IMPLICIT RUNGE–KUTTA METHODS AND DISCONTINUOUSGALERKIN DISCRETIZATIONS FOR LINEAR MAXWELL’S

EQUATIONS

MARLIS HOCHBRUCK, AND TOMISLAV PAZUR∗

Abstract. In this paper we consider implicit Runge–Kutta methods for the time-integration oflinear Maxwell’s equations. We first present error bounds for the abstract Cauchy problem whichrespect the unboundedness of the differential operators using energy techniques. The error boundshold for algebraically stable and coercive methods such as Gauß and Radau collocation methods.The results for the abstract evolution equation are then combined with a discontinuous Galerkindiscretization in space using upwind fluxes. For the case that permeability and permittivity arepiecewise constant functions, we show error bounds for the full discretization, where the constantsdo not deteriorate if the spatial mesh width tends to zero.

Key words. implicit Runge–Kutta methods, time integration, discontinuous Galerkin finiteelements, error analysis, evolution equations, Maxwell’s equations

AMS subject classifications. Primary: 65M12, 65M15. Secondary: 65M60, 65J10.

1. Introduction. In recent years, there has been a great interest in solving theMaxwell’s equations numerically using discontinuous Galerkin (dG) finite elementmethods for the spatial discretization, see the recent textbooks [6, 16]. This approachis particularly attractive if one is interested in the simulation of wave propagation incomposite materials, where the electric permittivity and the magnetic permeabilityare discontinuous. For the full discretization, discontinuous Galerkin methods haveto be supplemented with suitable time integration schemes. Explicit time integratorscan exploit the block diagonal structure of the mass matrix of discontinuous Galerkinschemes and thus lead to fully explicit schemes. The RKDG methods of [5] achievehigh-order convergence both in space and time by using a strong stability preservingRunge-Kutta schemes in time. We refer to [10] for more details on SSP-RK methods.

On the other hand, explicit methods suffer from step size restrictions due tostability requirements (CFL condition). Note that the CFL condition restricts thetime step size no matter how many small elements appear in the grid, i.e., for uniformlyrefined grids as well as for locally refined grids. However, for the latter, local timestepping methods are a very attractive alternative, see, e.g., [11, 19, 27, 28].

In contrast, implicit methods can be used with large time steps at the cost ofsolving linear or, in general, even nonlinear systems. Since this additional computa-tional effort only pays off if the time step size can be chosen significantly larger thanfor explicit methods, the advantage becomes more significant if the spatial grid issufficiently fine. However, the efficiency of implicit methods strongly depends on theavailability of fast linear solvers. For elliptic problems, it is well known that multigridpreconditioning allows to solve the linear systems in linear complexity. Such resultsare not yet available for the linear systems arising in hyperbolic problems. Never-theless, we showed in [17] that even with a standard parallel implementation whichwas not optimized for the particular application, implicit time integration of linearMaxwell’s equations outperforms explicit methods on a rather large test problem. Fordetails on the implementation we refer to [17, Sec. 5] and for details on the numericalexperiments we refer to [17, Sec. 6].

∗ Department of Mathematics, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany.marlis.hochbruck@kit.edu, tomislav.pazur@kit.edu, version of October 22, 2014

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Note that in the case that only a small part of the mesh contains small elements,then fast linear solvers are available by using precondioners which essentially solvethe problem on the fine grid only. We are thus convinced that implicit methods havea great potential for solving wave type problems numerically.

It is the aim of our paper to analyze the full discretization error of linear Maxwell’sequations with discontinuous Galerkin methods with upwind fluxes in space and im-plicit Runge–Kutta methods in time. For details on an extension to central fluxes werefer to [24]. Moreover, our analysis easily carries over to H(curl)-conforming finiteelement discretizations using Nedelec elements, cf. [22]. In fact, the situation for con-forming methods is much simpler than for discontinuous Galerkin methods, since thediscrete and the continuous bilinear forms coincide.

In a different framework and for more general first-order systems, error boundsfor explicit Runge–Kutta methods of order two and three have been proven in [4].For constant permittivity and permeability, an optimal convergence rate of k+ 1/2 inspace, where k denotes the degree of polynomials in the dG method, and s in time,where s denotes the number of stages of the explicit Runge-Kutta method, was shown.For dG methods with central fluxes and the leap-frog method, error bounds of orderk in space and order two in time have been shown in [9]. Also in the context of dGmethods with central fluxes, a locally-implicit time integration method was suggestedin [27] and analyzed in [7] and [23].

Implicit time integration for linear, abstract initial value problems

∂tu(t) +Au(t) = f(t), u(0) = u0, (1.1)

where the operator A is a generator of a bounded C0 semigroup, cf. [8, 25], has beenstudied in [1] generalizing earlier work in [2, 3] for the homogeneous case f ≡ 0.The elegant proofs are based on a Hille-Phillips operational calculus using Laplacetransformations. The results shown in these papers can be applied to our situationand then yield convergence of order s+ 1 in time for s-stage collocation methods, inparticular for Gauß and Radau methods. The error estimates for the full discretizationcan also be generalized to the application considered here. Using an elliptic projectionyields the same order of convergence as our analysis but requires more regularity ofthe solution while using an L2-projection leads to an order reduction in space.

To the best of our knowledge, the analysis used in [1, 2, 3] does not generalize tononlinear problems. Therefore, in this paper, we follow a different approach using en-ergy techniques, which were motivated by the analysis presented in [20] for quasilinearparabolic problems and L-stable Runge–Kutta methods and [21] for Gauß–Runge–Kutta time discretizations of wave equations on evolving surfaces. The assumptionsin [20] exclude Gauß collocation methods, which are particularly interesting for hy-perbolic problems. Our analysis applies to implicit Runge–Kutta methods which arealgebraically stable and coercive (see, [14, Sections IV.12 and IV.14]) and Section 3.1below). The analysis presented here for linear Maxwell’s equations will be a firststep to prove error bounds for more general problems such as quasilinear Maxwell’sequations, acoustic and elastic wave equations, etc. These results will be reportedelsewhere.

Our paper is organized as follows: In Section 2 we provide the analytical frame-work for Maxwell’s equations for linear isotropic materials with perfectly conductingboundary conditions.

In Section 3, we present error estimates for the time discretization of the abstractinitial value problem by s-stage implicit Runge–Kutta methods using an energy tech-nique motivated by [20]. We show full temporal order of convergence for the implicit

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Euler method and for the implicit midpoint rule while in general, the temporal orderwill suffer from order reduction to order s + 1. It is well known that full temporalorder will not be achieved for stiff problems without additional regularity assump-tions, which are often unnatural in the context of time-dependent partial differentialequations, cf. [1]. Our error bounds depend in an explicit form on the regularity ofthe solution. We also discuss the relation to the earlier work [1] showing the sameresult with a different technique.

