness, then develop two fully-discrete finite element methods for solving this model. Detailed stability analysis, error estimates and numerical results supporting the analysis are carried out.

In this paper,C >0 denotes a generic constant, which is independent of the finite element mesh sizehand time step sizeτ. Let(H^α(Ω))³be the standard Sobolev space equipped with the norm·αand semi-norm·α. Specifically,·0will mean the(L²(Ω))³-norm.

We also introduce some common notation [30]:

H (curl;Ω)= v∈

L²(Ω)3

; ∇ ×v∈

L²(Ω)3 , H0(curl;Ω)=

v∈H (curl;Ω);n×v=0 on∂Ω , H^α(curl;Ω)=

v∈

H^α(Ω)3

; ∇ ×v∈

H^α(Ω)3 ,

whereα≥0 is a real number, andΩis a bounded Lipschitz polyhedral domain inR³with connected boundary∂Ω. We equipH (curl;Ω)with normv0,curl=(v²₀+ curlv²₀)^1/2, andH^α(curl;Ω)with normvα,curl=(v²_α+ curlv²_α)^1/2. For clarity, we introduce the vector notation

L²(Ω)= L²(Ω)3

, H^α(Ω)=

H^α(Ω)3

, H^α(curl;Ω)=

H^α(curl;Ω)3

. The rest of the paper is organized as follows. In Sect.2, we present the governing equa- tions for the plasma-Lorentz metamaterial model. In Sect.3, we develop two fully-discrete

schemes, and prove the corresponding stability analysis and error estimates. Then in Sect.4, we present some numerical results justifying our theoretical analysis. Finally, we conclude our paper in Sect.5.

2 The Governing Equations

To simulate electromagnetic wave propagation, we have to solve the general Maxwell’s equations

∇ ×E= −∂B

∂t , ∇ ×H=∂D

∂t , (2.1)

whereE(x, t )and H(x, t )are the electric and magnetic fields,D(x, t ) and B(x, t )are the corresponding electric and magnetic flux densities. In a general medium,DandBare related toEandHthrough the constitutive relations

D=0E+P ≡E, B=μ0H+M≡μH, (2.2)

where0andμ0are the permittivity and permeability in free space, respectively, andP and Mare polarization and magnetization, respectively. Moreover,andμare the permittivity and permeability of the underlying medium, respectively.

The permittivity can be described by the cold electron plasma model [32,33]:

(ω)=0

1− ω²_p ω(ω+j ν)

, (2.3)

whereωis the excitation angular frequency,ωp>0 is the effective plasma frequency,ν≥0 is the loss parameter, andj=√

−1 is the imaginary unit. On the other hand, the permeabil- ity can be described by the Lorentz model [39,41]:

μ(ω)=μ0

1− F ω₀² ω²+j γ ω−ω₀²

, (2.4)

whereω0>0 is the resonant frequency,γ ≥0 is the loss parameter, andF ∈(0,1)is a parameter depending on the geometry of the unit cell of the metamaterial.

Using a time-harmonic variation of exp(j ωt ), and substituting (2.3) and (2.4) into (2.2), respectively, we obtain the time-domain equation for the polarization:

∂²P

∂t² +ν∂P

∂t =0ω²_pE, (2.5)

and the equation for the magnetization:

∂²M

∂t² +γ∂M

∂t +ω²₀M=μ0F ω²₀H. (2.6) To facility the mathematical study of the model, by introducing the induced electric cur- rentJ =^∂_∂t^P and magnetic currentK=^∂_∂t^M, we can write the time domain governing equa- tions for the plasma-Lorentz model as following:

0

∂E

∂t = ∇ ×H−J, (2.7)

μ0

∂H

∂t = −∇ ×E−K, (2.8)

1 μ0ω²₀F

∂K

∂t + γ

μ0ω²₀FK+ 1

μ0FM=H, (2.9)

1 μ0F

∂M

∂t = 1

μ0FK, (2.10)

1 0ω_p²

∂J

∂t + ν

0ω_p²J=E. (2.11)

To complete the model problem (2.7)–(2.11), we assume that the boundary ofΩis per- fect conducting so that

n×E=0 on∂Ω, (2.12)

wherenis the unit outward normal to∂Ω. Furthermore, we assume that the initial conditions are

E(x,0)=E0(x), H(x,0)=H0(x), (2.13)

K(x,0)=K0(x), M(x,0)=M0(x), J(x,0)=J0(x), (2.14) whereE0,H0,K0,M0andJ0are some given functions.

