Error Estimates for the Interior Penalty DG Method for Maxwell’s Equations in Cold Plasma

Optimal L

Error Estimates for the Interior Penalty DG Method for Maxwell’s Equations in Cold Plasma

Jichun Li^∗

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

Received 1 December 2009; Accepted (in revised version) 15 June 2010 Available online 24 October 2011

Abstract. In this paper, we consider an interior penalty discontinuous Galerkin (DG) method for the time-dependent Maxwell’s equations in cold plasma. In Huang and Li (J. Sci. Comput., 42 (2009), 321–340), for both semi and fully discrete DG schemes, we proved error estimates which are optimal in the energy norm, but sub-optimal in the L²-norm. Here by filling this gap, we show that these schemes are optimally conver- gent in theL²-norm on quasi-uniform tetrahedral meshes if the solution is sufficiently smooth.

AMS subject classifications: 65N30, 35L15, 78-08

Key words: Maxwell’s equations, cold plasma, discontinuous Galerkin method.

1 Introduction

Recently, there is a growing interest in the finite element modeling and analysis of Maxwell’s equations (see books [7, 14, 21] and references cited therein). However, most work are still limited to the simple medium (such as vacuum) case. On the other hand, dispersive media (whose physical parameters are wavelength dependent) are ubiqui- tous. Examples include human tissue, soil, snow, ice, plasma, optical fibers and radar- absorbing materials. Hence the study of how electromagnetic waves interacting with dispersive media becomes an important subject.

Though the original discontinuous Galerkin (DG) method has been known since its introduction in 1973 by Reed and Hill, it was only recently that DG regained its popu- larity in solving various differential equations. It is known that the DG method offers great flexibility in the mesh construction by allowing different types of elements, non- matching grids, and even varying polynomial orders. Due to the imposition of weak continuity across element interfaces, the DG method is easy for parallel implementation.

∗Corresponding author.Email address:[email protected](J. Li)

http://www.global-sci.com/ 319 2012 Global-Science Pressc

A detailed overview of the evolution of the DG methods from 1973 to 1999 is provided by Cockburn et al. [6]. More details and early references on DG can be found in [2, 6].

Some DG methods have been developed for Maxwell’s equations in the simple medium case [4, 5, 8, 9, 13, 15, 22] in the past decade. We like to remark that most of the DG methods are based on writing the Maxwell’s equations in first-order hyperbolic systems; while [9, 15] treated the Maxwell’s equations in second order vector wave equa- tion. Some most recent developments of DG methods for wave problems can be found in the Proceedings of Waves 2009 [3]. However, the study of DG method for Maxwell’s equations in dispersive media is quite limited. In 2004, a time-domain DG method was investigated in [20] for solving the first-order Maxwell’s equations in dispersive media.

In 2009, we [16] initiated the analysis of the interior penalty DG method for Maxwell’s equations in dispersive media. However, the error estimates obtained there is optimal in the energy norm, but sub-optimal in theL²-norm. In this paper, by borrowing many ideas from [9,11,12,15] originally developed for the curl-curl operator, we manage to prove the optimal error estimates in the L²-norm for both semi and fully discrete schemes. Note that our proof is slightly different from [9, 11, 12, 15] by considering that our problem is a differential-integral equation instead of the standard vector wave equation. For sim- plicity, we only consider the cold plasma model here, since analysis of other dispersive media models [17] can be carried out similarly.

By introducingcv= (√_ǫ

0µ0)⁻¹as the speed of wave propagation in vacuum, we can rewrite the governing equation for the isotropic nonmagnetized cold electron plasma model [16, Eq. (1)] as

E_tt+∇× ^c2v∇×^E+ω²_pE−^J(E) =0, in Ω×^{I, (1.1)} whereEis the electric field,ωpis the plasma frequency, andJ is the polarization current density represented as

J(x,t;E)≡^J(E) =νω²_p Z _t

0 e^−ν(t−s)E(x,s)ds, (1.2) hereν≥0 is the electron-neutral collision frequency. Incv, ǫ0 andµ0 represent the per- mittivity and permeability in vacuum, respectively. HereI=(0,T)is a finite time interval andΩis a bounded Lipschitz polyhedron inR³.

