A Locally Divergence-Free Interior Penalty Method for Two-Dimensional Curl-Curl Problems

A LOCALLY DIVERGENCE-FREE INTERIOR PENALTY METHOD FOR TWO-DIMENSIONAL CURL-CURL PROBLEMS

SUSANNE C. BRENNER∗, FENGYAN LI†, AND LI-YENG SUNG‡

Abstract. An interior penalty method for certain two-dimensional curl-curl problems is inves- tigated in this paper. This method computes the divergence-free part of the solution using locally divergence-free discontinuous P1 vector ﬁelds on graded meshes. It has optimal order convergence (up to an arbitrarily small ) for the source problem and the eigenproblem. Results of numerical experiments that corroborate the theoretical results are also presented.

Key words. curl-curl problem, Maxwell eigenproblem, locally divergence-free, interior penalty methods, graded meshes

AMS subject classiﬁcation. 65N30, 65N15, 35Q60

1. Introduction. Let Ω ⊂ R2 be a bounded polygonal domain. Consider the following weak curl-curl problem: Find u ∈ H0(curl; Ω) such that

(1.1) (∇× u, ∇× v)+ α(u, v) = (f, v) ∀ v ∈ H0(curl; Ω),

2 where α is a constant, (·, ·) denotes the inner product of L2(Ω) (or [L2(Ω)] ),

v1 2 ∂v2 ∂v1 H(curl; Ω) = v = ∈ [L2(Ω)] : ∇× v = − ∈ L2(Ω) n v2 ∂x1 ∂x2 o and

H0(curl; Ω) = {v ∈ H(curl; Ω) : n×v =0on ∂Ω}, with n being the unit outer normal on ∂Ω. The curl-curl problem (1.1) is related to electromagnetic problems. For α ≤ 0, it is the weak form of the time-harmonic (frequency-domain) Maxwell equations. For α> 0, it is related to the spatial problems appearing in implicit semi-discretizations of the time-dependent (time-domain) Maxwell equations. It is well-known [21, 22, 4] that H1 conforming nodal finite element methods can lead to a wrong solution of (1.1) if Ω is not convex. Many alternative approaches have been developed which include H(curl; Ω) conforming edge element methods [37, 38, 12, 29, 34, 36], H1 conforming nodal finite methods with weighted regularization [24, 26], the singular complement/field method [28, 4], and interior penalty methods [39, 32, 33, 31, 30]. In this paper we take a different approach. By the Helmholtz decomposition, the solution u ∈ H0(curl; Ω) of (1.1) can be written uniquely as (1.2) u = ˚u + ∇φ,

∗Department of Mathematics and Center for Computation and Technology, Louisiana State Uni- versity, Baton Rouge, LA 70803 ([email protected]). The work of this author was supported in part by the National Science Foundation under Grant No. DMS-03-11790. †Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180 ([email protected]). The work of this author was supported in part by the National Science Foundation under Grant No. DMS-06-52481. ‡Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803 ([email protected]). 1 2 Susanne C. Brenner, Fengyan Li and Li-yeng Sung

0 1 where ˚u ∈ H0(curl; Ω) ∩ H(div ; Ω), φ ∈ H0 (Ω),

v1 2 ∂v1 ∂v2 H(div; Ω) = v = ∈ [L2(Ω)] : ∇ · v = + ∈ L2(Ω) , n v2 ∂x1 ∂x2 o and

H(div0;Ω) = v ∈ H(div; Ω) : ∇ · v =0}. n It is easy to see that φ satisﬁes

1 (1.3) α(∇φ, ∇ψ) = (f, ∇ψ) ∀ ψ ∈ H0 (Ω). Since the Poisson problem (1.3) (when α 6= 0) can be solved by many standard methods under the assumption that f ∈ H(div; Ω), we will focus on the divergence- free part ˚u, which satisﬁes

0 (1.4) (∇× ˚u, ∇× v)+ α(˚u, v) = (f, v) ∀ v ∈ H0(curl; Ω) ∩ H(div ; Ω).

We shall refer to (1.4) as the weak form of the reduced curl-curl problem and assume in the case where α ≤ 0 that (1.4) has a unique solution (i.e. −α is not a Maxwell eigenvalue). Note that the strong form of the reduced curl-curl problem is given by

(1.5) ∇× (∇× ˚u)+ α˚u = Qf,

2 0 where Q : [L2(Ω)] −→ H(div ; Ω) is the orthogonal projection. Remark 1.1. The strong form (1.5) and the well-posedness of (1.4) imply immediately

(1.6) k∇× ˚ukL2(Ω) ≤ CΩ,kkQfkL2(Ω) ≤ CΩ,kkfkL2(Ω). Here and below we use C (with or without subscripts) to denote a generic positive constant that can take different values at different occurrences. The advantage of working with (1.4) is that it behaves like an elliptic problem, unlike the full curl-curl problem (1.1). In particular, the solution ˚u of (1.4) enjoys elliptic regularity under the assumption that f ∈ L2(Ω), which greatly simplifies the duality argument needed for the analysis. In [14] we developed a numerical scheme (in the case where α ≤ 0) for (1.4) using the Crouzeix-Raviart nonconforming P1 vector fields [27]. In this paper we investigate an interior penalty method that avoids the difficulties involved in the construction of local bases for the Crouzeix-Raviart vector fields at the cost of a larger number of unknowns. These two methods are very closely related, which simplifies the analysis of the new scheme. Since the solution operator of the interior penalty method converges uniformly to the solution operator of (1.4) with respect to the norm k·kL2(Ω), classical spectral approximation theory [6] implies that we can also apply the interior penalty method to the following Maxwell eigenproblem: 0 Find (λ, ˚u) ∈ R × H0(curl; Ω) ∩ H(div ; Ω) such that

0 (1.7) (∇× ˚u, ∇× v)= λ(˚u, v) ∀ v ∈ H0(curl; Ω) ∩ H(div ; Ω). An Interior Penalty Method for 2D Curl-Curl Problems 3

In particular, the spectral approximation based on our interior penalty method is free of spurious eigenvalues. The rest of the paper is organized as follows. We introduce the finite element space and graded meshes in Section 2. The interior penalty method and some preliminary estimates are presented in Section 3. The convergence analysis for the source problem (1.1) is carried out in Section 4. Application to the eigenproblem (1.7) is discussed in Section 5, followed by numerical results in Section 6, and we end with some concluding remarks in Section 7. 2. Finite Element Spaces and Graded Meshes. In this section we will introduce discontinuous finite element spaces defined on meshes graded around the corners c1,...,cL of Ω. We assume that the triangulation Th of Ω satisfies the following condition:

(2.1) hT ≈ hΦµ(T ) ∀ T ∈ Th, where h is the mesh parameter, hT = diam T , and µ = (µ1,...,µL) is the vector containing the grading parameters. The weight Φµ(T ) is deﬁned by

L 1−µ` (2.2) Φµ(T ) = Π`=1|c` − cT | ,

where cT is the center of T . Note that

(2.3) hT ≤ Ch.

The choices of the grading parameters are dictated by the singularities [5, 25, 23] of the solution ˚u of (1.4). In order to recover optimal convergence rates in both the energy norm and the L2 norm (cf. [14]), we take π µ = 1 if ω ≤ , ` ` 2 (2.4) π π µ` < if ω` > . 2ω` 2

The construction of Th satisfying (2.1) can be found for example in [1, 13]. Note that Th can be constructed so that it satisfies a minimum angle condition for any fixed choice of the grading parameters. The finite element space Vh is defined by v 2 v v 2 v Vh = { ∈ [L2(Ω)] : T = T ∈ [P1(T )] and ∇ · T =0 ∀ T ∈ Th}.