First, we have the following stability for our model problem (2.7)–(2.11).

Lemma 2.1 For the solution (E,H,K,M,J) of the model problem (2.7)–(2.11) with boundary condition (2.12) and initial conditions (2.13)–(2.14), the following stability holds true:

0E(t )²

0+μ0H(t )²

0+ 1

μ0ω₀²FK(t )²

0+ 1

μ0FM(t )²

0+ 1

0ω²_pJ(t )²

0

≤0E0²₀+μ0H0²₀+ 1

μ0ω²₀FK0²₀+ 1

μ0FM0²₀+ 1

0ω²_pJ0²₀. (2.15) Proof Note that the problem (2.7)–(2.11) can be rewritten as

∂

∂t^Au(t )=(B+C)u(t ), (2.16) where we denoteu(t )=(E,H,K,M,J), operators

A=diag

0I, μ0I, 1

μ0ω₀²FI, 1 μ0FI, 1

0ω²_pI

,

C=diag

0,0,− γ

μ0ω²₀FI,0,− ν 0ω²_pI

,

and

B=

⎛

⎜⎜

⎜⎜

⎝

0 ∇× 0 0 −I

−∇× 0 −I 0 0

0 I 0 −_μ¹

0FI 0

0 0 _μ¹

0FI 0 0

I 0 0 0 0

⎞

⎟⎟

⎟⎟

⎠,

hereI denotes a 3×3 identity matrix. Note that the operatorBis anti-symmetric.

To prove the stability, multiplying (2.16) byu, then integrating overΩ, and using the propertyuBu=0, we obtain

d dt

uAu

=uCu≤0,

integrating which with respect totleads to the stability (2.15).

Remark 2.1 The stability (2.15) can be proved in a direct way. Multiplying (2.7)–(2.11) by E,H,K,M,J and integrating overΩ, then adding the resultants together, we obtain

1 2

d dt

0E(t )²

0+μ0H(t )²

0+ 1

μ0ω²₀FK(t )²

0+ 1

μ0FM(t )²

0+ 1

0ω²_pJ(t )²

0

+ ν

0ω_p²J(t )²

0+ γ

μ0ω²₀FK(t )²

0=0,

integrating which from 0 tot leads to (2.15). This is the main reason why we write the last three governing equations (2.9)–(2.11) in this way.

Now, let us prove the existence for our model problem (2.7)–(2.11).

Lemma 2.2 The problem (2.7)–(2.11) has a unique solution (E,H) in H (curl;Ω)⊕ H (curl;Ω).

Proof From ordinary differential equation theory, we can prove that the solutions of (2.5) and (2.6) with zero initial conditions can be expressed as

P(x, t )=0ω²_p ν

t 0

1−e^−ν(t−s)

E(x, s)ds, (2.17) and

M(x, t )=μ0F ω²_{0 t}

0

g(t−s)H(x, s)ds, (2.18) respectively. Here the kernelg(t )=_α¹e^−γ2tsin(αt ), whereα=

ω²₀−(^γ₂)².

Using the definitionJ =^∂P_∂t andK=^∂M_∂t , then substituting (2.17) and (2.18) into (2.7) and (2.8), respectively, we can rewrite (2.7) and (2.8) as follows:

d

dt(AE+K∗E)=LE+F, (2.19)

where we denote_E=(E,H),∗for the convolution product,_F for source terms obtained by transforming a problem with non-zero initial conditions to a problem with zero initial conditions, and operators

A=

0I3 0₃ 0₃ μ0I3

, K=

1I3 0₃ 0₃ μ1I3

, L=

0₃ ∇×

−∇× 0₃

.

In the above,I3is the 3×3 unit matrix, 03is the zero matrix,1=⁰_ν^ω2p(u(t )−e^−νt)and μ1=μ0F ω²₀g(t ). Hereu(t )denotes the unit step function.

Hence our problem (2.19) is a special case of Problem I of [15], whose existence and

uniqueness is guaranteed by Theorem 3.1 of [15].

