Analysis of a mixed discontinuous Galerkin method for the time-harmonic Maxwell equations with minimal smoothness requirements

An error analysis of a mixed discontinuous Galerkin (DG) method with Brezzi numerical flux for the time-harmonic Maxwell equations with minimal smoothness requirements is presented. The key difficulty in the error analysis for the DG method is that the tangential or normal trace of the exact solution is not well-defined on the mesh faces of the computational mesh. We overcome this difficulty by two steps. First, we employ a lifting operator to replace the integrals of the tangential/normal traces on mesh faces by volume integrals. Second, optimal convergence rates are proven by using smoothed interpolations that are well-defined for merely integrable functions. As a byproduct of our analysis, an explicit and easily computable stabilization parameter is given.


Introduction
We consider the analysis of mixed discontinuous Galerkin approximations for the time-harmonic Maxwell equations with low regularity solutions: find u u u, p such that (1.1a) ∇ · (εu u u) = 0 in Ω , (1.1b) n n n × u u u = 0 0 0 on Γ , (1.1c) p = 0 on Γ . (1.1d) Here, u u u represents the electrical field, p the Lagrange multiplier used to enforce the divergence constraint (1.1b), k is the wave number and j j j ∈ L 2 (Ω ) 3 is the source term. The piecewise constant coefficients µ and ε are the magnetic permeability and electrical permittivity of the media, respectively. We assume et al. (2016) proposed an interior-penalty method with C 0 finite elements for the Maxwell equations with minimal smoothness requirements. Recently, Ern & Guermond (2019) analyzed a non-conforming approximation of elliptic PDEs with minimal regularity by introducing a generalized normal derivative of the exact solution at the mesh faces. They also showed that this idea can be extended to solve the time-harmonic Maxwell equations with low regularity solutions by introducing a more general concept for the tangential trace. Another technique that avoids the definition of generalized traces, which has been proposed by Gudi (2010) in the context of elliptic PDEs, is to use an enriching map to transform a non-conforming function into a conforming one.
In this paper, we analyse a mixed DG formulation for the Maxwell equations with low regularity solutions, which modifies the method of Houston et al. (2004) by employing Brezzi numerical fluxes (Brezzi et al., 2000). The main objective is to generalize the error analysis of Houston et al. (2004) to the non-smooth case and present optimal a priori error estimates for the low regularity solution in the broken Sobolev space H s (T h ), s 0 with T h the finite element partition. The proof of our a priori error analysis is different from (Buffa & Perugia, 2006;Bonito et al., 2013;Ern & Guermond, 2019) in that, first, it employs a lifting operator that allows us to replace integrals over faces by integrals over volumes and, thus, avoids the definition of a generalized tangential trace on mesh faces. Second, smoothed interpolations, which are well-defined for merely integrable functions, are used to prove optimal convergence rates. A further major benefit of using the lifting operator is that we obtain an explicit expression for stabilization parameters, which, compared to Houston et al. (2004), facilitates the implementation considerably.
The paper is organized as follows. We introduce notation and the variational formulation of the time-harmonic Maxwell equations in Section 2. The finite element spaces and the mixed discontinuous Galerkin method with the Brezzi numerical flux are presented in Section 3. We state the main results of the paper in Section 4. An auxiliary variational formulation in the spirit of Houston et al. (2004) and some interpolation error estimates are presented in Section 5. Next, we first derive an error estimate for the Maxwell equations (1.1a) with k = 0 in Section 6, and subsequently, we show in Section 7 the well-posedness and error estimates of the mixed DG method for the indefinite Maxwell equations, i.e., k = 0. Some auxiliary results are proven in the Appendix.

Variational formulation
Throughout the paper we assume that the coefficients µ, ε are piecewise constant matrix-valued functions such that there exist positive constants µ * , µ * , ε * , ε * with for a.e. x x x ∈ Ω , and all vectors ξ ξ ξ ∈ R d . More precisely, we assume that µ and ε are piecewise constant with respect to some partition T h of Ω into Lipschitz polyhedra. In the following, we also assume that k 2 is not an interior Maxwell eigenvalue, see (Monk, 2003, Section 1.4.2) or (Boffi et al., 2013, (11.2.6)) for a definition. Let V := H 0 (curl, Ω ) and Q := H 1 0 (Ω ). Define the bilinear forms a(·, ·) and b(·, ·) as The mixed variational formulation of the time-harmonic Maxwell equations is to find u u u ∈ V and p ∈ Q such that Due to that a(·, ·) is continuous and coercive on the kernel of b, and b(·, ·) is continuous and satisfies the inf-sup condition, see (Houston et al., 2005c, Section 2.3) or (Boffi et al., 2013, Theorem 11.2.1), the variational problem is well-posed.