Section 4 deals with the discontinuous Galerkin approximation of Maxwell’s equa-tions using upwind fluxes. It is known that the optimal convergence rate of dG meth-ods applied to first order hyperbolic equations is k+1/2 when polynomials of degree kare used [18, 26]. Upwind fluxes for Maxwell’s equation are considered in [15] (proofof order k) and [29] (proof of order k + 1/2 for dispersive media). For Maxwell’sequations with smooth coefficients written as second order PDE system and interiorpenalty dG method, order k+ 1 was proved [12, 13]. We present a convergence resultfor the error of the semidiscretization in a form required for the analysis of the fulldiscretization.

Section 5 contains error bounds for the full discretization for the implicit Eulerscheme and for general higher order implicit Runge–Kutta methods by generalizing theresults from Section 3 to the discrete Maxwell operator. For the sake of readability,some technical details are postponed to the appendix. We also show that for thehomogeneous Maxwell equations, the divergence is conserved in a weak sense, see also[9] for a related result for explicit Runge–Kutta methods.

2. Analytic framework. Let Ω be an open, bounded, Lipschitz domain in Rd,and let T > 0 be a finite time. We consider Maxwell’s equations for linear isotropicmaterials with perfectly conducting boundary conditons:

µ∂tH +∇×E = 0, Ω× (0, T ),

ε∂tE−∇×H = 0, Ω× (0, T ),

∇·(εE) = 0, ∇·(µH) = 0, Ω× (0, T ),

n×E = 0, n · (µH) = 0, ∂Ω× (0, T ),

E(0) = E0, H(0) = H0, Ω.

(2.1)

The differential operators and boundary conditions are understood in the sense ofdistributions and traces, respectively. For the coefficients we assume that

µ, ε ∈ L∞(Ω), ε, µ ≥ δ > 0. (2.2)

The Hilbert space V := L2(Ω)3 × L2(Ω)3 will be equipped with the weighted scalarproduct

(

(H1

E1

),

(H2

E2

))V =

∫Ω

µH1 ·H2 + εE1 ·E2. (2.3)

We define

V0 :=

(H,E) ∈ L2(Ω)6| ∇·(εE) = 0, ∇·(µH) = 0, n · (µH) = 0⊂ V.

The subspace V0 is closed in V because of the closedness of ∇· and the continuity ofthe normal trace since the differential operators are defined in distributional sense.The Maxwell operator

A

(HE

):=

(µ−1∇×E−ε−1∇×H

)(2.4)

3

is defined on its domain D(A) = H(curl,Ω) × H0(curl,Ω) and the graph norm isdenoted by

‖v‖D(A) := (‖v‖2V + ‖Av‖2V )1/2. (2.5)

With respect to the weighted inner product (2.3), the Maxwell operator satisfies thefollowing more general assumption for V and (·, ·)V given above:

Assumption 2.1. Let V be a Hilbert space with inner product (·, ·)V . We assumethat the linear operator A : D(A)→ V is skew adjoint, in particular

(Av,w)V = −(v,Aw)V , for all v, w ∈ D(A). (2.6)

In the following, we consider the abstract initial value problem

∂tu(t) +Au(t) = 0, u(0) =

(H0

E0

)(2.7)

for an operator A satisfying Assumption 2.1. Since A generates a unitary C0-group,‖u(t)‖V = ‖u(0)‖V . For the Maxwell operator, ‖·‖V corresponds to the electromag-netic energy, which is thus preserved.

Moreover, by Stone’s theorem [25, Theorem 1.10.8.], for (H0,E0)T ∈ D(A) ∩ V0

a unique solution u ∈ C1(R;V0) ∩ C(R;D(A) ∩ V0) of (2.7) exists. Therefore it issufficient that the initial values E0 and H0 satisfy the divergence condition and thatH0 satisfies the boundary condition to ensure that these conditions are valid for E(t)and H(t) for all t ∈ [0, T ].

More generally, we consider the inhomogeneous problem

∂tu(t) +Au(t) = f(t), u(0) =

(H0

E0

)(2.8)

for f ∈ C(0, T ;V ), see also [4].

3. Time discretization by implicit Runge–Kutta methods. We start bydiscretizing the abstract Cauchy problem (2.8) in time using implicit s-stage Runge–Kutta methods. This yields approximations Uni ≈ u(tn + ciτ) and un+1 ≈ u(tn+1)defined by

Uni +AUni = fni, fni = f(tn + ciτ), i = 1, . . . s,

Uni = un + τ

s∑j=1

aijUnj , i = 1, . . . s, (3.1)

un+1 = un + τ

s∑i=1

biUni.

3.1. Algebraically stable and coercive Runge–Kutta methods. Recallthat a Runge–Kutta method with s distinct nodes 0 ≤ ci ≤ 1 and weights Oι =(aij)

si,j=1 and b = (bi)

si=1 is called algebraically stable [14, Definition IV.12.5], if bi ≥ 0

for i = 1, . . . , s and

M = (mij)si,j=1, with mij = biaij + bjaji − bibj (3.2)

4

is positive semidefinite. It is well known that Gauß, Radau IA (c1 = 0), and Radau IIA(cs = 1) collocation methods are algebraically stable, cf. [14, Theorem IV.12.9].

For our analysis, we also need the coercivity condition that there exists a diagonal,positive definite matrix D ∈ Rs,s and a positive scalar α such that

uTDOι−1u ≥ αuTDu for all u ∈ Rs. (3.3)

This condition also plays an important role in proving the existence of Runge–Kuttaapproximations, cf. [14, Section IV.14]. For Gauß collocation methods, (3.3) is sat-isfied for D = B(C−1 − I), where B := diag(b1, . . . , bs)) and C := diag(c1, . . . , cs).For Radau IA and Radau IIA collocation methods it is satisfied for D = B(I − C)and D = BC−1, respectively. For Gauß and Radau methods, the constant α is givenexplicitly in terms of the nodes ci, i = 1, . . . , s, see [14, Theorem IV.14.5].

Remark 3.1. For A-stable collocation methods such as Gauß- and Radau meth-ods, a unique solution of the linear system defining the interior Runge-Kutta approx-imations Uni, i = 1, . . . , s exists because A is skew-adjoint. This can be easily seenusing the coercivity condition (3.3).

3.2. Error bounds. We will present error bounds for algebraically stableRunge–Kutta methods satisfying the coercivity condition (3.3).

Defects. We start by inserting the exact solution of (2.8) into the numericalscheme using the notation

un = u(tn), Uni = u(tn + ciτ), ˙Uni = u′(tn + ciτ).

This yields

˙Uni +AUni = fni,

Uni = un + τ

s∑j=1

aij˙Unj + ∆ni,

un+1 = un + τ

s∑i=1

bi˙Uni + δn+1.