Moreover, we assume that the boundary ofΩis a perfect conductor so that

n×^E=0, on ∂Ω×^{I, (1.3)}

wherendenotes the unit outward normal of∂Ω. Furthermore, we assume that the initial conditions for (1.1) are given as

E(x,0) =E₀(x) and E_t(x,0) =E₁(x), (1.4) whereE₀(x)andE₁(x)are some given functions.

The rest of the paper is organized as follows. In Section 2, we describe the semi- discrete DG formulation for the plasma model, and state the optimal error estimate in theL²-norm. Detailed proof of the error estimate is given in Section 2.1. In Section 3, a fully discrete DG scheme is constructed, and the optimal error estimate in theL²-norm.

In this paper, C (sometimes with subindex) denotes a generic constant which is in- dependent of both the time stepτand the mesh sizeh. Here we denoteH^α(^Ω)³ for the standard Sobolev space equipped with the normk·kα,Ω. Whenα=0, we just denotek·k0

for theL²(^Ω)³norm. For a time-dependent solutionu(x,t), we need the Bochner space L^p 0,T;H^α(^Ω)³=ⁿu:(0,T)→^Hα(^Ω)³; ^Z T

0 k^u(·^,t)k^pα,Ωdt¹_p

<_∞

o, 1≤^p<_∞, endowed with norm

k^ukL^p(0,T;H^α(Ω)³)=

Z _T

0 k^u(·^,t)k_α,^pΩdt¹_p .

Whenp=∞, we define the spaceL^∞(0,T;H^α(Ω)³)equipped with norm k^ukL^∞(0,T;H^α(Ω)³)= max

0≤^t≤^Tk^u(·^,t)kH^α(Ω)³.

2 Semi-discrete DG scheme

We consider a shape-regular meshTh that partitions the domainΩinto disjoint tetrahe- dral elements {^K}, such thatΩ=^S_K∈ThK. We denote the diameter ofKby hK, and the mesh sizehbyh=maxK∈ThhK. Furthermore, we denote the set of all interior faces byF_h^I, the set of all boundary faces by F_h^B, and the set of all faces byF_h=F_h^I∪^Fh^B. We want to remark that our optimal L²-norm error estimate is based on a duality argument and in- verse estimate, hence we need to assume that the mesh be quasi-uniform and the domain Ωbe convex.

We assume that the finite element space is given by V_h=ⁿv∈^L2(^Ω)³: v|K∈(Pl(K))³, K∈^Th

o

, l≥^{1, (2.1)}

wherePl(K)denotes the space of polynomials of total degree at mostlonK.

A semi-discrete DG scheme can be formed for (1.1): For anyt∈(0,T), findE^h(·^,t)∈^Vh

such that

(E^h_tt,φ)+ah(E^h,φ)+ω²_p(E^h,φ)−(J(E^h),φ) =0, ∀^φ∈^Vh, (2.2) subject to the initial conditions

E^h|^t=0=^Π₂E₀, E^h_t|^t=0=^Π₂E₁, (2.3)

whereΠ₂denotes the standard L2- projection ontoV_h. Moreover, the bilinear forma_h is defined onV_h×^Vhas

a_h(u,v) =

∑

K∈Th

Z

Kc²_v∇×^u·∇×^vdx−

∑

f∈^Fh

Z

f[[u]]T·{{^c2v∇×^v}}^dA

−

∑

f∈^Fh

Z

f[[v]]T·{{^c2v∇×^u}}^dA+

∑

f∈^Fh

Z

fa[[u]]T·[[v]]TdA.