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The following stability results, which were proven in Bonito et al. (2013), are very useful for our error estimates.
Note that in general the differentiability indices τ ε , τ µ are less than 1/2 for Lipschitz domains and discontinuous coefficients ε and µ (see Bonito et al., 2016).

Finite element spaces
Let T h be a shape regular partition of the domain Ω into tetrahedra, such that the coefficients are constant on each K ∈ T h . We denote by h K the diameter of an element K and denote h = max K∈T h h K . For an integer ℓ 0 and an element K ∈ T h , we define P ℓ (K) as the space of polynomials of total degree ℓ in K. Let F h be the union of interior faces F I h and boundary faces F B h . For piecewise smooth vector-or scalar-valued functions v v v and q, we define jumps and averages at faces in the mesh T h . Let F ∈ F I h be an interior face shared by two elements K + and K − and let n n n ± be the unit outward normal vectors on the boundaries ∂ K ± , then the tangential and normal jumps across F are, respectively, defined by v v v T = n n n + × v v v + + n n n − × v v v − , v v v N = v v v + · n n n + + v v v − · n n n − , and q N = q + n n n + + q − n n n − . We also define the averages by We define the broken Sobolev spaces with respect to the partition T h of Ω as Moreover, we define the finite element spaces without inter-element continuity condition as where R ℓ denotes the space of Nédélec functions of degree ℓ, i.e., for d = 3, R ℓ = (P ℓ−1 ) 3 ⊕ S ℓ , and S ℓ = {q q q ∈ ( P ℓ ) 3 : x x x · q q q = 0} with P ℓ being the homogeneous polynomials of degree ℓ, see (Monk, 2003, Remark 5.29) for d = 2.
We also define H(curl)-conforming subspaces with and without vanishing tangential trace on the boundary, respectively, as

Lifting operator
The following lifting operators are useful in the DG discretization by replacing the penalty terms over faces by volume integrals, which make sense also for low regularity functions. First, we define the local lifting operator R F : Since the right hand side is nonzero only when { {v v v} } has support on F, the support of the lifting R F (η) is limited to the elements adjacent to face F. Next, the local lifting operator can be used to define a global one. Define R : where h F denotes the diameter of face F and C 1 , C 2 are independent of v v v.

Mixed DG discretization
The mixed discontinuous Galerkin discretization with Brezzi numerical flux (see, e.g., Brezzi et al., 2000) for the time-harmonic Maxwell equations is: where, Here, ∇ h and ∇ h × denote the elementwise action of the differential operators ∇ and ∇×, respectively. We set the parameter γ F strictly positive for all F ∈ F h , and α F > 0 will be chosen later. The readers are referred to Houston et al. (2004) for the derivation of the DG formulation with the Brezzi numerical flux replaced by the interior penalty numerical flux.
REMARK 3.1 The main difference between the mixed DG formulation (3.6)-(3.7) and those discussed in (Houston et al., 2004;Lu et al., 2017) is the use of the lifting operator R F . Two main benefits of using the lifting operator are: (i) precise condition for α F can be computed from the mesh T h that ensures stability, see Proposition 4.1, while in practice for interior-penalty formulations, depending on the computational mesh, the penalty parameters in (Houston et al., 2004;Lu et al., 2017) need to be frequently adjusted; (ii) the bilinear forms a h and b h can be trivially extended to This avoids the generalization of the tangential trace on element faces for low regularity solutions, which causes great technical difficulties for nonconforming finite element methods for solving problems with low regularity solution, see, e.g., Buffa & Perugia (2006) REMARK 3.2 Since the coefficients µ and ε are piecewise constant, we note from the definition of the lifting operator R that the bilinear forms a h : Note that compared with the implementation of the DG discretization in Houston et al. (2004), we see that only the penalty terms need to be changed.

