on the schur complement form of the dirichlet-to-neumann operator

Available online at www.sciencedirect.com

Wave Motion 45 (2008) 309–324

www.elsevier.com/locate/wavemoti

On the Schur complement form of theDirichlet-to-Neumann operator

L. Knockaert *, D. De Zutter, G. Lippens, H. Rogier

Department of Information Technology, Ghent University, St. Pietersnieuwstraat 41, B-9000 Gent, Belgium

Received 14 December 2006; received in revised form 8 May 2007; accepted 10 July 2007Available online 28 July 2007

Abstract

The Dirichlet-to-Neumann operator relates the values of the normal derivative of a scalar field to the values of the fielditself on the boundary of a simply connected domain. Although it is easy to prove analytically that the Dirichlet-to-Neu-mann operator is self-adjoint, the discretization of the pertinent Green’s integral equation by means of the usual Galerkinapproach generally results in a non-symmetric matrix representation of that operator. In this paper we remedy this bymeans of a Hamiltonian Schur complement method and compare it with other symmetrization approaches. It is alsoshown that the problem of E-wave scattering by a lossy dielectric cylinder can be nicely simplified by means of the Dirich-let-to-Neumann operator. A numerical Galerkin implementation demonstrates the strength and versatility of the presentapproach.� 2007 Elsevier B.V. All rights reserved.

Keywords: Dirichlet-to-Neumann operator; E-wave scattering; Reciprocity; Hamiltonian Schur complement

1. Introduction

The Dirichlet-to-Neumann [1] or Poincare–Steklov [2] operator Y relates the values of the normal deriva-tive En of a scalar electric field E to the values of the field itself on the boundary c of a simply connecteddomain D. In other words we have En ¼ YE. Inside the domain D we suppose that the field E satisfies theHelmholtz equation for a homogeneous, non-magnetic material, i.e.,

0165-2doi:10.

* CoE-m

r2E þ k2E ¼ 0 ð1Þ

with k ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�jxl0ðrþ jx�Þ

p. Hence Y is dependent on the wavenumber k. For any two fields u, v satisfying (1),

we readily find by Green’s theorem that

0 ¼Z

cðvun � uvnÞds ¼

ZcðvYu� uYvÞds ð2Þ

125/$ - see front matter � 2007 Elsevier B.V. All rights reserved.1016/j.wavemoti.2007.07.004

rresponding author.ail address: [email protected] (L. Knockaert).

mailto:[email protected]

310 L. Knockaert et al. / Wave Motion 45 (2008) 309–324

which implies that the operator Y is self-adjoint. More precisely, we have that Y is a self-adjoint pseudo-dif-ferential operator of order 1 [1], with domain the space H1/2(c) and range the dual space H�1/2(c) [3,4]. One ofthe striking facts concerning the Dirichlet-to-Neumann operator is the observation [5] that the wavenumber inthe domain D may be replaced with the wavenumber k0 of the surrounding region, provided we introduce anequivalent surface current density |s related to the value of the electric field E on the boundary c by means of adifferential surface admittance operator defined by:

1 Anoperattransp

|s ¼ �E ¼def 1

jxl0

½Yk � Yk0�E ð3Þ

Note that in the sequel we will generally adopt the notation Y ¼ Yk and Y0 ¼ Yk0. It is shown in [5] that the

introduction of this equivalent surface current density greatly simplifies skin effect calculations of good (butnot perfect) conductors in the presence of external fields. A possible description of the Dirichlet-to-Neumannoperator is given by the well-known boundary integral equation

1

2EðrÞ ¼

Zc

o

on0gðr� r0Þ

� �Eðr0Þ � gðr� r0ÞEnðr0Þ

� �ds0 ð4Þ

where the Green’s function is gðr� r0Þ ¼ j4H ð2Þ0 ðkjr� r0jÞ. This can be compactly written in an easily under-

stood operator format:

1

2E ¼ gn0E � gEn ð5Þ

Hence, formally, from Eq. (5) we find

Y ¼ g�1ðgn0 � I=2Þ ð6Þ

where I is the identity operator. It is seen that the self-adjoint character of Y is not apparent in Eq. (6) at firstglance. Nevertheless, from the single layer potential representation

EðrÞ ¼Z

cgðr� r0Þnðr0Þds0 ð7Þ

with normal derivative [6]:

EnðrÞ ¼ �1

2nðrÞ þ

Zc

o

ongðr� r0Þ

� �nðr0Þds0 ð8Þ

which can be formally written as E = gn and En = �n/2 + gnn, we obtain:

Y ¼ ðgn � I=2Þg�1 ¼ Y 0 ð9Þ

thereby proving the self-adjointness of Y. Here we have utilized the fact that ðgnÞ0 ¼ gn0 and g 0 = g.

