J Sci Comput (2012) 51:394–420DOI 10.1007/s10915-011-9513-3

Commuting Smoothed Projectors in Weighted Normswith an Application to Axisymmetric Maxwell Equations

J. Gopalakrishnan · M. Oh

Received: 12 January 2011 / Revised: 27 April 2011 / Accepted: 28 June 2011 /Published online: 22 July 2011© Springer Science+Business Media, LLC 2011

Abstract We construct finite element projectors that can be applied to functions with lowregularity. These projectors are continuous in a weighted norm arising naturally when mod-eling devices with axial symmetry. They have important commuting diagram propertiesneeded for finite element analysis. As an application, we use the projectors to prove qua-sioptimal convergence for the edge finite element approximation of the axisymmetric time-harmonic Maxwell equations on nonsmooth domains. Supplementary numerical investiga-tions on convergence deterioration at high wavenumbers and near Maxwell eigenvalues andare also reported.

Keywords Interpolation · Finite element · Clement · Schöberl projector · Smoothedprojector · Bounded cochain projector, axisymmetric, Maxwell, weighted Sobolev,axisymmetry, quasioptimality, pollution

1 Introduction

Projectors (or interpolation operators) into finite element subspaces of Sobolev spaces are afundamental ingredient in finite element error analyses. Every finite element has a canoni-cal projector defined by its degrees of freedom. Often however, a technical problem arises,namely the unboundedness of the canonical projection in the Sobolev space where the so-lution is sought. To overcome this, many early analyses assumed that the solution is regularenough to be contained in the domain of the canonical projection. Clément [10] offered an

This work was supported in part by the NSF under DMS-1014817. The authors also gratefullyacknowledge the support from the IMA (Minneapolis) during their 2010-11 annual program.

J. Gopalakrishnan (�)Department of Mathematics, University of Florida, Gainesville, FL 32611-8105, USAe-mail: jayg@ufl.edu

M. OhDepartment of Mathematics and Statistics, James Madison University, Harrisonburg, VA 22807, USAe-mail: ohmx@jmu.edu

J Sci Comput (2012) 51:394–420 395

alternative, at least for variational problems set in the Sobolev space H 1. The Clément in-terpolant is uniformly bounded in the L2-norm and gives optimal approximation estimates.However, in the analysis of mixed methods, one needs projectors with further commutativ-ity properties the Clément interpolant does not have. The importance of such commutingprojectors has been evident early on [7, 23, 24] and has only been enhanced in more re-cent works [2]. The basic idea of Clément was generalized by Schöberl in [26–28] to obtainsimilar projectors with the additional commutativity properties. His generalization was sub-stantial, requiring several new ideas. In this paper, we will refer to operators obtained by hismethod as Schöberl projectors. Further refinements of Schöberl’s ideas have been recentlymade in [8] (where the operators were called “smoothed projectors”) and in [2] (where theywere called “bounded cochain projectors”), but they do not extend to the case we intend tostudy here.

Our aim in this paper is to construct Schöberl projectors in the weighted norms arisingin the study of axisymmetric Maxwell equations. Under axial symmetry, the time harmonicMaxwell equations in cylindrical coordinates (r, θ, z) decouple [3, 12] into two systemsin the rz-halfplane. Due to the Jacobian arising from the change of variables however, wemust work in weighted Sobolev spaces, where the (degenerate) weight function is the radialcoordinate r . Let L2

r and Hr(curl) denote the r-weighted analogues of the L2 and H(curl)-spaces (see their definitions in Sect. 2). To adapt the standard finite element techniques tothese weighted spaces, we need commuting projectors in the weighted norms, in particu-lar, the Hr(curl)-norm for treating axisymmetric electromagnetics. A commuting projectorbounded in a more regular subspace of Hr(curl) is already known [12]. A weighted Clémentoperator has been constructed in [4] for application to the axisymmetric Stokes problem.Even a commuting projector bounded in Hr(curl) is also already known [11]. But, all theseprojectors are insufficient for various axisymmetric electromagnetic applications requiringlow-regularity estimates, including the development of adaptive and multigrid algorithms.Hence we take up the task of constructing Schöberl projectors in Hr(curl). In fact, antici-pating other applications, we will do so for all the spaces in an exact sequence of weightedSobolev spaces. As an example of how to apply the projector to obtain new results, we in-clude a simple error analysis in weighted norms, under minimal regularity assumptions, forthe axisymmetric indefinite time-harmonic Maxwell approximation following [22].

The outline of this paper follows the main steps in the construction of Schöberl projec-tors, as laid out in [26–28].

(1) First, we recall existing nodal interpolation operators which are well defined for suf-ficiently regular functions in the weighted spaces (in Sect. 2) and summarize resultsof [11, 12, 16] in this direction.

(2) Second, we introduce mesh dependent smoothing operators that are bounded in theweighted spaces (in Sect. 3) adapting the techniques in [28] to weighted spaces.

(3) Third, we compose the above two operations to form quasi-interpolation operatorsbounded in L2

r (in Sect. 4) as in [26, 28].(4) The quasi-interpolation operators are not projectors. So in a final step, we compose with

a finite dimensional inverse to obtain a projector, as in [27]. This construction is givenin Sect. 5, where the main result (Theorem 5.1) appears.

In Sect. 6, as an application, we use the projectors to prove an error estimate for the finiteelement method applied to the axisymmetric Maxwell equations under minimal regularityassumptions.

396 J Sci Comput (2012) 51:394–420

2 Preliminaries

In this section, we recall the definitions of the weighted Sobolev spaces and the nodal inter-polants of their corresponding finite element spaces.

Let R2+ denote the rz-halfplane and D ⊆ R

2+ be a domain with Lipschitz boundary. Dueto the axisymmetric applications we have in mind, we will further assume that the revolutionof D about the z-axis, namely �, also has Lipschitz boundary. The weighted L2-space isdefined by

L2r (D) =

{u :

∫D

|u|2rdrdz < ∞}

.

This is a Hilbert space with the inner product (u1, u2)r = ∫D

u1u2rdrdz. For a general do-main G we denote the L2

r -inner product on G by (·, ·)r,G. Let H 1r (D) be the space of all

functions in L2r (D) whose first order distributional derivatives are also in L2

r (D). The normand the seminorm on H 1

r (D) are defined by

‖u‖H 1r (D) =

(∫D

(|u|2 + |gradrzu|2) r drdz

)1/2

,

|u|H 1r (D) =

(∫D

|gradrzu|2 r drdz

)1/2

,

where gradrzu = (∂ru, ∂zu). Furthermore, for any real number 0 < s < 1, Hsr (D) is defined

as the Hilbert interpolation space [H 1r (D),L2

r (D)]1−s of index 1 − s between the spacesH 1

r (D) and L2r (D) [6]. In general, we will denote ‖·‖X and | · |X to indicate the norm and

the semi-norm respectively in a Sobolev space X.Define the two-dimensional curl operator by

curlrz(vr , vz) = ∂zvr − ∂rvz, (2.1)

and set

H r (curl;D) = {v ∈ L2

r (D)2 : curlrzv ∈ L2r (D)

}.

H r (curl;D) is a Hilbert space with the inner product

�(v1,v2) = (v1,v2)r + (curlrzv1, curlrzv2)r .

The induced norm is denoted by ‖ · ‖�.Now, let ∂D = �0 ∪ �1 where �0 is the open segment forming the part of the bound-

ary of D on the axis of symmetry (z-axis), and �1= ∂D \ �0 denotes the remainder of theboundary. (These notations are also marked in a later illustration—see Fig. 2.) Then, it iswell-known that functions in H 1

r (D) have traces in L2r (�1), i.e., for u in H 1

r (D), the traceu|�1 makes sense as a function in L2

r (�1), but trace on �0 is not defined in general [19].Additionally, since ∂D is Lipschitz, the tangential trace operator on H r (curl;D), which wewill denote by v · t |�1 , is proved in [12, Proposition 2.2] to be well-defined. Therefore, wecan define the following closed subspaces of H 1

r (D) and H r (curl;D):

H 1r,�(D) = {

u ∈ H 1r (D) : u|�1 = 0

},

H r,�(curl;D) = {v ∈ H r (curl;D) : v · t |�1 = 0

},

where t denotes the unit tangent vector oriented counter-clockwise.