The main results
We now give explicit bounds on the stabilization parameter α F that ensures well-posedness of (3.6)-(3.7). Subsequently, we present an a priori error estimate for low regularity solutions. We start with defining V (h) = V +V h and Q(h) = Q + Q h endowed with the semi-norms and norms The following proposition shows a h (·, ·) is coercive on V (h) respect to the semi-norm | · | V (h) for a simple and explicit choice of the parameter α F , which facilitates the implementation of the DG method and is essential in the proof of the well-posedness of the mixed DG discretization.
Proof. From the definition of the lifting operator R F , we have for any Recall that the support of the local lifting operator R F ( v v v T ), denoted by ω F , consists of the element(s) adjacent to face F. By using the Cauchy-Schwarz and Young's inequality, there holds where n K is the number of faces of an element K ∈ T h . Hence, Hence, by setting δ = 1 4n K , we can get the coercivity with constant 1 2 if α F 1 2 + 2n K . The following two theorems state the well-posedness of the mixed DG method (3.6)-(3.7) and a priori error estimates.
where the constant C > 0 is independent of the mesh size and the solution (u u u h , p h ).
Then, for all mesh sizes h small enough, there exists a unique solution (u u u h , p h ) ∈ V h × Q h to (3.6)-(3.7) ANALYSIS OF A MIXED DG METHOD FOR MAXWELL EQUATIONS 9 of 25 and it satisfies the a priori error estimates where the constant C > 0 depends on the bounds (2.2), wave number k and polynomial degree ℓ, but is independent of the mesh size. Here, χ(s) = 1 if s 1 2 and χ(s) = 0 otherwise, and D K : Note that in the estimates D K can be replaced by K for all s > 1 2 . REMARK 4.1 We note that all above results also hold true with the choice of Nédélec elements of the second type for V h and a full polynomial space of order ℓ + 1 for Q h , see (Buffa & Perugia, 2006, Section 7.1) and (Monk, 2003, Section 8.2) for more details.
Theorem 4.2 will be proved for k = 0 in the next section using an auxiliary formulation in the spirit of Houston et al. (2004). The case k = 0 is treated in Section 7. Theorem 4.3 will be proven in Section 6 for k = 0 and in Section 7 for k = 0, respectively.

Auxiliary mixed formulation
The variational problem (3.6)-(3.7) is a saddle-point problem with penalty, to facilitate its analysis we follow (Houston et al., 2004, Section 4 and 5) and introduce an equivalent auxiliary mixed formulation, that is a saddle-point problem without penalty. To do so, let us introduce the discrete auxiliary space with semi-norm and norm defined as: We state the auxiliary mixed formulation as follows: LEMMA 5.1 The mixed DG formulation (3.6)-(3.7) is equivalent to (5.1)- Hence, λ h = p h N . This shows that (u h , p h ) solves (3.6)-(3.7). The other direction follows immediately by setting λ h = p h N . Define the kernel of the form B h (·, ·) as The following three lemmas form the basis for the proof of Theorem 4.2 for k = 0.
LEMMA 5.2 (Continuity) There exists a constant C independent of the mesh size and the coefficients µ and ε such that This lemma follows directly from an application of the Cauchy-Schwarz inequality.
LEMMA 5.3 (Ellipticity on the kernel) For α F given by Proposition 4.1 and γ F 1 2 , there holds where κ A > 0 depends on the coefficients µ and ε but independent of the mesh size.
Recalling the definition of A h and using the coercivity of a h stated in Proposition 4.1, there holds From the discrete Friedrichs inequality in Appendix A.2, we have which for δ > 0 leads to

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By setting δ = which, together with (5.5), completes the proof with κ A = 1 2(1+c 2 F ) . The following stability result follows with similar arguments as in (Houston et al., 2004, Proposition 5.4) and using the stability of the lifting operator (3.5).
LEMMA 5.4 (Inf-sup condition) There holds where κ B > 0 depends on the coefficients µ and ε but is independent of the mesh size. Now, we are ready to prove the well-posedness of (3.6)-(3.7) for k = 0. Proof of Theorem 4.2 for k = 0. From the classical theory of mixed FEM (see, e.g., Boffi et al., 2013), Lemma 5.2, 5.3 and 5.4 imply that the auxiliary formulation (5.1)-(5.2) with k = 0 has a unique solution where C > 0 is independent of the mesh size. From Lemma 5.1, (u u u h , p h ) ∈ V h × Q h also solves (3.6)-(3.7) and the uniqueness of (3.6)-(3.7) follows from the a priori estimate (5.7).

Conforming approximation
In the error analysis, we shall use the conforming projection Π c h , introduced in (Houston et al., 2005a, Proposition 4.5), which states that the approximation of a discontinuous function in V h by a H(curl) averaging operator can be quantified in terms of certain jumps. The following lemma is actually a byproduct of the proof of (Houston et al., 2005a, Proposition 4.5), see (Houston et al., 2005a, Appendix) for more details.