Here[[v]]_T and{{^v}}are the standard notation for the tangential jumps and averages of vacross an interior face f=∂K⁺∩^∂K−between two neighboring elementsK⁺andK⁻:

[[v]]_T=n⁺×^v++n⁻×^v−, {{^v}}=(v⁺+v⁻)

2 , (2.4)

wherev^±denote the traces ofvfrom withinK^±, andn^±denote the unit outward normal vectors on the boundaries∂K^±, respectively. While on a boundary face f=∂K∩^{∂Ω, we} define[[v]]_T=n×^vand{{^v}}=v. Finally,ais a penalty function, which is defined on each face f∈^Fhas:

a|f=γc²_vh¯⁻¹,

where ¯h|f=min{^hK⁺,h_K−}for an interior face f=∂K⁺∩^{∂K−and ¯h}|f=hKfor a boundary face f =∂K∩^∂Ω. The penalty parameter γ is a positive constant and has to be chosen sufficiently large, which might negatively affect the CFL condition given in (3.6).

Furthermore, we denote the spaceV(h) =H0(curl;Ω)+V_hand define the semi-norm

|^v|²h=

∑

K∈Fh

k^cv∇×^vk²0,K+

∑

f∈F_h

a¹²[[v]]_T²_0,f and the DG energy norm by

k^vk²h=k^ωpvk²0,Ω+|^v|²h.

In order to carry out the error analysis, we introduce an auxiliary bilinear form ˜ahon V(h)×^V(h)defined as [11, 15]

˜

ah(u,v) =

∑

K∈Th

Z

Kc²_v∇×^u·∇×^vdx−

∑

f∈Fh

Z

f[[u]]_T·{{^c2vΠ₂(∇×^v)}}^dA

−

∑

f∈^Fh

Z

f[[v]]_T·{{^c2vΠ₂(∇×^u)}}^dA+

∑

f∈^Fh

Z

fa[[u]]_T·[[v]]_TdA,

where Π₂ is the L2-projection onto V_h. Note that ˜ah equals ah on V_h×^Vh and is well defined onH0(curl;Ω)×^H0(curl;Ω). To prove the error estimate for our DG scheme, we need some basic results developed for the curl-curl operator.

Lemma 2.1. (see[9, Lemma 5])Forγlarger than a positive constantγmin, independent of the local mesh sizes, we have

|^a˜h(u,v)| ≤^Ccont|^u|h|^v|h, a˜h(v,v)≥^Ccoer|^v|²h, u,v∈^V(h), where Ccont=√

2and Ccoer=1/2.

For an elementKand anyu∈(Pl(K))³, we have the standard inverse estimate k∇×^uk0,K≤^Ch−K¹k^uk0,K

and the trace estimate

k^uk0,∂K≤^Ch−K¹²k^uk0,K, which, along with Lemma 2.1, yields the following lemma.

Lemma 2.2. For a quasi-uniform mesh Th, there holds

|^a˜h(u,u)| ≤^Cbh⁻²k^uk²0, u∈^Vh,

where the constant C_b>₀depends on the quasi-uniformity constant of the mesh and polynomial degree l, but is independent of the mesh size h.

For the standardL²-projection, it is known that Lemma 2.3. For anyv∈^Hα(K)³,α≥^{0, K}∈^Th, there holds

k^v−^Π2vk0,K≤^Chmin_K ^{α,l+1}k^vkα,K.

Our proof for the optimal L² error estimates depends on a Galerkin-type projection Π_hintroduced by Grote et al. [11]. Letu∈^Hα(^Ω)³with∇×^u∈^Hα(^Ω)³, forα>_{1/2, then} the projectionΠ_hu∈^Vhis the solution of

˜

a_h(Π_hu−^u,v)+ω²_p(Π_hu−^u,v) =−^rh(u;v), ∀^v∈^Vh. (2.5) Here the operatorrhwas introduced by Houston et al. [15] as follows: For any v∈^V(h) and anyuwith∇×^u∈^Hα(^Ω)³,α>_1/2,

rh(u;v) =

∑

f∈F_h

Z

f[[v]]_T·{{^µ−₀¹∇×^u−^µ−₀^1Π2(∇×^u)}}^{dA. (2.6)} Lemma 2.4. (see[11, Lemma 5.3])For anyu∈^Hα+σE(^Ω)³with∇×^u∈^Hα(^Ω)³forα>_1/2, whereσE∈(1/2,1]is closely related to the regularity properties of the Laplacian in polyhedra[1].