However, discretizing and solving the integral equation (4) by means of a Galerkin approach generallyresults in a non-symmetric matrix representation of the operator Y, and hence the reciprocity relation (2)at first sight does not seem to have a discrete equivalent. This is of course counter-intuitive and un-physical.In Section 2 we remedy this formally (in an involutive algebra setting1, see [7]) by means of a HamiltonianSchur complement approach. In Section 3 we successfully apply this to the problem of E-wave or transversemagnetic (TM) scattering by a lossy dielectric cylinder: it is shown that the single integral equation treatmentof [8] can be nicely simplified by means of the Dirichlet-to-Neumann operator. In Sections 4 and 5 pertinentGalerkin schemes are proposed, documented and implemented in order to find the Schur and other explicitlysymmetric discretized matrix formulations of the Dirichlet-to-Neumann operator. Finally, typical low fre-

involutive algebra is an operator algebra together with an involution0

such that (a0) 0 = a, (a + b) 0 = a0 + b0, (ab) 0 = b 0a0. Theor a 0 is called the adjoint operator of a. When a0 = a the operator a is called self-adjoint. In a matrix context a0 can stand for theose aT (real involution) or the Hermitian transpose aH (complex involution).

L. Knockaert et al. / Wave Motion 45 (2008) 309–324 311

quency wave propagation calculations are presented in order to demonstrate the versatility of the presentapproach.

2. A Hamiltonian Schur complement approach

Following [2,9] we obtain an explicitly self-adjoint expression for Y by adjoining to (4) the derived integralequation

1

2EnðrÞ ¼

Zc

o2

onon0gðr� r0Þ

� �Eðr0Þ � o

ongðr� r0Þ

� �Enðr0Þ

� �ds0 ð10Þ

which can formally be written as

1

2En ¼ gnn0E � gnEn ð11Þ

By construction, it is seen that gnn0 is self-adjoint. Eqs. (5) and (11) together can be written in block formatas

EEn

� �¼ K �G

J �K 0

� �EEn

� �¼def

HEEn

� �ð12Þ

with G = G 0 = 2g, K ¼ 2gn0 , K 0 = 2gn and J ¼ J 0 ¼ 2gnn0 . The operator H thus defined exhibits a Hamiltonianstructure [10], i.e., we have

0 I�I 0

� �H ¼ J �K 0

�K G

� �¼ 0 I

�I 0

� �H

� �0ð13Þ

We know that En ¼ YE with Y ¼ Y 0. From Eq. (12) it is also clear that Y satisfies the operatorequations

I ¼ K � GY ð14ÞY ¼ J � K 0Y ð15Þ

Furthermore, the square of the operator H is

H2 ¼ K2 � GJ �KGþ GK 0

JK � K 0J ðK2 � GJÞ0� �

ð16Þ

From Eq. (14) we obtain G ¼ KG� GYG, and from the adjoint of Eq. (14), i.e., I ¼ K 0 � YG, we obtainG ¼ GK 0 � GYG, implying that KG � GK 0 = 0. Also, pre-multiplying Eq. (14) with Y we obtainY ¼ YK � YGY, thereby showing that K 0Y ¼ YK. Similarly, from Eq. (15) we obtain YK ¼ JK � K 0YK, andfrom the adjoint of Eq. (15), i.e.,Y ¼ J � YK, we obtain K 0Y ¼ K 0J � K 0YK, implying that J K � K 0J = 0. Hence

H2 ¼ K2 � GJ 00 ðK2 � GJÞ0

� �ð17Þ

Eqs. (14) and (15) can be written in block format as:

IY

� �¼ H

IY

� �ð18Þ

Hence, applying H twice we obtain

IY

� �¼ H2 I

Y

� �¼ K2 � GJ 0

0 ðK2 � GJÞ0� �

IY

� �ð19Þ


implying that K2 � GJ = I. In other words H2 ¼ I , i.e., H is a Hamiltonian square root of the identity, imply-ing that the so-called Calderon operators P� ¼ ðI �HÞ=2 are oblique projectors in general2. The explicitlyself-adjoint form of Y can be found as the Schur complement [11,12]:

2 Sel

Y ¼ 1

2J � ðI � K 0ÞG�1ðI � KÞ� �

ð20Þ

For Z ¼ Y�1 we have a similar self-adjoint Schur complement:

Z ¼ 1

2�Gþ ðI þ KÞJ�1ðI þ K 0Þ� �

ð21Þ

Note that Y is a self-adjoint solution of the algebraic operator Riccati equation

2Y þ YGY � J ¼ 0 ð22Þ

Remark 1. Since K2 � GJ = I and KG = GK 0, implying G�1K = K 0G�1, we easily obtain (see also a similarargument in [13]):

J ¼ K 0G�1K � G�1 ð23Þ

Inserting this in equation (20) we find:

Y ¼ 1

2K 0G�1K � G�1 � ðI � K 0ÞG�1ðI � KÞ

¼ 1

2ðK 0 � IÞG�1 þ G�1ðK � IÞ

ð24Þ

This is exactly the symmetrized form of the solution of equation (14), which is Y ¼ G�1ðK � IÞ.

Remark 2. Consider the associated Hamiltonian ~H, obtained by replacing K with �K, i.e.,

~H ¼def �K �GJ K 0

� �ð25Þ

It is clear that we also have ~H2 ¼ I . The associated Hamiltonian ~H therefore also admits an associated Dirich-let-to-Neumann operator ~Y defined by the equations:

I ¼ �K � G~Y ð26Þ~Y ¼ J þ K 0 ~Y ð27Þ

yielding the self-adjoint Schur complement representations:

~Y ¼ 1

2J � ðI þ K 0ÞG�1ðI þ KÞ� �

ð28Þ

~Z ¼ ~Y�1 ¼ 1

2�Gþ ðI � KÞJ�1ðI � K 0Þ� �

ð29Þ

It is a simple matter to prove the following important identities:

� 2G�1 ¼ Y þ ~Y; 2J�1 ¼ Z þ ~Z ð30Þ2K ¼ GðY � ~YÞ ¼ ðZ � ~ZÞJ ð31Þ

Note that, given two self-adjoint invertible operators Y and ~Y, with respective inverses Z and ~Z, the identities(30) imply (31) and are sufficient to define a Hamiltonian H such that H2 ¼ I .

