Engineering Analysis with Boundary Elements 35 (2011) 667–677

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Engineering Analysis with Boundary Elements

0955-79

doi:10.1

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(108710� Corr

Yat-sen

E-m

(H.-T. H

journal homepage: www.elsevier.com/locate/enganabound

Stability analysis via condition number and effective condition number for thefirst kind boundary integral equations by advanced quadrature methods,a comparison$

Jin Huang a, Hung-Tsai Huang b, Zi-Cai Li c,d,�, Yimin Wei e,f

a College of Applied Mathematics, University of Electronic & Science Technology of China, ChengDu, Chinab Department of Applied Mathematics, I-Shou University, Kaohsiung County, Taiwanc Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwand Department of Computer Science and Engineering, National Sun Yat-sen University, Kaohsiung, Taiwane School of Mathematical Sciences, Fudan University, Shanghai 200433, PR Chinaf Key Laboratory of Mathematics for Nonlinear Sciences, (Fudan University) Ministry of Education, Shanghai 200433, PR China

a r t i c l e i n f o

Article history:

Received 15 August 2010

Accepted 6 November 2010Available online 30 December 2010

Keywords:

Stability analysis

Condition number

Effective condition number

First kind boundary integral equation

Advanced (i.e., mechanical) quadrature

method

Numerical partial differential equations

97/$ - see front matter & 2010 Elsevier Ltd. A

016/j.enganabound.2010.11.006

work is supported by the National Natural

34).

esponding author at: Department of Applied

University, Kaohsiung, Taiwan.

ail addresses: huangjin12345@163.com (J. Hu

uang), zcli@math.nsysu.edu.tw (Z.-C. Li), ym

a b s t r a c t

In our previous study [Huang et al., 2008, 2009, 2010 [21,24,20]; Huang and Lu, 2004 [22,23]; Lu and

Huang, 2000 [38]], we have proposed advanced (i.e., mechanical) quadrature methods (AQMs) for solving

the boundary integral equations (BIEs) of the first kind. These methods have high accuracy O(h3), where

h¼max1pmpdhm and hm (m¼1,y,d) are the mesh widths of the curved edge Gm. The algorithms are

simple and easy to carry out, because the entries of discrete matrix are explicit without any singular

integrals. Although the algorithms and error analysis of AQMs are discussed in Huang et al. (2008, 2009,

2010) [21,24,20], Huang and Lu (2004) [22,23], Lu and Huang (2000) [38], there is a lack of systematic

stability analysis. The first aim of this paper is to explore a new and systematic stability analysis of AQMs

based on the condition number (Cond) and the effective condition number (Cond_eff) for the discrete

matrix Kh. The challenging and difficult lower bound of the minimal eigenvalue is derived in detail for the

discrete matrix of AQMs for a typical BIE of the first kind. We obtain Cond¼O(hmin�1 ) and Cond_eff¼O(hmin

�1 ),

where hmin ¼min1pmpdhm , to display excellent stability. Note that Cond_eff ¼ O(Cond) is greatly distinct

to the case of numerical partial differential equations (PDEs) in Li et al. (2007, 2008, 2009, 2010) [26,31–

37], Li and Huang (2008) [27–30], Huang and Li (2006) [19] where Cond_eff is much smaller than Cond.

The second aim of this paper is to explore intrinsic characteristics of Cond_eff, and to make a comparison

with numerical PDEs. Numerical experiments are carried out for three models with smooth and

singularity solutions, to support the analysis made.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The advanced (i.e., mechanical) quadrature methods (AQMs) areproposed in [20–24,38] for the first kind boundary integralequations (BIEs) of Laplace’s equation. The AQMs provide highaccuracy O(h3), accompanied with low computation complexity,where h¼max1rmrdhm and hm (m¼1,y,d) are the mesh widthsof the curved edge Gm. Note that the entries of discrete matrix Kh

are explicit without any singular integrals. Especially, for concavepolygons O, the solution at concave corners of @O has singularities,to heavily damage accuracy of numerical solutions. The accuracy of

ll rights reserved.

Science Foundation of China

Mathematics, National Sun

ang), huanght@isu.edu.tw

wei@fudan.edu.cn (Y. Wei).

Galerkin methods (GMs) [44,45] is only Oðh1þ eÞ ð0oeo1Þ and theaccuracy of collocation methods (CMs) [47] is even lower. Incontrast, the accuracy of AQMs for singularities is as high asO(h3). In fact, the quadrature method was first proposed for anintegral equation with a logarithmic kernel in Christiansen [11] in1971, called the modified quadrature method, and its analysis wasgiven in Saranen [41], to yield only the O(h2) convergence rate. Inprevious study [20–24,38], we propose the new quadraturemethods called the AQMs, to yield the high O(h3) convergencerate. Moreover, for AQMs, by extrapolations and splitting extra-polations methods (SEMs), the higher precision of numericalsolutions and a posteriori error estimates can be achieved.

This paper is devoted to its stability analysis. Consider the linearalgebraic equations:

Khx¼ b, ð1:1Þ

resulting from the first kind BIE, where the xARn and bARn are theunknown and known vectors, respectively. The condition number

J. Huang et al. / Engineering Analysis with Boundary Elements 35 (2011) 667–677668

is defined by

Cond¼jl1j

jlnj, ð1:2Þ

where li ði¼ 1, . . . ,nÞ are the eigenvalues of matrix KhARn�n in thedescending order in magnitude: jl1jZ � � �Z jlnj40. When thereoccurs a perturbation of b, the errors Dx of x also satisfy1

KhðxþDxÞ ¼ bþDb:

The values of Cond are used to measure the relative errors of x,given by

JDxJ

JxJr Cond �

JDbJ

JbJ, ð1:3Þ

where JxJ is the Euclidean norm. Note that the equality of (1.3)occurs only at very rare cases. In practical applications, the vector bvaries within a certain region, and the true relative errors from theperturbation of b or Kh may be smaller, or even much smaller thanCond given in (1.2).