1 s 2 For any s > 2 there is a natural weak interpolation operator ΠT : [H (T )] −→ 2 [P1(T )] deﬁned by 1 (2.5) (ΠT ζ)(mei )= ζ ds for i =1, 2, 3, |ei| Zei where mei is the midpoint of the edge ei of T . It follows immediately from (2.5), the midpoint rule and Green’s theorem that

s 2 (2.6) ∇× (ΠT ζ)dx = ∇× ζ dx ∀ ζ ∈ [H (T )] , ZT ZT s 2 (2.7) ∇ · (ΠT ζ)dx = ∇ · ζ dx ∀ ζ ∈ [H (T )] . ZT ZT 4 Susanne C. Brenner, Fengyan Li and Li-yeng Sung

Furthermore, given s ∈ (1/2, 2], we have the following interpolation error estimates [27]:

min(s,1) s min(s,1) s (2.8) kζ − ΠT ζkL2(T ) + hT |ζ − ΠT ζ|H (T ) ≤ CT hT |ζ|H (T )

s 2 for all ζ ∈ [H (T )] , where the positive constant CT depends on the minimum angle of T (and also on s when s is close to 1/2). Since the solution ˚u of the reduced curl-curl problem belongs to [Hs(Ω)]2 for 1 some s> 2 (cf. [36]), we can deﬁne a global interpolant of ˚u by

(Πh˚u)T = ΠT ˚uT ∀ T ∈ Th, u u u u where ˚T = ˚ T . It follows from (2.7) and ∇ · ˚ = 0 that Πh˚ ∈ Vh. Remark 2.1. Note that Π ˚u is the same interpolation operator used in [14] and h Πh˚u belongs to the space of locally divergence-free Crouzeix-Raviart nonconforming P1 vector fields [27]. Since the vector fields in Vh are discontinuous, their jumps across the edges of Th play an important role in the development of interior penalty methods. We will i i denote by Eh (resp. Eh) the set of the edges (resp. interior edges) of Th. Let e ∈ Eh be shared by the two triangles T1,T2 ∈ Th (cf. Figure 2.1) and n1 (resp. n2) be the unit normal of e pointing towards the outside of T1 (resp. T2). We define, on e, n v n v n v (2.9a) [[ × ]] = 1 × T 1 e + 2 × T 2 e, n v n v n v (2.9b) [[ · ]] = 1 · T 1 e + 2 · T 2 e.

e T2

n1 n2

Fig. 2.1. Triangles and normals in the deﬁnitions of [[n × v]] and [[n · v]]

For an edge e along ∂Ω, we take ne to be the unit normal of e pointing towards the outside of Ω and deﬁne

n v n v (2.10) [[ × ]] = e × e.

We will also denote the piecewise deﬁned curl and div operators by ∇h× and ∇h·, i.e.,

v v v v (∇h × ) T = ∇× T and (∇h · ) T = ∇ · T .

3. Discretization and Preliminary Error Estimates. In this section we present the interior penalty method for the reduced curl-curl problem and some preliminary estimates. We will denote by |e| the length of an edge e. The discrete problem is to ﬁnd ˚uh ∈ Vh such that

(3.1) ah(˚uh, v) = (f, v) ∀ v ∈ Vh, An Interior Penalty Method for 2D Curl-Curl Problems 5 where

ah(w, v) = (∇h × w, ∇h × v)+ α(w, v) [Φ (e)]2 + µ [[n × w]][[n × v]] ds |e| Ze eX∈Eh [Φ (e)]2 (3.2) + µ [[n · w]][[n · v]] ds i |e| Ze eX∈Eh

−2 1 0 0 + h Πe[[n × w]] Πe[[n × v]] ds |e| Ze eX∈Eh −2 1 0 0 + h Πe[[n · w]] Πe[[n · v]] ds, i |e| Ze eX∈Eh

0 where Πe is the orthogonal projection from L2(e) onto P0(e) (the space of constant functions on e). The weight Φµ(e) in (3.2) is deﬁned by

L 1−µ` Φµ(e) = Π`=1|c` − me| , where me is the midpoint of e. Remark 3.1. The weight Φµ(e) is closely related to the weight Φµ(T ) in (2.1). In fact, we have

(3.3) Φµ(e) ≈ Φµ(T ) if e ⊂ ∂T.

The inclusion of Φµ(e) in (3.2) is crucial for the derivation of optimal convergence rates on graded meshes. Remark 3.2. The weak problem (1.4) for the reduced curl-curl problem can be written as

0 ah(˚u, v) = (f, v) ∀ v ∈ H0(curl; Ω) ∩ H(div ; Ω).

Remark 3.3. For nonconforming Crouzeix-Raviart P1 vector ﬁelds w and v, we have

ah(w, v) = (∇h × w, ∇h × v)+ α(w, v) [Φ (e)]2 + µ [[n × w]][[n × v]] ds |e| Ze eX∈Eh [Φ (e)]2 + µ [[n · w]][[n · v]] ds, i |e| Ze eX∈Eh which is the variational form used in [14]. The two additional sums in (3.2) compen- sate for the lack of weak continuity for the vector ﬁelds in Vh. Let the mesh-dependent energy norm k·kh be deﬁned by v 2 v 2 v 2 k kh = k∇h × kL2(Ω) + k kL2(Ω) [Φ (e)]2 [Φ (e)]2 (3.4) + µ k[[n × v]]k2 + µ k[[n · v]]k2 |e| L2(e) |e| L2(e) e∈E i Xh eX∈Eh 6 Susanne C. Brenner, Fengyan Li and Li-yeng Sung

1 1 + h−2 kΠ0[[n × v]]k2 + kΠ0[[n · v]]k2 . |e| e L2(e) |e| e L2(e) e∈E i Xh eX∈Eh Note that

0 (3.5) kvkL2(Ω) ≤kvkh ∀ v ∈ H0(curl; Ω) ∩ H(div ;Ω)+ Vh, and from (3.2) and (3.4),

0 (3.6) |ah(w, v)|≤ (|α| + 1)kwkhkvkh ∀ w, v ∈ [H0(curl; Ω) ∩ H(div ;Ω)]+ Vh.

For α> 0, ah(·, ·) is coercive, i.e.,

2 0 (3.7) ah(v, v) ≥ min(1, α)kvkh ∀ v ∈ H0(curl; Ω) ∩ H(div ;Ω)+ Vh. In this case the discrete problem is well-posed and we have the following abstract error estimate. Lemma 3.4. Let α be positive, β = min(1, α), ˚u be the solution of (1.4) and ˚uh satisfy (3.1). It holds that

1+ α + β 1 ah(˚u − ˚uh, w) (3.8) k˚u − ˚uhkh ≤ inf k˚u − vkh + max . 0 β v∈Vh β w∈Vh\{ } kwkh

Proof. Let v ∈ Vh be arbitrary. It follows from (3.7) and the triangle inequality that

k˚u − ˚uhkh ≤k˚u − vkh + kv − ˚uhkh

1 ah(v − ˚uh, w) ≤k˚u − vkh + max β w∈Vh\{0} kwkh

1+ α + β 1 ah(˚u − ˚uh, w) ≤ k˚u − vkh + max 0 β β w∈Vh\{ } kwkh which implies (3.8). For α ≤ 0, the following G˚arding equality holds:

2 0 (3.9) ah(v, v) + (|α| + 1)(v, v)= kvkh ∀ v ∈ [H0(curl; Ω) ∩ H(div ;Ω)] + Vh. In this case the discrete problem is indeﬁnite and the following lemma provides an abstract error estimate for the scheme (3.1) under the assumption that it has a solution. 0 Lemma 3.5. Let ˚u ∈ H0(curl; Ω) ∩ H(div ; Ω) satisfy (1.4) and ˚uh be a solution of (3.1). It holds that

ah(˚u − ˚uh, w) (3.10) k˚u − ˚uhkh ≤ (2|α| + 3) inf k˚u − vkh + max v∈Vh w∈Vh\{0} kwkh

+ (|α| + 1)k˚u − ˚uhkL2(Ω).