Finally, we can prove that the Gauss’s Law holds true for our model.

Lemma 2.3 Assume that the initial fields are divergence free:

∇ ·(0E0)=0, ∇ ·(μ0H0)=0, ∇ ·K0=0, ∇ ·M0=0, ∇ ·J0=0, then for any timet >0, the fields are still divergence free, i.e., we have

∇ · 0E(t )

=0, ∇ ·

μ0H(t )

=0, ∇ ·K(t )=0,

∇ ·M(t )=0, ∇ ·J(t )=0.

Proof Taking the divergence of (2.7), we have

∂

∂t

∇ ·(0E)

+ ∇ ·J=0. (2.20) Similarly, taking the divergence of (2.11), we have

∂

∂t(∇ ·J)+ν∇ ·J=0ω_p²∇ ·E. (2.21) Substituting (2.20) into (2.21), we have

∂²

∂t²(∇ ·0E)+ν∂

∂t(∇ ·0E)+ω_p²(∇ ·0E)=0. (2.22) By the assumed initial conditions and (2.20), we have initial conditions for∇ ·0E:

∇ ·0E(0)=0, ∂

∂t(∇ ·0E)(0)= −∇ ·J(0)=0. (2.23) It is easy to see that the ordinary differential equation (2.22) with initial conditions (2.23) only has zero solution, i.e.,∇ ·0E(t )≡0.

Other divergence free conditions can be proved similarly.

3 Fully-Discrete Schemes

In this section, we will develop two fully-discrete finite element methods for solving (2.7)–

(2.11). We assume that the domainΩ is partitioned by a family of regular tetrahedral (or

cubic) meshesT^hwith maximum mesh sizeh. Depending upon the regularity of the solution of (2.7)–(2.11), we can use a proper order Raviart-Thomas-Nédélec (RTN) mixed finite element space ([30,31]): For anyl≥1, on a tetrahedral element, we can choose

Uh=

uh∈H (div;Ω):uh|K∈(p_l−1)³⊕ ˜p_l−1x, ∀K∈T^h , Vh=

vh∈H (curl;Ω):vh|K∈(p_l−1)³⊕Sl,∀K∈T^h , Sl=

p∈(p˜l)³,x· p=0 , while on a cubic element we choose

U_h=

u_h∈H (div;Ω):u_h|K∈Q_l,l−1,l−1×Q_l−1,l,l−1×Q_l−1,l−1,l,∀K∈T^h , V_h=

v_h∈H (curl;Ω):v_h|K∈Q_l−1,l,l×Q_l,l−1,l×Q_l,l,l−1,∀K∈T^h .

Herep˜k denotes the space of homogeneous polynomials of degree k, andQi,j,k denotes the space of polynomials whose degrees are less than or equal toi, j, kin variablesx, y, z, respectively. To impose the boundary condition (2.12), we denoteV⁰_h= {v∈V_h:v×n=0 on∂Ω}. It is easy to see that

∇ ×Vh⊂Uh. (3.1)

To carry out the error analysis below, we need to define two operators. The first one is the standardL²-projection operatorPh: For anyH∈L²(Ω),PhH∈Uhsatisfies

(PhH−H,ψ_h)=0, ∀ψh∈Uh.

Another one is the standard Nédélec interpolation operatorΠhmapped fromH (curl;Ω)to Vh. It is known thatΠhsatisfies the following interpolation error estimate [30,31]:

E−ΠhE0+∇ ×(E−ΠhE)

0≤Ch^lEl,curl, ∀E∈H^l(curl;Ω),1≤l, (3.2) andPhhas the projection error estimate:

H−PhH0≤Ch^lHl, ∀H∈H^l(Ω), 0≤l. (3.3) To define a fully discrete scheme, we divide the time interval (0, T ) intoN uniform subintervals by points 0=t0< t1<· · ·< tN=T, wheretk=kτ,τ=T /N, and denote the k-th subinterval byIk= [tk−1, tk]. Moreover, we defineu^k=u(·, kτ )for 0≤k≤N, and the notation:

δτu^k=

u^k−u^k−1

/τ, u^k=1 2

u^k+u^k−1 .