Here we briefly provide the algorithms for the effective conditionnumber. Details are given in [26,27,31]. Let the matrix KhARn�n bereal symmetric. The eigenvectors ui satisfy Khui ¼ liui, where {ui} areorthogonal, with uT

i uj ¼ dij, where dij ¼ 1 if i¼ j and dij ¼ 0 if ia j.In [12,26,27,31] the effective condition number is defined by

Cond_eff ¼JbJ

jlnjJxJ¼

1

jlnj

JbJffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPni ¼ 1

b2i

l2i

s , ð1:4Þ

where bi ¼ uTi b.

When the matrix Kh in (1.1) is singular with RankðKhÞ ¼ rrn,the corresponding traditional and the effective condition numbersare defined by

Cond¼jl1j

jlr j, Cond_eff ¼

JbJ

jlrjJxJ, jlrj40:

For the linear algebraic equations in (1.1), the traditional conditionnumber in (1.2) provides the stability for all b; but the effectivecondition number in (1.4) pursues the stability for the specificvector b given. A similar idea of the stability for the specific solutionx obtained is discussed in Chen et al. [9], called model participationfactor in engineering for source excitation.

Stability is a severe issue for numerical solutions of the first kindBIEs. Although stability analysis for the modified quadrature methodwas given in [12,13], it is important to provide a systematic stabilityanalysis for the AQMs. In this paper, the effective condition number isalso applied to the first kind BIEs for Laplace’s equation on arbitraryplane domains by AQMs, and the bounds of both Cond_eff and Condare derived in detail. In this paper, we obtain that Cond ¼ O(hmin

�1 ) andCond_eff¼O(hmin

�1 ), where hmin ¼min1rmrdhm. This displays anexcellent stability of AQMs for smooth and singularity problems.

In [20], the AQMs are applied to Laplace’s equation withdiscontinuity solutions, and the Cond ¼ O(h�1) remains. This isa remarkably advantage over the FEM, FDM, etc. for PDEs, wherelarge or even huge Cond always occur for corner and discontinuitysingularity.

On the other hand, the fact that Cond_eff ¼ O(Cond) indicatesthe insignificant improvements of Cond_eff over Cond for stability.However, in [26–37,19] for numerical PDEs, the Cond is large oreven huge, but the Cond_eff is much smaller than Cond. Why is theperformance of Cond_eff so different? What are the rationalesbehind? To find their answers, in this paper we will exploreintrinsic characteristics of Cond_eff for the first kind BIEs, andmake a comparison with numerical PDEs. Such comprehensiveanalysis and comparison may provide an objective and

1 More discussions are given in [27,31] for a perturbation of both b and Kh.

comprehensive evaluation on Cond_eff. In fact, the improvementsof Cond_eff for perturbation errors are also insignificant for solvinglinear algebraic equations by Gaussian elimination and QR factor-ization. Numerical examples were first given in Banoczi et al. [4] in1998, and a theoretical justification is supplied in [27] recently.

Let us briefly review the references of condition number andeffective condition number. The definition of condition numberwas first given in Wilkinson [46], and then often used (e.g., Goluband van Loan [16]). Other discussions of normwise distance forstructure perturbation are given in Rump [40]. The conditionnumber is used to provide bounds of relative errors from aperturbation of Kh and all b. However, in practical applications,the true relative errors may be smaller, or even much smaller thanthe worst Cond. Such a case was first studied in Rice [39] in 1981,and then in Chan and Foulser [5], Christiansen and Hansen [12],Christiansen and Saranen [13], Axelsson and Kaporin [2,3], and Liet al. in [26–37,19] for numerical PDEs recently. This paper providesa concise review on the effective condition number, and explainsthe fact that why the effective condition number is not developeduntil its applications to numerical PDEs.

This paper is organized as follows. In the next section, the bounds ofCond_eff and Cond are derived in detail for typical BIEs of the first kind.In Section 3, bounds of Cond_eff and Cond are provided for closedsmooth curvesG, curved polygonsG and open contoursG. In Section 4,analysis and comparisons of Cond_eff are made for between numericalBIEs and PDEs, and in Section 5, three numerical examples are reportedto support the stability analysis given, and to show significance ofAQMs. In the last section, a few remarks are made.

2. Typical BIEs of the first kind

There are the Galerkin methods (GMs) and the collocationmethods (CMs). Since the AQMs are superior to GMs and CMs, thestability analysis of the AQMs may be regarded as a representationof that for numerical first kind BIE. To catch the intrinsic char-acteristics of stability easily, we stat with typical BIEs.

2.1. Description of algorithms

We have proposed the AQMs in [38] for solving

Av¼ f , ð2:1Þ

where the unknown v(t) and known f(t) are smooth periodicfunctions on ½0,2p� with the period 2p, and the boundary integraloperator A is defined by

ðAvÞðtÞ ¼

Z 2p

0aðt,tÞvðtÞ dt, tA ½0,2p�, ð2:2Þ

where the integral kernel aðt,tÞ ¼�1=2plnj2e�1=2sinðt�tÞ=2j.Denote h¼ 2p=n,nAN and let ftj ¼ jh,j¼ 1, . . . ,ng be the mesh

set. Using the quadrature rules [43], the Nystrom approximateoperator is given by

ðAhvÞðtÞ ¼ hXn

j ¼ 1t a tj

aðt,tjÞvðtjÞ�1

2p lne�1=2h

2p

� �vðtÞh, ð2:3Þ

with the errors

EnðAÞ ¼ 2X‘�1

m ¼ 1

1

ð2mÞ! xuð�2mÞvð2mÞðtÞh2mþ1þOðh2‘Þ as h-0, ð2:4Þ

where

EnðAÞ ¼ ðAhvÞðtÞ�ðAvÞðtÞ, vA ~C2‘½0,2p�,

and xðtÞ is the Riemann zeta function. The notation vA ~C2‘½0,2p�

denotes the periodic function v with period 2p having 2‘ order

J. Huang et al. / Engineering Analysis with Boundary Elements 35 (2011) 667–677 669

derivatives, defined by

vA ~C2‘½0,2p� ¼ fvðtÞjvðmÞðtÞAC½0,2p�

and

vðmÞðtþ2pÞ ¼ vðmÞðtÞ, m¼ 0,1, . . . ,2‘g:

Using quadrature rules (2.3), we obtain the linear algebraic equations

ðAhvhÞðtiÞ ¼ f ðtiÞ, i¼ 1, . . . ,n: ð2:5Þ

In [38] we have proved that there exist the unique solutions for(2.5) such that

jvhðtiÞ�vðtiÞj ¼Oðh3Þ, i¼ 1, . . . ,n as vðtÞA ~C4½0,2p�:

Moreover, we have derived in [38]

vhðtÞ�vðtÞ ¼X2

m ¼ 1

wmðtÞh2mþ1þOðh6Þ, tAftig as vðtÞA ~C

6½0,2p�,

where wmðtÞA ~C ½0,2p� ðm¼ 1,2Þ are independent of h, and vh(t) andv(t) are the solutions of (2.5) and (2.1) at tAfti,i¼ 1,2, . . . ,ng,respectively. Hence, for AQMs the superconvergence O(h6) canbe achieved by Richardson’s extrapolations [38].

2.2. Condition number and effective condition number

From (2.3) and (2.5), the discrete matrix Ah is symmetric andcirculant, with the entries

a0 ¼�1

nln

e�1=2

n, aj ¼�

1

nln 2e�1=2sin

jpn

��������, j¼ 1, . . . ,n�1,

where h¼ 2p=n. Based on the theory of circulant matrix [14],eigenvalues lk of matrix Ah can be expressed by

lk ¼Xn�1

j ¼ 0

ajejk, ek ¼ expð2kp

ffiffiffiffiffiffiffi�1p

=nÞ, k¼ 0,1, . . . ,n�1: ð2:6Þ

We have the following theorem.

Theorem 2.1. Let the discrete matrix Ah in (2.5) from (2.3) be

symmetric and circulant. Then the condition number has the bound

Cond¼maxkjlkj

minkjlkjr

1

n¼ Oðh�1Þ: ð2:7Þ

Proof. First, we estimate the upper bound of lk. For the leadingeigenvalue of matrix Ah, we have from (2.6),

l0 ¼�1

nln

e�1=2

nþXn�1

j ¼ 1

ln 2e�1=2sinjpn

��������

8<:

9=;

¼�1

n�

1

2�ln nþðn�1Þ ln 2�

1

2

� �þ ln Pn�1

j ¼ 1sinjpn

� �� �: ð2:8Þ

From the equality in [24]

Pn�1j ¼ 1sin

jpn¼

n

2n�1, ð2:9Þ

we have

ln Pn�1j ¼ 1sin

jpn

� �¼ ln n�ðn�1Þln 2: ð2:10Þ

Combining (2.8) and (2.10) gives

l0 ¼12: ð2:11Þ

Next, from (2.6) other eigenvalues lkð ¼ 1,2, . . . ,n�1Þof matrix Ah

are given by

lk ¼�1

nln

e�1=2

nþXn�1

j ¼ 1

cos2kpj

nln 2e�1=2sin

jpn

��������

8<:

9=;,

k¼ 1,2, . . . ,n�1: ð2:12Þ

We have from (2.9)

jlkjr1

nln

e�1=2

n

��������þ lnPn�1

j ¼ 1 2e�1=2sinjpn

��������

� �

¼1

n

1

2þ ln nþ

1

2þ ln n�

n

2

��������

� �¼

1

2: ð2:13Þ

Combining (2.13) and (2.11) gives

maxkjlkj ¼

12: ð2:14Þ

Below, we derive the lower bound of lk. Since

Xn�1

j ¼ 1

cos2kpj

n¼�1, ð2:15Þ

we have from (2.12)

lk ¼1

nln n�

Xn�1

j ¼ 1

cos2kpj

nln 2sin

jpn

��������

8<:

9=;: ð2:16Þ

Let us introduce the cðzÞ special function, defined by the logarith-

mic derivative of the Gamma function (see Gradshteyan and Ryzhik

[17], p. 943 (item 1 in 8.362)),

cðzÞ ¼d

dzlnGðzÞ ¼�g�1

zþzX1j ¼ 1

1

jðjþzÞ, za0,1,2, . . . , ð2:17Þ

where gð ¼ 0:5772 . . .Þ is the Euler constant. Moreover, there exist

other expansions for cðzÞ (see Remark 3.1 below),

ck

n

� �¼�g�ln n�

p2

cotkpn

� �þXn�1

j ¼ 1

cos2kpj

nln 2sin

jpn

��������: ð2:18Þ

Then we have from (2.17) and (2.18)

Xn�1

j ¼ 1

cos2kpj

nln 2sin

jpn

��������¼ ln n�

n

kþp2

cotkpn

� �þ

k

n

X1j ¼ 1

1

j jþk

n

� � :ð2:19Þ

Hence Eq. (2.16) leads to

lk ¼1

k�

p2n

cotkpn

� ��

k

n2

X1j ¼ 1

1

j jþk

n

� � : ð2:20Þ

Moreover, there exists the series expansion for cotx in [17], p. 35

(item 7 in 1.411),

cotx¼1

x�

x

3� � � ��

22jjB2jj

ð2jÞ!x2j�1� � � � , jxjop, ð2:21Þ

where Bj is the Bernoulli number. Substituting (2.21) into (2.20)

gives

lk ¼1

k�

p2n

n

kp�kp3n� � � � �

22jjB2jj

2j!

kpn

� �2j�1

� � � �

( )�

k

n2

X1j ¼ 1

1

j jþk

n

� �

Z1

k�

1

2kþ

kp2

6n2�

k

n2

X1j ¼ 1

1

j jþk

n

� �

¼1

2kþ

k

n2

X1j ¼ 1

1

j2�

1

j jþk

n

� �8>><>>:

9>>=>>;Z

1

2kZ

1

2n, k¼ 1,2, . . . ,n�1,

ð2:22Þ

2 For the mixed (i.e., Robin) boundary condition, the AQMs with extrapolation

techniques are explored in Huang et al. [21].