Proof. It follows from (3.5) and (3.9) that, for v ∈ Vh \{0},

ah(v, v) (v, v) (3.11) kvkh ≤ + (|α| + 1) kvkh kvkh An Interior Penalty Method for 2D Curl-Curl Problems 7

ah(v, w) ≤ max + (|α| + 1)kvkL2(Ω). w∈Vh\{0} kwkh

Let v ∈ Vh be arbitrary. We ﬁnd, using (3.5), (3.6), (3.11) and the triangle inequality, k˚u − ˚uhkh ≤k˚u − vkh + kv − ˚uhkh

ah(v − ˚uh, w) ≤k˚u − vkh + max + (|α| + 1)kv − ˚uhk 0 L2(Ω) w∈Vh\{ } kwkh

ah(˚u − ˚uh, w) ≤ (2|α| + 3)k˚u − vkh + max + (|α| + 1)k˚u − ˚uhk , 0 L2(Ω) w∈Vh\{ } kwkh which implies (3.10). From here on we consider α to be fixed and drop the dependence on α in our estimates. Remark 3.6. The first term on the right-hand side of (3.8) and (3.10) measures the approximation property of Vh with respect to the norm k·kh. The second term measures the consistency error. The third term on the right-hand side of (3.10) addresses the indefiniteness of the reduced curl-curl problem when α ≤ 0. As mentioned in Remark 2.1, the interpolation operator Πh is also the one used in [14]. Therefore in our analysis we can use the following two results from that paper (cf. Lemma 5.1 and Lemma 5.2 of [14]), which were obtained using (2.4), (2.8) and a representation of ˚u as the sum of a regular part and a singular part. 0 Lemma 3.7. Let ˚u ∈ H0(curl; Ω) ∩ H(div ; Ω) be the solution of (1.4). It holds that

2− (3.12) k˚u − Πh˚ukL2(Ω) ≤ Ch kfkL2(Ω) for any > 0.

0 Lemma 3.8. Let ˚u ∈ H0(curl; Ω) ∩ H(div ; Ω) be the solution of (1.4). It holds that 2 [Φµ(e)] 2 2− 2 (3.13) k[[˚u − Πh˚u]]k ≤ Ch kfk |e| L2(e) L2(Ω) eX∈Eh for any > 0, where [[˚u − Πh˚u]] is the jump of ˚u − Πhu across the interior edges of Th and [[˚u − Πh˚u]] = ˚u − Πh˚u on the boundary edges of Th. Lemma 3.9. It holds that

−2 2 2 2 1 1 |e| [Φµ(e)] kη − η¯Te kL2(e) ≤ Ch |η|H (Ω)) ∀ η ∈ H (Ω), eX∈Eh where 1 (3.14)η ¯Te = η dx |Te| ZTe is the mean of η over Te, one of the triangles in Th that has e as an edge. Proof. This is the consequence of (2.1), (3.3), the trace theorem (with scaling) and a standard interpolation error estimate [19, 15]:

−2 2 |e| [Φµ(e)] kη − η¯Te kL2(e) eX∈Eh 8 Susanne C. Brenner, Fengyan Li and Li-yeng Sung

0 Recall that Q is the L2 orthogonal projection operator onto H(div ;Ω). The following result will be useful in addressing the consistency error caused by the ap- pearance of Q in (1.5). Lemma 3.10. The following estimate holds:

0 (3.15) kv − QvkL2(Ω) ≤ Chkvkh ∀ v ∈ [H0(curl; Ω) ∩ H(div ;Ω)] + Vh.

0 Proof. Let v ∈ [H0(curl; Ω) ∩ H(div ;Ω)]+ Vh be arbitrary. Since v − Qv belongs 1 0 2 to ∇H0 (Ω), the orthogonal complement of H(div ; Ω) in [L2(Ω)] , we have, by duality, (v − Qv, ∇η) (v, ∇η) (3.16) kv − QvkL2(Ω) = sup = sup . 1 k∇ηk 1 k∇ηk η∈H0 (Ω)\{0} L2(Ω) η∈H0 (Ω)\{0} L2(Ω)

1 Let η ∈ H0 (Ω) be arbitrary. Since ∇ · v = 0 on each triangle T ∈ Th, we ﬁnd using integration by parts and the fact that η ∂Ω = 0,

0 (3.17) (v, ∇η)= (η − η¯Te )[[n · v]] ds + η¯Te Πe[[n · v]] ds = S1 + S2, i Ze i Ze eX∈Eh eX∈Eh

whereη ¯Te is deﬁned in (3.14). By the Cauchy-Schwarz inequality, (3.4) and Lemma 3.9, we have

1/2 2 1/2 −2 2 [Φµ(e)] 2 (3.18) S1 ≤ |e|[Φµ(e)] kη − η¯ k k[[n · v]]k Te L2(e) |e| L2(e) h i i h i i eX∈Eh eX∈Eh

≤ Chk∇ηkL2(Ω)kvkh.

On the other hand, from (3.4), the Cauchy-Schwarz inequality, and a Poincar´e- Friedrichs inequality, we ﬁnd

1/2 −1/2 0 S2 ≤ |e| kη¯Te kL2(e) |e| kΠe[[n · v]]kL2(e) i eX∈Eh 1/2 1 1/2 (3.19) ≤ Ch kη¯ k2 h−2 kΠ0[[n · v]]k2 Te L2(Te) |e| e L2(e) h i i h i i eX∈Eh eX∈Eh

≤ ChkηkL2(Ω)kvkh ≤ Chk∇ηkL2(Ω)kvkh.

Here we have also used the simple fact that, if e is an edge of a triangle T , then

2 2 (3.20) |e|kqkL2(e) ≤ CT kqkL2(T ) for any constant function q, where the positive constant CT depends only on the shape regularity of T . The estimate (3.15) follows from (3.16)–(3.19). An Interior Penalty Method for 2D Curl-Curl Problems 9

4. Convergence Analysis. We begin with three lemmas that provide estimates to the three terms on the right-hand side of (3.8) and (3.10). 0 Lemma 4.1. Let ˚u ∈ H0(curl; Ω) ∩ H(div ; Ω) be the solution of (1.4). It holds that

1− (4.1) inf k˚u − vkh ≤k˚u − Πh˚ukh ≤ Ch kfkL2(Ω) v∈Vh for any > 0. 0 0 Proof. Since Πe[[n × (˚u − Πh˚u)]] = 0 for all e ∈Eh and Πe[[n · (˚u − Πh˚u)]] = 0 for i all e ∈Eh, we have ˚u ˚u 2 ˚u ˚u 2 ˚u ˚u 2 k − Πh kh = k∇h × ( − Πh )kL2(Ω) + k − Πh kL2(Ω) 2 [Φµ(e)] 2 (4.2) + k[[n × (˚u − Πh˚u)]]k |e| L2(e) eX∈Eh [Φ (e)]2 µ n u u 2 + k[[ · (˚ − Πh˚)]]kL2(e). i |e| eX∈Eh

The second term on the right-hand side of (4.2) has been estimated in Lemma 3.7, and the third and fourth terms can be estimated using Lemma 3.8. Therefore it only remains to estimate the ﬁrst term. Observe that (2.6) implies