J Sci Comput (2012) 51:394–420 397

We are interested in constructing commuting projectors on H 1r,�(D), H r,�(curl;D), and

L2r (D) that require lower-order regularity. Assume that D is meshed by a finite element

triangulation Th satisfying the usual geometrical conformity conditions [9]. We assume thatTh is quasiuniform with a representative mesh size h. The corresponding lowest order finiteelement subspace of H 1

r,�(D) is the space of continuous functions that are linear on eachmesh element, denoted by Vh,�. It has one nodal basis function φv for every mesh vertexv not on �1. The lowest order Nédélec subspace [23] of H r,�(curl;D), on the same meshis denoted by Wh,�, with its corresponding edge basis functions {φe}. And finally, let Sh

denote the L2r (D)-subspace with basis functions {φK} that are indicator functions of each

mesh element K . That the sequence

0 Vh,�gradrz

W h,�curlrz

Sh 0 (2.2)

is exact is proved in [11, Appendix A] under the further assumptions that �1 is connectedand D is simply connected. We also make the same assumptions throughout this paper.

We will need the following inequalities on polynomial spaces obtained by local homo-geneity arguments. Let K denote a triangle, r(y) denote the value of the radial coordinateat a point y ∈ R

2+,

hK = diam(K), rK = maxx∈K

r(x),

ρK denote the diameter of the largest circle inscribed in K . Finally, let P denote the spaceof polynomials of degree at most (for some ≥ 0). Throughout this paper, we use C

to denote a generic positive constant that is independent of {hK}. Its value may differ atdifferent occurrences and can depend on shape regularity ratio hK/ρK , but not on hK byitself. A proof of the following proposition is indicated in Appendix.

Proposition 2.1 For all v ∈ P,

‖gradrzv‖2L2

r (K)≤ Ch−2

K ‖v‖2L2

r (K), ‖gradrzv‖2

L∞(K) ≤ Ch−2K ‖v‖2

L∞(K),

rKh2K‖v‖2

L∞(K) ≤ C‖v‖2L2

r (K),

on any K ∈ Th.

For smooth functions, the canonical interpolation operators of the finite element spacesVh,�,W h,�, and Sh, namely, I

g

h , I ch , and I o

h , resp., are such that the following diagram com-mutes:

H 1r,�(D) ∩ C∞(D)

gradrz

Igh

V

Hr,�(curl,D) ∩ C∞(D)curlrz

I chV

L2r (D) ∩ C∞(D)

IohV

Vh,�gradrz

W h,�curlrz

Sh

(2.3)

The projectors Ig

h , I ch , and I o

h are defined by the natural degrees of freedom (see [12, 16] forthe modifications required in the degrees of freedom in weighted spaces), namely

(Ig

h ω)(x) =∑v∈V

ω(v)φv(x), (2.4)

398 J Sci Comput (2012) 51:394–420

(I chz)(x) =

∑e∈E

(∫e

z · t ds

)φe(x), (2.5)

(I oh s)(x) =

∑K∈Th

(1

|K|∫

K

s dx

)φK, (2.6)

where V denotes the set of vertices in Th not on �1, E denotes the set of edges in Th noton �1, |K| denotes the measure of K , and K ∈ Th denotes all triangles in Th. Note that I

g

h

and I ch cannot be applied, however, to all functions in H 1

r,�(D) and H r,�(curl,D), resp. Theycan at best be extended to more regular function spaces where the degrees of freedom makesense.

Our goal is to construct projectors, analogous to Ig

h and I ch , satisfying the same commuta-

tivity properties in (2.3), but bounded in L2r (D) or L2

r (D)2. Therefore, in the next section, wedefine mesh dependent smoothers for functions in L2

r (R2+) so that we can apply the classical

nodal interpolation operators after we apply these smoothers to L2r -functions.

3 Smoothing Operators

The purpose of this section is to define smoothing operators Sgu,Scv and Sow, which wewill use later. We introduce the notations and an intermediate result that will lead to Defini-tion 3.1.

Let a = (ar , az) be a point in R2+ and let Da be a closed disk of radius ρ, or its half,

centered around a or a, as shown in each of the three cases delineated in Fig. 1. We needfunctions supported within these disks that will act as kernels within the integral smoothingoperations to be defined. This will be obtained using the next proposition.

Proposition 3.1 Let ≥ 0. In all the three cases in Fig. 1, there exists a function ηa(r, z) ∈P such that

(ηa,p)r,Da = p(a), ∀p ∈ P, (3.1)

‖ηa‖2L2

r (Da )≤ C

ρ2ra

, where ra ={

ρ, in Case 1,miny∈Da r(y), in Cases 2 and 3,

(3.2)

‖ηa‖2L1

r (Da )≤ C

ra + ρ

ra

, where ‖ηa‖L1r (Da ) =

∫Da

|ηa | r(y) dy. (3.3)

A proof of Proposition 3.1 is included in Appendix.Next, let us define the “smoothing domains” Dh

a for each mesh vertex a ∈ Th. Let δ > 0be a global parameter. (We will need to choose it sufficiently small shortly.)

(1) If a is in (open) �0, then Dha is set to the Da in Case 1 of Fig. 1 with ρ = hδ.

(2) If a is in the interior of D, then Dha is set to Da of Case 2 in Fig. 1 with ρ = hδ.

Fig. 1 Domains Dacorresponding to point a

J Sci Comput (2012) 51:394–420 399

Fig. 2 A domain, a few vertexpatches, and associatedsmoothing domains

(3) If a is on �1, then Dha is set to Da of Case 3 in Fig. 1 with ρ = hδ and a = a +

c hδ(cos θ, sin θ).

These smoothing domains are illustrated in Fig. 2. We have to choose δ, c, and θ , notingthat δ and c are global constants, while θ ≡ θa can vary depending on a. Let Da denote the“vertex patch” of a, i.e., the domain formed by the union of all triangles in Th connectedto the mesh vertex a (see Fig. 2). For a not on �1, due to quasiuniformity, there exists0 < δ0 < 1 such that Dh

a ⊂ Da for all a and the smoothing domains of the distinct meshvertices do not intersect. We choose the parameters such that the following holds: (i) Forall a not on �1, we have Dh

a ⊂ Da and ra ≥ δh. (This can be ensured by choosing a δ ≤ δ0

and making δ even smaller if necessary.) (ii) For all a on �1, we have Dha ⊂ R

2+\D andra ≥ δh. (This can be ensured by choosing θ and c appropriately, due to the uniform coneproperty [18] implied by the Lipschitz regularity of the boundary.) Note that with thesesettings, we have

ra ≥ δh (3.4)

for all mesh vertices in Th, including vertices falling into Case 1.We need some more notations to define the smoothing operators. We use [. . .] to denote

the convex hull of its arguments. Accordingly, a triangle K with vertices a1, a2, and a3 isK = [a1,a2,a3]. Its three edges are e1 = [a2,a3], e2 = [a3,a1], and e3 = [a1,a2]. Let Dh

ai

be the smoothing domains introduced above for the vertices ai (i = 1,2,3), and let yi ∈ Dhai

.Set

κi(yi ) = r(yi )ηai(yi ), (3.5)

where ηaiis the function given by Proposition 3.1. We write κ123 = κ1κ2κ3 and κ12 = κ1κ2,

etc. For each x ∈ K , let λi(x) denote its barycentric coordinates in K so that

x =3∑

i=1

λi(x)ai .

Following [27], we now define xy by

xy(x,y1,y2,y3) =3∑

i=1

λi(x)yi (3.6)

and introduce these mesh dependent smoothers:

400 J Sci Comput (2012) 51:394–420

Definition 3.1 For all u,w ∈ L2r (R

2+) and v ∈ L2r (R

2+)2, define

Sgu(x) =∫

Dha1

∫Dh

a2

∫Dh

a3

κ123 u(xy) dy3dy2dy1,

Scv(x) =∫

Dha1

∫Dh

a2

∫Dh

a3

κ123

(dxy

dx

)T

v(xy) dy3dy2dy1,

Sow(x) =∫

Dha1

∫Dh

a2

∫Dh

a3

κ123 det

(dxy

dx

)w(xy) dy3dy2dy1,

for all x ∈ K and for each K ∈ Th.