Remark 3. If we have the Dirichlet eigenvalues km and eigenfunctions nm(r) for the region D at our disposal,we can also write down the integral kernel of Y [5] as:

Yðr; r0Þ ¼X1m¼1

nmn ðrÞn

mn ðr0Þ

k2 � km

ð32Þ

f-adjoint idempotents are called orthogonal projectors, while non-self-adjoint idempotents are called oblique projectors.


Similarly, with the Neumann eigenvalues ~km and eigenfunctions ~nmðrÞ for the region D, we can write down theintegral kernel of Z:

Zðr; r0Þ ¼ �X1m¼1

~nmðrÞ~nmðr0Þk2 � ~km

ð33Þ

Interestingly enough, the representations (32) and (33) are the respective solutions to the algebraic operatorRiccati Eq. (22) in the degenerate cases G = 0 (Dirichlet Green’s function) and J = 0 (Neumann Green’sfunction).

3. TM scattering by a dielectric cylinder

As an application, we consider the TM scattering problem of a lossy dielectric cylinder with wavenumber k

and cross-section D, in an incoming field Ei(r). The pertinent integral equations are [8]:

1

2EðrÞ ¼

Zc

o

on0gðr� r0Þ

� �Eðr0Þ � gðr� r0ÞEnðr0Þ

� �ds0 ð34Þ

EiðrÞ � 1

2EðrÞ ¼

Zc

o

on0g0ðr� r0Þ

� �Eðr0Þ � g0ðr� r0ÞEnðr0Þ

� �ds0 ð35Þ

where g0ðr� r0Þ ¼ j4H ð2Þ0 ðk0jr� r0jÞ with k0 the wavenumber outside the cylinder. The integral equations (34)

and (35) can be written in an easily understood shorthand notation:

1

2E ¼ gn0E � gEn ð36Þ

Ei � 1

2E ¼ g0;n0E � g0En ð37Þ

� 1

2Esc ¼ g0;n0E

sc � g0Escn ð38Þ

where the scattered field Esc = E � Ei. Note that we have used the fact:

1

2Ei ¼ g0;n0E

i � g0Ein ð39Þ

which is equivalent with Ein ¼ Y0Ei. The scattered field Esc can be written as a single layer potential Esc = g0 f,

yielding the single integral equation for f:

1

2Ei � gn0E

i þ gEin ¼ � 1

2g0 �

1

2g þ gn0g0 � gg0;n

� �f ð40Þ

Inside the scatterer we can write the single layer potential E = gn, yielding the single integral equation for n:

Ei ¼ 1

2g0 þ

1

2g þ g0;n0g � g0gn

� �n ð41Þ

The integral equations (40) and (41), which were first introduced in [8] and later also documented in [14], canbe further reduced to a simpler form. With the notations of Section 2, the integral equations (36) and (37) canbe written as

E ¼ KE � GEn ð42Þ2Ei � E ¼ K0E � G0En ð43Þ

The integral equation (42) can be replaced with En ¼ YE, which inserted in (43) yields

Ei ¼ �g0~Y0 þ Y

E ð44Þ

The simple integral equation (44) allows us to find E on c directly. The relationships E = Ei + g0f and E = g n(7) then yield f and n and therefore the complete solutions inside and outside the scatterer. The integral equa-


tion (44) presents an interesting twist: since the first identity of (30) tells us that Y0 þ ~Y0 ¼ �g�10 , Eq. (44) can

be written as

Ei ¼ E þ g0 Y0 � Y½ �E ð45Þ

which is a Fredholm equation of the second kind. Eq. (45), which exhibits the differential Dirichlet-to-Neu-mann operator [5] Y0 � Y, can also be found by another means, for instance by introducing the additional‘phantom’ integral equation:

1

2E ¼ g0;n0E � g0

~En ð46Þ

We can always do so provided ~En ¼ Y0E. Subtracting (46) from (37) we obtain

Ei � E ¼ g0~En � En

ð47Þ

which is precisely integral equation (45). Note that Eq. (45) implies that

Esc ¼ g0 Y � Y0½ �E ð48Þ

resulting in f ¼ ½Y � Y0�E. Finally, let us effectively prove that the formidable integral equations (40) and (41)can be reduced to the very simple integral equation (44). Regarding integral equation (41), it is relatively easyto show that

1

2g0 þ

1

2g þ g0;n0g � g0gn ¼ �g0

~Y0 þ Y

g ð49Þ

which is consistent with integral equation (44). We can also find an explicit expression for n, i.e.,

n ¼ � ~Y þ Y0

Ei � ~Y � ~Y0

Esc ð50Þ

The case of integral equation (40) is more difficult; however, it is not too hard to prove that

1

2Ei � gn0E

i þ gEin ¼ g Y0 � Y½ �Ei ð51Þ

and

� 1

2g0 �

1

2g þ gn0g0 � gg0;n ¼ g ~Y0 þ Y

g0 ð52Þ

With the proviso that nullspace(g) = {0}, integral equation (40) is equivalent with

Y0 � Y½ �Ei ¼ ~Y0 þ Y

Esc ð53Þ

which is easily transformed into Eq. (48). The above derivations also show the conceptual superiority of theDirichlet-to-Neumann approach as compared to the usual integral equation techniques.