3.1 The Crank-Nicolson Scheme

We start with a Crank-Nicolson type scheme for solving (2.7)–(2.11): Fork=1,2, . . . , N, findE^k_h∈V⁰_h,J^k_h∈Vh,H^k_h,K^k_h,M^k_h∈Uhsuch that

0

δτE^k_h,φ_h

−

H^k_h,∇ ×φ_h +

J^k_h,φ_h

=0, (3.4)

μ0

δτH^k_h,ψ_h +

∇ ×E^k_h,ψ_h +

K^k_h,ψ_h

=0, (3.5)

1 μ0ω₀²F

δτK^k_h,ψ˜_1h + γ

μ0ω₀²F

K^k_h,ψ˜_1h + 1

μ0F

M^k_h,ψ˜_1h

=

H^k_h,ψ˜_1h , (3.6)

1 μ0F

δ_τM^k_h,ψ˜_2h

= 1 μ0F

K^k_h,ψ˜_2h

, (3.7)

1 0ω²_p

δτJ^k_h,φ˜_h + ν

0ω_p²

J^k_h,φ˜_h

= E^k_h,φ˜_h

, (3.8)

hold true for anyφ_h∈V⁰_h,ψ_h,ψ˜_1h,ψ˜_2h∈Uh,φ˜_h∈Vh, and are subject to the initial ap- proximations

E⁰_h(x)=ΠhE0(x), H⁰_h(x)=PhH0(x),

K⁰_h(x)=PhK0(x), M⁰_h(x)=PhM0(x), J⁰_h(x)=ΠhJ0(x).

First, we can show that our scheme (3.4)–(3.8) satisfies a discrete stability, which has exactly the same form as the continuous case proved in Lemma2.1.

Lemma 3.1 For anyk≥1, we have 0E^k_h²

0+μ0H^k_h²

0+ 1

0ω²_pJ^k_h²

0+ 1

μ0ω²₀FK^k_h²

0+ 1

μ0FM^k_h²

0

≤0E⁰_h²

0+μ0H⁰_h²

0+ 1

0ω²_pJ⁰_h²

0+ 1

μ0ω²₀FK⁰_h²

0+ 1

μ0FM⁰_h²

0. (3.9) Proof Choosingφ_h=τE^k_h,ψ_h=τH^k_h,ψ˜_1h=τK^k_h,ψ˜_2h=τM^k_h,φ˜_h=τJ^k_hin (3.4)–(3.8), respectively, then adding the resultants together, we have

0

2E^k_h²

0−E^k−1_h ²

0

+μ0

2 H^k_h²

0−H^k−1_h ²

0

+ 1

20ω_p²J^k_h²

0−J^k−1_h ²

0

+ τ ν 0ω²_pJ^k_h²

0+ 1

2μ₀ω²₀FK^k_h²

0−K^k−1_h ²

0

+ τ γ

μ0ω₀²FK^k_h²

0

+ 1

2μ0FM^k_h²

0−M^k−1_h ²

0

=0,

which easily leads to (3.9).

For the Crank-Nicolson scheme (3.4)–(3.8), we can prove the following optimal error estimate.

Theorem 3.2 Let (E^m,H^m,K^m,M^m,J^m) and (E^m_h,H^m_h,K^m_h,M^m_h,J^m_h) be the analytic and numerical solutions of (2.7)–(2.11) and (3.4)–(3.8) at timetm, respectively. Under the following regularity assumptions:

E_t,J_t∈L²

0, T;H^l(curl;Ω) ,

E_{t t},J_{t t},H_{t t},K_{t t},M_{t t},∇ ×E_{t t},∇ ×H_{t t}∈L²

0, T;L²(Ω) , E,J,∇ ×E∈L^∞

0, T;H^l(curl;Ω) , H,K,M∈L^∞

0, T;H^l(Ω) ,

there exists a constantC >0 independent of the mesh sizehand time stepτ, such that maxn≥1

0Eⁿ−Eⁿ_h²

0+μ0Hⁿ−Hⁿ_h²

0+ 1

0ω²_pJⁿ−Jⁿ_h²

0

+ 1

μ0ω²₀FKⁿ−Kⁿ_h²

0+ 1

μ0FMⁿ−Mⁿ_h²

0

≤C

h^2l+τ⁴ ,

wherel≥1 is the order of the basis functions in spacesUhandVh.