0 (4.3) ∇h × (Πh˚u) = Πh(∇× ˚u),

0 where Πh is the orthogonal projection from L2(Ω) onto the space of piecewise constant functions with respect to Th. It then follows from (1.6), (2.3), (4.3) and a standard interpolation error estimate [19, 15] that

u u 2 u 0 u 2 (4.4) k∇h × (˚ − Πh˚)kL2(Ω) = k∇ × ˚ − Πh(∇× ˚)kL2(Ω) 2 2 2 2 u 1 f ≤ Ch |∇ × ˚|H (Ω) ≤ Ch k kL2(Ω). The estimate (4.1) follows from (4.2), (4.4) and Lemmas 3.7–3.8. 0 Lemma 4.2. Let ˚u ∈ H0(curl; Ω)∩H(div ; Ω) be the solution of (1.4) and ˚uh ∈ Vh satisfy (3.1). It holds that

a (˚u − ˚u , w) (4.5) max h h ≤ Chkfk . 0 L2(Ω) w∈Vh\{ } kwkh

Proof. Let w ∈ Vh be arbitrary. Using integration by parts, the discrete problem (3.1), the strong form of the reduced curl-curl problem (1.5), and the fact that ∇h·w = 0, we ﬁnd

(4.6) ah(˚u − ˚uh, w)= ah(˚u, w) − (f, w)

= (Qf − f, w)+ (∇× ˚u)(nT × wT ) ds, Z∂T TX∈Th

n where wT = w T and T is the unit outer normal along ∂T . We see from (4.6) that there are two sources for the inconsistency of the scheme deﬁned by (3.1), namely the projection Q that appears in (1.5) and the discontinuity of the vector ﬁelds in Vh. 10 Susanne C. Brenner, Fengyan Li and Li-yeng Sung

The equation (4.6) can be rewritten as

ah(˚u − ˚uh, w) = (f,Qw − w)+ (∇× ˚u)[[n × w]] ds Ze eX∈Eh f w w ˚u ˚u n w (4.7) = ( ,Q − )+ (∇× − (∇× )Te )[[ × ]] ds Ze eX∈Eh ˚u 0 n w + (∇× )Te Πe[[ × ]] ds Ze eX∈Eh

˚u ˚u where (∇× )Te is the mean of ∇× on one of the triangles Te ∈ Th that has e as an edge. The ﬁrst two terms on the right-hand side of (4.7) satisfy the estimate

f w w ˚u ˚u n w f w (4.8) ( ,Q − )+ (∇× − (∇× )Te )[[ × ]] ds ≤ Chk kL2(Ω)k kh. Ze eX∈Eh

The derivation of (4.8), which is based on Lemma 3.9, Lemma 3.10 and (1.6), can be found in the proof of Lemma 6.2 in [14]. Using the Cauchy-Schwarz inequality, (1.6), (3.4) and (3.20), the third term on the right-hand side of (4.7) can be estimated as follows:

˚u 0 n w (∇× )Te Πe[[ × ]] ds Ze eX∈Eh 1/2 ˚u −1/2 0 n w (4.9) ≤ |e| k(∇× )Te kL2(e) |e| kΠe[[ × ]]kL2(e) eX∈Eh 1/2 1 1/2 ≤ Ch k(∇× ˚u) k2 h−2 kΠ0[[n × w]]k2 Te L2(Te) |e| e L2(e) eX∈Eh eX∈Eh

≤ Chk∇× ˚ukL2(Ω)kwkh ≤ ChkfkL2(Ω)kwkh.

The estimate (4.5) follows from (4.7)–(4.9). 0 Lemma 4.3. Let ˚u ∈ H0(curl; Ω)∩H(div ; Ω) be the solution of (1.4) and ˚uh ∈ Vh satisfy (3.1). It holds that

2− 1− (4.10) k˚u − ˚uhkL2(Ω) ≤ C h kfkL2(Ω) + h k˚u − ˚uhkh for any > 0. 0 Proof. We use a duality argument. Let ˚z ∈ H0(curl; Ω) ∩ H(div ; Ω) satisfy the reduced curl-curl problem

0 (4.11) ah(v,˚z) = (v, ˚u − ˚uh) ∀ v ∈ H0(curl; Ω) ∩ H(div ; Ω).

The strong form of (4.11) is

(4.12) ∇× (∇× ˚z)+ α˚z = Q(˚u − ˚uh), and we have the following analog of (1.6):

1 (4.13) k∇× ˚zkH (Ω) ≤ CΩk˚u − ˚uhkL2(Ω). An Interior Penalty Method for 2D Curl-Curl Problems 11

It follows from (4.12) and integration by parts that

(4.14) ah(˚uh,˚z) = (˚uh,Q(˚u − ˚uh)) + (∇× ˚z)[[n × ˚uh]] ds. Ze eX∈Eh From (4.11) and (4.14), we have ˚u ˚u 2 ˚u ˚u ˚u ˚u ˚u ˚u k − hkL2(Ω) = ( , − h) − ( h, − h)

(4.15) = ah(˚u − ˚uh,˚z)+ ah(˚uh,˚z) − (˚uh, ˚u − ˚uh)

= ah(˚u − ˚uh,˚z) − (˚uh, (I − Q)(˚u − ˚uh))

+ (∇× ˚z)[[n × ˚uh]] ds. Ze eX∈Eh We will estimate the three terms on the right-hand side of (4.15) separately. Using (4.7) and the fact that [[n × (Πh˚z)]] vanishes at the midpoints of all e ∈Eh, we can rewrite the ﬁrst term as

ah(˚u − ˚uh,˚z)= ah(˚u − ˚uh,˚z − Πh˚z)+ ah(˚u − ˚uh, Πh˚z)

(4.16) = ah(˚u − ˚uh,˚z − Πh˚z) + (f,QΠh˚z − Πh˚z)

˚u ˚u n ˚z + (∇× − (∇× )Te )[[ × (Πh )]] ds, Ze eX∈Eh ˚u ˚u where (∇× )Te is the mean of ∇× on one of the triangles Te ∈ Th that has e as an edge. Using the expression (4.16), the following estimate was obtained in the proof of Lemma 6.5 of [14]: 2− 1− (4.17) ah(˚u − ˚uh,˚z) ≤ C h kfkL2(Ω) + h k˚u − ˚uhkh)k˚u − ˚uhkL2(Ω). Since (I − Q)˚u = 0, the second term on the right-hand side of (4.15) can be estimated by Lemma 3.10:

(4.18) − ˚uh, (I − Q)(˚u − ˚uh) = ˚u − ˚uh, (I − Q)(˚u − ˚uh) ≤ Ck˚u − ˚uhkL2(Ω) hk˚u − ˚uhkh . Finally we estimate the third term on the right-hand side of (4.15). We have

˚z n ˚u ˚z ˚z n ˚u (4.19) (∇× )[[ × h]] ds = ∇× − (∇× )Te [[ × h]] ds Ze Ze eX∈Eh eX∈Eh ˚z 0 n ˚u + (∇× )Te Πe[[ × h]] ds, Ze eX∈Eh ˚z ˚z where (∇× )Te is the mean of ∇× on a triangle Te ∈ Th that has e as an edge. The following estimate can be found in the proof of Lemma 6.5 of [14]:

˚z ˚z n ˚u ˚u ˚u ˚u ˚u (4.20) ∇× − (∇× )Te [[ × h]] ds ≤ Chk − hkL2(Ω)k − hkh. Ze eX∈Eh On the other hand, we obtain by the Cauchy-Schwarz inequality, (3.4), (3.20) and (4.13),

˚z 0 n ˚u (∇× )Te Πe[[ × h]] ds Ze eX∈Eh 12 Susanne C. Brenner, Fengyan Li and Li-yeng Sung

˚z 0 n ˚u ˚u (4.21) = (∇× )Te Πe[[ × ( h − )]] ds Ze eX∈Eh

≤ Chk∇× ˚zkL2(Ω)k˚u − ˚uhkh ≤ Chk˚u − ˚uhkL2(Ω)k˚u − ˚uhkh. The estimate (4.10) follows from (4.15) and (4.17)–(4.21). In the case where α > 0, the following theorem is an immediate consequence of Lemma 3.4 and Lemmas 4.1–4.3. Theorem 4.4. Let α be positive. The following estimates hold for the solution ˚uh of (3.1):

1− (4.22) k˚u − ˚uhkh ≤ Ch kfkL2(Ω) for any > 0, 2− (4.23) k˚u − ˚uhkL2(Ω) ≤ Ch kfkL2(Ω) for any > 0.