J. Huang et al. / Engineering Analysis with Boundary Elements 35 (2011) 667–677670

where we have used the bound, p2=6¼P1

j ¼ 1 1=j2. Furthermore,

combining (2.11) and (2.22) gives

minkjlkjZ

1

2n: ð2:23Þ

The desired result (2.7) follows from (2.14) and (2.23). This

completes the proof of Theorem 2.1. &

Below, we derive the bounds of effective condition number.From [1,44,48], when the solution vðtÞA ~C

4½0,2p� of (2.1), we have

jvhðtiÞ�vðtiÞj ¼Oðh3Þði¼ 1, . . . ,nÞ, to give

JxJ2 ¼Xn

i ¼ 1

v2hðtiÞ

( )1=2

�Xn

i ¼ 1

v2ðtiÞ

( )1=2

� Oðh�1=2Þ, ð2:24Þ

where the notation a � bða � OðbÞÞ,b40, denotes that there existtwo constants c1 and c2 independent of n such that c1br jajrc2b.Next, from [1,44,48], if vðtÞA ~C

k½0,2p�, then f ðtÞA ~C

kþ1½0,2p�.

We have

JbJ2 ¼Xn

j ¼ 1

½f ðtjÞ�2

8<:

9=;

1=2

¼ h�1=2Xn

j ¼ 1

h½f ðtjÞ�2

8<:

9=;

1=2

Ch�1=2

Z 2p

0f 2ðtÞdt

( )1=2

� O h�1=2�

: ð2:25Þ

From (2.24), (2.25) and (2.23), we have

Cond_eff ¼JbJ

minkjlkjJxJ¼Oðh�1Þ, ð2:26Þ

to give the following theorem.

Theorem 2.2. Let the discrete matrix Ah in (2.5) from (2.3) be

symmetric and circulant. Then the effective condition number has

the bound (2.26).

Remark 2.1. For the lower bound (2.23) of lk, Eq. (2.18) is a keyexpansion we need to double-check. We cite from [17], p. 944 (item6 in 8.363),

ck

n

� �¼�C�ln n�

p2

cotkpn

� �þ2

XEððnþ1Þ=2Þ�1

j ¼ 1

cos2kpj

nln sin

jpn

��������, ð2:27Þ

where C is a constant, and E(x) is the integer part of real x. We havefrom (2.15)

2XEððnþ1Þ=2Þ�1

j ¼ 1

cos2kpj

nln sin

jpn

��������¼Xn�1

j ¼ 1

cos2kpj

nln sin

jpn

��������

¼Xn�1

j ¼ 1

cos2kpj

nln 2sin

jpn

���������ln 2

Xn�1

j ¼ 1

cos2kpj

n

¼ ln 2þXn�1

j ¼ 1

cos2kpj

nln 2sin

jpn

��������: ð2:28Þ

Combining (2.27) and (2.28) gives

ck

n

� �¼�Cþ ln 2�ln n�

p2

cotkpn

� �þXn�1

j ¼ 1

cos2kpj

nln 2sin

jpn

��������: ð2:29Þ

Compared (2.29) with (2.18), we find the constant

C ¼ gþ ln 2: ð2:30Þ

There may have the other choice of constant as C ¼ g. By using somespecial values of cðzÞ in [17], p. 945 (8.366),

c1

2

� �¼�g�2ln 2¼�1:963510026, c

1

4

� �¼�g�p

2�3ln 2, ð2:31Þ

we confirm (2.18) as well as the constant (2.30).

3. Dirichlet’s problems of Laplace’s equation

3.1. Closed smooth curve G

By the layer potential theory, Dirichlet’s problems of Laplace’sequation2:

Du¼ 0 in O,

u¼ f on G¼ @O,

uðxÞ ¼OðlnjxjÞ as jxj-1,

8><>: ð3:1Þ

are converted into the first kind BIEs [1,44]

�1

2p

ZG

vðxÞlnjx�yj dsx ¼ f ðyÞ, yAG, ð3:2Þ

where O� R2 is a bounded domain with a closed smooth edge G, and

jx�yj ¼ fðx1�y1Þ2þðx2�y2Þ

2g1=2:

In (3.2) the unknown function

vðxÞ ¼@uðxÞ

@n� �@uðxÞ

@nþ ,

where n is a unit outward normal at a point xAG. From the knownresults [1,44,47], when the logarithmic capacity (transfinite diameter)CGa1, there exists a unique solution of (3.2). As soon as v(x) is solvedfrom (3.2), the solutions of (3.1) at interior or exterior points can beevaluated by

uðyÞ ¼�1

2p

ZG

vðxÞlnjx�yj dsx, yAR2\G:

Assume that CGa1 andG can be described by the parameter mapping

xðtÞ ¼ ðx1ðtÞ,x2ðtÞÞA ~C‘½0,2p� : ½0,2p�-G

with

�mZ jxuðtÞj2 ¼ jxu1ðtÞj2þjxu2ðtÞj

2Zm40,

where �m and m are two constants. Define the boundary integraloperator

ðKvÞðtÞ ¼

Z 2p

0kðt,tÞvðtÞ dt, tA ½0,2p�,

where kðt,tÞ ¼ � 12p lnjxðtÞ�xðtÞj and vðtÞ ¼ vðxðtÞÞjxuðtÞj. Then Eq. (3.2)

is converted into

Kv¼ AvþBv¼ f , ð3:3Þ

where B¼K�A and ðBvÞðtÞ ¼R 2p

0 bðt,tÞvðtÞ dt with

bðt,tÞ ¼�

1

2plne1=2ðxðtÞ�xðtÞÞ

2sint�t

2

� ���������

��������for t�t =2 2pZ,

�1

2pln e1=2xuðtÞ�� �� for t�tA2pZ,

8>>>>>>><>>>>>>>:

where Z ¼ f0,71,72, . . .g.Using the trapezoidal or the midpoint rule [15], Nystrom’s

approximate operator Bh of B is given by

ðBhvÞðtÞ ¼Xn

tj a t

j ¼ 1

hbðt,tjÞvðtjÞþ�h

2p ln e1=2xuðtÞ�� ��vðtÞ:

Hence, we obtain the approximate equations of (3.3)

AhvhðtiÞþBhvhðtiÞ ¼ f ðtiÞ, i¼ 1, . . . ,n: ð3:4Þ

J. Huang et al. / Engineering Analysis with Boundary Elements 35 (2011) 667–677 671

Lemma 3.1. Let G with CGa1 be an arbitrarily closed smooth curve.

Assume that vðtÞAC6½0,2p� and kðt,tÞvðtÞ is periodic with period 2p,and has six order differentiable on ð�1,1Þ\ftþ2pmg1m ¼ �1.

(1)

There exists a unique solution in (3.4) with the errors

jvhðtiÞ�vðtiÞj ¼ Oðh3Þ, i¼ 1, . . . ,n:

(2)

There exist the functions wmðtÞA ~C ½0,2p� ðm¼ 1,2Þ independent of

h such that

vhðtÞ�vðtÞ ¼X2

m ¼ 1

wmðtÞh2mþ1þOðh6Þ, tAftig,

where vh(t) and v(t) are the solutions of (3.4) and (3.3) at t¼tj,respectively.

Lemma 3.1 implies that for the closed smooth curve G with CGa1,the superconvergence O(h6) can also be achieved by Richardson’sextrapolations.

3.2. Closed curved polygons G or open contours G

Let G¼Sd

m ¼ 1 Gmðd41Þ be curved polygons or open contourswith CGa1, and Gm be a piecewise smooth curve. Define theboundary integral operators on Gm,

ðKqmvmÞðyÞ ¼�1

2p

ZGm

vmðxÞlnjy�xj dsx, yAGq, m,q¼ 1, . . . ,d,

ð3:5Þ

where vmðxÞ ¼ @umðxÞ=@n��@umðxÞ=@nþ . Then Eq. (3.2) can be con-verted into a matrix operator equation

Kv¼ F, ð3:6Þ

where K¼ ½Kqm�dq,m ¼ 1, v¼ ðv1ðxÞ, . . . ,vdðxÞÞ

T and F¼ ðf1ðyÞ, � � � ,fdðyÞÞT .

Here, let K be symmetric operators.Assume that Gm can be described by the parameter mapping

xmðsÞ ¼ ðxm1ðsÞ,xm2ðsÞÞ : ½0,1�-Gm

with

�mZ jxumðsÞj ¼ ½jxum1ðsÞj2þjxum2ðsÞj

2�1=2Zm40,

where �m and m are two constants. Using the sinp�transformation in

[42]

s¼jpðtÞ : ½0,1�-½0,1�, pAN, ð3:7Þ

with jpðtÞ ¼ WpðtÞ=Wpð1Þ and WpðtÞ ¼R t

0ðsinpyÞpdy, then the integraloperators (3.5) can be converted into integral operators on [0,1] asfollows:

ðAqqwqÞðtÞ ¼

Z 1

0aqðt,tÞwqðtÞ dt, tA ½0,1�, ð3:8Þ

ðBqmwmÞðtÞ ¼

Z 1

0bqmðt,tÞwmðtÞ dt, tA ½0,1�, ð3:9Þ

where

aqðt,tÞ ¼ � 1

2pln 2e�1=2sinpðt�tÞ�� ��, ð3:10Þ

wmðtÞ ¼ vmðxmðjpðtÞÞÞjxumðjpðtÞÞjjupðtÞ, ð3:11Þ

bqmðt,tÞ ¼�

1

2pln

xqðtÞ�xqðtÞ2e�1=2sinpðt�tÞ

�������� for q¼m,

�1

2plnjxqðtÞ�xmðtÞj for qam:

8>>><>>>:

ð3:12Þ

In (3.8) and (3.9),

xmðtÞ ¼ ðxm1ðjpðtÞÞ, xm2ðjpðtÞÞÞ ðm¼ 1, . . . ,dÞ,

jxqðtÞ�xmðtÞj ¼ ½ðxq1ðtÞ�xm1ðtÞÞ2þðxq2ðtÞ�xm2ðtÞÞ2�1=2:

Hence Eq. (3.6) becomes

ðAþBÞW¼ G, ð3:13Þ

where A¼ diagðA11, . . . ,AqqÞ and B¼ ½Bqm�dq,m ¼ 1 are symmetric

operators, and W¼ ðw1, . . . ,wdÞT and G¼ ðg1, . . . ,gdÞ

T withgmðtÞ ¼ fmðxmðtÞÞ. Let

hm ¼1

nm, nmAN, m¼ 1, . . . ,d,

be mesh widths for the nodes,

tj ¼ tj ¼ ðj�12Þhm, j¼ 1, . . . ,nm:

By the trapezoidal or the midpoint rule [15] we have Nystrom’sapproximate operator Bqm

h of Bqm. For the weakly singular operatorsAmm, by the quadrature formula (2.3) (see [43]), we can also havethe Nystrom approximate operator Aqq

h . Setting t¼ti ði¼ 1, . . . ,nqÞ,we obtain the following approximate equations of (3.13)

KhWh ¼ ðAhþBhÞWh ¼Gh, ð3:14Þ

where

Wh ¼ ðwh1ðt1Þ, . . . ,w

h1ðtn1Þ, . . . ,wh

dðt1Þ, . . . ,whdðtndÞÞ

T ,

Ah ¼ diagðAh11, . . . ,Ah

ddÞ, Ahqq ¼ ½aqðtj,tiÞ�

nq

j,i ¼ 1,

Bh ¼ ½Bhqm�

dq,m ¼ 1, Bh

qm ¼ ½bqmðtj,tiÞ�nq ,nm

j,i ¼ 1,

Gh ¼ ðg1ðt1Þ, . . . ,g1ðtn1Þ, . . . ,gdðt1Þ, . . . ,gdðtnd

ÞÞT ,

and

aqðtj,tiÞ ¼

�hqlnj2e�1=2sinpðti�tjÞj

2p as ia j,

�hqlnje�1=2hqj

2pas i¼ j:

8>>><>>>:

ð3:15Þ

Obviously, (3.14) is a linear system with n-unknowns, wheren¼ n1þ � � � þnd. We cite the results of [23,24] as a lemma.