It is easy to see that an equivalent way to write the last two operators is

Scv(x) =∫

Dha1

∫Dh

a2

∫Dh

a3

κ123

(3∑

i=1

yi · v(xy)gradrzλi(x)

)dy3dy2dy1, (3.7)

Sow(x) =∫

Dha1

∫Dh

a2

∫Dh

a3

κ123

(3∑

m=1

(∂rλm)ym

)×

(3∑

n=1

(∂zλn)yn

)w(xy) dy3dy2dy1, (3.8)

where for two dimensional vectors c = (cr , cz) and d = (dr , dz), the (wedge) cross productyields the scalar c × d = crdz − czdr . We will need to use the following properties of thesesmoothing operators.

Proposition 3.2 Let u ∈ H 1(R2+), v ∈ H r (curl,R2+) and w ∈ L2

r (R2+). Then the commuta-

tivity properties

Sc(gradrzu) = gradrz(Sgu), (3.9)

So(curlrzv) = curlrz(Scv) (3.10)

hold. Moreover, if (i, j, k) is a permutation of (1,2,3), the following identities hold:

Sgu(ai ) =∫

Dhai

κi u(yi ) dyi , (3.11)

∫ek

Scv · t ds =∫

Dhaj

∫Dh

ak

κij

∫[yj ,yk ]

v · t dsdykdyj , (3.12)

∫K

Sow dx =∫

Dha1

∫Dh

a2

∫Dh

a3

κ123

∫[y1,y2,y3]

w(z) dzdy3dy2dy1. (3.13)

Proof The commutativity properties (3.9) and (3.10) hold by construction. It is easy to seethis from Definition 3.1 and the Piola transformation.

To prove (3.11), we use (3.1) of Proposition 3.1, which implies that (ηa2 ,1)r,Dha2

=(ηa3 ,1)r,Dh

a3= 1. Since xy = y1 whenever x = a1 (see (3.6)) we have

Sgu(a1) =∫

Dha1

∫Dh

a2

∫Dh

a3

κ123u(y1)dy3dy2dy1 = (ηa2 ,1)r,Dha2

(ηa3 ,1)r,Dha3

∫Dh

a1

κ1u(y1)dy1.

This proves (3.11) (since similar identities obviously hold for i = 2,3 as well).

J Sci Comput (2012) 51:394–420 401

To prove (3.12), let q(s) = (1 − s)a2 + sa3,0 ≤ s ≤ 1, and e1 = [a2,a3]. Then, we havethat

∫e1

Scv · tds =∫ 1

0Scv(q(s))q ′(s)ds,

=∫ 1

0

∫Dh

a1

∫Dh

a2

∫Dh

a3

κ123

(3∑

i=1

yi · v((1 − s)y2 + sy3)gradrzλi

)

× (a3 − a2) dy3dy2dy1ds,

=∫ 1

0

∫Dh

a1

∫Dh

a2

∫Dh

a3

κ123(y3 − y2) · v((1 − s)y2 + sy3) dy3dy2dy1ds,

=∫

Dha1

∫Dh

a2

∫Dh

a3

κ123

∫[y2,y3]

v · t ds dy3dy2dy1,

=∫

Dha2

∫Dh

a3

κ23

∫[y2,y3]

v · t ds dy3dy2.

Here we have used the obvious identities gradrzλ1 · (a3 − a2) = 0, gradrzλ2 · (a3 − a2) =−1, and gradrzλ3 · (a3 − a2) = 1. The identities on the other edges follow similarly.

Finally, to prove (3.13), we use similar manipulations, with the additional observationthat

1

(∑3

m=1(∂rλm(x))ym) × (∑3

n=1(∂zλn(x))yn)

is the Jacobian arising from change of variables from x to xy . �

4 Quasi-interpolation Operators

The next step is to study the quasi-interpolation operators that are compositions of the canon-ical interpolants and the smoothers of the previous section. From here on, let u,w ∈ L2

r (D)

and v ∈ L2r (D)2. The trivial extension (by zero) of these functions to L2

r (R2+) and L2

r (R2+)2

will be denoted by u, w, and v, resp.

Definition 4.1 Define the quasi-interpolation operators Rg

h , Rch, and Ro

h, by

Rg

hu = Ig

h Sgu (Rg

h : L2r (D) �−→ Vh,�),

Rchv = I c

hScv (Rch : L2

r (D)2 �−→ Wh,�),

Rohw = I o

h Sow (Roh : L2

r (D)2 �−→ Sh).

Note that these operators are not projectors as they do not preserve functions that arealready in the finite element spaces. In the remainder of this section, we will establish twocategories of properties of these operators. The first consist of commutativity identities,collected in Lemma 4.1. The second category, collected in Lemma 4.2, consists of normestimates. In particular, we will prove that these quasi-interpolation operators are uniformlybounded in the weighted L2

r -norm.

402 J Sci Comput (2012) 51:394–420

Lemma 4.1 Rg

h , Rch, and Ro

h satisfy the following commutativity properties:

gradrz(Rg

hu) = Rch(gradrzu), ∀u ∈ H 1

r,�(D), (4.1)

curlrz(Rchv) = Ro

h(curlrzv), ∀v ∈ H r,�(curl,D). (4.2)

Proof This is obvious from the commutativity properties in Proposition 3.2 and (2.3). �

Lemma 4.2 There exists a constant C independent of h and δ such that

∥∥Rg

hu∥∥2

L2r (D)

≤ C

δ3‖u‖2

L2r (D)

, ∀u ∈ L2r (D), (4.3)

∥∥Rg

huh − uh

∥∥2

L2r (D)

≤ Cδ2 ‖uh‖2L2

r (D), ∀uh ∈ Vh,�, (4.4)

∥∥Rchv

∥∥2

L2r (D)

≤ C

δ3‖v‖2

L2r (D)

, ∀v ∈ L2r (D)2, (4.5)

∥∥Rchvh − vh

∥∥2

L2r (D)

≤ Cδ2 ‖vh‖2L2

r (D), ∀vh ∈ W h,�, (4.6)

∥∥Rohw

∥∥2

L2r (D)

≤ C

δ‖w‖2

L2r (D)

, ∀w ∈ L2r (D), (4.7)

∥∥Rohwh − wh

∥∥2

L2r (D)

≤ Cδ2 ‖wh‖2L2

r (D), ∀wh ∈ Sh. (4.8)

Proof Let K = [a1,a2,a3] be a fixed triangle in Th, and let DK denote the union of Dha1

,Dh

a2, and Dh

a3.

Proof of (4.3). Due to the shape regularity of Th and the fact that u is the extension of u

by zero, it suffices to prove local estimate

∥∥Rg

hu∥∥2

L2r (K)

≤ C

δ3‖u‖2

L2r (DK)

.

The nodal values of Rg

hu on K equal Sgu(ai ), which can be estimated by

|Sgu(ai )| =∣∣∣∣∫

Dhai

κi u(yi )dyi

∣∣∣∣ by (3.11),

= |(ηai, u)r,Dh

ai| by (3.5),

≤ ∥∥ηai

∥∥L2

r (Dhai

)‖u‖L2

r (Dhai

) ,

≤ C√(hδ)2rai

‖u‖L2r (Dh

ai) by Proposition 3.1, (3.2). (4.9)

Therefore,

∥∥Rg

hu∥∥2

L2r (K)

=∫

K

∣∣∣∣3∑

i=1

Sgu(ai )λi(x)

∣∣∣∣2

r(x)dx

≤ C

3∑i=1

|Sgu(ai )|2∫

K

|λi(x)|2r(x)dx,

≤ C

3∑i=1

1

(hδ)2rai

‖u‖2L2

r (Dhai

)h2rK by (4.9).

J Sci Comput (2012) 51:394–420 403

Since rK ≤ rai+ Ch, the ratio rK/rai

can be bounded by

rK

rai

≤ 1 + Ch

rai

≤ 1 + C

δ(4.10)

where we have used (3.4). This proves (4.3).

Proof of (4.4). Again, we only need to prove the local estimate∥∥R

g

huh − uh

∥∥L2

r (K)≤ Cδ ‖uh‖L2

r (DK) , (4.11)

because uh is the extension of uh by zero and Th is quasiuniform. First observe that

|(Sguh)(ai ) − uh(ai )|=

∫Dh

ai

κi(uh(yi ) − uh(ai ))dyi ,

≤ maxyi∈Dh

ai

|ai − yi |∥∥gradrzuh

∥∥L∞(Dh

ai)‖ηai

‖L1r (Dh

ai)

≤ Chδ∥∥gradrzuh

∥∥L∞(Dh

ai)

(rai

+ hδ

rai

)1/2

by (3.3),

≤ Chδ∥∥gradrzuh

∥∥L∞(Dh

ai)

by (3.4).