In the case where α ≤ 0, we have the following convergence theorem for the scheme (3.1). Theorem 4.5. Assume that −α ≥ 0 is not a Maxwell eigenvalue. There exists a positive number h∗ such that the discrete problem (3.1) is uniquely solvable for all h ≤ h∗, in which case the following discretization error estimates are valid:

1− (4.24) k˚u − ˚uhkh ≤ Ch kfkL2(Ω) for any > 0, 2− (4.25) k˚u − ˚uhkL2(Ω) ≤ Ch kfkL2(Ω) for any > 0.

Proof. We follow the approach of Schatz [40] for indeﬁnite problems. Assuming ˚uh satisﬁes (3.1), it follows from (3.10) and Lemmas 4.1–4.3 that

1− (4.26) k˚u − ˚uhkh ≤ Ch kfkL2(Ω) + k˚u − ˚uhkh for any > 0. 1/(1−∗) By choosing an ∗ > 0, we deduce from (4.26) that, for h ≤ h∗ =1/(2C∗ ) ,

1−∗ k˚u − ˚uhkh ≤ C∗ h kfkL2(Ω) + k˚u − ˚uhkh 1−∗ 1−∗ ≤ C∗ h kfkL2(Ω) + C∗ h∗ k˚u − ˚uhkh

1−∗ 1 ≤ C ∗ h kfk + k˚u − ˚u k , L2(Ω) 2 h h and hence

1−∗ (4.27) k˚u − ˚uhkh ≤ 2C∗ h kfkL2(Ω).

Therefore, any solution ˚zh ∈ Vh of the homogeneous discrete problem

(4.28) ah(˚zh, v)=0 ∀ v ∈ Vh, which corresponds to the special case where f = 0 = ˚z, will satisfy the following special case of (4.27):

k˚zhkh ≤ 0. Hence the only solution of (4.28) is the trivial solution and the discrete problem (3.1) is uniquely solvable for h ≤ h∗. The energy error estimate (4.24) now follows (4.27), and the L2 error estimate (4.25) follows from Lemma 4.3 and (4.24). An Interior Penalty Method for 2D Curl-Curl Problems 13

2 5. Application to the Maxwell Eigenproblem. Given any f ∈ [L2(Ω)] , we 0 deﬁne T f ∈ H0(curl; Ω) ∩ H(div ; Ω) by the condition that

0 (5.1) (∇× (T f), ∇× v) + (T f, v) = (f, v) ∀ v ∈ H0(curl; Ω) ∩ H(div ; Ω).

0 2 Since H0(curl; Ω)∩H(div ;Ω) is a compact subspace of [L2(Ω)] (cf. [20]), the operator 2 2 T : [L2(Ω)] −→ [L2(Ω)] is symmetric, positive and compact. Moreover, (λ, ˚u) satisfy the Maxwell eigenproblem (1.7) if and only if 1 (5.2) T˚u = ˚u. 1+ λ

Similarly, let Thf ∈ Vh be deﬁned by the condition

(5.3) ah,1(Thf, v) = (f, v) ∀ v ∈ Vh, where ah,1(·, ·) is the bilinear form ah(·, ·) in (3.2) with α = 1, then 1 (5.4) Th˚uh = ˚uh 1+ λh is equivalent to

(5.5) ah,0(˚uh, v)= λh(˚u, v) ∀ v ∈ Vh, where ah,0(·, ·) is the bilinear form ah(·, ·) in (3.2) with α = 0. From Theorem 4.4 we have

2 (5.6) k(T − Th)fkL2(Ω) ≤ Ch kfkL2(Ω)

2 for all f ∈ [L2(Ω)] and

1− 1− (5.7) k(T − Th)fkh ≤ Ch kfkL2(Ω) ≤ Ch kfkh

0 for all f ∈ H0(curl; Ω) ∩ H(div ;Ω)+ Vh. Thus the symmetric ﬁnite rank operator Th converges uniformly to the symmetric positive compact operator T as h ↓ 0, which implies that the classical theory of spectral approximation [35, 6] can be applied to the approximation of the eigenvalues and eigenfunctions of T by the eigenvalues and eigenfunctions of Th. Hence we have the following result through the connections between (1.7) and (5.2) and between (5.5) and (5.4). Theorem 5.1. Let 0 ≤ λ1 ≤ λ2 ≤ . . . be the eigenvalues of (1.7), λ = λj = λj+1 = λj+m−1 be an eigenvalue with multiplicity m, and Vλ ⊂ H0(curl; Ω) ∩ 0 H(div ; Ω) be the corresponding m dimensional eigenspace. Let λh,1 ≤ λh,2 ≤ . . . be the eigenvalues of (5.5). Then, as h ↓ 0, we have

2− (5.8) |λh,` − λ|≤ Cλ,h ` = j,...,j + m − 1.

Furthermore, if Vh,λ ⊂ Vh is the space spanned by the eigenfunctions corresponding to λh,j ,...,λh,j+m−1, then the gap between Vλ and Vh,λ goes to zero at the rate 2− 1− of Ch in the L2 norm and at the rate of Ch in the norm k·kh. Proof. Let Eλ (resp. Eh,λ) be the L2 orthogonal projection onto Vλ (resp. Vh,λ). It follows from (5.6)–(5.7) and the classical theory of spectral approximation of compact operators [35, 6] that

2− (5.9) k(Eλ − Eh,λ)wkL2(Ω) ≤ Ch kwkL2(Ω) 14 Susanne C. Brenner, Fengyan Li and Li-yeng Sung

2 for all w ∈ [L2(Ω)] and

1− (5.10) k(Eλ − Eh,λ)wkh ≤ Ch kwkh

0 for all w ∈ H0(curl; Ω) ∩ H(div ;Ω)+ Vh. Let x ∈ Vh,λ be a unit eigenfunction of λh,` for some ` between j and j + m − 1. Then we have

(5.11) (Thx, x)= µh,` and kxkL2(Ω) =1, where (cf. (5.4))

−1 (5.12) µh,` = (1+ λh,`) .

Let xˆ = Eλx and yˆ = xˆ − x. Then xˆ and yˆ are orthogonal with respect to both the L2 inner product and the inner product (T ·, ·), and the estimate (5.9) implies that

2− (5.13) kyˆkL2(Ω) = k(Eλ − Eh,λ)xkL2(Ω) ≤ Ch .

In particular, it follows from Pythagoras’ theorem with respect to the L2 inner product that

xˆ 2 x 2 xˆ 2 yˆ 2 4− (5.14) 1 −k kL2(Ω) = k kL2(Ω) −k kL2(Ω) = k kL2(Ω) ≤ Ch .

Let eˆ = xˆ/kxˆkL2(Ω). Then eˆ is a unit eigenfunction of T corresponding to the eigenvalue (cf. (5.2))

(5.15) µ = (1+ λ)−1, and we have

(5.16) |µ − µh,`| = (T eˆ, eˆ) − (Thx, x) ≤ ((T − Th)x, x) + (T eˆ, eˆ) − (T x, x) .

From (5.6) we have

2− (5.17) ((T − Th)x, x) ≤ Ch , and it follows from Pythagoras’ theorem with respect to the inner product (T ·, ·) that

(5.18) (T eˆ, eˆ) − (T x, x) = (T eˆ, eˆ) − (T xˆ, xˆ) + (T yˆ, yˆ) x 2 e e y y ≤ (1 −kˆkL2(Ω))(T ˆ, ˆ) + (T ˆ , ˆ). Combining (5.13), (5.14) and (5.18), we ﬁnd

4− (T eˆ, eˆ) − (T x, x) ≤ Ch , which together with (5.16)–(5.17) implies

2− (5.19) |µ − µh,`|≤ Ch .