(3.4)

For Gauss collocation methods with s ≥ 1, Radau collocation methods with s ≥ 2and u(s+1) ∈ L2(0, T ;D(A)), u(s+2) ∈ L2(0, T ;V ) the defects are given by

∆ni = τstn+1∫tn

u(s+1)(t)κi

(t− tnτ

)dt = O(τs+1), (3.5a)

δn+1 = τs+1

tn+1∫tn

u(s+2)(t)κ

(t− tnτ

)dt = O(τs+2). (3.5b)

Here κi and κ denote the Peano kernels corresponding to the quadrature rules definingthe Runge–Kutta method. They are uniformly bounded with constants depending onthe Runge–Kutta coefficients only. Hence we have

τ

N∑n=1

(s∑i=1

∥∥∆ni∥∥2

D(A)+

∥∥∥∥1

τδn+1

∥∥∥∥2

V

)≤ Cτ2(s+1)B(u, s, T ), (3.6)

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where

B(u, s, T ) =

∫ T

0

∥∥∥u(s+1)(t)∥∥∥2

D(A)dt+

∫ T

0

∥∥∥u(s+2)(t)∥∥∥2

Vdt. (3.7)

Remark 3.2. The order of the defect δn+1 is not sharp, if the solution is moreregular. More precisely, for Gauß collocation methods and u(2s+1) ∈ L2(0, T, V ) wehave δn+1 = O(τ2s+1), while for Radau collocation methods and u(2s) ∈ L2(0, T, V )we have δn+1 = O(τ2s).

However, we cannot exploit this in our convergence analysis, since the global orderis determined by the stage order, which is s for all collocation methods.

By subtracting (3.4) from (3.1), the time integration errors

en := un − un, Eni := Uni − Uni

satisfy

Eni +AEni = 0, i = 1, . . . s, (3.8a)

Eni = en + τ

s∑j=1

aijEnj −∆ni, i = 1, . . . s, (3.8b)

en+1 = en + τ

s∑i=1

biEni − δn+1. (3.8c)

Let ∆n =(∆n1 . . .∆ns

)T, En =

(En1 . . . Ens

)T, En =

(En1 . . . Ens

)T. Then, (3.8)

can be written in a more compact form as

En + (I ⊗A)En = 0,

En = 1l⊗ en + τ(Oι⊗ I)En −∆n,

en+1 = en + τ(bT ⊗ I)En − δn+1,

(3.9)

where 1l = [1 . . . 1]T .

Energy techniques. The following analysis uses an energy technique motivatedby [20].

Lemma 3.3. The error en = un − u(tn) satisfies

∥∥en+1∥∥2

V− ‖en‖2V ≤

C

T + 1τ

(‖en‖2V +

s∑i=1

∥∥Eni∥∥2

V

)

+ C(T + 1)τ

(s∑i=1

∥∥∆ni∥∥2

D(A)+

∥∥∥∥1

τδn+1

∥∥∥∥2

V

).

(3.10)

Here, the constant C only depends on Oι, b, and s.Proof. Taking the inner product of (3.8c) with itself we obtain

∥∥en+1∥∥2

V=

∥∥∥∥∥en + τ

s∑i=1

biEni

∥∥∥∥∥2

V

− 2(δn+1, en + τ

s∑i=1

biEni)V +

∥∥δn+1∥∥2

V. (3.11)

6

We estimate each of these three terms separately. For the first term we have∥∥∥∥∥en + τ

s∑i=1

biEni

∥∥∥∥∥2

V

= ‖en‖2V + 2τ

s∑i=1

bi(en, Eni)V + τ2

s∑i,j=1

bibj(Eni, Enj)V (3.12)

Using (3.8b), we write en as

en = Eni − τs∑j=1

aijEnj + ∆ni.

Inserting this identity into the second term of (3.12) we obtain∥∥∥∥∥en + τ

s∑i=1

biEni

∥∥∥∥∥2

V

= ‖en‖2V + 2τ

s∑i=1

bi(Eni + ∆ni, Eni)V

+ τ2s∑

i,j=1

(bibj − biaij + bjaji)(Eni, Enj)V .

Since the method is algebraically stable, the last term is not positive and we end upwith ∥∥∥∥∥en + τ

s∑i=1

biEni

∥∥∥∥∥2

V

≤ ‖en‖2V + 2τ

s∑i=1

bi(Eni, Eni + ∆ni)V . (3.13)

The skew-symmetry (2.6) of the operator A implies

(Eni, Eni)V = −(AEni, Eni)V = 0,

(Eni,∆ni)V = −(AEni,∆ni)V = (Eni, A∆ni)V ≤∥∥Eni∥∥

V

∥∥A∆ni∥∥V.

A∆ni is well defined because of u(s+1) ∈ L2(0, T ;D(A)). For arbitrary γ > 0, Young’sinequality gives∥∥∥∥∥en + τ

s∑i=1

biEni

∥∥∥∥∥2

V

≤ ‖en‖2V + τ

s∑i=1

bi

(1

γ

∥∥Eni∥∥2

V+ γ

∥∥A∆ni∥∥V

). (3.14)

To bound the second term in (3.11) first observe that

(δn+1, en)V ≤ τ∥∥∥∥1

τδn+1

∥∥∥∥V

‖en‖V ≤1

2γτ ‖en‖2V +

γ

2τ

∥∥∥∥1

τδn+1

∥∥∥∥2

V

. (3.15)

In order to bound (δn+1, Eni)V we use the compact form (3.9). From the secondequation of (3.9) we have

En =1

τ(Oι−1 ⊗ I)(En + ∆n − 1l⊗ en).

If we denote the inverse of the Runge–Kutta matrix by

Oι−1 = (ωij)i,j , (3.16)

7

then

Eni =1

τ

s∑j=1

ωij(Enj + ∆nj − en). (3.17)

Hence we have

τ(δn+1, Eni)V ≤ τ∥∥∥∥1

τδn+1

∥∥∥∥V

s∑j=1

|ωij |(‖en‖V +

∥∥Enj∥∥V

+∥∥∆nj

∥∥V

)≤ γ

2τ

∥∥∥∥1

τδn+1

∥∥∥∥2

V

+C

γτ(‖en‖2V +

s∑j=1

∥∥Enj∥∥2

V+

s∑j=1

∥∥∆nj∥∥2

V

),

where C = C(Oι, b, s). Since the right-hand side does not depend on i and∑i bi = 1,

bi ≥ 0, we conclude from (3.15) that

(δn+1, en + τ

s∑i=1

biEni)V ≤

C

γτ(‖en‖2V +

s∑j=1

∥∥Enj∥∥2

V+

s∑j=1

∥∥∆nj∥∥2

V

)+ γτ

∥∥∥∥1

τδn+1

∥∥∥∥2

V

.

(3.18)

Inserting (3.14) and (3.18) for γ = 1 + T into (3.11), and writing∥∥δn+1

∥∥2

V=

τ2∥∥δn+1/τ

∥∥2

Vfinally proves the Lemma.

In order to use the Gronwall inequality, we have to bound∥∥Eni∥∥

Vin terms of

‖en‖V and the defects.Lemma 3.4. The error of the inner stages satisfies

s∑i=1

∥∥Eni∥∥2

V≤ C

(‖en‖2V +

s∑i=1

∥∥∆ni∥∥2

V

), (3.19)

where, C = C(Oι, s,D, α).Proof. From (3.9) we conclude

En = 1l⊗ en − τ(Oι⊗A)En −∆n.

Multiplying by DOι−1 ⊗ I, where D = diag(d1, . . . , ds) is the diagonal matrix arisingin the coercivity property (3.3), and taking the inner product with En yields

(En, (DOι−1⊗ I)En)V s = −τ(En, (D⊗A)En)V s + (En, (DOι−1⊗ I)(1l⊗ en−∆n))V s .