LEMMA 5.5 (Conforming approximation) There exists an operator
Here, the constant C > 0 depends on the shape-regularity of the mesh and the polynomial degree ℓ, but not on the mesh size.
By using the stability of the lifting operator (3.5), Lemma 5.5 immediately implies the following approximation and stability result.
Here, the constant C > 0 depends on the shape-regularity of the mesh and the polynomial degree ℓ, but not on the mesh size.

Smoothed interpolation
The idea of combining the canonical finite element interpolation operators with some mollification technique for low regularity functions has been introduced in many papers, e.g., by Schöberl (2001Schöberl ( , 2008, Arnold et al. (2006), Christiansen & Winther (2008), Falk & Winther (2014) and Ern & Guermond (2016). In this section, we combine the shrinking-based mollification in (Ern & Guermond, 2016, Section 3) with canonical finite element interpolations to prove the convergence of the DG approximation to solutions of the Maxwell equations with low regularity requirement.
To prove the local approximation properties, stated in Proposition 5.1 and 5.2 below, we will employ a family of smoothing operators developed in (Ern & Guermond, 2016, Section 6.1).

the families of mollification operators introduced in (Ern & Guermond, 2016, Section 3.2 and 5.2). Let
be the canonical finite element interpolation operators, i.e., I g h the Lagrange interpolation, I c h the standard Nédélec interpolation of the first kind (see Monk, 2003, Section 5.5), I d h the divergence conforming interpolation (see Monk, 2003, Section 5.4) and I b h the L 2 projection (see Monk, 2003, Section 5.7), which enjoy a commuting diagram property (see Monk, 2003, (5.59)).
Combining the mollification operators K δ with the canonical interpolation operators, we obtain the smoothed interpolation operators I h := I h K δ . (5.10) We note that the canonical interpolation operators require sufficient smoothness while the smoothed operators requires merely L 1 -integrability. From the definition of the smoothed interpolation and commuting diagrams (see Monk, 2003, (5.59)) and (Ern & Guermond, 2016, (3.7)), we deduce the following lemma.
LEMMA 5.7 (Commuting properties) Let I h be defined as in (5.10). There hold: . We finish this section by a family of approximation results. Since the proof is technical, especially some new local properties of K δ are needed, we postpone it to Appendix A.
PROPOSITION 5.1 (Local approximation) Let I h be defined as in (5.10) and s ∈ [0, 1 2 ). There exists a constant c > 0 independent of the mesh size such that for all where D K is the macro element defined in Theorem 4.3. Similarly, there hold

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Smoothed interpolation with boundary conditions
Similar to last section, we establish the approximation properties of the smoothed interpolation with boundary restriction, which will be used to prove the best approximation given in Theorem 6.3.
be the families of mollification operators introduced in (Ern & Guermond, 2016, Section 4.2). Then the smoothed interpolations satisfy the following commuting properties.
LEMMA 5.8 (Commuting properties) Let I h0 be defined as in (5.11). There hold: By following the proof of Proposition 5.1, we can conclude the following approximation results.
PROPOSITION 5.2 (Local approximation) Let I h0 be defined as in (5.11) and s ∈ [0, 1 2 ). There exists a constant c > 0 independent of the mesh size such that for all

Definite Maxwell equations
The error estimates of the mixed DG discretization (3.6)-(3.7) for the definite Maxwell equations with k = 0 will greatly facilitate the analysis for the indefinite problem discussed in Section 7.