Proof Integrating (2.7)–(2.11) from t_k−1 to t_k, then multiplying test functions φ_h ∈ V⁰_h,ψ_h,ψ˜_1h,ψ˜_2h∈Uh,φ˜_h∈Vh and integrating the resultant equations over domainΩ, we obtain

0

δ_τE^k,φ_h

− 1

τ

Ik

H,∇ ×φ_h

+ 1

τ

Ik

J,φ_h

=0, (3.10)

μ0

δτH^k,ψ_h +

1 τ

Ik

∇ ×E,ψ_h

+ 1

τ

Ik

K,ψ_h

=0, (3.11)

1 μ0ω²₀F

δτK^k,ψ˜_1h + γ

μ0ω²₀F 1

τ

Ik

K,ψ˜_1h

+ 1 μ0F

1 τ

Ik

M,ψ˜_1h

= 1

τ

Ik

H,ψ˜_1h

, 1

μ0F

δτM^k,ψ˜_2h

= 1 μ0F

1 τ

Ik

K,ψ˜_2h

,

(3.12)

1 0ω²_p

δτJ^k,φ˜_h + ν

0ω²_p 1

τ

Ik

J,φ˜_h

= 1

τ

Ik

E,φ˜_h

. (3.13)

Let us denote

ξ_h^k=ΠhE^k−E^k_h, η^k_h=PhH^k−H^k_h, ξ˜_1h^k =PhK^k−K^k_h, ξ˜_2h^k =PhM^k−M^k_h, ζ_h^k=ΠhJ^k−J^k_h.

Subtracting (3.10)–(3.13) from (3.4)–(3.8) gives us the error equations 0

δτξ_h^k,φ_h

−

η^k_h,∇ ×φ_h +

ζ^k_h,φ_h

=0

δτ

ΠhE^k−E^k ,φ_h

−

PhH^k−1 τ

Ik

H,∇ ×φ_h

+

ΠhJ^k−1 τ

Ik

J,φ_h

, (3.14)

μ0

δτη^k_h,ψ_h +

∇ ×ξ^k_h,ψ_h +ξ˜

k 1h,ψ_h

=μ0

δτ

PhH^k−H^k ,ψ_h

+

∇ ×

ΠhE^k−1 τ

Ik

E

,ψ_h

+

PhK^k−1 τ

Ik

K,ψ_h

, (3.15)

1 μ0ω₀²F

δ_τξ˜_1h^k,ψ˜_1h + γ

μ0ω²₀F

ξ˜^k_1h,ψ˜_1h + 1

μ0F

ξ˜^k_2h,ψ˜_1h

− η^k_h,ψ˜_1h

= 1 μ0ω₀²F

δτ

PhK^k−K^k ,ψ˜_1h

+ γ μ0ω²₀F

PhK^k−1 τ

Ik

K,ψ˜_1h

+ 1 μ0F

PhM^k−1 τ

Ik

M,ψ˜_1h

−

PhH^k−1 τ

Ik

H,ψ˜_1h

, (3.16)

1 μ0F

δτξ˜_2h^k,ψ˜_2h

− 1 μ0F

ξ˜

k 1h,ψ˜_2h

= 1 μ0F

δτ

PhM^k−M^k ,ψ˜_2h

− 1 μ0F

PhK^k−1 τ

Ik

K,ψ˜_2h

, (3.17) 1

0ω²_p

δτζ_h^k,φ˜_h + ν

0ω_p²

ζ^k_h,φ˜_h

− ξ^k_h,φ˜_h

= 1 0ω²_p

δτ

ΠhJ^k−J^k ,φ˜_h

+ ν 0ω²_p

ΠhJ^k−1 τ

Ik

J,φ˜_h

−

ΠhE^k−1 τ

Ik

E,φ˜_h

. (3.18)

Choosingφ_h=τ ξ^k_h,ψ_h=τ η^k_h,ψ˜_1h=τξ˜

k

1h,ψ˜_2h=τξ˜

k

2h,φ˜_h=τ ζ^k_hin (3.14)–(3.18), re- spectively, then adding the resultants together and using the projection property, we have