Then we have the optimal projection error

k^u−^Πhuk0≤^{Chmin(α,l)+σE} k^ukα+σE+k∇×^ukα

,

where the constant C>₀is independent of the mesh size h.Furthermore,σE=1whenΩis convex.

Theorem 2.1. Let E andE^h be the solutions of (1.1) and (2.2), respectively. Then under the following regularity assumptions

E,Et∈^L∞(0,T;H^α+σE(^Ω)³), ∇×^E,∇×^Et∈^L∞(0,T;H^α(^Ω)³), forα>1/2, there holds

k^E−^EhkL^∞(0,T;L²(Ω)³)≤^{Chmin(α,l)+σE,}

where l≥¹ is the degree of the polynomial function in the finite element space (2.1), and the constant C>₀is independent of h.

Remark 2.1. For smooth solutions on convex domains (so σE=1), Theorem 2.1 gives optimal error estimate in theL²-norm:

k^E−^EhkL^∞(0,T;L²(Ω)³)≤^Chl+1. 2.1 Proof of Theorem 2.1

For the analytical solutionEof (1.1), using integration by parts and the fact that[[E]]_T=0 across all faces, it is easy to see that [9, pp. 381]:

˜

a_h(E,v) = ∇×(c²_v∇×^E),v

+r_h(E;v), ∀^v∈^V(h). Hence by (1.1), we obtain

(E_tt,v)+a˜_h(E,v)+ω²_p(E,v) = (J(E),v)+r_h(E;v), ∀^v∈^V(h). (2.7) Denote the errore=E−^Eh. Subtracting (2.2) from (2.7), and using the fact that ˜ahcoincides witha_honV_h×^Vh, we can obtain the error equation

(ett,v)+a˜h(e,v)+ω²_p(e,v) = (J(e),v)+rh(E;v), ∀^v∈^Vh, (2.8) which leads to

r_h(E;v)−^a˜h(E−^Eh,v) = (ett,v)+ω²_p(e,v)−(J(e),v), ∀^v∈^V(h). (2.9) Using (2.9) and the definition ofΠ_h, we obtain

((E^h−^ΠhE)tt,v)+a˜h(E^h−^ΠhE,v)

=((E^h−^ΠhE)_tt,v)+a˜h(E−^ΠhE,v)−^a˜h(E−^Eh,v)

=((E^h−^ΠhE)tt,v)−^ω2p(E−^ΠhE,v)+rh(E;v)−^a˜h(E−^Eh,v)

=((E^h−^ΠhE)tt,v)−^ω2p(E−^ΠhE,v)+(ett,v)+ω²_p(e,v)−(J(e),v)

=((E−^ΠhE)tt,v)+ω²_p(^Π_hE−^Eh,v)−(J(e),v). (2.10)

Integration of (2.10) over[0,t]yields

((E^h−^ΠhE)_t,v)−((E^h−^ΠhE)_t(0),v)+

Z _t

0 a˜h(E^h−^ΠhE,v)dt

=((E−^ΠhE)t,v)−((E−^ΠhE)t(0),v)+ω²_p Z _t

0 (^Π_hE−^Eh,v)dt−

Z _t

0(J(e),v)dt. (2.11) Choosingv=E^h−^ΠhEin (2.11) and using the fact that((E−^Eh)t(0),v)=0, we can rewrite (2.11) as

1 2

d

dtk^Eh−^ΠhEk²0+

Z _t

0 a˜_h(E^h−^ΠhE,E^h−^ΠhE)dt

=((E−^ΠhE)_t,E^h−^ΠhE)−^ω2p

Z _t

0(E^h−^ΠhE,E^h−^ΠhE)dt−

Z _t

0(J(e),E^h−^ΠhE)dt.