Residual operators
Following (Houston et al., 2004, Section 6.1), we introduce two consistency-related residual operators, which play a key role in deriving an a priori error estimate under minimal smoothness requirements. Suppose that (u u u, p) ∈ V × Q is the exact solution of continuous variational problem (2.3)-(2.4). We define the residuals for all (v v v, η) ∈ W h and q ∈ Q h . We also define norms of the residual operators as The analysis of Houston et al. (2004) relies crucially on the the smoothness H s (T h ), s > 1 2 . In this section, we will extend the analysis to s ∈ [0, 1 2 ).
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LEMMA 6.1 Let (u u u, p) ∈ V × Q be the solution of (2.3)-(2.4), then Proof. The identities are actually direct results of (2.3)-(2.4). In fact, for all v v v ∈ V c h0 , there holds from the definition of R 1 and (2.3) The other identity follows from (2.4) The following lemma estimates the residuals for the smooth case s > 1 2 . LEMMA 6.2 (Residual estimates) Let (u u u, p) ∈ V × Q be the solution of (2.3)-(2.4) with Then, there hold where the constant C > 0 is independent of the mesh size.
Proof. See the proof of (Houston et al., 2004, Proposition 6.2). Now, we are ready to state our main results about the residuals.
Then, there hold where the constant C > 0 is independent of the mesh size.
Step 1: Estimate of R 1 . For any (v v v, η) ∈ W h , from the definition of R 1 and (6.1), one gets To treat the last term, we employ the smoothed interpolation I c h : H(curl, Ω ) → V c h from (5.10), i.e., By using the definition of the lifting operator R and integration by parts, we infer for Substituting this identity into (6.2), we have where we have used the conforming approximation in Lemma 5.6 and the commuting property (i) of Lemma 5.7. Finally, the estimate for R 1 follows directly from Proposition 5.1.

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Step 2: Estimate of R 2 . Similarly, we have for q ∈ Q h (6.4) Since I d h εu u u belongs to the divergence conforming finite element space V d h and V d h ⊂ V h , using the definition of lifting operator R and integration by parts, we obtain From the commuting property (ii) of Lemma 5.7 and ∇ · εu u u = 0, one arrives at Substituting (6.5)-(6.6) into (6.4) leads to where the estimate for R 2 is a direct result of Proposition 5.1. Combining Lemma 6.2 and Proposition 6.1, we obtain the following residual estimates.
COROLLARY 6.1 For k = 0, let (u u u, p) ∈ V × Q be the solution of (2.3)-(2.4) with Then, there hold where the constant C > 0 is independent of the mesh size, χ(s) = 1 if s 1 2 and χ(s) = 0 otherwise. Note that D K in the estimates can be replaced by K for s > 1 2 .

Error estimates
The framework for the abstract error estimates, combined with the stability conditions given in Section 5.1, leads to the abstract error estimates for the auxiliary formulation (5.1)-(5.2) with k = 0. THEOREM 6.2 For k = 0, let (u u u, p) be the solution of the weak formulation of the time-harmonic Maxwell equations (2.3)-(2.4), and let (u u u h , λ h ; p h ) be the solution of discontinuous Galerkin discretization (5.1)-(5.2). Then, there exists a constant C > 0, independent of mesh size, such that Proof. The estimates follows from a extension of the standard mixed finite theory, see the proof of (Houston et al., 2004, Theorem 6.1) combined with the stability conditions in Lemma 5.2-5.4. The next results quantifies the best-approximation error.
THEOREM 6.3 Suppose that u u u ∈ H 0 (curl, Ω ) such that εu u u, µ −1 ∇ × u u u ∈ H s (T h ) and p ∈ Q such that ∇p ∈ H s (T h ), s 0. There exists a constant C > 0 independent of the mesh size such that Proof. By taking v v v = I c h0 u u u, which is in H 0 (curl, Ω ), and using commuting property (i) of Lemma 5.8, we have Then, (6.7) follows from the approximation properties given in Proposition 5.2. We note (6.8) is the standard approximation result of Clément interpolation (Monk, 2003, Section 5.6.1), or the ScottZhang interpolation (see, e.g., Scott & Zhang, 1990;Brenner & Scott, 2008). Now, we are ready to prove Theorem 4.3 in the special case of k = 0. Proof of Theorem 4.3 for k = 0.
From the abstract error estimates in Theorem 6.2, the polynomial approximation results (6.7)-(6.8) and the residual estimates in Corollary 6.1, we easily derive the a priori error bounds in Theorem 4.3 for k = 0.

Indefinite Maxwell equations
We will first discuss existence and uniqueness properties of the mixed DG method (3.6)-(3.7) for the indefinite Maxwell equations (1.1a), and subsequently provide error estimates for k = 0, k 2 not a Maxwell eigenvalue, under minimal regularity requirements. Instead of using Fredholm alternatives to show well-posedness of the mixed DG system (3.6)-(3.7), we shall prove an inf-sup condition for A h on the kernel of B h for k = 0.