4. The projected Hamiltonian matrix

As was said in Section 1, Y is a self-adjoint pseudo-differential operator of order 1 mapping H1/2(c)to H�1/2(c), where c is the boundary contour of the relevant region D. It should be noted that H1/2(c) is aSobolev–Hilbert space with norm [15] p. 208:

kEk1=2 ¼Z

cjEðsÞj2dsþ ‘

Zc

Zc

jEðsÞ � Eðs0Þj2

R2dsds0

!12

ð54Þ

In Eq. (54) R stands for the Euclidian distance between the two points parametrized by s and s 0 on the contourc, and ‘ is a positive parameter proportional to the length of the contour c in order to make formula (54)dimensionally correct. The dual norm kEk�1/2 is defined via the usual procedure (see [4] p. 320), but is virtuallyimpossible to compute in any practical situation. Fortunately, we know [3] that


H 1=2ðcÞ � L2ðcÞ � H�1=2ðcÞ ð55Þ

with H1/2(c) dense in L2(c). For instance, as a consequence of the proper subset equation (55), there are func-tions in L2(c) that are not in H1/2(c), and functions in H�1/2(c) that are not in L2(c).

In the context of a Galerkin procedure for obtaining an approximation to Y, this is important, since wehave to develop E into basis functions in H1/2(c) and En into basis functions in H�1/2(c). Note that the secondintegral in formula (54) is a so-called hypersingular integral, which diverges for piecewise constant functions Ebut converges for piecewise linear functions E [16]. This having been specified, recall that our pertinent equa-tions are

E ¼ KE � GEn ð56ÞEn ¼ JE � K 0En ð57ÞEn ¼ YE ð58Þ

The fields E and En are expanded as

E ¼X1k¼1

akbkðsÞ ð59Þ

En ¼X1k¼1

bktkðsÞ ð60Þ

where the set of real basis functions {bk(s)} and {tk(s)} are complete, respectively, in H1/2(c) and H�1/2(c).Inserting this in Eqs. (56) and (58) we obtain
X1
k¼1

akbk ¼X1k¼1

akKbk �X1k¼1

bkGtk ð61Þ

X1k¼1

bktk ¼X1k¼1

akJbk �X1k¼1

bkK 0tk ð62Þ

X1k¼1

bktk ¼X1k¼1

akYbk ð63Þ

A self-consistent Galerkin procedure is obtained by testing Eq. (61) with the set {tk}, while testing Eqs. (62)and (63) by means of the set {bk}, resulting in
X1
k¼1

akhtl; bki ¼X1k¼1

akhtl;Kbki �X1k¼1

bkhtl;Gtki ð64Þ

X1k¼1

bkhbl; tki ¼X1k¼1

akhbl; Jbki �X1k¼1

bkhbl;K 0tki ð65Þ

X1k¼1

bkhbl; tki ¼X1k¼1

akhbl;Ybki ð66Þ

Note that, the operator J being a hypersingular operator [16], the self-patch elements hbk,Jbki are finite if andonly if the basis functions bk belong to H1/2(c), since the hypersingular integral present in the norm (54) exhib-its the same singularity as J. In the same context, it is known that operator J can alternatively be written as anintegro-differential operator [9], i.e.,

J ¼ k2nðrÞ � nðr0Þgðr� r0Þ þ ogðr� r0Þos0

o

osð67Þ

resulting in the Galerkin elements:

hbk; Jbli ¼ k2

Zc

Zc

bkðsÞblðs0ÞnðrÞ � nðr0Þgðr� r0Þdsds0 �Z

c

Zc

b0kðsÞb0lðs0Þgðr� r0Þdsds0 ð68Þ


Of course, the derivation of formula (68) requires the slightly more stringent assumption that the basis func-tions {bk} belong to H1(c), which is, e.g., the case when the {bk} are piecewise boundedly differentiable, say inpractice when one uses the well-known linear rooftop functions [16].

Eqs. (64)–(66) can be written in the easily understood infinite matrix format

Ta ¼ Ka� Gb ð69ÞTTb ¼ Ja� KTb ð70ÞTTb ¼ Ya ð71Þ

where

Tkl ¼ htk; bli Kkl ¼ htk;Kbli Gkl ¼ htk;Gtli Jkl ¼ hbk; Jbli Ykl ¼ hbk;Ybli ð72Þ

Eliminating the unknowns a and b, we obtain the projected infinite block matrix equations

IY

� �¼ H

IY

� �ð73Þ

with projected Hamiltonian matrix

H ¼ T�1K �T�1GT�T

J �KT T�T

� �¼def bK �bGbJ �bKT

� �ð74Þ

The explicitly symmetric form of Y can be found as the Schur complement:

Y ¼ 1

2bJ � ðI� bKTÞbG�1ðI� bKÞh i

¼ 1

2J� ðT� KÞTG�1ðT� KÞh i

ð75Þ

It is seen that, exactly in parallel with the operator formalism of Section 2, the requirement H2 = I is needed,

implying that bKbG ¼ bG bKT, bJ bK ¼ bKTbJ and bK2 � bGbJ ¼ I. Also, following Remark 1 of Section 2, it follows thatY can similarly be written in the symmetrized form

Y ¼ 1

2ðbKT � IÞbG�1 þ bG�1ðbK � IÞh i

¼ 1

2ðK� TÞTG�1Tþ TTG�1ðK� TÞh i

ð76Þ

and hence the hypersingular Galerkin matrix J ¼ bJ can be written as

J ¼ bKTbG�1 bK � bG�1 ¼ KTG�1K� TTG�1T ð77Þ
A similar (although in a different context) expression for J was found in [13]. This means that we only have tocalculate the entries of the matrices G, K and T. A still ‘cheaper’ approach can be found in the Theorems A andB of the Appendix, where it is shown that one only needs the eigen-decomposition of the matrix bK and thediagonal entries of one of the matrices bG or bJ. Of course, it should be noted that all this will only be approx-imately valid when the bases {bk} and {tk} are truncated, i.e., when we have 1 6 k 6M for some sufficientlylarge, but finite M.
5. Verification and numerical simulations

The different expressions for the discretized Dirichlet-to-Neumann operator Y will be constructed for anumber of two-dimensional examples, where the cross-section D of the cylinder is a rectangle. The parametersof the lossy cylinder are �, l0, r and the working frequency is f = x/2p, resulting in the wavenumbers

k0 = x/c = 2p/k and k ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�jxl0ðrþ jx�Þ

p: From the derivation of formula (68) we need {bk} � H1(c),

and for all practical purposes this means that one may use the well-known linear rooftop functions [16]. Inorder to avoid unnecessary complications we also consider the ‘full’ Galerkin case {tk} = {bk}. The linearrooftop basis {bk} is shown in Fig. 1. It is seen that the boundary of the cross-section of the cylinder is dividedinto 32 overlapping segments of width 2d, where d = dx = dy is the discretization step. The number of discret-izations in the x- and y-directions are nx = 12 and ny = 4 for the configuration shown in Fig. 1.


In this section, we shall first calculate a number of projected Dirichlet-to-Neumann operators and verify thesymmetry and convergence properties. The entries of these Dirichlet-to-Neumann operators are subsequentlycompared with the results obtained from the Dirichlet eigenfunction expansion method [5]. Finally, we willdiscuss the performance of the different methods with respect to the modelling of low frequency waves.The methods presented in the previous sections, that will be applied below, will be referred to as the Schur

complement method (see Eq. (75)), the symmetrization approach (see Eq. (76)), the asymmetric description (uti-lizing only the two Eqs. (69) and (71))) and the eigen-decomposition method (see Theorems A and B of theAppendix).

5.1. Comparison of the projected operators with results from other methods

In this subsection, we compare the different Y matrices, obtained by applying a Galerkin projection inthe case of a rectangular cylinder with dimensions a = 6 mm and b = 3 mm. The lossless (r = 0) cylinderhas a permittivity of �r = 6. The results obtained with the Schur complement approach (75) are firstcompared with those resulting from the Dirichlet method [5] and with those obtained from the asymmet-ric approach (76). The interactions with a rooftop in the upper right corner (row 19 of the admittancematrix) of the cylinder are plotted as a function of frequency in Fig. 2 (imaginary part) and in Fig. 3(real part). Figs. 2 and 3 indicate that the Schur complement method matrix elements coincide with thematrix entries obtained with the Dirichlet method [5]. The asymmetric approach yields mostly correctresults, although the self-interaction (located at the 19th coordinate) is not truly accurate. When observ-ing the real part (Fig. 3) of the three methods under scrutiny, which should be zero (r = 0), the resultobtained with the asymmetric approach is largely unsatisfactory. The real part is small, but non-negli-gible. The overall impact could be unacceptable, should an admittance model be used in a globalsimulation.

The interactions of a rooftop in the middle of the upper edge of the cylinder (10th row of the Y matrix) areshown in Fig. 4 (imaginary part) and in Fig. 5 (real part). All methods now yield correct results with respect tothe imaginary part, but the discrepancies in the real part are still important. The other non-Schur methodsresult in comparable discrepancies of the real part. In Fig. 6, the real parts of the corner interactions (19throw of the Y matrix) are presented, for three non-Schur approaches: the asymmetric method, the symmetri-zation approach and the eigen-decomposition algorithm. For this particular example, the symmetrizationapproach performs better than the other two methods. The residual dissipation is more or less equal forthe three examples.

In Fig. 7, the real parts of the interactions with a rooftop in the middle of the upper edge (row 19) areshown, for the three non-Schur approaches discussed above. Comparing with the corner case, we can concludethat the behavior is similar, whereas the residual dissipation is somewhat smaller on the edges than on thecorners. In the next subsection, it will be shown that the residual dissipation resulting from the non-Schurapproaches significantly alters the results when a conductive material is used.

17

131

29

32

7

2 d

x

y

Fig. 1. Galerkin discretization of a rectangular cylinder. Two rooftop basis functions are shown. One of them is positioned on an edge(number 7) and the other one is located on a corner (number 13).