0

2ξ_h^k2

0−ξ_h^k−12

0

+μ0

2 η^k_h²

0−η_h^k−12

0

+ 1

2₀ω_p²ζ_h^k2

0−ζ_h^k−12

0

+ τ ν 0ω²_pζ^k_h²

0

+ 1

2μ0ω²₀F˜ξ_1h^k2

0−˜ξ_1h^k−12

0

+ τ γ

μ0ω²₀F˜ξ^k_1h²

0+ 1

2μ0F˜ξ_2h^k2

0−˜ξ_2h^k−12

0

=τ 0

δτ

ΠhE^k−E^k , ξ^k_h

−τ

H^k−1 τ

Ik

H,∇ ×ξ^k_h

+τ

ΠhJ^k−1 τ

Ik

J, ξ^k_h

+τ

∇ ×

ΠhE^k−1 τ

Ik

E

, η^k_h

+τ

K^k−1 τ

Ik

K, η^k_h

+τ γ μ0ω²₀F

K^k−1

τ

Ik

K,ξ˜

k 1h

+τ 1

μ0F

M^k−1 τ

Ik

M,ξ˜

k 1h

−τ

H^k−1 τ

Ik

H,ξ˜

k 1h

−τ 1 μ0F

K^k−1

τ

Ik

K,ξ˜

k 2h

+τ 1 0ω_p²

δ_τ

Π_hJ^k−J^k , ζ^k_h

+τ ν 0ω²_p

Π_hJ^k−1 τ

Ik

J, ζ^k_h

−τ

ΠhE^k−1 τ

Ik

E, ζ^k_h

. (3.19)

The rest proof is standard (see our early work [24]) by estimating each term on the right hand side of (3.19), and using the triangle inequality and the estimates (3.2)–(3.3).

3.2 The Leap-Frog Scheme

We construct a leap-frog type scheme for solving (2.7)–(2.11): Fork=1,2, . . ., findE^k_h∈ V⁰_h,J^k+

12

h ∈Vh,H^k+

12

h ,K^k_h,M^k+

12

h ∈Uhsuch that 0

E^k_h−E^k−1_h τ ,φ_h

− H^k−

12

h ,∇ ×φ_h +

J^k−

12 h ,φ_h

=0, (3.20)

μ0

H^k+

1 2 h −H^k−

1 2 h

τ ,ψ_h

+

∇ ×E^k_h,ψ_h +

K^k_h,ψ_h

=0, (3.21)

1 μ0ω²₀F

K^k_h−K^k−1_h τ ,ψ˜_1h

+ γ

μ0ω²₀F

K^k_h+K^k−1_h 2 ,ψ˜_1h

+ 1

μ0F M^k−

12 h ,ψ˜_1h

= H^k−

1 2 h ,ψ˜_1h

, (3.22)

1 μ0F

M^k+

1 2 h −M^k−

1 2 h

τ ,ψ˜_2h

= 1 μ0F

K^k_h,ψ˜_2h ,

1 0ω²_p

J^k+

12 h −J^k−

12 h

τ ,φ˜_h

+ ν 0ω_p²

J^k+

12 h +J^k−

12 h

2 ,φ˜_h

= E^k_h,φ˜_h

, (3.23)

hold true for anyφ_h∈V⁰_h,ψ_h,ψ˜_1h,ψ˜_2h∈U_h,φ˜_h∈V_h, and are subject to the initial ap- proximations

E⁰_h(x)=ΠhE₀(x), H

12

h(x)=Ph

H0(x)−τ 2μ⁻¹₀

∇ ×E0(x)+K0(x) ,

(3.24)

K⁰_h(x)=PhK0(x), M

1 2

h(x)=Ph

M0(x)+τ 2K0(x)

,

J

12

h(x)=Πh

J0(x)+τ 2

0ω_p²E0(x)−νJ0(x) .

(3.25)

First, let us prove the following discrete stability for the leap-frog scheme (3.20)–(3.23).

Lemma 3.3 Denotecv=√μ¹₀₀ for the speed of light in vacuum, and a constantcinv>0 in the standard inverse estimate

∇ ×u_h0≤cinvh⁻¹u_h0, ∀u_h∈V_h. (3.26) Under the time step constraint

τ=min 1

2ω0

√F, 1 2ω0

, 1 2ω_p, h

2c_vc_inv

, (3.27)