Hence,

∥∥Rg

huh − uh

∥∥2

L2r (K)

≤ C

3∑i=1

|(Sguh)(ai ) − uh(ai )|2∫

K

|λi(x)|2r(x)dx,

≤ C

3∑i=1

(hδ)2∥∥gradrzuh

∥∥2

L∞(Dhai

)h2rK

≤ C(hδ)2∥∥gradrzuh

∥∥2

L2r (DK)

≤ Cδ2 ‖uh‖2L2

r (DK),

where we have used Proposition 2.1 in the last step. This proves (4.11).

Proof of (4.5). Let CK denote the convex hull of Dha1

, Dha2

, and Dha3

. For the same reasonas in the previous proofs, it suffices to prove the local estimate

∥∥Rchv

∥∥2

L2r (K)

≤ C

δ3‖v‖2

L2r (CK )

. (4.12)

Let ei denote the edge connecting aj and ak . Recall that throughout, (i, j, k) denotes apermutation of (1,2,3). Since Rc

hv|K = ∑3i=1(

∫ei

Scv · tds)φei, with φei

= ±(λj gradrzλk −λkgradrzλj ), and since ‖φei

‖2L2

r (K)≤ CrK , we have

∥∥Rchv

∥∥2

L2r (K)

=∫

K

∣∣∣∣3∑

i=1

φei(x)

∫ei

Scv · t ds

∣∣∣∣2

r(x)dx,

≤ C

3∑i=1

∣∣∣∣∫

ei

Scv · t ds

∣∣∣∣2

‖φei‖2

L2r (K)

≤ CrK

3∑i=1

∣∣∣∣∫

ei

Scv · t ds

∣∣∣∣2

. (4.13)

404 J Sci Comput (2012) 51:394–420

Fig. 3 Illustration of the changeof variable in the proof of (4.5)

Let us now bound summands. By (3.12),

∣∣∣∣∫

ei

Scv · t ds

∣∣∣∣ =∣∣∣∣∣∫

Dhaj

∫Dh

ak

κjk

∫ 1

0v((1 − s)yj + syk) · (yk − yj ) ds dykdyj

∣∣∣∣∣≤ Ch

∫Dh

aj

∫Dh

ak

|κjk|∫ 1

0|v((1 − s)yj + syk)|ds dykdyj

≤ Ch

∫Dh

aj

∫Dh

ak

|κjk|(∫ 1/2

0+

∫ 1

1/2

)|v((1 − s)yj + syk)|ds dykdyj = A + B.

Here we have broken the integral with respect to s into two integrals, one over s ∈ [0,1/2](named A) and the other over s ∈ [1/2,1] (named B).

The first can be bounded as follows:

A :=∫

Dhak

h|κk|∫ 1/2

0

∫Dh

aj

|κj | |v((1 − s)yj + syk)|dyj ds dyk

≤∫

Dhak

h|κk|∫ 1/2

0

∥∥ηaj

∥∥L2

r (Dhaj

)

∥∥v((1 − s)yj + syk)∥∥

L2r (Dh

aj)ds dyk. (4.14)

Now, consider the change of variable z = (1 − s)yj + syk . Whenever 0 ≤ s ≤ 12 , we have

r(yj ) ≤ 2(1 − s)r(yj ) ≤ 2((1 − s)r(yj ) + sr(yk)) = 2r(z).

Hence, by Cauchy-Schwarz inequality,

∫ 1/2

0

∥∥v((1 − s)yj + syk)∥∥

L2r (Dh

aj)ds

=∫ 1/2

0

(∫Dh

aj

r(yj )|v((1 − s)yj + syk)|2 dyj

)1/2

ds

≤ C

(∫ 1/2

0

∫Zkj

r(z) |v(z)|2 dz

(1 − s)2ds

)1/2

,

where Zkj is the transformed domain under the change of variable—see Fig. 3. Clearly,(1 − s)−2 ≤ 4 and Zkj ⊂ CK . Therefore, continuing from (4.14) and using (3.2),

A ≤ Ch√(hδ)2raj

‖v‖L2r (CK) ‖ηak

‖L1r (Dh

ak).

By (3.3),

‖ηak‖L1

r (Dhak

) ≤ C

(1 + hδ

rak

)≤ C, (4.15)

J Sci Comput (2012) 51:394–420 405

Fig. 4 A domain of integrationby parts in the proof of (4.6)

where we have also used (3.4). Hence

A ≤ C

δ√

raj

‖v‖L2r (CK) . (4.16)

By a similar argument, we can bound the other integral B as well. Thus, returning to (4.13),we have

∥∥Rchv

∥∥2

L2r (K)

≤ C

δ2

(rK

raj

+ rK

rak

)‖v‖2

L2r (CK )

.

Estimating the ratios as before—see (4.10)—we prove (4.12).

Proof of (4.6). To perform a similar argument leading to (4.6), we now need to bound| ∫

e1(Scvh − vh) · t ds|. To this end, we will use an integration by parts over the area Ljk

enclosed by the line segments [aj ,ak], [ak,yk], [yk,yj ] and [yj ,aj ] (see Fig. 4), namely

∣∣∣∣∫

L

curlrzvhdx

∣∣∣∣ =∣∣∣∣∣∫

[aj ,ak ]+[ak ,yk ]+[yk ,yj ]+[yj ,aj ]vh · tds

∣∣∣∣∣ .

Beginning with (3.12) and using the above,

∣∣∣∣∫

ei

(Scvh − vh) · tds

∣∣∣∣

=∣∣∣∣∣∫

Dhaj

∫Dh

ak

κjk

(∫[yj ,yk ]

vh · t ds −∫

[aj ,ak ]vh · t ds

)dykdyj

∣∣∣∣∣≤

∫Dh

aj

∫Dh

ak

|κjk|(∣∣∣∣

∫Ljk

curlrzvh dx

∣∣∣∣ +∣∣∣∣∫

[aj ,yj ]vh · t ds

∣∣∣∣ +∣∣∣∣∫

[ak ,yk ]vh · t ds

∣∣∣∣)

dykdyj

≤∫

Dhaj

∫Dh

ak

|κjk|Chδ(h‖curlrzvh‖L∞(Ljk) + ‖vh‖L∞(CK)

)dykdyj ,

where we have used that |Ljk| ≤ Ch(hδ), |[aj ,yj ]| ≤ Chδ, and |[ak,yk]| ≤ Chδ. Thus,

∣∣∣∣∫

ei

(Scvh − vh) · tds

∣∣∣∣ ≤ Chδ(h‖curlrzvh‖L∞(CK) + ‖vh‖L∞(CK)

)‖ηaj‖L1

r (Dhaj

)‖ηak‖L1

r (Dhak

).

The L1r -norms above are uniformly bounded—see (4.15). Hence, using Proposition 2.1 and

proceeding as in (4.13), we reach

∥∥Rchvh − vh

∥∥2

L2r (K)

≤ CrKh2δ2 ‖vh‖2L∞(CK ) ≤ Cδ2 ‖vh‖2

L2r (CK )

,

thus completing the proof of (4.6).

406 J Sci Comput (2012) 51:394–420

Fig. 5 K is mapped to Ky underthe map x �→ xy for each choiceof {y}

Proof of (4.7). Let

J = det

(dxy

dx

)=

(3∑

m=1

(∂rλm)ym

)×

(3∑

n=1

(∂zλn)yn

)

be the Jacobian appearing in the definition of So—see (3.8). Let Ky denote the image of K

under the map x �→ xy (see Fig. 5). Obviously,

|J | = |Ky ||K| ≤ C. (4.17)

Then, by (2.6),

‖Rohw‖2

L2r (K)

= ‖I oh Sow‖2

L2r (K)

=∫

K

r(y)

∣∣∣∣φK

1

|K|∫

K

Sow dx

∣∣∣∣2

dy

≤ CrK

h2K

∣∣∣∣∫

K

Sow dx

∣∣∣∣2

, (4.18)

so now we proceed to estimate the last term. Let T = {x ∈ K : λ(x) > 13 }, for 1 ≤ ≤ 3.