Lemma 3.2. Assume that Gmðm¼ 1, . . . ,dÞ are smooth curves, G¼[d

m ¼ 1Gm with CGa1, and h¼max1rmrdhm is sufficiently small. Also

let u0AC4ðGÞ � C4ðGÞ and pZ6 in (3.7). Then there exists a unique

solution Wh of (3.14) such that

Wh�W¼ diagðh31, . . . ,h3

dÞfþOðh4Þ, ð3:16Þ

at node points, where a vector functionf¼ ðf1, . . . ,fdÞT AðC0½0,1�Þd is

independent of h¼ ðh1, . . . ,hdÞT , and the subspace

C0½0,1� ¼ fvðtÞAC½0,1� :vðtÞ

sin2ðptÞ

AC½0,1�g,

with the norm JvJ ¼max0r tr1jvðtÞ=sin2ðptÞj.

From Lemma 3.2 we can obtain the approximate solutions witha higher order accuracy o(h3) by solving some coarse grid discreteequations in parallel. The algorithms of the splitting extrapolationalgorithms (SEAs) are described as follows.

Step 1. Choose h(0)¼(h1,y,hd) and hðmÞ ¼ ðh1, . . . ,hm=2, . . . ,hdÞ,

solve (3.14) for h(m) (m¼1,y,d) in parallel, and then obtain thesolutions WhðmÞ ðtiÞ.

Step 2. Compute the h3-Richardson extrapolation, based on thenumerical solutions on coarse grids

WðtiÞ ¼8

7

Xd

m ¼ 1

WhðmÞ ðtiÞ� d�7

8

� �Whð0Þ ðtiÞ

" #, ð3:17Þ

Table 1The errors en and en

E, Cond and Cond_eff for (5.1).

n en enE

jl1j jlnj Cond Cond_eff

23 3.820(�2) 3.156 1.088 2.899 1.500

24 4.737(�3) 4.361(�5) 3.143 5.444(�1) 5.774 2.899

25 5.885(�4) 4.086(�6) 3.141 2.722(�1) 1.154(+1) 5.7742

26 7.342(�5) 1.695(�7) 3.141 1.361(�1) 2.308(+1) 1.154(+1)

27 9.172(�6) 5.955(�9) 3.141 6.805(�2) 4.616(+1) 2.308(+1)

28 1.146(�6) 1.964(�10) 3.141 3.402(�2) 9.233(+1) 4.616(+1)

29 1.432(�7) 6.293(�12) 3.141 1.701(�2) 1.846(+2) 9.233(+1)

210 1.791(�8) 2.978(�14) 3.141 8.506(�3) 3.693(+2) 1.846(+2)

Table 2

The values of lk , bk and yk at n¼25.

k lk bk yk k lk bk yk

1 3.141 �1.001(�15) �1.570 17 0.408 �8.081(�16) �1.570

2 3.141 �2.053(�15) �1.570 18 0.369 6.993(�16) 1.570

3 3.141 3.162(�15) 1.570 19 0.369 �2.224(�16) �1.570

4 1.571 �6.249 �0.102 20 0.339 �1.055(�15) �1.570

5 1.571 �6.456(�1) 1.467 21 0.339 3.333(�16) 1.570

J. Huang et al. / Engineering Analysis with Boundary Elements 35 (2011) 667–677672

and then obtain uh(y) ðyAO\GÞ by

uhðyÞ ¼�1

2pXd

m ¼ 1

Xnm

i ¼ 1

hmflnjy�xmðtiÞjgjxumðtiÞjwhmðtiÞ: ð3:18Þ

Step 3. From (3.17) we have

WðtiÞ�1

d

Xd

m ¼ 1

WhðmÞ ðtiÞ

����������r WðtiÞ�

8

7

Xd

m ¼ 1

WhðmÞ ðtiÞ� d�7

8

� �Whð0Þ ðtiÞ

" #����������

þ8d�7

7

� �1

d

Xd

m ¼ 1

WhðmÞ ðtiÞ�Whð0Þ ðtiÞ

����������

r8d�7

7

� �1

d

Xd

m ¼ 1

WhðmÞ ðtiÞ�Whð0Þ ðtiÞ

����������þoðh3

0Þ: ð3:19Þ

Note that the most right-hand side of (3.19) provides the aposteriori error estimates.

3.3. Condition number and effective condition number

Take the first kind BIEs in Section 3.2 for example. Under theabove assumptions, the operators A and B in (3.13) are symmetric,and the matrices Ah and Bh in (3.14) are also symmetric. FollowingSection 2.2, we can obtain the similar bounds of condition numberand effective condition number (see [24]).

Theorem 3.1. Assume that Gm ðm¼ 1, . . . ,dÞ is smooth curve and

G¼Sd

m ¼ 1 Gm with CGa1. Let Ah be defined by rules (3.15) and Bh be

defined by the trapezoidal or the midpoint rule [15]. Then the condition

number for (3.14) has the bound

Cond¼ Oðh�1minÞ, hmin ¼ min

1rmrdhm, ð3:20Þ

and effective condition number for (3.14)

Cond_eff ¼JbJ

minkjlkjJxJ¼Oðh�1

minÞ: ð3:21Þ

Eqs. (3.20) and (3.21) imply

Cond_eff ¼OðCondÞ: ð3:22Þ

Hence, the numerical stability of the AQMs is excellent for the firstkind BIEs, which also agrees with [13]. The new stability analysis inthis paper enhances the AQMs, whose error analysis has alreadybeen explored in [20–24,38]. Note that the improvements ofCond_eff for stability are insignificant, since the Cond itself isnot large in practical applications, due to not small h in computa-tion. However, when CG-1 (see [13]) the values of Cond are large,and the Cond_eff may be much smaller than Cond.