Integrating the previous identity over[0,t], and using the facts that

˜

ah E^h−^ΠhE,E^h−^ΠhE

≥^{0, Eh}−^ΠhE,E^h−^ΠhE

≥^0, we have

k(E^h−^ΠhE)(t)k²0−k(E^h−^ΠhE)(0)k²0

≤²

Z _t

0

((E−^ΠhE)_t,E^h−^ΠhE)dt−²

Z _t

0

(J(e),E^h−^ΠhE)dt

=Err1+Err2. (2.12)

By Cauchy-Schwarz inequality and Lemma 2.4, we have Err1≤

Z _t

0k(^E−^ΠhE)tk²0dt+ Z _t

0 k^Eh−^ΠhEk²0dt

≤^Ch2(min(α,l)+σ_E) k^Etk²_L2(I;H^α+σE(Ω)³)+k∇×^Etk²_L2(I;H^α(Ω)³)

+ Z t

0 k^Eh−^ΠhEk²0dt. (2.13) By the definition ofJ, we have

k^J(e(t))k²0=ν²ω⁴_p Z

Ω

Z _t

0 e^−ν(t−s)e(s)ds

2dΩ

≤^ν2ω4p

Z

Ω

^Z t

0 e^{−2ν(t−s)}ds^Z t

0 |^e(s)|^2dsdΩ

≤^νω

4p

2 Z _t

0 k^e(s)k²0ds,

using which, along with the Cauchy-Schwarz inequality and Lemma 2.4, we have Err2≤

Z _t

0 k^J(e)k²0dt+

Z _t

0 k^Eh−^ΠhEk²0dt

≤^νω

4pT 2

Z _t

0 k^E−^ΠhE+^Π_hE−^Ehk²0dt+

Z _t

0 k^Eh−^ΠhEk²0dt

≤^νω4pT·^{Ch2(min(α,l)+σE)} k^Ek²L²(I;H^α+σE(Ω)³)+k∇×^Ek²L²(I;H^α(Ω)³)

+(νω⁴_pT+1)

Z _t

0 k^Eh−^ΠhEk²0dt. (2.14)

Substituting (2.13) and (2.14) into (2.12) and using Gronwall inequality, we obtain k(E^h−^ΠhE)(t)k²0≤^Ck(E^h−^ΠhE)(0)k²0+Ch^{2(min(α,l)+σE)},

which, along with the triangle inequality, Lemmas 2.3 and 2.4, and the estimate k(E^h−^ΠhE)(0)k²0≤²(k(E^h−^E)(0)k²0+k(E−^ΠhE)(0)k²0)

≤^{Ch2(min(α,l+1)}k^E0k^α+Ch^{2(min(α,l)+σE)}(k^E0k²α+σE+k∇×^E0k²α), (2.15) which completes the proof.

2.2 Existence and uniqueness

In this subsection, we prove the existence and uniqueness of solutions for both the con- tinuous model (1.1) and the discrete model (2.2).

First, let us consider the discrete model. Choosingφ=E^h_t in (2.2), we obtain 1

2 d

dt k^Ehtk²0+ah(E^h,E^h)+ω²_pk^Ehk²0

−(J(E^h),E^h_t) =0.

Integrating the above, we have

k^Eht(t)k²0+a_h(E^h(t),E^h(t))+ω²_pk^Eh(t)k²0

=k^Eht(0)k²0+a_h(E^h(0),E^h(0))+ω²_pk^Eh(0)k²0+2 Z _t

0 (J(E^h),E^h_t)dt. (2.16) Using the Cauchy-Schwarz inequality and the definition ofJ(E), we have

2 Z _t

0(J(E^h),E^h_t)dt≤

Z _t

0 k^J(E^h(t))k²0dt+

Z _t

0 k^Eh(t)k²0dt

≤

Z _t

0

νω⁴_p 2

Z _t

0 k^Eh(s)k²0dsdt+

Z _t

0 k^Eh(t)k²0dt

≤^νω

4pt 2

Z _t

0 k^Eh(t)k²0dt+

Z _t

0 k^Eh(t)k²0dt.

Substituting the above estimate to (2.16), then using Lemma 2.1 and the discrete Gronwall inequality, we have the following stability

k^Eht(t)k²0+|^Eh(t)|²h+k^Eh(t)k²0≤ k^Eht(0)k²0+|^Eh(0)|²h+k^Eh(0)k²0,

which implies the uniqueness of solution for (2.2). Since (2.2) is a finite dimensional linear system, the uniqueness of solution gives the existence immediately.