0 5 10 15 20 25 30 35 40 45 50−10

−8

−6

−4

−2

0

2

4

6

8x 10−10

Schur complementDirichletAsymmetric

Fig. 3. Real part of the 19th row (corner interactions) of the Schur complement, eigenfunction-based and symmetrization Y matrices.

0 5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

6

7x 10−8


Fig. 2. Imaginary part of the 19th row (corner interaction) of the Schur complement, Dirichlet-based and symmetrization Y matrices.


5.2. Verification of the DC resistance

When a low frequency transverse magnetically (TM) polarized wave impinges on a homogeneous conduct-ing cylinder or when a low frequency signal propagates through the cylinder, a uniform current distributionpattern develops inside the cylinder. The associated DC resistance per unit of length (p.u.l.) is given by Pouil-let’s law:

0 5 10 15 20 25 30 35 40 45 50−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10−10


Fig. 5. Real part of the 10th row (middle of upper edge) of the Schur complement, Dirichlet-based and symmetrization Y matrices.

0 5 10 15 20 25 30 35 40 45 500

0.5

1

1.5

2

2.5

3x 10−7


Fig. 4. Imaginary part of the 10th row (middle of upper edge) of the Schur complement, Dirichlet-based and symmetrization Y matrices.


RDC ¼1

r1

abð78Þ

where a and b are the dimensions of the rectangular cylinder. The calculation of the DC p.u.l. resistance fromthe admittance matrix is straightforward:

0 5 10 15 20 25 30 35 40 45 50−10

−8

−6

−4

−2

0

2

4

6

8x 10−10

AsymmetricSymmetrizationEigendecomposition based

Fig. 6. Real part of the 19th row (corner interactions) of three non-Schur-based Y matrices.

0 5 10 15 20 25 30 35 40 45 50−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10−10

AsymmetricSymmetrizationEigendecomposition based

Fig. 7. Real part of the 10th row (middle of upper edge) of three non-Schur-based Y matrices.


RDC ¼ RX

i

Xj

Yij

( )�1

ð79Þ

It is of course very important that RDC is accurately predicted by any numerical method. We will now take alook at the performance of the various methods discussed above with regard to this DC resistance. We con-


sider a copper cylinder (r = 5.72 107 S/m) with aspect ratio r = a/b = 3/2. The number of discretizations in thex, resp. y-direction, is nx, resp. ny.

In Table 1, the theoretical DC resistance (in X/m) is shown for five simulations with subsequent meshrefinements, together with the results obtained with the Schur complement method and with those obtainedwith three non-Schur methods. These are the asymmetric approach, the symmetrized approach and theeigen-decomposition method, which turn out to be equivalent when calculating DC resistances. This explainswhy only a single value is given in the last column of Table 1. It should be noted that the DC resistances ofTable 1 are remarkably close to the theoretical values given by Eq. (78). The non-Schur results deviate onlyslightly for some simulations. When simulating semiconductor type cylinders, the conclusions are verydifferent.

In Table 2, the admittances YDC = 1/RDC of a 14.4 lm (nx = 12) by 9.6 lm (ny = 8) cylinder are shown fora total of 20 different r values. YDC is in units of 10�12 Sm. Only the Schur results are displayed; the admit-tances obtained from the non-Schur methods were several orders of magnitude too large and negative, i.e.,YDC (non-Schur) in the range [�0.3283 · 10�9,�0.1442 · 10�9] Sm for all non-Schur simulations. The resultsobtained with the Schur complement method are correct when a sufficient number of basis functions is chosen.The results for the lower conductivity values are slightly less accurate, probably due to the residual dissipationwhich starts to play a role at these low conductivity values.

5.3. Numerical verification of symmetry

In order to verify the accuracy of the expressions (75) and (76) in a realistic context, we consider how the Ymatrices of a rectangular cylinder evolve as a function of frequency, the number of basis functions and thematerial parameters. We use the same notation as in the previous example for the dimensions and discretiza-tions. Here we consider good conductors, semiconductors and dielectric materials: copper [Cu](r = 5.72 · 107 S/m, �r = 1.0), silicon [Si] (r = 1.6 10�3 S/m, �r = 11.0) and PECVD Silicon Nitride [SiN](r = 10�10 S/m, �r = 7.0). The material name abbreviations between square brackets are used in Table 3,where the relevant parameters for the subsequent simulations are shown.

For the simulations with Si and SiN, a frequency sweep between fl = 1 MHz and fh = 1 THz is performed.For the copper example, the simulation bandwidth is chosen such that q = 0.5(a + b)/d 2 [10�2, 104], wherethe skin depth d is defined as:

TableDC p.

a

15.014.415.014.716.2

d ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2

2pf rl0

sð80Þ

Because of this particular choice of a set of frequency points, the transition region between the low frequencysituation (the current flows uniformly in the longitudinal direction) and the high frequency case (the current isforced to the border of the cylinder), is completely covered. The numerical symmetry of the Schur complementformulation is compared with the results of the asymmetric and eigen-decomposition formulations. In order tomeasure the departure from symmetry, we take the relative 2-norm of the asymmetric part: e(f) = kY � YTk2/kYk2 as a function of frequency. The maximum and minimum values over the chosen frequency range for theSchur approach are listed in Table 3 as emax and emin for a total of twelve simulations. For the asymmetric andeigen-decomposition formulations, e(f) as a function of frequency remains acceptable at the higher part of thespectrum, but very rapidly increases for the lower frequencies to reach values of the order of 0.5. The observed