The coercivity implies the lower bound

(En, (DOι−1 ⊗ I)En)V s ≥ αs∑i=1

di∥∥Eni∥∥2

V.

Note that

(En, (D ⊗A)En)V s =

s∑i=1

di(Eni, AEni)V = 0

8

since A is skew-symmetric, cf. (2.6). Using the notation (3.16) for the entries of Oι−1

we obtain

(En, (DOι−1 ⊗ I)(1l⊗ en −∆n))V s =

s∑i,j=1

diωij(Eni, en −∆nj)V

≤ γs∑i=1

di∥∥Eni∥∥2

V+C

γ

(‖en‖2V +

s∑i=1

∥∥∆ni∥∥2

V

).

(3.20)

Choosing γ = α2 completes the proof.

Main result (time discretization error). From the bounds on the defectsand the energy technique we are now able to state and prove our main result for theerror of the time integration scheme.

Theorem 3.5. Let u be the solution of (2.8) and assume that u(s+1) ∈L2(0, T ;D(A)) and u(s+2) ∈ L2(0, T ;V ). Then the error of an s-stage algebraicallystable and coercive Runge–Kutta method of order at least two satisfies

‖eN‖V ≤ C(T + 1)1/2τs+1B(u, s, T )1/2,

where B is defined in (3.7). The constant C = C(Oι, b) is independent of u.Proof. Inserting (3.19) into (3.10) we get

∥∥en+1∥∥2

V− ‖en‖2V ≤

Cτ

T + 1‖en‖2V + C(T + 1)τ

(s∑i=1

∥∥∆ni∥∥2

D(A)+

∥∥∥∥1

τδn+1

∥∥∥∥2

V

).

Applying a discrete Gronwall lemma in differential form, we have for τ sufficientlysmall

‖eN‖2V ≤ CeN C

T+1 τ (T + 1)τ

N∑n=1

(s∑i=1

∥∥∆ni∥∥2

D(A)+

∥∥∥∥1

τδn+1

∥∥∥∥2

V

)The assumptions on the regularity of u and the bound (3.6) for the defects prove thedesired result.

Remark 3.6. In [1, Theorem 1] (with order and strict order s+1), the followingclosely related result is proved in a completely different way:

‖eN‖V ≤ Cτs+1

(∫ T

0

∥∥∥u(s+1)(t)∥∥∥D(A)

dt+

∫ T

0

∥∥∥u(s+2)(t)∥∥∥Vdt

). (3.21)

We nevertheless presented our proof here since it will provide the basis for our analysisof the fully discrete problem.

Remark 3.7. For the implicit Euler method (Radau IIA for s = 1), the followingconvergence result can be shown with a simple proof∥∥eN∥∥

V≤ C(T + 1)1/2τ

(∫ T

0

‖u′′(t)‖2V dt)1/2

.

Details of the proof can be found in Section 5.1 for the fully discrete case.Remark 3.8. For the Maxwell operator (2.4) and f ≡ 0, the Runge–Kutta

solution preserves the divergence, i.e,

∇ · (εEn) = ∇ · (εE0), ∇ · (µHn) = ∇ · (µH0), n = 1, 2, . . . .

9

4. Spatial discretization of Maxwell’s equations. We discretize (2.8) inspace using a discontinuous Galerkin (dG) method, see, e.g., [6, 16].

4.1. Discontinous Galerkin approximations. For the purpose of presenta-tion, we restrict ourselves to simplicial meshes. However, all our results hold for moregeneral meshes as well.

We use the following notation: by Pk we denote the set of all multivariate poly-nomials of degree at most k. Th = K is a simplicial mesh of a polyhedron Ω ⊂ Rdconsisting of elements K, i.e., Ω =

⋃K. hK denotes the diameter of K and rK de-

notes the radius of the largest ball inscribed in K. The index h refers to the maximumdiameter of all elements of Th. The set of faces is denoted by Fh = F int

h ∪Fexth , where

F inth and Fext

h consist of all interior and all exterior faces, respectively. If F ∈ F inth is

an interior face, then there exists a neighboring element KF of K with ∂K∩∂KF = F .By nF we denote the unit normal of a face F , where the orientation of nF is fixedonce and forever for each inner face. For a boundary face F , nF is an outward normalvector. With vK := vh|K we denote the restriction of a function vh to an element K.Jumps of vh on an interior face F with normal vector nF pointing from K to KF aredefined as

JvhKF := (vKF)|F − (vK)|F .

Note that the sign of the jump on face F is fixed by the direction of the normal vectornF . The discontinuous Galerkin space w.r.t. the mesh Th is defined as the space ofpiecewise polynomials with fixed polynomial degree k:

Vh =vh ∈ L2(Ω) | vh|K ∈ Pk(K)

6. (4.1)

In general, Vh 6⊂ D(A), so that the method is nonconforming.As in [6, Def. 1.38], the following properties of the mesh are required:Assumption 4.1. We suppose shape and contact regularity of the mesh, which

means that there exists a constant ρ > 0, such that all elements K and their neigh-boring elements KF satisfy

hKF

hK≥ ρ, rK

hK≥ ρ. (4.2)

Assumption 4.2. We suppose that µK := µ|K and εK := ε|K are constant foreach K ∈ Th.

By πh : V → Vh we denote the L2-orthogonal projection on Vh which satisfies

(vh, u− πhu)0,Ω = 0 for all u ∈ V, vh ∈ Vh. (4.3)

Note that for piecewise constant coefficients, we have

(vh, u− πhu)V = 0 for all u ∈ V, vh ∈ Vh. (4.4)

Additionally, we use broken Sobolev spaces

Hq(Th) :=v ∈ L2(Ω) | v|K ∈ Hq(K), ∀K ∈ Th

, q ∈ N, (4.5)

which are Hilbert spaces with seminorm and norm defined via

|u|2Hq(Th) :=∑K∈Th

|u|2Hq(K) , ‖u‖2Hq(Th) :=

q∑j=0

|u|2Hq(Th) ,

10

respectively.After discretization by a discontinuous Galerkin method with upwind fluxes, we

end up with the space semidiscrete problem

∂tuh +Ahuh = fh, uh(0) = πhu0, (4.6)

where uh ∈ C1(0, T ;Vh) is the semidiscrete solution and fh = πhf . Given uh =(Hh,Eh) and wh = (φh, ψh) ∈ Vh, the dG operator Ah : Vh → Vh is given as

(Ahuh, wh)V :=∑K

((∇×Eh, φh)0,K − (∇×Hh, ψh)0,K

)+

∑F∈F int

h

((nF × JEhKF , αKφK + αKF

φKF)0,F

− (nF × JHhKF , βKψK + βKFψKF

)0,F (4.7)

+ γF (nF × JEhKF , nF × JψhKF )0,F

+ δF (nF × JHhKF , nF × JφhKF )0,F

)+

∑F∈Fext

h

(− (n×Eh, φh)0,F + 2γF (n×Eh, n× ψh)0,F

).