Now we prove the existence for (1.1). Taking the Laplace transformation of (1.1) and denoting ˜E(s)as the Laplace transformation ofE(t), we have

s²E˜−^sE(0)−^Et(0)+∇×(c²_v∇×^E˜)+ω²_pE˜−^νω2p

1

s+νE˜ =0, which can be rewritten as

s(s²+νs+ω²_p)E^˜+(s+ν)∇×(c²_v∇×^E˜) =s(s+ν)E(0)+(s+ν)E_t(0). (2.17) The weak formulation of (2.17) can be formulated as: Find ˜E∈^H0(curl;Ω)such that

s(s²+νs+ω²_p)(E^˜,φ)+(s+ν)(c²_v∇×^E˜,∇×^φ) = s(s+ν)E(0)+(s+ν)E_t(0),φ

(2.18) holds true for anyφ∈^H0(curl;Ω). The existence of a unique weak solution ˜Eis guaranteed by the Lax-Milgram lemma. Taking the inverse Laplace transformation of ˜Eleads to the existence of a unique solutionEfor (1.1).

3 Fully discrete DG scheme

To define a fully discrete scheme, we divide the time interval(0,T)intoNuniform subin- tervals by points 0=t0<_t1<···<_tN=T, wheret_k=kτ, and denote thek-th subinterval by I^k= [tk,tk+1]. Moreover, we defineu^k=u(·^,kτ)for 0≤^k≤^Nand denote the second-order central difference operator:

δ²u^k=(u^k+1−^2uk+u^k−1)

τ² .

Now we can formulate a fully explicit scheme for (1.1): For any 1≤ⁿ≤^N−^{1, findEn}_h⁺¹∈^Vh

such that

(δ²_τEⁿ_h,v)+ah(Eⁿ_h,v)+ω²_p(Eⁿ_h,v)−(Jⁿ_h,v) =0, ∀^v∈^Vh (3.1) subject to the initial approximation

E⁰_h=^Π₂E₀, E¹_h=E⁰_h+τΠ₂E₁+^τ

2

2 Π₂E_tt(0), (3.2) where E_tt(0) =−[∇×(c²_v∇×^E0)+ω²_pE₀] is obtained by setting t=0 in the governing equation (1.1). Furthermore,Jⁿ_h is obtained from the following recursive formula

J⁰_h=0, Jⁿ_h=e^−ντJⁿ_h⁻¹+^νω

2p

2 τ(e^−ντEⁿ_h⁻¹+Eⁿ_h), n≥^{1, (3.3)} whereEⁿ_hdenotes the finite element solution ofEat timet=tn.

Lemma 3.1. Let J^k≡^J(E(·^,tk)) be defined by(1.2) and J^k_h be defined by (3.3). Then for any 1≤^k≤^{N,we have}

k^Jk−^Jkhk²0≤^Cτ

k

∑

j=0

k^Ejh−^ΠhE^jk²0+Ch^{2(min(α,l)+σE)} k^Ek²_L^∞(I;H^α+σE(Ω)³)

+k∇×^Ek²L^∞(I;H^α(Ω)³)

+Cτ⁴ Z _tk

0 (k^Ek²0+k^Etk²0+k^Ettk²0)dt, (3.4) where the constant C>₀is independent of both h andτ.

Proof. From [18, Lemma 3.3], we have

|^Jk−^Jkh| ≤^Cτ

k

∑

j=0

|^Ej_h−^Ej|+Cτ² Z _tk

0 (|^E|+|^Et|+|^Ett|)dt, which easily leads to

k^Jk−^Jkhk²0≤^Cτ2

k

∑

j=0

1^{2 k}

∑

j=0

k^E_h^j−^ΠhE^jk²0+k^ΠhE^j−^Ejk²0

+Cτ^4Z t_k

0 1²dt^Z t_k

0 k^Ek²0+k^Etk²0+k^Ettk²0

dt

≤^CTτ

k

∑

j=0

k^Ej_h−^ΠhE^jk²0+CT²h^{2(min(α,l)+σE)} k^Ek²_L^∞_(I;H^α+σE(Ω)³)

+k∇×^Ek²_L^∞(I;H^α(Ω)³)

+CTτ⁴ Z _tk

0 (k^Ek²0+k^Etk²0+k^Ettk²0)dt, (3.5) where we used the fact thatkτ≤^Tand Lemma 2.4.