Then we may overestimate the integral by

∫K

Sow dx ≤3∑

=1

∣∣∣∣∫

T

Sow dx

∣∣∣∣ . (4.19)

We will now bound the first summand (and the others will be similarly bounded). By Fu-bini’s theorem and Cauchy-Schwarz inequality,∣∣∣∣

∫T1

Sow dx

∣∣∣∣≤

∫T1

∫Dh

a1

∫Dh

a2

∫Dh

a3

|κ123J | ∣∣w(xy)∣∣ dy3dy2dy1 dx

=∫

Dha2

∫Dh

a3

|κ23|(∫

T1

∫Dh

a1

r(y1)∣∣ηa1(y1)w(xy) J

∣∣dy1 dx

)dy3 dy2

≤ ‖ηa1‖L2r (Dh

a1)

√|T1| |J |∫

Dha2

∫Dh

a3

|κ23|(∫

T1

∫Dh

a1

r(y1)|w(xy)∣∣2|J |dy1dx

)1/2

dy3dy2.

(4.20)

J Sci Comput (2012) 51:394–420 407

To estimate the integral in parenthesis, we first observe that since 1/λ1(x) < 3 for all x ∈ T1,the inequality

r(y1) = 1

λ1(x)λ1(x) r(y1) < 3r(λ1(x)y1)

≤ 3(r(λ1(x)y1) + r(λ2(x)y2) + r(λ3(x)y3)

) = 3r(xy)

holds, so∫

T1

∫Dh

a1

r(y1)|w(xy)|2|J | dy1 dx ≤ C

∫T1

∫Dh

a1

r(xy)|w(xy)|2|J | dy1 dx

= C

∫Dh

a1

∫T1

r(z)|w(z)|2 dzdy1

≤ C(hδ)2‖w‖2L2

r (CK),

where T1 is the image of T1 under the map x �→ xy and we have used the bound for itsJacobian in (4.17). Thus, returning to (4.20), using the above estimate together with Propo-sition 3.1,

∣∣∣∣∫

T1

Sow dx

∣∣∣∣ ≤ C√(hδ)2ra1

h(hδ)‖w‖L2r (CK )

∫Dh

a2

∫Dh

a3

|κ23|dy3 dy2.

Due to (4.15), this estimate simplifies to∣∣∣∣∫

T1

Sow dx

∣∣∣∣ ≤ Ch√ra1

‖w‖L2r (CK ).

A similar estimate holds for all the three integrals in (4.19).Therefore, returning to (4.18) we find that

‖Rohw‖2

L2r (K)

≤ C

(rK

ra1

+ rK

ra2

+ rK

ra3

)‖w‖2

L2r (CK)

and the proof is finished by the estimate (4.10).

Proof of (4.8). On the element K , by virtue of (3.1), we can write

(Rohwh − wh)

∣∣K

= 1

|K|∫

K

∫Dh

a1

∫Dh

a2

∫Dh

a3

κ123

(J wh(xy) − wh(x)

)dy3dy2dy1 dx

= 1

|K|∫

Dha1

∫Dh

a2

∫Dh

a3

κ123

(∫Ky

wh(z) dz −∫

K

wh(x) dx

)dy3dy2dy1

by a change of variables. The difference of the integrals within the parenthesis can be writtenas integrals over small domains. Indeed,

∫Ky

wh(z) dz −∫

K

wh(x) dx =∫

Ky∩K

(wh(z) − wh(z)) dz

+∫

Ky\Kwh(z) dz −

∫K\Ky

wh(x) dx,

408 J Sci Comput (2012) 51:394–420

where the first integral on the right hand side vanishes, and the remaining can be boundedusing |(Ky \ K) ∪ (K \ Ky)| ≤ Ch(hδ) – see Fig. 5. Thus,

‖Rohwh − wh‖2

L2r (K)

≤ Ch2rK

|K|2 h2(hδ)2‖wh‖2L∞(CK )

≤ Cδ2‖wh‖2L2

r (K)

by Proposition 2.1. This proves the last estimate (4.8). �

5 The Main Result

In this section we state and prove our main theorem on the existence of the commutingprojectors bounded in the weighted L2

r -norms. As already mentioned, the quasi-interpolationoperators obtained in the previous section are not projectors. So in this section, we modifythese operators to obtain projections. The definitions of the final projections appear belowin Definition 5.1 and the main result is Theorem 5.1.

The basic idea, again due to [27], stems from the observation that each of the op-erators R

g

h , Rch and Ro

h, when restricted to their respective range finite element spaces(Vh,�,Wh,�, Sh, resp.) are invertible for small δ.

Lemma 5.1 There are operators

Jg

h : Vh,� → Vh,�, J ch : Wh,� → Wh,�, J o

h : Sh → Sh,

and a δ1 > 0 such that for all 0 < δ < δ1, the operators Rg

h |Vh,� ,Rch|Wh,� , and Ro

h|Share

invertible, their inverses are Jg

h , J ch , and J o

h , resp., and their operator norms satisfy

‖J g

h ‖L2r (D) ≤ 2, ‖J c

h‖L2r (D) ≤ 2, ‖J o

h ‖L2r (D) ≤ 2.

Proof Considering any one of the three operators, say Rch|Wh,� , we note that by (4.6), there

exists a δ1 such that the L2r -operator norm of (I − Rc

h)|Wh,� is less than 1/2 for all δ < δ1.Consequently, by a standard Neumann series argument, the series

J ch := (

I − (I − Rch)|Wh,�

)−1 =∞∑

m=0

(I − Rch|Wh,�)

m, (5.1)

converges in L2r (D)-norm, the inverse (Rc

h|Wh,�)−1 = (I − (I −Rc

h)|Wh,�)−1 exists (which we

denote by J ch ), and moreover the norm bound

‖J ch‖L2

r (D) ≤ 1

1 − ‖(I − Rch)|Wh,�‖L2

r (D)

≤ 1

1 − (1/2)= 2,

holds. Similarly the other two inverses exist and the same bound holds. �

For the rest of the paper, we fix a δ ∈ (0, δ1], where δ1 is as given by Lemma 5.1. Fromnow on, we will let our generic constant C depend on (this fixed) δ. It will continue toremain independent of the mesh size h and the functions being estimated. We can now givethe final definition of the smoothed projectors.

J Sci Comput (2012) 51:394–420 409

Definition 5.1 Define �g

h : L2r (D) → Vh,�, �c

h : L2r (D)2 → Wh,�, and �o

h : L2r (D) → Sh by

�g

h = Jg

h Rg

h, �ch = J c

hRch, �o

h = J oh Ro

h.

Theorem 5.1 The above operators are projectors and have the following properties:

(1) Continuity. There exists a C > 0 such that

‖�g

hu‖L2r (D) ≤ C‖u‖L2

r (D), ∀u ∈ L2r (D),

‖�chv‖L2

r (D) ≤ C‖v‖L2r (D), ∀v ∈ L2

r (D)2,

‖�ohw‖L2

r (D) ≤ C‖w‖L2r (D), ∀w ∈ L2

r (D).

(2) Commutativity. The operators satisfy the following commuting diagram:

H 1r,�(D)

gradrz

�ghV

Hr,�(curl,D)curlrz

�chV

L2r (D)

�ohV

Vh,�gradrz

W h,�curlrz

Sh

(5.2)

(3) Approximation. For all 0 ≤ s ≤ 1,a) ‖u − �

g

hu‖r ≤ Chs‖u‖Hsr (D) for all u ∈ Hs

r (D).b) ‖v − �c

hv‖r ≤ Chs‖v‖Hsr (D) for all v ∈ Hs

r (D)2.c) ‖w − �o

hw‖r ≤ Chs‖w‖Hsr (D) for all w ∈ Hs

r (D).

Proof First, to verify that the operators are projectors, we observe that by Lemma 5.1,

(�g

h)2u = J

g

h Rg

hJg

h Rg

hu = Jg

h (Rg

h |Vh,�Jg

h )Rg

hu = Jg

h Rg

hu = �g

hu,

so �g

h is a projector. Similarly, the other two operators are also projectors.The continuity estimates follow from the estimates of Lemma 4.2 and Lemma 5.1. For

instance, ‖�ch‖L2

r (D) ≤ ‖J ch‖L2

r (D)‖Rch‖L2

r (D) ≤ 2Cδ−3/2.