If cK = (εKµK)−1/2 denotes the speed of light in the media of element K, then

αK,F =cKF

εKF

cKFεKF

+ cKεK=

1

1 +(εKµKF

µKεKF

)1/2,

βK,F =cKF

µKF

cKFµKF

+ cKµK=

1

1 +(µKεKF

εKµKF

)1/2,

γF =1

cKFµKF

+ cKµK, δF =

1

cKFεKF

+ cKεK.

Note that

αK,F + αKF ,F = 1, βK,F + βKF ,F = 1, αK,F = βKF ,F . (4.8)

By (4.7), Ah is also well defined as an operator from Vh +(D(A) ∩H1(Th)6

)to

Vh. This allows to show the following consistency property.Lemma 4.3. For u ∈ D(A) ∩H1(Th)6 we have

Ahu = πhAu. (4.9)

Proof. For u = (H,E) ∈ D(A) ∩H1(Th)6 we have nF × JEKF = nF × JHKF = 0for F ∈ F int

h and nF × E = 0 for F ∈ Fexth ⊂ ∂Ω. Hence the sum over the faces

vanishes and for all wh = (ψh, φh) ∈ Vh we have

(Ahu,wh)V =∑K

((∇×E, φh)0,K − (∇×H, ψh)0,K

)= (∇×E, φh)0,Ω − (∇×H, ψh)0,Ω = (Au,wh)V .

11

This is equivalent to (4.9).Lemma 4.4. For all uh = (Hh,Eh) ∈ Vh we have

(Ahuh, uh)V =∑

F∈F inth

(γF ‖nF × JEhKF ‖20,F + δF ‖nF × JHhKF ‖20,F

)+∑

F∈Fexth

2γF ‖nF ×Eh‖20,F ≥ 0.(4.10)

In particular, −Ah is dissipative on Vh.Proof. Integration by parts yields∑

K

((∇×Eh,Hh)0,K − (∇×Hh,Eh)0,K

)=

∑F∈F int

h

((nF ×EK ,HK)0,F + (nKF

×EKF,HKF

)0,F

)+

∑F∈Fext

h

(nF ×Eh,Hh)0,F .

Inserting this into the definition of Ah and using αK + βK = 1, we obtain

(Ahuh, uh)V =∑

F∈F inth

(αK(nF ×EKF

,HK)0,F − αKF(nF ×EK ,HKF

)0,F

− βK(nF ×HKF,EK)0,F + βKF

(nF ×HK ,EKF)0,F

+ γF ‖nF × JEhKF ‖20,F + δF ‖nF × JHhKF ‖20,F)

+∑

F∈Fexth

2γF ‖nF ×Eh‖20,F .

By (4.8), the first and the fourth term sum to zero, and so do the second and thethird term. This shows (4.10).

The previous lemma implies that the electromagnetic energy is non-increasing iff = 0:

∂t ‖uh‖2V ≤ 0 (4.11)

The following integration by parts formula will be used frequently later. It allowsto move all derivatives and tangential jumps to the test functions.

Lemma 4.5. For u = (H,E) ∈ Vh +(D(A) ∩H1(Th)6

)and wh = (φh, ψh) ∈ Vh

the following relation holds:

(Ahu,wh)V =∑K

((E,∇×φh)0,K − (H,∇×ψh)0,K

)+∑

F∈F inth

((αKEKF

+ αKFEK + δFnF × JHKF , nF × JφhKF )0,F

− (βKHKF+ βKF

HK − γFnF × JEKF , nF × JψhKF )0,F

)−∑

F∈Fexth

(H, nF × ψh)0,F − 2γF (nF ×E, nF × ψh)0,F .

(4.12)

Proof. Integration by parts and (4.8).

12

4.2. Error of spatial discretization. Applying the L2-projection πh to thecontinuous problem (2.8) and using the consistency property of Lemma 4.3, the exactsolution satisfies

∂tπhu+Ahu = fh, πhu(0) = πhu0. (4.13)

We define errors

e(t) = uh(t)− u(t) = eh(t)− eπ(t), (4.14a)

where

eh(t) = uh(t)− πhu(t), eπ(t) := u(t)− πhu(t). (4.14b)

For the projection error we have the following result:Lemma 4.6. For u ∈ Hk+1(K)6, the following error bounds hold:

‖eπ‖0,K ≤ Chk+1K |u|Hk+1(K)6 (4.15)

and

‖eπ,K‖0,F ≤ Chk+1/2K |u|Hk+1(K)6 . (4.16)

where C is independent of K (and thus also of hK).Proof. See [6, Lemma 1.58 and Lemma 1.59].The error eh satisfies the following error bound:Theorem 4.7. Let u ∈ C0

(0, T ;D(A) ∩Hk+1(Th)6

)∩ C1(0, T ;V ) be a solution

of (2.8) and uh ∈ C1(0, T ;Vh) be a solution of the semidiscrete problem (4.6) Then,the error eh = uh − u satisfies

‖eh(T )‖2V +

T∫0

(Aheh(t), eh(t))V dt ≤ Ch2k+1

T∫0

|u(t)|2Hk+1(Th)6 dt, (4.17)

where C is independent of u and h.Proof. Subtracting (4.13) from (4.6), we obtain

∂teh(t) +Aheh(t) = Aheπ(t). (4.18)

Taking the V -inner product with eh and integrating from 0 to T we obtain

1

2

T∫0

d

dt‖eh(t)‖2V dt+

T∫0

(Aheh(t), eh(t))V dt =

∫ T

0

(Aheπ(t), eh(t))V dt.

For the right-hand side we use Lemma 4.5. Since eπ is the projection error we have

(eEπ ,∇× eHh )0,K = 0, (eHπ ,∇× eEh )0,K = 0.

Hence, the first sum in (4.12) vanishes and we obtain

(Aheπ, eh)V =∑

F∈F inth

((αKe

Eπ,KF

+ αKFeEπ,K + δFnF × JeHπ KF , nF × JeHh KF )0,F

13

− (βKeHπ,KF

+ βKFeHπ,K − γFnF × JeEπ KF , nF × JeEh KF )0,F

)−

∑F∈Fext

h

(eHπ , nF × eEh )0,F − 2γF (nF × eEπ , nF × eEh )0,F .

Using Cauchy-Schwarz and Young’s inequalities and then Lemmas 4.4 and 4.6, we get

(Aheπ, eh)V ≤1

2(Aheh, eh)V + Ch2k+1 |u|2Hk+1(Th)6 . (4.19)

Since eh(0) = 0 this proves (4.17).Corollary 4.8. If the assumptions of Theorem 4.7 are satisfied, then the

semidiscrete error e = eh − eπ is bounded by

‖e(T )‖2V +

T∫0

(Ae(t), e(t))V dt ≤ Ch2k+1( T∫

0

|u(t)|2Hk+1(Th)6 dt+ h |u(T )|2Hk+1(Th)6

),

where C is independent of u and h.Proof. We have (Ahv, eπ)V = 0 for all v ∈ Vh + D(A) by (4.4). This yields the

identity

(Ah(eh − eπ), eh − eπ)V = (Aheh, eh)V − (Aheπ, eh)V .

The estimate now follows directly from Lemma 4.6 and Theorem 4.7.