1u.l. resistance RDC (in X/m) of a copper cylinder (r = 5.72 · 107S/m) (a and b are in lm)

b nx ny RDC (Pouillet) RDC (Schur) RDC (non-Schur)

10.0 6 4 116.6 116.6 116.79.6 12 8 126.5 126.5 126.6

10.0 15 10 116.6 116.6 116.79.8 21 14 121.4 121.4 121.4

10.8 27 18 99.9 99.9 100.0

Table 2DC admittance YDC = 1/RDC of a semiconducting cylinder (in units of 10�12 Sm)

r (S/m) YDC (Pouillet) YDC (12 · 8) YDC (24 · 16) YDC (36 · 24)

0.0008 0.11 0.14 0.12 0.120.0016 0.22 0.25 0.23 0.230.0024 0.33 0.36 0.34 0.340.0032 0.44 0.47 0.45 0.450.0040 0.55 0.58 0.56 0.560.0048 0.66 0.69 0.67 0.670.0056 0.77 0.80 0.79 0.780.0064 0.88 0.91 0.89 0.890.0072 0.99 1.02 1.01 1.000.0080 1.11 1.13 1.12 1.110.0088 1.21 1.24 1.23 1.220.0096 1.33 1.36 1.34 1.330.0104 1.44 1.47 1.45 1.440.0112 1.55 1.57 1.56 1.560.0120 1.66 1.69 1.67 1.670.0128 1.77 1.79 1.78 1.780.0136 1.88 1.91 1.89 1.890.0144 1.99 2.02 2.00 2.000.0152 2.10 2.13 2.11 2.11

Theory compared with Schur complement method (nx · ny between parentheses).

Table 3Symmetry assessment of Dirichlet-to-Neumann operators (mat.: material, a and b are in lm)

mat. a b nx ny emin (Schur) emax (Schur) emin (non-Schur)

Cu 15.0 10.0 6 4 1.1 · 10�16 8.8 · 10�8 3.8 · 10�4

Cu 14.4 9.6 12 8 4.4 · 10�16 5.3 · 10�7 7.8 · 10�4

Cu 15.0 10.0 15 10 7.1 · 10�16 9.1 · 10�7 1.0 · 10�3

Cu 14.7 9.8 21 14 9.0 · 10�16 1.6 · 10�6 9.9 · 10�4

Cu 14.4 9.6 27 18 1.2 · 10�15 9.1 · 10�7 9.5 · 10�4

Si 15.0 10.0 6 4 4.0 · 10�14 5.8 · 10�7 5.5 · 10�3

Si 14.4 9.6 12 8 1.8 · 10�13 3.0 · 10�6 4.0 · 10�3

Si 15.0 10.0 15 10 2.9 · 10�13 5.9 · 10�6 3.7 · 10�3

Si 14.7 9.8 21 14 6.1 · 10�13 1.5 · 10�5 3.6 · 10�3

Si 14.4 9.6 27 18 1.2 · 10�12 1.9 · 10�5 3.0 · 10�3

SiN 15.0 10.0 6 4 7.4 · 10�14 7.1 · 10�7 6.6 · 10�3

SiN 14.4 9.6 12 8 4.9 · 10�13 1.3 · 10�5 3.9 · 10�3

SiN 15.0 10.0 15 10 5.6 · 10�13 7.8 · 10�6 4.5 · 10�3

SiN 14.7 9.8 21 14 1.5 · 10�12 1.7 · 10�5 5.2 · 10�3

SiN 14.4 9.6 27 18 1.5 · 10�12 2.8 · 10�5 3.7 · 10�3


minimum values emin are also given in Table 3. These values clearly show that the Schur approach is the onlyone assuring that the physically required symmetry property of the Dirichlet-to-Neumann operator is well-preserved in its discretized version.

Acknowledgements

H. Rogier is a Postdoctoral Researcher of the FWO-V. This work was partly supported by a grant of theInstitute for the Promotion of Innovation by Science and Technology in Flanders for a joint project with Agi-lent Technologies.


Appendix A

Theorem A. Let the complex matrices G = GT, J = JT and K be such that

(1)K is diagonalizable with all its eigenvalues different.3

(2)KG = GKT, JK = KTJ and K2 � GJ = I.

Then K, G and J can be written as

3 No

K ¼ PKKP�1 G ¼ PKGP T J ¼ P�TKJ P�1 ðA:1Þ
where KK, KG and KJ are diagonal matrices satisfying K2
K � KGKJ ¼ I .