The commutativity identities follow from the commutativity properties of the Rh-operators stated in Lemma 4.1 and those of the Jh-operators. The latter follows from theformer. For example, let J c

hvh = wh for vh,wh ∈ Wh,�. Then, Rchwh = vh and curlrzvh =

curlrz(Rchwh) = Ro

hcurlrzwh. Thus,

J oh (curlrzvh) = J o

h (Rohcurlrzwh) = curlrzwh = curlrz(J

chvh), (5.3)

for all vh ∈ Wh,�. Hence,

curlrz(�chv) = curlrz(J

chRc

hv) = J oh curlrz(R

chvh) by (5.3)

= J oh Ro

hcurlrzvh by Lemma 4.1

= �ohcurlrzvh.

This proves the commutativity property in the right end of the diagram (5.2). The remainingidentities are proved similarly.

410 J Sci Comput (2012) 51:394–420

To prove the approximation estimates, consider a general vh ∈ Wh,�. Since �ch is a pro-

jection �chvh = vh. Hence,

‖v − �chv‖L2

r (D) = ‖(v − vh) − �ch(v − vh)‖L2

r (D)

≤ (1 + C)‖v − vh‖L2r (D)

for any vh ∈ Wh,�. Hence taking the infimum over vh,

‖v − �chv‖L2

r (D) ≤ C infvh∈Wh,�

‖v − vh‖L2r (D).

Similar inequalities hold for the other two projectors as well.It remains to bound the best approximation error using approximants with known conver-

gence rates in the weighted norms. In the case of Sh, this is standard (see e.g., [4, Lemma 5]).In the case of Wh,� and Vh,�, this follows from [12]. Namely, the interpolants constructedin [12, Lemma 5.3] show that for all v ∈ H 1

r (D)2 and u ∈ H 1r (D), there are vh ∈ Wh,� and

uh ∈ Vh,� such that

‖v − vh‖L2r (D) ≤ Ch|v|H 1

r (D),

‖u − uh‖L2r (D) ≤ Ch|u|H 1

r (D).

Thus the estimates of item ((3)) are proved for the case s = 1. The same estimates in thecase s = 0 trivially follow item ((1)). For all intermediate values of s, the estimates followby the standard theory of interpolation of operators [5]. �

Remark 5.1 We considered the weight function r because of its many applications in ax-isymmetric problems. But the techniques are generalizable to handle other weight functions.The crucial estimates are those of Lemma 4.2. One would need to generalize them to thecase of the particular weight function of interest.

Remark 5.2 The quasi-interpolants of the previous section themselves have approximationproperties (even though they are not projectors). It may be possible to quantify these prop-erties in the higher order case by varying in Proposition 3.1. But note that in our analysiswe have only used the = 0 case of the proposition.

6 Application to Axisymmetric Maxwell Equations

In this section, we will use the commuting projectors of Sect. 5 to prove a convergence re-sult for the edge finite element approximation of the so-called “meridian” subproblem of theaxisymmetric Maxwell system. The three-dimensional (3D) time harmonic Maxwell equa-tions decouples into two two-dimensional (2D) problems: one called the azimuthal problemand the other the meridian problem [3, 12]. The meridian problem is posed on the right halfof the rz-plane (sometimes called the meridian half-plane). It finds the r and z componentsof the electric field, i.e., Erz = Erer + Ezez. The components Er and Ez are functions of r

and z alone (as there is no θ dependence due to axial symmetry).The meridian problem is to find Erz satisfying

curlrz

(1

μcurlrzErz

)− κ2εErz = F , (6.1)

J Sci Comput (2012) 51:394–420 411

where the scalar-valued curlrz is as defined in (2.1), the vector-valued curl is defined by

curlrzψ = (−∂zψ, r−1∂r(rψ)),

the material coefficient μ represents magnetic permeability, ε is the dielectric constant, F

represents given sources, all of which are axisymmetric, and κ is the wavenumber. Recallthat D ⊂ R

2+ denotes the restriction of the original axisymmetric 3D domain, which wewill call � ⊂ R

3, to the meridian half-plane, i.e., � is obtained by rotating D about thez-axis. Recall that we have assumed that D has Lipschitz boundary, �1 is connected, and D

is simply connected. This implies that ∂� is Lipschitz. We are able to perform the analysiswith this minimal regularity on ∂� due to the low-regularity projectors constructed earlier.

Perfect electric boundary conditions on ∂� translates to the boundary condition Erz · t =0 on �1, where t denotes the unit tangent vector. For error analysis, we consider a modelproblem for real-valued functions with this boundary condition, together with unit materialproperties. Its weak formulation is to find u ∈ H r,�(curl,D) such that

A(u,v) = (F ,v)r , ∀v ∈ H r,�(curl,D), (6.2)

where A : H r,�(curl,D) × H r,�(curl,D) �→ R is the bilinear form defined by

A(u,v) = (curlrzu, curlrzv)r − κ2(u,v)r .

There is a countable set of real values κ for which (6.2) does not have a unique solution [21].Throughout this paper, we will assume that κ is chosen so that κ2 is not a Maxwell eigen-value. Then (6.2) is uniquely solvable.

The corresponding discrete problem is to find uh ∈ Wh,� such that

A(uh,vh) = (F ,vh)r , (6.3)

for all vh ∈ Wh,�. For brevity, let us denote the H r (curl,D)-inner product by

(u,v)� = (curlrzu, curlrzv)r + (u,v)r ,

and the corresponding norm by ‖v‖� = (v,v)1/2� . The following error estimate is the main

new result we will prove in this section. Note that an error estimate for the positive defi-nite problem (obtained by replacing A(·, ·) by (·, ·)� in (6.3)) follows from the analysis in[12, Theorem 6.1]. However, the analysis of our indefinite problem is more involved.

Theorem 6.1 (Quasioptimality) Suppose (6.2) has a unique solution u ∈ H r,�(curl,D).Then, there are constants h0 and C such that, for all 0 < h < h0, (6.3) also has a uniquesolution uh, and

‖u − uh‖� ≤ C infwh∈Wh,�

‖u − wh‖� . (6.4)

The constants h0 and C depend on κ but are independent of u,uh, and h.

Similar results are well known for the standard (unweighted) Maxwell system. One of thefirst such results for the time-harmonic 3D system was proved by Monk in [20] using a varia-tion of the Schatz [25] duality argument. These techniques were refined and used in [15, 17]for preconditioning purposes using the new tools introduced in [1]. These works in turnprompted the development of a cleaner error analysis for the 3D Maxwell equations [22].

412 J Sci Comput (2012) 51:394–420

All these developments are summarized in the book [21, §7.2] which also details what isnow considered the standard proof of quasioptimality of edge element approximations ofthe 3D Maxwell system. The technique we will employ to prove Theorem 6.1 follows alongthe lines of this standard proof. We make a few further simplifications, possible due to theavailability of the Schöberl projectors (constructed in the previous section). However, weneed a few new ingredients to handle the additional complications resulting from our degen-erate weight function. We begin with a regularity result for the meridian problem (6.2).

Lemma 6.1 Suppose F ∈ L2r (D)2 satisfies

(F ,gradrzφ)r = 0 ∀φ ∈ H 1r,�(D), (6.5)

and suppose u solves (6.2). Then

‖u‖H

12

r (D)

+ ‖curlrzu‖H

12

r (D)

≤ C ‖F‖r .

Proof We use the available 3D regularity results. For any Sobolev space X(�), we willuse X(�) to indicate its subspace of axisymmetric (scalar or vector) functions. Given a 2Dvector field v(r, z) = (vr(r, z), vz(r, z)) on D, we define its revolution as a 3D vector fieldv�(x) on � by v�(r, θ, z) = vr(r, z)er + 0eθ + vz(r, z)ez.

By the isomorphisms of axisymmetry in [3] (such as between L2(�) and L2r (D)), we

know that u� ∈ H 0(curl,�) and F� ∈ L2(�)2. Moreover, it was shown in [12] that condi-tion (6.5) implies that (F�,grad ζ )r = 0 for all ζ ∈ H 1

0 (�). Therefore, by taking derivativein the sense of distributions, (divF�, ζ ) = 0 for all ζ ∈ D(�), and so divF� = 0 in L2(�).Similarly a direct calculation from (6.2) shows that curl curlu� −κ2u� = F�. All together,we have that u� ∈ H 0(curl,�) ∩ H (div,�). Therefore, since ∂� is Lipschitz, applying the3D result of [13, Theorem 2], we obtain

‖u�‖H

12 (�)

+ ‖ curlu�‖H

12 (�)

≤ ‖F �‖L2(�).