5. Full discretization of Maxwell’s equations.

5.1. Error of full discretization for the implicit Euler method. In thissection we consider the implicit Euler method for the time integration of (4.6):

un+1h = unh + τ(−Ahun+1

h + fn+1h ). (5.1)

We treat this scheme separately because its analysis simplifies considerably comparedto general higher order methods. Moreover, since stage order and order of the implicitEuler scheme coincide, it does not fit into the assumptions of Theorem 5.4 below.

Theorem 5.1. Let u ∈ C(0, T ;D(A) ∩Hk+1(Th)6

)denote the solution of (2.8)

and assume that u′′ ∈ L2(0, T, V ). Then, for τ ≤ 3/4(T + 1), the error of the implicitEuler method is bounded by

∥∥eNh ∥∥2

V+τ

N∑n=0

(Ahen+1h , en+1

h )V

≤C(T + 1)(τ2

∫ T

0

‖u′′(t)‖2V dt+ h2k+1 maxt∈[0,T ]

|u(t)|2Hk+1(Th)6

),

where the constant C = C(Oι, b, k, ρ) is independent of h and u.Proof. The exact solution satisfies

un+1 = un + τ(−Aun+1 + fn+1) + δn+1,

where δn+1 is given in (3.5b) for s = 0. Projecting onto L2 and subtracting from (5.1)yields the error recursion

en+1h = enh − τAh(en+1

h − en+1π )− πhδn+1. (5.2)

14

Taking the V -inner product with en+1h and using (4.19) we obtain

(en+1h − enh, en+1

h )V +1

2τ(Ahe

n+1h , en+1

h )V

≤ Cτh2k+1∣∣un+1

∣∣2Hk+1(Th)6

− (πhδn+1, en+1

h )V .(5.3)

We sum from 0 to N − 1 and use the following representation of the left-hand side

N−1∑n=0

(en+1h − enh, en+1

h )V =1

2

∥∥eNh ∥∥2

V+

1

2

∥∥eNh ∥∥2

V− (eN−1

h , eNh )V +1

2

∥∥eN−1h

∥∥2

V

+1

2

∥∥eN−1h

∥∥2

V− (eN−2

h , eN−1h )V +

1

2

∥∥eN−2h

∥∥2

V+ . . .

+1

2

∥∥e1h

∥∥2

V− (e0

h, e1h)V +

1

2

∥∥e0h

∥∥2

V− 1

2

∥∥e0h

∥∥2

V

≥ 1

2

∥∥eNh ∥∥2

V− 1

2

∥∥e0h

∥∥2

V.

For the right-hand side of (5.3) we have

N−1∑n=0

(πhδn+1, en+1

h )V ≤τ

2

N−1∑n=0

((T + 1)

∥∥∥∥δn+1

τ

∥∥∥∥2

V

+1

T + 1

∥∥en+1h

∥∥2

V

).

For τ ≤ 3/4(T + 1), the result follows by a discrete Gronwall inequality.

5.2. Error of full discretization for higher order Runge–Kutta methods.An implicit s-stage Runge–Kutta method applied to (4.6) yields the approximations

Unih +AhUnih = fnih ,

Unih = unh + τ

s∑j=1

aijUnjh ,

un+1h = unh + τ

s∑i=1

biUnih .

(5.4)

For A-stable collocation methods such as Gauß- and Radau methods, a unique so-lution of the linear system defining the interior Runge-Kutta approximations Unih ,i = 1, . . . , s exists because Ah is dissipative.

Defects. Inserting the exact solution un = u(tn), Uni = u(tn + ciτ), and ˙Uni =u′(tn + ciτ) into the numerical scheme yields

πh˙Uni +AhU

ni = fnih ,

Uni = un + τ

s∑j=1

aij˙Unj + ∆ni,

un+1 = un + τ

s∑i=1

bi˙Uni + δn+1,

(5.5)

where the defects are given in (3.5). We define errors as

enh := unh − πhun, enπ := un − πhun,15

Enih := Unih − πhUni, Eniπ := Uni − πhUni, (5.6)

Enih := Unih − πh˙Uni.

Subtracting (5.5) from (5.4) yields

Enih +AhEnih = AhE

niπ , (5.7a)

Enih = enh + τ

s∑j=1

aijEnjh − πh∆ni, (5.7b)

en+1h = enh + τ

s∑i=1

biEnih − πhδn+1. (5.7c)

For

∆n =

∆n1

...∆ns

, Enh =

En1h...

Ensh

, Enπ =

En1π...

Ensπ ,

, Enh =

En1h...

Ensh

,

we can write (5.7) in compact form as

Enh + (I ⊗Ah)Enh = (I ⊗Ah)Enπ (5.8a)

Enh = 1l⊗ enh + τ(Oι⊗ I)Enh − πh∆n (5.8b)

en+1h = enh + τ(bT ⊗ I)Enh − πhδn+1. (5.8c)

Energy technique. To analyze the error we use the same technique as in thecontinuous case. Taking the inner product of en+1

h with itself using (5.7c) yields

∥∥en+1h

∥∥2

V=

∥∥∥∥∥enh + τ

s∑i=1

biEnih

∥∥∥∥∥2

V

− 2(πhδn+1, enh + τ

s∑i=1

biEnih )V +

∥∥πhδn+1∥∥2

V. (5.9)

Theorem 5.2. The errors (5.6) of the Runge–Kutta method (5.4) applied to(4.6) satisfy∥∥en+1

h

∥∥2

V−‖enh‖

2V + τ

s∑i=1

bi(AhEnih , E

nih )V

≤ C

T + 1τ

(‖enh‖

2V +

s∑i=1

∥∥Enih ∥∥2

V

)+ Cτh2k+1

s∑i=1

bi |u(tn + ciτ)|2Hk+1(Th)6

+ C(T + 1)τ

(s∑i=1

(∥∥πh∆ni∥∥2

V+∣∣∆ni

∣∣2H1(Th)6

)+

∥∥∥∥1

τπhδ

n+1

∥∥∥∥2

V

).

Here the constant C = C(Oι, b, k, ρ) is independent of h and u.Proof. We estimate each of the three terms in (5.9) separately. The second and

the third term can be handled completely analogously to the continuous case. For thesecond term, (3.18) now reads

(πhδn+1, enh + τ

s∑i=1

biEnih )V ≤

C

γτ(‖enh‖

2V +

s∑j=1

∥∥∥Enjh ∥∥∥2

V+

s∑j=1

∥∥πh∆nj∥∥2

V

)+ γτ

∥∥∥∥1

τπhδ

n+1

∥∥∥∥2

V

.

(5.10)

16

For the first term of (5.9) we have to work harder. The reason is the additionalterm in (5.7a) and the fact that Ah is not skew adjoint. However, the derivation ofthe estimate (3.13) remains true, so that we can start from∥∥∥∥∥enh + τ

s∑i=1

biEnih

∥∥∥∥∥2

V

≤ ‖enh‖2V + 2τ

s∑i=1

bi(Enih , E

nih + πh∆ni)V .