Theorem 3.1. LetEandEⁿ_h be the solutions of the problem(1.1) and the finite element scheme (3.1)-(3.3)at time t and tⁿ, respectively. Under the CFL condition

τ<_2h q

Cb+ω²_ph²₋1

, (3.6)

where C_bis the constant of Lemma 2.2. Furthermore, we assume that

E,Et∈^L∞(0,T;H^α+σE(^Ω)³), ∇×^E,∇×^Et∈^L∞(0,T;H^α(^Ω)³), E_t,E_t²,E_t³∈^L∞(0,T;L²(^Ω)³), E_t4∈^L2(0,T;L²(^Ω)³).

Then there is a constant C>_0,independent of both the time stepτand mesh size h,such that

1max≤n≤Nk^Enh−^Enk⁰≤^C(τ²+h^{min(α,l)+σE}), l≥^1.

Remark 3.1. For smooth solutions on convex domain, we have the optimalL²error esti- mate:

1max≤n≤Nk^Enh−^Enk0≤^C(τ²+h^l+1), l≥^1.

3.1 Proof of Theorem 3.1 Denote the error

eⁿ=Eⁿ−^Enh= (Eⁿ−^ΠhEⁿ)+(^Π_hEⁿ−^Enh) =ηⁿ+φⁿ, whereΠ_hEis the Galerkin projection ofEdefined by (2.5).

Using the equivalence ofa_hand ˜a_honV_h×^Vh, and subtracting (3.1) from (2.7) att=tn, we obtain

(Eⁿ_tt−^δ2ΠhEⁿ+δ²Π_hEⁿ−^δ2Enh,v)+a˜h(Eⁿ−^ΠhEⁿ+^Π_hEⁿ−^Enh,v) +ω²_p(Eⁿ−^ΠhEⁿ+^Π_hEⁿ−^Enh,v)−(Jⁿ−^Jnh,v) =rh(Eⁿ;v), from which and the definition of the Galerkin projection (2.5), we have

(δ²φⁿ,v)+a˜h(φⁿ,v)+ω²_p(φⁿ,v) = (rⁿ,v)+(Jⁿ−^Jnh,v), ∀^v∈^Vh, (3.7) where we denote

rⁿ=δ²(^Π_hEⁿ)−^Entt, 1≤ⁿ≤^N−^1.

Summing (3.7) fromn=1 ton=m, 1≤^m≤^N−1, and multiplying the resultant byτ, we have

φ^m+1−^φm τ ,v

−^φ

1−^φ0 τ ,v

+τ

m

∑

n=1

˜

a_h(φⁿ,v)+τω²_p

m

∑

n=1

(φⁿ,v)

=τ

m

∑

n=1

(rⁿ,v)+τ

m

∑

n=1

(Jⁿ−^Jnh,v), which can be rewritten as

φ^m+1−^φm τ ,v

+a˜h(^Φm,v)+ω²_p(^Φm,v) = (R^m,v)+τ

m

∑

n=1

(Jⁿ−^Jnh,v), (3.8) where we introduced notations

Φ⁰=0, Φ^m=τ

m

∑

n=1

φⁿ, R^m=τ

m

∑

n=0

rⁿ, r⁰=(φ¹−^φ0)

τ² . (3.9)

Choosingv=τ(φ^m+φ^m+1)∈^Vhin (3.8), and summing the resultant fromm=0 tom=n−^1, 1≤ⁿ≤^{N, we have}

k^φnk²0−k^φ0k²0+τ

n−¹

∑

m=0

˜

ah(^Φm,φ^m+φ^m+1)+τω²_p

n−¹

∑

m=0

(^Φm,φ^m+φ^m+1)

=τ

n−1

∑

m=0

(R^m,φ^m+φ^m+1)+τ²

n−1

∑

m=0 m

∑

k=1

(J^k−^Jkh,φ^m+φ^m+1). (3.10)