Now, it is known that [3] the spaces H12 (�) and H

12

r (D) × H12

r (D) × H12

r (D) are isomor-phic. Since the θ -component of curlu� is curlrzu, this implies the stated result. �

The next lemma concerns a “solenoidal” projection operator S. For any wh ∈ Wh,� defineSwh ∈ H r,�(curl,D), together with Pwh ∈ L2

r (D), as the unique solution of the dual mixedvariational equations

(Swh,v)r − (Pwh, curlrzv)r = 0, ∀v in H r,�(curl,D), (6.6)

(s, curlrzSwh)r = (s, curlrzwh)r , ∀s in L2r (D). (6.7)

(Note that setting v to gradients, we can conclude that Swh is solenoidal.)

Lemma 6.2 Let wh ∈ Wh,� satisfy (wh,gradrzφh)r = 0 for all φh ∈ Vh,�. Then

‖Swh − wh‖r ≤ Ch12 ‖curlrzwh‖r .

Proof By the exactness of the sequence (2.2), we know that wh = curl′rzph, for someph ∈ Sh, where curl′rz : Sh → Wh,� denotes the L2

r -adjoint of curlrz : Wh,� → Sh, i.e.,

(wh,vh)r − (ph, curlrzvh)r = 0, ∀vh ∈ Wh,�.

J Sci Comput (2012) 51:394–420 413

Subtracting this from (6.6), we obtain

(Swh − wh,vh)r − (Pwh − ph, curlrzvh)r = 0, ∀vh ∈ Wh,�. (6.8)

Now let vh = �chSwh − wh. Then curlrzvh = �o

hcurlrz(Swh) − curlrzwh by Theorem 5.1.Moreover, by (6.7), curlrz(Swh) = curlrzwh. Since �o

h is a projector, this implies thatcurlrzvh = 0. Thus (6.8) implies

‖Swh − wh‖r ≤ ∥∥Swh − �chSwh

∥∥r≤ Ch

12 ‖Swh‖

H

12

r (D)

.

Now, by a minor modification of [11, Theorem 3.2] to domains with Lipschitz boundary(using [13]) we obtain ‖Swh‖

H

12

r (D)

≤ C‖curlrzwh‖r . This completes the proof. �

Proof of Theorem 6.1 If the result holds, then the wellposedness of problem (6.3) follows,so we only need to prove the error estimate for small enough h. Let e = u − uh, and letwh ∈ Wh,� be arbitrary. Then A(e,wh) = 0, so

‖e‖2� = �(e,u − wh) + �(e,wh − uh),

≤ ‖e‖� ‖u − wh‖� + A(e,wh − uh) + (1 + κ2)(e,wh − uh)r ,

= ‖e‖� ‖u − wh‖� + (1 + κ2)(e,wh − uh)r . (6.9)

Next, we approximate (e,wh − uh)r in two parts. Let e = gradrzψ + β be the uniqueHelmholtz decomposition in the weighted spaces, i.e., ψ ∈ H 1

r,�(D) and β ∈ H r,�(curl,D).A discrete Helmholtz decomposition in weighted spaces is also available from [11], whichwe use to decompose wh − uh = gradrzξh + curl′rzsh with ξh ∈ Vh,� and sh ∈ Sh. Then(gradrzψ,gradrzξh)r = (e,gradrzξh)r = −κ−2A(e,gradrzξh) = 0. Hence,

(gradrzψ,wh − uh)r = (gradrzψ, curl′rzsh)r = (gradrzψ, curl′rzsh − S(curl′rzsh))r ,

≤ ‖e‖r

∥∥curl′rzsh − S(curl′rzsh)∥∥

r,

≤ Ch12 ‖e‖r

∥∥curlrz(curl′rzsh)∥∥

r

by Lemma 6.2. Thus, we have

(gradrzψ,wh − uh)r ≤ Ch12 ‖e‖r ‖curlrz(wh − uh)‖r (6.10)

which bounds a part of (e,wh − uh)r .

To bound the β-component, let z ∈ H r,�(curl,D) be the solution of

A(z,v) = (β,v)r ∀v ∈ H r,�(curl,D).

Then,

‖β‖2r = (β, e)r = A(z, e) = A(z − �c

hz, e) ≤ C∥∥z − �c

hz∥∥

�‖e‖� .

By Theorem 5.1, ‖z − �chz‖r ≤ Ch1/2‖z‖

H

12

r (D)

. Moreover, by the commutativity,

‖curlrz(z − �chz)‖r = ‖(I − �o

h)curlrzz‖r ≤ Ch1/2‖curlrzz‖H

12

r (D)

.

414 J Sci Comput (2012) 51:394–420

Hence,

‖β‖2r ≤ Ch

12

(‖z‖

H

12

r (D)

+ ‖curlrzz‖H

12

r (D)

)‖e‖�

≤ Ch12 ‖β‖r ‖e‖�

by Lemma 6.1. This together with (6.10) implies that

(e,wh − uh)r ≤ Ch12 ‖e‖� ‖wh − uh‖�. (6.11)

Therefore, returning to (6.9),

‖e‖2� ≤ ‖e‖� ‖u − wh‖� + Ch

12 ‖e‖� ‖wh − uh‖�

≤ ‖e‖� ‖u − wh‖� + Ch12 ‖e‖� ‖wh − u‖� + Ch

12 ‖e‖2

� .

Thus, we have proved that for any wh ∈ Wh,�,

‖u − uh‖� ≤ 1 + Ch12

1 − Ch12

‖u − wh‖�,

whenever 1 − Ch12 > 0, a condition satisfied for all 0 < h < h0 if we choose h0 < 1/C2. �

To conclude the discussion of this application, we present results from a numerical ex-periment. The above theory does not tell us the value of the quasioptimality constant C inTheorem 6.1. To get some indication of how large this constant is in a typical example, welet D be the unit square on the meridian half-plane, split by a uniform mesh of triangles(with positively sloped diagonals) of mesh size h = 1/128. We compute the approximatesolution by solving (6.3), but with the non-homogeneous boundary condition u · t = g. Weset F = (0,0) and choose g so that the exact solution is u = κJ0(κr)ez ≡ curlrz(J1(κr)).Let DE denote the discretization error ‖u − uh‖�. We are interested in comparing this to theerror in best approximation, namely the infimum appearing on the right hand side of (6.4),which we denote by BAE. Theorem 6.1 asserts that DE/BAE is bounded by C. To see thepractical manifestation of this result, we plot DE/BAE as a function of κ in Fig. 6(a). Notethat here, as we increase the wavenumber κ , we adjust the mesh size h so that κh = 0.78,resulting in approximately 8 points per wavelength. When κh is held fixed, the relative bestapproximation error (BAE/‖u‖�) remains approximately constant (about 18%) indepen-dent of κ , as seen from the second curve in Fig. 6(a). However, the first curve shows that theratio DE/BAE increases from the optimal value of 1 as κ is increased. We therefore expectthe quasioptimality constant C to also increase with κ . This is evidence of the well-knownpollution effect.

For cavity problems like (6.2), we also expect C to grow as we approach a cavity res-onance. To see this, we study the dependence with κ , in finer resolution, in the smallerinterval [0.2,6], for the same problem. This time, we fix h = 1/128, so there are meshpoints aplenty per wavelength. The relative BAE (shown in Fig. 6(b)) ranges from 0.1% to1% for κ in [0.2,6]. In this interval, there are 6 cavity resonances—see e.g., [14, Appendix]for the TM Maxwell eigenvalues on a cylinder. As seen from the two spikes in Fig. 6(b), ourdata seems to excite two of these modes more than the others. The spikes are near two ofthe eigenvalues.

J Sci Comput (2012) 51:394–420 415

Fig. 6 The ratio of discretization and best approximation errors vs. wavenumber

Appendix: Proof of Propositions 2.1 and 3.1

Both proofs involve scaling arguments where the weight function must be explicitly mapped.Due to the degeneracy of the weight function we must work with more than one referencedomain, as we will see.