The estimate (5.8) follows from (5.12), (5.15) and (5.19). Recall that the gap δˆ(M,N) between two subspaces M and N of a normed linear space (X, k·kX ) is deﬁned by (cf. [35])

δˆ(M,N) = max(δ(M,N),δ(N,M)), An Interior Penalty Method for 2D Curl-Curl Problems 15 where

δ(M,N) = sup inf kx − ykX . x∈M y∈N kxkX =1

Therefore the statements about the gap between Vλ and Vh,λ follow immediately from (5.9) and (5.10). Remark 5.2. The compactness of the solution operator T and the existence of the uniform estimates (5.6) and (5.7) greatly simplify the analysis of the Maxwell eigenproblem. These ingredients are absent from Maxwell spectral approximations based on the full curl-curl problem and hence their justiﬁcations are much more involved [9, 10, 7, 17, 18, 8, 16, 11]. 6. Numerical Results. In this section, we report a series of numerical examples that corroborate our theoretical results. Besides the L2 error k˚u − ˚uhkL2(Ω) and the energy error k˚u − ˚uhkh, we also report the errors measured in the semi-norm | · |curl deﬁned by

|v|curl = k∇h × vkL2(Ω).

6.1. Source Problems. We ﬁrst demonstrate the performance of our scheme for the source problems where we take α to be −k2 for all the computations in this subsection. In the ﬁrst experiment, we check the convergence behavior of our numerical scheme (3.1) on the square (0, 0.5)2 with uniform meshes, where the exact solution is

(6.1) ˚u = [y(y − 0.5) sin(ky), x(x − 0.5) cos(kx)] for k = 0, 1 and 10. The results tabulated in Table 6.1 confirm the estimates (4.24) and (4.25). Observe also that, as expected, finer meshes are needed for computing satisfactory approximate solutions when the wave numbers become larger. Since our numerical scheme is designed for solving the reduced curl-curl problem (1.5), its performance should not be affected by the addition to the right-hand side of a 1 gradient term ∇G where G ∈ H0 (Ω). This is demonstrated by our second experiment. Here we add ∇G with

(6.2) G(x, y)= xy(x − 0.5)(y − 0.5) sin(x + y) to the right-hand side

(6.3) f = ∇× (∇× ˚u) − ˚u of the problem in the ﬁrst experiment (with k = 1) and compare their performance. The results are reported in Table 6.2. Note that the scheme (3.1) diﬀers from the one proposed in [14] because of the last two stabilizing terms. In the third experiment, we show that these two terms are necessary for the convergence of our method due to the lack of the continuity of the approximating functions across the element interfaces. See Table 6.3 for the numerical evidence. The last example in this subsection demonstrates the convergence behavior of our scheme on the L-shaped domain (−0.5, 0.5)2 \ [0, 0.5]2 with corners (0.5, 0), (0, 0), 16 Susanne C. Brenner, Fengyan Li and Li-yeng Sung

Table 6.1 Convergence of the scheme (3.1) on the square (0, 0.5)2 with uniform meshes and exact solution ˚u given by (6.1)

k˚u − ˚uhkL (Ω) ||˚u − ˚u || |˚u − ˚u | h 2 order h h order h curl order k˚ukL2(Ω) ||˚u||h |˚u|curl k =0 1/10 8.11E−02 − 3.61E−01 − 1.70E−01 − 1/20 1.85E−02 2.13 1.76E−01 1.04 8.52E−02 0.99 1/40 4.43E−03 2.07 8.69E−02 1.02 4.25E−02 1.00 1/80 1.08E−03 2.03 4.30E−02 1.01 2.12E−02 1.00 1/160 2.68E−04 2.02 2.15E−02 1.00 1.06E−02 1.00 k =1 1/10 7.19E−02 − 3.45E−01 − 1.64E−01 − 1/20 1.65E−02 2.12 1.68E−01 1.04 8.14E−02 1.01 1/40 3.93E−03 2.07 8.32E−02 1.02 4.09E−02 1.00 1/80 9.60E−04 2.04 4.13E−02 1.01 2.05E−02 1.00 1/160 2.37E−04 2.01 2.06E−02 1.01 1.02E−02 1.00 k = 10 1/10 4.82E−01 − 9.11E−01 − 4.33E−01 − 1/20 7.62E−02 2.66 3.65E−01 1.32 2.00E−01 1.12 1/40 1.70E−02 2.17 1.75E−01 1.06 9.89E−02 1.02 1/80 4.07E−03 2.06 8.63E−02 1.02 4.93E−02 1.01 1/160 9.99E−04 2.03 4.32E−02 1.00 2.47E−02 1.00

(0, 0.5), (−0.5, 0.5), (−0.5, −0.5) and (0.5, −0.5). We take k = 1 and the exact solution is chosen to be 2 π (6.4) ˚u = ∇× r2/3 cos θ − φ(r/0.5) , 3 3 where (r, θ) are the polar coordinates at the origin and the cut-oﬀ function is given by

1 r ≤ 0.25  −16(r − 0.75)3 φ(r)=  2  × 5+15(r − 0.75) + 12(r − 0.75) 0.25 ≤ r ≤ 0.75 0 r ≥ 0.75   The meshes are graded around the re-entrant corner with the grading parameter equal to 1/3. The results are tabulated in Table 6.4 and they agree with the estimates (4.24) and (4.25). 6.2. Maxwell Eigenproblem. In this subsection, we report the numerical results for the Maxwell eigenproblem. In the first experiment, the computation is carried out on the square domain (0, π)2 with uniform meshes. In this case, the exact eigenvalues are r2 + s2, r, s = 0, 1, 2, 3, 4, · · · with r2 +s2 6= 0. For instance, the first ten eigenvalues are 1, 1, 2, 4, 4, 5, 5, 8, 9, 9. In Figure 6.1, we plot the first twenty numerical eigenvalues versus the parameter n = π/h. The symbol “o” on the right side denotes the exact eigenvalue, An Interior Penalty Method for 2D Curl-Curl Problems 17

Table 6.2 Comparison of the performance of the scheme (3.1) on (0, 0.5)2 with k = 1 for the right-hand sides f and f + ∇G, where f is given by (6.3) and G is given by (6.2)

h rhs = f rhs = f + ∇G

k˚u − ˚uhkL2(Ω) 1/10 0.001643092661 0.001643020363 1/20 0.000376001607 0.000375992406 1/40 0.000089765112 0.000089764564 1/80 0.000021922450 0.000021922530 1/160 0.000005415103 0.000005417204

k˚u − ˚uhkh 1/10 0.051109229470 0.051109792169 1/20 0.024919760832 0.024919911664 1/40 0.012296880765 0.012296952986 1/80 0.006107323566 0.006107362343 1/160 0.003043400530 0.003043424464