Note that the analysis in this paper is under the condition of thelogarithmic capacity (transfinite diameter) CGa1, where CG is just thediameter of the circular boundaryG. Interestingly, even for the Dirichletproblem (3.1), when CG ¼ 1 the solutions may not exist, or not unique ifexisting, to cause a singularity of the discrete algebraic equations. Theproblem with CG ¼ 1 in the BEM is called the degenerate scaleproblems; the treatments for such a difficulty in computation arereported for circular domains with circular holes in Chen et al. [10].

6 1.049 �2.155(�15) �1.570 22 0.316 �7.770(�16) �1.570

7 1.049 �1.836(�15) �1.570 23 0.316 �4.167(�17) �1.570

8 0.789 �1.199(�15) �1.570 24 0.299 2.677(�16) 1.570

9 0.789 3.882(�15) 1.570 25 0.299 �4.991(�15) �1.570

10 0.634 �3.711(�16) �1.570 26 0.287 �4.814(�16) �1.570

11 0.634 1.119(�16) 1.570 27 0.287 1.118(�16) 1.570

12 0.532 1.944(�15) 1.570 28 0.278 8.322(�16) 1.570

13 0.532 3.536(�16) 1.570 29 0.278 2.778(�16) 1.570

14 0.460 1.117(�15) 1.570 30 0.273 �3.600(�16) �1.570

15 0.460 1.278(�15) 1.570 31 0.273 �5.557(�17) �1.570

16 0.408 �1.668(�16) �1.570 32 0.272 �9.434(�16) �1.570

4. Analysis of effective condition number

In [26,27], the effective condition number is applied to the finitedifference method (FDM) for solving Poisson’s equation withDirichlet boundary conditions,

�Du¼ f in O,

u¼ g on G,

(ð4:1Þ

where D¼ @2=@x2þ@2=@y2, and O is a polygon with the boundary@O¼G. The traditional condition number of the difference matrixis well known as

Cond¼Oðh�2minÞ, ð4:2Þ

where hmin is the minimal meshspacing of difference grids. In [26],the bounds of Cond_eff were derived, to give

Cond_effrc Jf J0,Oþh�1=2h�1minJgJ0,G

n o, ð4:3Þ

where Jf J0,O and JgJ0,G are the Sobolev norms, h is the maximalmeshspacing, and c is a constant independent of h. Evidently,Cond_eff in (4.3) is smaller than Cond in (4.2). In particular, whenthe boundary conditions are homogeneous, i.e., g¼0, we obtain

Cond_eff ¼Oð1Þ, ð4:4Þ

which is significantly smaller than Cond in (4.2).In [31,36], the effective condition number is applied to the

collocation Trefftz method (CTM) for Laplace’s and biharmonicequations with crack singularities. The condition number growsexponentially with respect to the number of singular solutions used,but the effective condition number grows polynomially (even linearly[31]). The improvements of Cond_eff for stability are most significant.

In Sections 2 and 3, for numerical BIEs of the first kind by AQMs,when CGa1, Cond_eff and Cond have the same growth rates ash-0 (see (3.22)). Also from the data in Tables 1–3 given in Section 5later, we can see

Cond

Cond_eff¼ cAð1:5,3:5Þ:

Hence, the improvements of Cond_eff over Cond are insignificantfor stability analysis. For numerical partial differential equations(PDEs) and numerical BIEs, why is the performance of Cond_eff so

Table 3The errors en and en

E, Cond and Cond_eff for (5.7).

n en enE

jl1j jlnj Cond Cond_eff

24 2.477(�3) 2.458(+1) 5.474(�1) 4.489(+1) 2.929(+1)

25 2.441(�4) 7.481(�5) 2.458(+1) 2.722(�1) 9.030(+1) 5.879(+1)

26 3.045(�5) 7.296(�8) 2.458(+1) 1.361(�1) 1.806(+2) 1.173(+2)

27 3.810(�6) 4.061(�9) 2.458(+1) 6.805(�2) 3.612(+2) 2.346(+2)

28 4.765(�7) 1.529(�10) 2.458(+1) 3.402(�2) 7.224(+2) 4.693(+2)

29 5.956(�8) 5.048(�12) 2.458(+1) 1.701(�2) 1.444(+3) 9.387(+2)

210 7.445(�9) 1.593(�13) 2.458(+1) 8.506(�3) 2.889(+3) 1.877(+3)

J. Huang et al. / Engineering Analysis with Boundary Elements 35 (2011) 667–677 673

different? What are the rationales behind? Below, we will exploresuch distinct characteristics of Cond_eff, based on matrix analysis.

4.1. Effective condition number for first kind BIEs

Denote the real eigenvalues li of the discrete stiffness matrix Kh

(or Ah) in Sections 2 and 3 as a descent order,

jl1j4 jl2jZ � � �Z jlnj40: ð4:5Þ

Based on the analysis of Sections 2 and 3 (see (2.14) and (2.23)),we assume that

jl1j � Oð1Þ, jlnj � OðhpÞ, p40, ð4:6Þ

where p¼1. Hence, the traditional condition number has the bound

Cond¼jl1j

jlnj� Oðh�pÞ:

Based on Sections 2 and 3 (see (2.24) and (2.25)), there exist the bounds

JxJ � JbJ � O h�1=2�

: ð4:7Þ

We have the following theorem.