Proof. K = PKKP�1 is simply the eigen-decomposition of K where supposedly kK = diag{kk} with all kk

different. Since P is non-singular, the symmetric matrix G can be written uniquely as G = P WPT, where W

symmetric is given by W = P�1GP�T. Since KG = PKKWPT has to be symmetric, it is necessary that KKW issymmetric. Hence we need

ðki � kjÞW ij ¼ 0 8 i; j ðA:2Þ
implying that W must be a diagonal matrix. Regarding J, the same reasoning applied to the matrix V = PTJP
results in V being diagonal, and the proof is complete. h

Note that if we put KG = diag{lk}, the matrix G can be written as

Gij ¼X

k

P ikP jklk ðA:3Þ

implying that the diagonal entries Gii are sufficient to obtain the values lk by solving the linear equations

Gii ¼X

k

P 2iklk ðA:4Þ

Obtaining KJ = diag{sk}, is then simply a matter of substitution (if none of the lk = 0) in the equation

k2k � lksk ¼ 1 ðA:5Þ

Similarly, putting Q = P�T, the values sk can be obtained by solving the linear equations

J ii ¼X

k

Q2iksk ðA:6Þ

and the lk follow from Eq. (A.5). The matrix Y = G�1(K � I) is also symmetric, and can be written as

Y ¼ QK�1G KK � Ið ÞQT ¼ QKJ KK þ Ið Þ�1QT ðA:7Þ

In the case where the eigenvalues of K are algebraically degenerate, the following weaker theorem can be used.

Theorem B. Let the complex matrices G = GT, J = JT and K be such that

(1) K is diagonalizable.(2) KG = GKT, JK = KTJ and K2 � GJ = I.

Then the matrix Y = G�1(K � I) is symmetric and can be constructed from the matrices K and HG alone, or,equivalently, the matrices K and HJ alone, where

K ¼ PKKP�1 G ¼ PHGP T J ¼ P�THJ P�1 ðA:8Þ

similar or other restrictions are necessary regarding the eigenvectors.


Proof. K = PKKP�1 is again the eigen-decomposition of K, but here we do not require that all eigenvalues aredifferent. G can be written as G = PHGPT, where the symmetric matrix HG = P�1GP�T. Similarly, for J, weobtain the symmetric matrix HJ = PTJP. Substituting this in K2 � GJ = I, we obtain K2

K �HGHJ ¼ I . PuttingQ = P�T, the matrix Y = G�1(K � I) can be written as

Y ¼ QH�1G KK � Ið ÞQT ¼ QHJ KK þ Ið Þ�1QT ðA:9Þ

From the premises KG = GKT and JK = KTJ it is clear that the matrices KKHG and HJKK are symmetric andhence also the matrix

H�1G KK ¼ KK HGKKð Þ�1KK ðA:10Þ

implying that Y is symmetric. h

References

[1] L. Friedlander, Remarks on Dirichlet and Neumann eigenvalues, Am. J. Math. 117 (1995) 257–262.[2] W. Hackbusch, B.N. Khoromskij, R. Kriemann, Direct Schur complement method by domain decomposition based on H-matrix

approximation, Comput. Vis. Sci. 8 (2005) 179–188.[3] V. Isakov, Inverse problems for partial differential equationsApplied Mathematical Sciences, vol. 127, Springer, New York, 2006.[4] G.C. Hsiao, R.E. Kleinman, Mathematical foundations for error estimation in numerical solutions of integral equations in

electromagnetics, IEEE Trans. Antennas Propagat. 45 (1986) 316–328.[5] D. De Zutter, L. Knockaert, Skin effect modeling based on a differential surface admittance operator, IEEE Trans. Microw. Theory

Technol. 53 (2005) 2526–2538.[6] J. Van Bladel, Electromagnetic Fields, McGraw-Hill, New York, 1964 (revised printing, Hemisphere, Washington DC, (1985)).[7] R. Dautray, J.-L. Lions, Mathematical analysis and numerical methods for science and technologySpectral Theory and Applications,

vol. 3, Springer, Berlin, 1990.[8] L.F. Knockaert, D. De Zutter, Integral equation for the fields inside a dielectric cylinder immersed in an incident E-wave, IEEE

Trans. Antennas Propagat. 34 (1986) 1065–1067.[9] S.P. Shipman, S. Venakides, Resonance and bound states in photonic crystal slabs, SIAM J. Appl. Math. 64 (2003) 322–342.

[10] W.W. Lin, C.S. Wang, On computing stable Lagrangian subspaces of Hamiltonian matrices and symplectic pencils, SIAM J. MatrixAnal. Appl. 18 (1997) 590–614.

[11] G. Corach, A. Maestripieri, D. Stojanoff, Generalized Schur complements and oblique projections, Lin. Alg. Appl. 341 (2002) 259–272.

[12] H. Bart, I. Gohberg, M.A. Kaashoek, A.C.M. Ran, Schur complements and state space realizations, Lin. Alg. Appl. 399 (2005) 203–224.

[13] G.Y. Yu, Relationship among coefficient matrices in symmetric Galerkin boundary element method for two-dimensional scalarproblems, J. Appl. Mech. 70 (2003) 479–486.

[14] R.E. Kleinman, P.A. Martin, On single integral equations for the transmission problem of acoustics, SIAM J. Appl. Math. 48 (1988)307–325.

[15] M. Renardy, R.C. Rogers, An Introduction to Partial Differential Equations, second ed., Springer, New York, 2004.[16] L.J. Gray, Evaluation of singular and hypersingular Galerkin boundary integrals: direct limits and symbolic computation, in: V.

Sladek, J. Sladek (Eds.), Singular Integrals in the Boundary Element Method, Advances in Boundary Elements, ComputationalMechanics Publishers, Southampton, UK, 1998, pp. 33–84.

on the schur complement form of the dirichlet-to-neumann operator

Documents