Proof of Proposition 2.1 If K is an element that has no vertex on �0, all the stated inequali-ties follow easily from their standard (unweighted) analogues, so we only need to prove thatthey hold also on the remaining K ∈ Th. These remaining elements can be classified in twotypes: For n = 1 or 2, we say that a triangle K is of type n if K has exactly n vertices on �0.We define two reference triangles in the r z-plane, namely, K1 = [(0,0), (1,1), (1,−1)] andK2 = [(0,−1), (1,0), (0,1)] (see e.g., [12, Fig. 1]). Clearly, type n triangles are in affinehomeomorphism with Kn.

We will prove only the last inequality as the others are similar. Let K be of type 1 withvertices ai such that a1 is on the z-axis. Let F be the affine map that maps K1 one-one ontoK such that a1 is mapped to (0,0). Let v ∈ P be mapped to v on K1 by v(r, z) = v(r, z).Clearly, by the equivalence of norms on finite dimensional spaces, there exists a C dependingonly on such that

‖v‖L∞(K1) ≤ C‖v‖L2r(K1). (A.1)

Setting ri = r(ai ), let us note that r = r2λ2 + r3λ3 is mapped under F to r . Hence, the righthand side can be bounded by

∫K1

r|v|2 dx =∫

K

(λ2 + λ3)|v|2 |K1||K| dx ≤ max(r−1

2 , r−13 )

∫K

r|v|2 |K1||K| dx.

Since the L∞-norm is unchanged under the F -mapping, (A.1) implies

rKh2K‖v‖2

L∞(K) ≤ C2 max

(rK

r2,rK

r3

)h2

K |K1||K| ‖v‖2

L2r (K)

. (A.2)

416 J Sci Comput (2012) 51:394–420

By the shape regularity of Th, neither r2 nor r3 can be smaller than Ch. Since K has a vertexon the z-axis, we also know that rK ≤ Ch. Hence (A.2) implies rKh2

K‖v‖2L∞(K) ≤ C‖v‖2

L2r (K)

and the proof is complete for type 1 triangles.For type 2 triangles, the proof uses similar arguments using a map F that maps the edge

of K2 on the z-axis to the edge of K on the z-axis. We omit the details. �

Proof of Proposition 3.1 We prove the result in each of the three cases of Fig. 1. In eachcase, we have a different “reference” domain.

Case 1. In this case, a = (0, az) and Da = {(r, z) : r2 + (z − az)2 ≤ ρ2 and r ≥ 0}. The

reference domain in this case is

D1 = {(r, z) : r2 + z2 ≤ 1, and r ≥ 0}.

Consider the mapping

r = r

ρ, z = z − az

ρ(or r = rρ, z = ρz + az).

This map takes D1 one-one onto to Da and the Jacobian is ρ−2.On the reference domain, define η1 ∈ P by

∫D1

r η1p drdz = p(0) ∀p ∈ P, (A.3)

where P denotes the space of polynomials in r and z of degree at most . Set

ηa(r, z) = 1

ρ3η1(r, z). (A.4)

We will prove that this ηa satisfies all the stated properties.To prove (3.1), we observe that by change of variables p(r, z) = p(r, z),

∫Da

r ηa(r, z)p(r, z) drdz =∫

D1

(rρ)1

ρ3η1(r, z)p(r, z)ρ2 drdz by (A.4)

=∫

D1

r η1p drdz

= p(0) = p(a), by (A.3),

for all p ∈ P.The estimate (3.2) follows from (3.1). Indeed, since ηa ∈ P,

‖ηa‖2r,Da

=∫

Da

ηa(r, z)2r drdz = ηa(a) = η1(0)

ρ3≤ C

ρ2ra

.

Here we have used (A.4) and the fact that η1 is a fixed function on the reference domain andra = ρ in Case 1.

J Sci Comput (2012) 51:394–420 417

The last estimate (3.3) follows from (3.2) and Cauchy-Schwarz inequality:

‖ηa‖2L1

r (Da )≤

(∫Da

r dx

)‖ηa‖2

L2r (Da )

≤ maxx∈Da

r(x)|Da| C

ρ2ra

≤ C(ra + ρ)

ra

(A.5)

since |Da| = πρ2 and ra = ρ.

Case 2. Now a is a point in R2+ not on the z-axis, and Da = {(r, z) : (r − ar)

2 +(z − az)

2 ≤ ρ2}. The transformation

r = r − ar

ρand z = z − az

ρ(or r = ar + ρr, and z = az + ρz) (A.6)

maps Da one-one onto the “reference” domain D2 = {(r, z) : r2 + z2 ≤ 1}. Define η2(r, z) ∈P and η2,α(r, z) ∈ P by

∫D2

α η2,αp drdz = p(0) for all p ∈ P, (A.7)

for any positive function α(r, z) bounded above and below on D2. Define ηa ∈ P by

ηa(r, z) = 1

ρ2η2,α(r, z), with α = ar + ρr. (A.8)

Note that when the linear function r in the rz-plane is mapped over to the r z-plane by (A.6),we obtain the above α(r, z). Hence, for any polynomial p(r, z) = p(r, z),

∫Da

rηa(r, z)p(r, z)drdz =∫

D2

(ar + ρr)1

ρ2η2,α(r, z)p(r, z)ρ2 drdz =

∫D2

α η2,αp drdz

= p(0) = p(a),

for all p ∈ P. This proves (3.1).Next, to prove (3.2), we will first show that

η2,α(0) ≤ C

miny∈D2

α(y). (A.9)

Let η2 = η2,1, i.e., η2 equals η2,α defined by (A.7) with α set to 1. Then, since η2α ∈ P,

∥∥η2,α

∥∥2

L2α(D2)

= η2,α(0) =∫

D2

η2η2,αdrdz

≤ ∥∥η2

∥∥L2(D2)

(miny∈D2

(α(y)))−1/2 ∥∥η2,α

∥∥L2

α(D2).

Thus,

η2,α(0) = ∥∥η2,α

∥∥2

L2α(D2)

≤ C

miny∈D2(α(y))

,

418 J Sci Comput (2012) 51:394–420

where C = ‖η2‖2L2(D2)

depends only on the fixed reference domain. This proves (A.9).Hence,

‖ηa‖2L2

r (Da )= ηa(a) by (3.1), since ηa ∈ P

= 1

ρ2η2,α(0) by (A.8)

≤ C

ρ2 miny∈D2

(α(y))by (A.9)

= C

ρ2 miny∈Da

(r(y)).

The last equality holds, since α(r, z) = r at any point (r, z) mapped to (r, z). This completesthe proof of (3.2) for Case 2.

The proof of (3.3) proceeds as in Case 1—see (A.5)—using (3.2).

Case 3. Now Da is the closed disk with center a and radius ρ, where a = (ar , az) isobtained by (

ar

az

)=

(ar

az

)+ c

(cos θ − sin θ

sin θ cos θ

)(ρ

0

),

for some fixed angle θ ≥ 0. Consider the mapping

(r

z

)= 1

ρ

(cos θ sin θ

− sin θ cos θ

)(r − ar

z − az

),

or, equivalently, (r

z

)=

(ar

az

)+ ρ

(cos θ − sin θ

sin θ cos θ

)(r

z

). (A.10)

It is straightforward to show that this map sends the disk Da in the rz-plane one-one ontothe disk

D3 = {(r, z) : (r − c)2 + z2 ≤ 1}in the r z-plane. Since c is fixed, D3 forms our third fixed “reference” domain. Note that theJacobian of the change of variables from (r, z) to (r, z) is again ρ2.

As in the previous case, we now define η3,α ∈ P by∫

D3

α η3,αp drdz = p(c,0) for all p ∈ P, (A.11)

for any positive bounded function α(r, z) on D3. Next, we define ηa ∈ P by

ηa(r, z) = 1

ρ2η3,α(r, z), (A.12)

after setting

α(r, z) = ar + ρr cos θ − ρz sin θ.

J Sci Comput (2012) 51:394–420 419

It is obvious from (A.10) that this choice of α(r, z) is obtained by mapping the linear func-tion r to the r z-plane. Therefore,

∫Da

rηa(r, z)p(r, z) drdz =∫

D3

α(r, z)1

ρ2η3,α(r, z)p(r, z)ρ2 drdz = p(c,0) = p(a).

This proves (3.1).The proofs of (3.2) and (3.3) proceed similarly as in the previous cases, after one proves

that

η3,α(c,0) ≤ C

miny∈D3

α(y), (A.13)

which is the analogue of (A.9) of Case 2. �

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