Table 6.3 Loss of convergence without the last two stabilizing terms in the scheme (3.1). The computation is on the square (0, 0.5)2 with uniform meshes and the exact solution ˚u is given by (6.1) with k = 1

k˚u − ˚uhk |˚u − ˚u | h L2(Ω) order h curl order k˚ukL2(Ω) |˚u|curl 1/10 5.17E−00 − 2.61E−01 − 1/20 4.91E−00 0.22 1.71E−01 0.61 1/40 4.78E−00 0.04 1.33E−01 0.37 1/80 4.69E−00 0.03 1.20E−01 0.15 and “(2)” indicates that the multiplicity of the eigenvalue is 2. All the numerical approximations converge at a second order rate. To save space, only the first four numerical eigenvalues are included in Table 6.5. Furthermore, there is no spurious eigenvalue in our results. In the second experiment, we compute the eigenvalues for the L-shaped domain (−0.5, 0.5)2 \[0, 0.5]2 and the meshes are graded around the re-entrant corner with the grading parameter equal to 1/3. Table 6.6 contains the first four numerical eigenvalues, which show second order convergence of our method. This agrees with our analysis. In this case the “exact eigenvalues” are derived from numerical results of Dauge, which can be found at http://perso.univ-rennes1.fr/monique.dauge/core/index.html. In Figure 6.2, we plot the first ten numerical eigenvalues versus the parameter n = 1/2h. Again, there is no spurious eigenvalue. 7. Concluding Remarks. We have analyzed an interior penalty method with weak over-penalization for the two-dimensional reduced curl-curl problem and the Maxwell eigenproblem. This method is inconsistent but stable for arbitrary choices of the penalty parameter. We expect other discontinuous Galerkin schemes will be discovered along this line. An advantage of this new approach is that fast solvers for the resulting schemes can be constructed using techniques developed for elliptic problems. 18 Susanne C. Brenner, Fengyan Li and Li-yeng Sung

Table 6.4 Convergence of the scheme (3.1) with graded meshes on the L-shaped domain (−0.5, 0.5)2 \ [0, 0.5]2 with k = 1 and exact solution ˚u given by (6.4)

k˚u − ˚uhkL (Ω) k˚u − ˚u k |˚u − ˚u | h 2 order h h order h curl order k˚ukL2(Ω) k˚ukh |˚u|curl 1/4 1.77E+02 - 2.20E+01 - 9.13E−00 - 1/8 4.22E+01 2.07 7.76E−00 1.50 4.52E−00 1.02 1/16 3.42E−00 3.63 2.24E−00 1.79 7.86E−01 2.52 1/32 7.10E−01 2.27 1.09E−00 1.04 4.27E−01 0.88 1/64 1.78E−01 2.00 5.55E−01 0.97 2.30E−01 0.89

eig1 eig2 20 eig3 eig4 eig5 18 eig6 eig7 (2) eig8 16 (2) eig9 eig10 eig11 eig12 14 eig13 (2) eig14 eig15 12 eig16 eig17 eig18 10 (2) eig19 eig20 (2) exact

eigenvalue 8

6 (2) 4 (2)

2 (2) 0 20 40 60 n=pi/h

Fig. 6.1. First twenty numerical eigenvalues versus n = π/h for Maxwell operator on (0,π)2.

The scheme developed in this paper can be extended to three dimensional domains. In the case of uniform meshes, our two dimensional analysis can be easily generalized to establish convergence at a sub-optimal rate. It is likely that quasi- optimal convergence rates can be recovered using graded meshes, which is of course much more complicated in three dimensions [2, 3]. This will be one of the topics of our ongoing research. Acknowledgement. The first author would like to thank the Alexander von Humboldt Foundation for support through her Humboldt Research Award. Part of the research in this paper was carried out while the first and third authors were visiting the Humboldt Universität zu Berlin and they would like to thank the members of the An Interior Penalty Method for 2D Curl-Curl Problems 19

Table 6.5 First four numerical eigenvalues on (0,π)2

n = π/h 1st order 2nd order 3rd order 4th order 11 0.7524 − 0.7575 − 1.4971 − 3.0011 − 12 0.7823 1.54 0.7867 1.54 1.5580 1.55 3.1211 1.54 13 0.8076 1.60 0.8115 1.60 1.6096 1.61 3.2227 1.59 14 0.8291 1.65 0.8325 1.64 1.6534 1.65 3.3092 1.64 15 0.8475 1.69 0.8505 1.68 1.6907 1.70 3.3830 1.68 16 0.8632 1.72 0.8659 1.72 1.7227 1.73 3.4464 1.71 17 0.8767 1.75 0.8792 1.75 1.7503 1.76 3.5010 1.74 18 0.8885 1.77 0.8906 1.77 1.7741 1.78 3.5483 1.77 19 0.8987 1.80 0.9006 1.80 1.7948 1.80 3.5895 1.79 20 0.9076 1.81 0.9094 1.81 1.8130 1.82 3.6256 1.81 21 0.9154 1.83 0.9171 1.83 1.8289 1.84 3.6572 1.83 22 0.9223 1.85 0.9238 1.85 1.8429 1.85 3.6852 1.84 23 0.9284 1.86 0.9298 1.86 1.8553 1.87 3.7099 1.86 24 0.9339 1.87 0.9352 1.87 1.8664 1.88 3.7320 1.87 25 0.9387 1.88 0.9399 1.88 1.8762 1.89 3.7517 1.88 26 0.9431 1.89 0.9442 1.89 1.8851 1.90 3.7693 1.89 27 0.9470 1.90 0.9481 1.90 1.8930 1.91 3.7852 1.90 28 0.9506 1.91 0.9515 1.91 1.9002 1.91 3.7995 1.90 29 0.9538 1.91 0.9547 1.91 1.9067 1.92 3.8125 1.91 30 0.9567 1.92 0.9575 1.92 1.9125 1.93 3.8242 1.92 31 0.9593 1.93 0.9601 1.93 1.9179 1.93 3.8349 1.92 32 0.9617 1.93 0.9625 1.93 1.9228 1.94 3.8447 1.93 33 0.9639 1.93 0.9646 1.93 1.9272 1.94 3.8537 1.93 34 0.9659 1.94 0.9666 1.94 1.9313 1.95 3.8619 1.94 35 0.9678 1.94 0.9684 1.94 1.9351 1.95 3.8694 1.94 36 0.9695 1.95 0.9701 1.95 1.9386 1.95 3.8764 1.95 37 0.9711 1.95 0.9717 1.95 1.9418 1.96 3.8828 1.95 38 0.9726 1.95 0.9731 1.95 1.9447 1.96 3.8887 1.95 39 0.9739 1.96 0.9744 1.96 1.9475 1.96 3.8942 1.95 40 0.9752 1.96 0.9757 1.97 1.9500 1.96 3.8993 1.96 Exact 1 1 2 4

Institut f¨ur Mathematik for their hospitality.

REFERENCES

[1] Th. Apel. Anisotropic Finite Elements. Teubner, Stuttgart, 1999. [2] Th. Apel and S. Nicaise. The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges. Math. Methods Appl. Sci., 21:519–549, 1998. [3] Th. Apel, S. Nicaise, and J. Schöberl. Crouzeix-Raviart type finite elements on anisotropic meshes. Numer. Math., 89:193–223, 2001. [4] F. Assous, P. Ciarlet, Jr., S. Labrunie, and S. Lohrengel. The singular complement method. In N. Debit, M. Garbey, R. Hoppe, D. Keyes, Y. Kuznetsov, and J. Périaux, editors, Domain Decomposition Methods in Science and Engineering, pages 161–189. CIMNE, Barcelona, 2002. [5] F. Assous, P. Ciarlet, Jr., and E. Sonnendrücker. Resolution of the Maxwell equation in a domain with reentrant corners. M2AN Math. Model. Numer. Anal., 32:359–389, 1998. 20 Susanne C. Brenner, Fengyan Li and Li-yeng Sung

eig1 eig2 eig3 eig4 110 eig5 eig6 100 eig7 eig8 eig9 90 eig10 exact 80

50 eigenvalue

40 (2)

0 10 20 30 40 50 60 70 n=1/(2h)

Fig. 6.2. First ten numerical eigenvalues versus n = 1/2h for the Maxwell operator on (−0.5, 0.5)2 \ [0, 0.5]2.