Eliminating Enih by (5.7a) yields

(Enih , Enih + πh∆ni)V =(AhE

niπ , E

nih )V − (AhE

nih , E

nih )V

+ (AhEniπ , πh∆ni)V − (AhE

nih , πh∆ni)V .

For the first two terms we have by (4.19)

(AhEniπ , E

nih )V − (AhE

nih , E

nih )V

≤ −1

2(AhE

nih , E

nih )V + Ch2k+1

∣∣∣Uni∣∣∣2Hk+1(Th)6

.(5.11)

The bounds for the last two terms are more involved and their proof is postponed toLemma A.4 below. This lemma yields for γ = T + 1

(AhEnih , πh∆ni)V ≤

C

T + 1

∥∥Enih ∥∥2

V+ C(T + 1)

∣∣∆ni∣∣2H1(Th)6

,

and for γ = 1 using (4.15)

(AhEniπ , πh∆ni)V ≤ Ch2k+2

∣∣∣Uni∣∣∣2Hk+1(Th)6

+ C∣∣∆ni

∣∣2H1(Th)6

.

This finally gives∥∥∥∥∥enh + τ

s∑i=1

biEnih

∥∥∥∥∥2

V

≤ ‖enh‖2V + 2τ

s∑i=1

bi

(− 1

2(AhE

nih , E

nih )V

+ Ch2k+1∣∣∣Uni∣∣∣2

Hk+1(Th)6+

C

T + 1

∥∥Enih ∥∥2

V+ C(T + 1)

∣∣∆ni∣∣2H1(Th)6

),

which shows the desired bound.

Bound on the inner stages. As in the continuous case, we need to bound theerror of the inner stages Enih in order to apply a Gronwall lemma.

Lemma 5.3. The error of the inner stages satisfies

s∑i=1

(∥∥Enih ∥∥2

V+ τ(AhE

nih , E

nih )V

)≤ C

(‖enh‖

2V +

∑i

∥∥πh∆ni∥∥2

V+ τh2k+1

∑i

|u(tn + ciτ)|2Hk+1(Th)6

),

(5.12)

where the constant C = C(Oι, b, k, ρ) is independent of h and u.Proof. We start from (5.8) and write

Enh = 1l⊗ enh + τ(Oι⊗Ah) (Enπ − Enh )− πh∆n.

17

Multiplying by DOι−1 ⊗ I and taking the inner product with Enh gives

(Enh , (DOι−1 ⊗ I)Enh )V s = τ (Enh , (D ⊗Ah)(Enπ − Enh ))V s

+ (Enh , (DOι−1 ⊗ I)(1l⊗ enh − πh∆n))V s .(5.13)

From the coercivity condition (3.3) we conclude

(Enh , (DOι−1 ⊗ I)Enh )V s ≥ αs∑i=1

di∥∥Enih ∥∥2

V.

Since D is diagonal we have by (5.11)

(Enh , (D ⊗Ah)(Enπ − Enh ))V s ≤∑i

(−di

2(AhE

nih , E

nih )V + Ch2k+1

∣∣∣Uni∣∣∣2Hk+1(Th)6

).

Treating the last term as in (3.20) for the continuous case and choosing γ = α2 shows

the result.

Main result (full discretization error). Our main result is contained in thefollowing theorem.

Theorem 5.4. Let u ∈ C(0, T ;D(A) ∩Hk+1(Th)6

)be the solution of (2.8) and

assume that u(s+1) ∈ L2(0, T ;D(A) ∩ H1(Th)6) and u(s+2) ∈ L2(0, T ;V ). Then theerror of an s-stage algebraically stable and coercive Runge–Kutta method of order atleast two satisfies

∥∥eNh ∥∥V +

(τ

N∑n=1

s∑i=1

bi(AhEnih , E

nih )V

)1/2

≤ C(T + 1)1/2(τs+1Bh(u, s, T )1/2 + hk+1/2 max

t∈[0,T ]|u(t)|Hk+1(Th)6

),

where

Bh(u, s, T ) =

∫ T

0

∥∥∥u(s+1)(t)∥∥∥2

H1(Th)6dt+

∫ T

0

∥∥∥u(s+2)(t)∥∥∥2

Vdt.

The constant C = C(Oι, b, k, ρ) is independent of h and u.Proof. The orthogonal projection is stable, i.e.,∥∥πh∆ni

∥∥V≤∥∥∆ni

∥∥V,∥∥πhδn+1

∥∥V≤∥∥δn+1

∥∥V.

By Lemma 5.3 we obtain

∥∥en+1h

∥∥2

V− ‖enh‖

2V + τ

s∑i=1

bi(AhEnih , E

nih )V

≤ C

T + 1τ ‖enh‖

2V + Cτh2k+1

s∑i=1

∣∣∣Uni∣∣∣2Hk+1(Th)6

+ C(T + 1)τ

(s∑i=1

∥∥∆ni∥∥2

H1(Th)6+

∥∥∥∥1

τδn+1

∥∥∥∥2

V

).

18

The regularity assumptions on u and (3.5) imply

τ

N∑n=1

(s∑i=1

∥∥∆ni∥∥2

H1(Th)6+

∥∥∥∥1

τδn+1

∥∥∥∥2

V

)≤ Cτ2(s+1)Bh(u, s, T ).

Applying a discrete Gronwall lemma in differential form we end up with

∥∥eNh ∥∥2

V+ τ

N∑n=1

s∑i=1

bi(AhEnih , E

nih )V ≤ C(T + 1)τ2s+1Bh(u, s, T )

+ CTh2k+1 maxt∈[0,T ]

|u(t)|2Hk+1(Th)6 ,

from which the result is easily obtained.Remark 5.5. The convergence results for the fully discrete scheme can be also

obtained by applying results from [1] but they require more regularity or give a lowerorder of convergence.

Corollary 5.6. Under the assumptions of Theorem 5.4, the error is alsobounded by

N∑n=1

τ ‖enh‖2V ≤ CT (T + 1)

(τ2s+2Bh(u, s, T ) + h2k+1 max

t∈[0,T ]|u(t)|2Hk+1(Th)6

).

Remark 5.7. A similar result can be proven if we use central fluxes instead ofthe upwind fluxes (4.19). The resulting dG operator Acf

h then satisfies

(Acfh eh, eh)V = 0

(Acfh eπ, eh)V ≤ C ‖eh‖2V + Ch2k |u|2Hk+1(Th)6 .

In [9] it was shown that the spatial discretization error is in O(hk) for central fluxes.Under the assumptions of Theorem 5.4, the full discretization error is bounded by∥∥eNh ∥∥V ≤ C(T + 1)1/2

(τs+1Bh(u, s, T )1/2 + T 1/2hk max

t∈[0,T ]|u(t)|Hk+1(Th)6

).

We refer to [24] for details.

Divergence error. We next study the divergence error of the numerical approxi-mation. From the inverse inequality [6, Lemma 1.44] and Theorem 5.4 we immediatelyobtain∥∥∇· eNh ∥∥V ≤ C(T + 1)1/2

(h−1τs+1Bh(u, s, T )1/2 + hk−1/2 max

t∈[0,T ]|u(t)|Hk+1(Th)6

).