Uniform convergence and spectral theory
The spectral properties of the solution operator are essential for the proof of the existence and uniqueness of the mixed DG method for the time-harmonic Maxwell equations. Because of the existence of the unique solution (u u u, p) to (2.3)-(2.4) for the Maxwell equations with k = 0 and the a priori estimate given in Lemma 2.1, we can uniquely define the bounded solution operators T : L 2 ε (Ω ) 3 → V and T p : 3)-(2.4) by T j j j := u u u, T p j j j := p. (7.1) Similarly, from the uniqueness of the solution (u u u h , p h ) to (3.6)-(3.7) for the Maxwell equations (1.1) with k = 0 and the a priori estimate (5.7), we can uniquely define the bounded discrete solution operators From the abstract estimates in Theorem 6.2 together with Corollary 6.1 and the consistency of the finite element spaces we obtain pointwise convergence of T h to T : for any fixed j j j ∈ L 2 (Ω ) d , T j j j − T h j j j V (h) → 0 as h → 0. The following proposition states the uniform convergence of T h , see Appendix C for a proof. By using (7.3), one can derive that for any z ∈ ρ(T ), the resolvent set of T , the resolvent operator R z (T h ) = (z − T h ) −1 : V h → V h exists and is bounded for all h sufficiently small. In fact, we have the following lemma (Descloux et al., 1978, Lemma 1). LEMMA 7.1 Let (7.3) hold and let F ⊂ ρ(T ) be closed. Then, for all h small enough, there holds where C > 0 depending on F is independent of the mesh size h.

Existence and uniqueness
By using the results of Lemma 7.1, it is straightforward to prove Theorem 4.2.
Proof of Theorem 4.2. We prove the uniqueness by proving the a priori bound in the theorem. It is obvious that there exists a unique element j j j h such that Hence, we can rewrite (3.6)-(3.7) as follows: From the definition of the solution operators T h and T p,h , we infer that u u u h = T h j j j h + k 2 T h u u u h , p h = T p,h j j j h + k 2 T p,h u u u h . (7.4) With z := 1 k 2 , the first equation becomes (z − T h )u u u h = zT h j j j h .
Since k 2 is not a Maxwell eigenvalue by assumption, i.e., z is not an eigenvalue of T , Lemma 7.1 shows that, for h small enough, the operator R z (T h ) : V h → V h exists and is bounded uniformly in h. Hence, u u u h is uniquely determined by From definition of T h (7.2), It follows that T h : L 2 ε (Ω ) 3 → V h is also bounded and there exists a constant C > 0 independent of the mesh size such that u u u h V (h) C R z (T h ) T h ε 1 2 j j j h 0,Ω C j j j 0,Ω . (7.5) The uniqueness of p h directly follows from the uniqueness of u u u h , and there exists a constant C > 0 independent of mesh size such that p h Q(h) C ε 1 2 j j j h 0,Ω + ε 1 2 u u u h 0,Ω C j j j 0,Ω . (7.6)

Error estimates
For the indefinite Maxwell equations (k = 0), we can similarly define the residuals In the same way as in Section 6, one can show that the estimates for the residuals in Proposition 6.1 still hold true. The next proposition provides the inf-sup condition of A h on the kernel of B h for k = 0, which is a crucial ingredient for the error estimates. The idea of the proof is classical (see, e.g., Melenk, 1995;Buffa & Perugia, 2006 for a positive constant κ A , which depends on the coefficients µ and ε but independent of h. Here, Ker(B h ) is the kernel of B h defined in (5.3).
Proof. For any (v v v, η) ∈ Ker(B h ), let (u u u h , λ h ) = (v v v, η) + k 2 (ũ u u h ,λ h ) with (ũ u u h ,λ h ;p h ) be the solution of DG method (5.1)-(5.2) with j j j = εv v v. Thus, by using the ellipticity of A h on Ker(B h ) in Lemma 5.3 and (5.1), we obtain From Lemma 5.1, we knowλ h = p h N . Hence, by using the stability estimates (7.5)-(7.6), we have Now, we conclude from the above two inequalities which is equivalent to (7.7) since A h (·, ·) is symmetric. Now, we are ready to prove Theorem 4.3. Proof of Theorem 4.3. Since k 2 is not a Maxwell eigenvalue, the inf-sup condition (7.7) holds true for all h small enough. Together with the inf-sup condition of B h (5.6), one can get the same abstract error estimates stated in Theorem 6.2 for k = 0 also for the indefinite time-harmonic Maxwell equations (1.1) with k 2 not a Maxwell eigenvalue. Thus, the a priori error bound follows directly from the polynomial approximation results (6.7)-(6.8) and residual estimates in Corollary 6.1.