[6] I. Babuˇska and J. Osborn. Eigenvalue Problems. In P.G. Ciarlet and J.L. Lions, editors, Handbook of Numerical Analysis II, pages 641–787. North-Holland, Amsterdam, 1991. [7] D. Boffi. Fortin operators and discrete compactness for edge elements. Numer. Math., 87:229– 246, 2000. [8] D. Boffi. Approximation of eigenvalues in mixed form, discrete compactness property, and application to hp mixed finite elements. preprint, 2005. [9] D. Boffi, F. Brezzi, and L. Gastaldi. On the convergence of eigenvalues for mixed formulations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 25:131–154, 1997. [10] D. Boffi, P. Fernandes, L. Gastaldi, and I. Perugia. Computational models of electromagnetic resonators: Analysis of edge element approximation. SIAM J. Numer. Anal., 36:1264–1290, 1999. [11] D. Boffi, F. Kikuchi, and J. Schöberl. Edge element computation of Maxwell’s eigenvalues on general quadrilateral meshes. Math. Models Methods Appl. Sci., 16:265–273, 2006. [12] A. Bossavit. Computational Electromagnetism. Academic Press, San Diego, 1998. [13] S.C. Brenner and C. Carstensen. Finite Element Methods. In E. Stein, R. de Borst, and T.J.R. Hughes, editors, Encyclopedia of Computational Mechanics, pages 73–118. Wiley, Weinheim, 2004. [14] S.C. Brenner, F. Li, and L.-Y. Sung. A locally divergence-free nonconforming finite element method for the reduced time-harmonic Maxwell equations. preprint, Department of Math- ematics, University of South Carolina, 2005 (to appear in Math. Comp.). [15] S.C. Brenner and L.R. Scott. The Mathematical Theory of Finite Element Methods (Second Edition). Springer-Verlag, New York-Berlin-Heidelberg, 2002. [16] A. Buffa and I. Perugia. Discontinuous Galerkin approximation of the Maxwell eigenproblem. preprint, 2006. [17] S. Caorsi, P. Fernandes, and M. Raffetto. On the convergence of Galerkin finite element approximations of electromagnetic eigenproblems. SIAM J. Numer. Anal., 38:580–607, An Interior Penalty Method for 2D Curl-Curl Problems 21

Table 6.6 The ﬁrst four numerical eigenvalues for the L-shaped domain (−0.5, 0.5)2 \ [0, 0.5]2 with graded meshes

n = π/h 1st order 2nd order 3rd order 4th order 2 2.6260 − 3.4424 − 5.0868 − 5.1743 − 3 4.0845 1.45 6.2091 0.74 7.4168 0.17 7.5007 0.17 4 4.7469 1.58 9.9625 2.23 11.2326 0.45 11.3161 0.44 5 5.1156 1.72 11.5421 2.13 17.1516 1.05 17.1606 1.04 6 5.3368 1.81 12.3228 1.96 24.8702 2.33 25.0031 2.37 7 5.4785 1.87 12.7934 1.95 28.4347 1.81 34.1115 6.44 8 5.5739 1.91 13.1020 1.96 30.6314 1.66 36.6531 4.81 9 5.6410 1.94 13.3157 1.97 32.2624 1.73 37.2948 2.19 10 5.6897 1.96 13.4698 1.98 33.4993 1.79 37.7336 2.13 11 5.7262 1.97 13.5844 1.98 34.4544 1.83 38.0505 2.10 12 5.7542 1.99 13.6720 1.99 35.2043 1.88 38.2878 2.09 13 5.7761 2.00 13.7404 1.99 35.8020 1.88 38.4702 2.08 14 5.7935 2.00 13.7948 2.00 36.2851 1.90 38.6137 2.07 15 5.8076 2.01 13.8387 2.00 36.6805 1.92 38.7286 2.07 16 5.8191 2.00 13.8747 2.00 37.0079 1.93 38.8220 2.06 17 5.8287 2.02 13.9045 2.00 37.2818 1.94 38.8990 2.06 18 5.8367 2.01 13.9296 2.01 37.5131 1.95 38.9632 2.05 19 5.8435 2.02 13.9508 2.00 37.7101 1.95 39.0173 2.05 20 5.8494 2.05 13.9689 2.00 37.8791 1.96 39.0633 2.05 21 5.8544 2.03 13.9845 2.01 38.0252 1.96 39.1028 2.05 Exact 5.9025 14.1361 39.4784 39.4784

2000. [18] S. Caorsi, P. Fernandes, and M. Raffetto. Spurious-free approximations of electromagnetic eigenproblems by means of Nédélec-type elements. M2AN, 35:331–358, 2001. [19] P.G. Ciarlet. The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1978. [20] M. Costabel. A remark on the regularity of solutions of Maxwell’s equations on Lipschitz domains. Math. Methods Appl. Sci., 12:36–368, 1990. [21] M. Costabel. A coercive bilinear form for Maxwell’s equations. J. Math. Anal. Appl., 157:527– 541, 1991. [22] M. Costabel and M. Dauge. Maxwell and Laméeigenvalues on polyhedra. Math. Methods Appl. Sci., 22:243–258, 1999. [23] M. Costabel and M. Dauge. Singularities of electromagnetic fields in polyhedral domains. Arch. Ration. Mech. Anal., 151:221–276, 2000. [24] M. Costabel and M. Dauge. Weighted regularization of Maxwell equations in polyhedral domains. Numer. Math., 93:239–277, 2002. [25] M. Costabel, M. Dauge, and S. Nicaise. Singularities of Maxwell interface problems. M2AN, 33:627–649, 1999. [26] M. Costabel, M. Dauge, and C. Schwab. Exponential convergence of hp-FEM for Maxwell equations with weighted regularization in polygonal domains. Math. Models Methods Appl. Sci., 15:575–622, 2005. [27] M. Crouzeix and P.-A. Raviart. Conforming and nonconforming finite element methods for solving the stationary Stokes equations I. RAIRO Anal. Numér., 7:33–75, 1973. [28] C. Hazard and S. Lohrengel. A singular field method for Maxwell’s equations: Numerical aspects for 2D magnetostatics. SIAM J. Numer. Anal., 40:1021–1040, 2002. [29] R. Hiptmair. Finite elements in computational electromagnetism. Acta Numer., 11:237–339, 2002. [30] P. Houston, I. Perugia, A. Schneebeli, and D. Schötzau. Interior penalty method for the indefinite time-harmonic Maxwell equation. Numer. Math., 100:485–518, 2005. 22 Susanne C. Brenner, Fengyan Li and Li-yeng Sung

[31] P. Houston, I. Perugia, A. Schneebeli, and D. Schötzau. Mixed discontinuous Galerkin approximation of of the Maxwell operator: the indefinite case. M2AN, 39:727–753, 2005. [32] P. Houston, I. Perugia, and D. Schötzau. Mixed discontinuous Galerkin approximation of of the Maxwell opertor. SIAM J. Numer. Anal., 42:434–459, 2004. [33] P. Houston, I. Perugia, and D. Schötzau. Mixed discontinuous Galerkin approximation of of the Maxwell operator: non-stabilized formulation. J. Sci. Comput., 22/23:315–346, 2005. [34] J. Jin. The Finite Element Method in Electromagnetics (Second Edition). John Wiley & Sons, Inc., New York, 2002. [35] T. Kato. Perturbation Theory of Linear Operators. Springer-Verlag, Berlin, 1966. [36] P. Monk. Finite Element Methods for Maxwell’s Equations. Oxford University Press, New York, 2003. [37] J.-C. Nédélec. Mixed finite elements in R3. Numer. Math., 35:315–341, 1980. [38] J.-C. Nédélec. A new family of mixed finite elements in R3. Numer. Math., 50:57–81, 1986. [39] I. Perugia, D. Schötzau, and P. Monk. Stabilized interior penalty methods for the time-harmonic Maxwell equations. Comput. Methods. Appl. Mech. Engrg., 191:4675–4697, 2002. [40] A. Schatz. An observation concerning Ritz-Galerkin methods with indefinite bilinear forms. Math. Comp., 28:959–962, 1974.