In a weak sense, we can even prove that the discrete divergence is preserved exactlyif f = 0. As in [9] we define a test space Xh ⊂ H1

0 (Ω) as the space of continuous,elementwise polynomial functions:

Xh =v ∈ C0(Ω) | v|K ∈ Pk+1(K)6, K ∈ Th

∩H1

0 (Ω).

By 〈·, ·〉−1 we denote the duality product between H−1(Ω) and H10 (Ω), in which

〈∇·u, ψ〉−1 = −(u,∇ψ)0,Ω for all u ∈ L2(Ω)3, ψ ∈ H10 (Ω).

19

Theorem 5.8. Let f ≡ 0. Then the Runge–Kutta solution (5.4) satisfies〈∇· εEn+1

h , ψ〉−1 = 〈∇· εEnh, ψ〉−1,

〈∇·µHn+1h , ψ〉−1 = 〈∇·µHn

h, ψ〉−1,for all ψ ∈ Xh.

Moreover, if the initial data is divergence free, then

〈∇· εEnh, ψ〉−1 = 〈∇·µHnh, ψ〉−1 = 0, n = 0, 1, 2, . . . .

Proof. For ψ ∈ Xh ⊂ H10 (Ω), integration by parts shows

〈∇·(ε(En+1

h −Enh)), ψ〉−1 = −(ε(En+1

h −Enh),∇ψ)0,Ω = −(un+1h − unh,

(0∇ψ

))V .

Using (5.4) and Lemma 4.5 we obtain

〈∇·(ε(En+1

h −Enh)), ψ〉−1 = τ

s∑i=1

bi(AhUnih ,

(0∇ψ

))V = 0,

since for functions in Xh we have ∇×∇ψ = 0, nF × J∇ψhKF = 0 for F ∈ F inth and

n×∇ψh = 0 on ∂Ω. The result for H is proved analogously.The second part follows from

〈∇· ε(E0h), ψ〉−1 = −

∫Ω

πhE0 · ε∇ψ = −

∫Ω

εE0 · ∇ψ =

∫Ω

∇·(εE0)ψ = 0

and similar for H.

Acknowledgements. This work was supported by the Deutsche Forschungsge-meinschaft (DFG) via RTG 1294.

The authors thank Dhia Mansour for fruitful discussions on the error analysis forfull discretizations of partial differential equations.

Appendix A. Auxiliary results.For the proof of Lemma A.4 below we need some auxiliary results:Lemma A.1. If Assumption 4.1 is satisfied, then for v ∈ H1(Th)3, w ∈ L2(Ω)3

and arbitrary γ > 0 we have∑K

|(w,∇×πhv)0,K | ≤1

2γ‖w‖20,Ω + Cγ

∑K

|v|21,K ,

where C = C(ρ, k) is independent of K.Proof. The Cauchy-Schwarz inequality and Young’s inequality yield

|(w,∇×πhv)0,K | ≤1

2γ‖w‖20,K +

γ

2‖∇×πhv‖20,K . (A.1)

From [6, Lemma 1.58] we have

‖∇×πhv‖0,K = ‖∇×(v + πhv − v)‖0,K≤ ‖∇× v‖0,K + ‖∇×(πhv − v)‖0,K≤ C |v|1,K

20

for v ∈ H1(K)3. Inserting this bound into (A.1) and summing over all elements Kshows the desired bound.

Lemma A.2. Let Assumption 4.1 be satisfied and let γ > 0 be arbitrarily chosen.If F is an interior face connecting the elements K and KF and nF is the unit normalvector pointing from K to KF , then for v ∈ H(curl,K ∪KF ) ∩H1(K)3 ∩H1(KF )3,w ∈ H1(K)3, we have

|(w, nF × JπhvKF )0,F | ≤1

γ

(h2K |w|

21,K + 3 ‖w‖20,K

)+ Cγ

(|v|21,K + |v|21,KF

). (A.2)

If F ⊂ ∂K is an exterior face with outward normal vector nF , w ∈ H1(K)3, andv ∈ H0(curl,Ω) ∩H1(K)3 then

|(w, nF × πhv)0,F | ≤1

2γ

(h2K |w|

21,K + 3 ‖w‖20,K

)+ Cγ |v|21,K . (A.3)

In both estimates C = C(ρ, k) is independent of K and KF .Proof. By assumption, we have nF × JvKF = 0. Thus we can write

(w, nF × JπhvKF )0,F = (w, nF × (πhvKF− vKF

))0,F − (w, nF × (πhvK − vK))0,F .

Using the continuous trace inequality [6, Lemma 1.49] and the polynomial approxi-mation properties on the faces [6, Lemma 1.59] we have

|(w, nF × (πhvK − vK))0,F | ≤ ‖w‖0,F ‖nF × (πhvK − vK)‖0,F

≤ C(|w|1,K ‖w‖0,K + h−1

K ‖w‖20,K

)1/2

h1/2K |v|1,K

≤ 1

2γ

(h2K |w|

21,K + 3 ‖w‖20,K

)+ Cγ |v|21,K .

Analogously, using (4.2), we obtain

|(wK , nF × (πhvKF− vKF

))0,F | ≤1

2γ

(h2K |w|

21,K + 3 ‖w‖20,K

)+ Cγ |v|21,KF

.

This proves (A.2). To prove (A.3), note that v ∈ H0(curl,Ω) implies nF × v = 0 onthe exterior face F , so that nF × πhv = nF × (πhv − v).

Lemma A.3. Let Assumption 4.1 be satisfied and let w ∈ Pk(K)3 and γ > 0 bearbitrarily chosen. If F is an interior face connecting the elements K and KF andnF is the unit normal vector pointing from K to KF , then for v ∈ H(curl,K ∪KF )∩H1(K)3 ∩H1(KF )3, we have

|(w, nF × JπhvKF )0,F | ≤1

γ‖w‖20,K + Cγ

(|v|21,K + |v|21,KF

). (A.4)

If F is an exterior face of K with outward normal vector nF and v ∈ H0(curl,Ω) ∩H1(K), then

|(w, nF × πhv)0,F | ≤1

γ‖w‖20,K + Cγ |v|21,K . (A.5)

In both estimates C = C(ρ, k) independent of K and KF .

21

Proof. Analogously to the previous lemma using the discrete trace inequality [6,Lemma 1.52].

Lemma A.4. Suppose that Th satisfies Assumption 4.1. Let v ∈ D(A) ∩H1(Th)6

and γ > 0 be arbitrarily chosen. Then, for u ∈ H1(Th)6 we have

|(Ahu, πhv)V | ≤C

γ

(‖u‖2V +

∑K

h2K |u|

21,K

)+ Cγ

∑K

|v|21,K . (A.6)

For uh ∈ Vh, we have

|(Ahuh, πhv)V | ≤C

γ‖uh‖2V + Cγ

∑K

|v|21,K . (A.7)

The constants C = C(ρ, k) are independent h and K.Proof. We use (4.12) and bound the terms separately. The bound on the sum over

the elements follows from Lemma A.1 and the bounds on the sums over the interiorand exterior faces from Lemmas A.2 and A.3, respectively.

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