Theorem 4.1. Let (4.6) and (4.7) hold. Then there exists the

equivalence

Cond_eff � OðCondÞ:

Proof. We have from (4.5) and (4.7)

Cond_eff ¼JbJ

jlnjJxJ� O

1

jlnj

� �� O

jl1j

jlnj

� �� OðCondÞ: &

Denote the eigenpairs ðli,uiÞ of Kh, and the angles yi between band ui by

cosyi ¼ cosðb,uiÞ ¼ðb,uiÞ

JbJ¼

bi

JbJ, ð4:8Þ

where bi ¼ ðb,uiÞ ¼ uTi b. We have the following lemmas.

Lemma 4.1. Let ðli,uiÞ be eigenpairs of matrix Kh. Suppose that matrix

Kh is symmetric and nonsingular, and that from (4.7)

JxJ � JbJ: ð4:9Þ

Then there exists the bound,

Xn

i ¼ 1

cos2yi ¼ 1, ð4:10Þ

and

Xn

i ¼ 1

cos2yi

l2i

� Oð1Þ, ð4:11Þ

where the angles yi are given in (4.8).

Proof. We have b¼Pn

i ¼ 1 biui. Then we obtain from (4.8)

JbJ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiXn

i ¼ 1

b2i

vuut ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXn

i ¼ 1

cos2yi

vuut8<:

9=;JbJ,

to give the first desired result (4.10). Next, from x¼K�1h b we have

x¼Pn

i ¼ 1bi

liui. Then there exists an equality,

JxJ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiXn

i ¼ 1

b2i

l2i

vuut ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXn

i ¼ 1

cos2yi

l2i

vuut JbJ:

We obtain from (4.9)

Xn

i ¼ 1

cos2yi

l2i

¼JxJ2

JbJ2� Oð1Þ: ð4:12Þ

This is the second result (4.11), and completes the proof of Lemma4.1. &

From (4.5) and (4.6), we may denote the eigenvalues by

jlij ¼ cihpi � Oðhpi Þ, ð4:13Þ

where ci are positive constants, and the powers pi are given by

0¼ p1rp2o � � �rpn ¼ p:

Lemma 4.2. Let the conditions in Lemma 4.1 and (4.13) hold. Then

there exist the bounds

jcosyij ¼ Oðhpi Þ ¼OðjlijÞ, ð4:14Þ

where yi are defined in (4.8).

Proof. We have from Lemma 4.1

jcosyij ¼ jlij

ffiffiffiffiffiffiffiffiffiffiffiffiffifficos2yi

l2i

sr jlij

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXn

i ¼ 1

cos2yi

l2i

vuut rcjlijrchpi ,

to give the desired result (4.14). This completes the proof ofLemma 4.2. &

In fact, u1 in eigenpair ðl1,u1Þ is the lowest frequency eigen-vector, and un in eigenpair ðln,unÞ is the highest frequencyeigenvector. The ui is said the low frequency eigenvector if itscorresponding eigenvalue li satisfying jlij � Oðjl1jÞ � Oð1Þ, or thehigh frequency eigenvector if jlij � OðjlnjÞ. Also b has a portioneddistribution on ui if

jcosyij ¼ jcosðb,uiÞjZc040, ð4:15Þ

where c0 is a constant independent of h. Hence, Lemma 4.2 impliesthat b must not have a portioned distribution on high frequencyeigenvectors, because jcosyij-0 when h is small.

Lemma 4.3. Let the conditions in Lemma 4.1 hold. Then the solution

vector x has a portioned distribution on low frequency eigenvector uk if

and only if b has a portioned distribution on uk.

Proof. First suppose that b has a portioned distribution on the lowfrequency eigenvector uk with jlkj � Oðjl1jÞ � Oð1Þ, where k is a smallinteger. Eq. (4.15) gives jcosðb,ukÞjZc040. Since JbJ=JxJZc140from (4.9), where c1 is a constant independent of h, we have

jcosðx,ukÞj ¼jðx,ukÞj

JxJ¼jðK�1

h b,ukÞj

JxJ¼jðb,K�1

h ukÞj

JxJ

¼jðb,ukÞj

jlkjJxJ¼jcosðb,ukÞj

jlkj

JbJ

JxJZ

c0c1

jlkj¼ c140, ð4:16Þ

where we have used jlkj � Oð1Þ, and c1 is a constant independent of h.This implies that x has a portioned distribution on uk.

On the other hand, suppose that jcosðx,ukÞjZc040, we have

jcosðb,ukÞj ¼jðb,ukÞj

JbJ¼jðKhx,ukÞj

JbJ¼jðx,KhukÞj

JbJ

¼jlkjjðx,ukÞj

JbJ¼ jlkjjcosðx,ukÞj

JxJ

JbJZ jlkj

c0

c2Zc240,

where we have used the bound, JxJ=JbJZc2 from (4.9), where c2

and c2 are constants independent of h. This implies that b

J. Huang et al. / Engineering Analysis with Boundary Elements 35 (2011) 667–677674

has a portioned distribution on uk, and completes the proof of

Lemma 4.3. &

Theorem 4.2. Let all conditions in Lemma 4.1 hold. Suppose that x has

a portioned distribution on a low frequency eigenvector uk. Then there

exists a constant c0 with 0oc0r1 independent of h such that

Cond_effZc0Cond: ð4:17Þ

Proof. By the assumption, we have jcosðx,ukÞjZc040. Denote thesolution vector x¼ x1þx2, where x1 ¼ ðx,ukÞuk. Then we have

Jx1J¼ jðx,ukÞj ¼ jcosðx,ukÞjJxJZc0JxJ,

to give

JxJr1

c0Jx1J: ð4:18Þ

Since b¼Khx, we have b¼ b1þb2, where b1 ¼Khx1 ¼ lkx1. Hence,we obtain

Jx1J¼1

jlkjJb1J: ð4:19Þ

Combining (4.18) and (4.19) yields

JxJr1

c0

1

jlkjJb1J:

Now we obtain

Cond�eff ¼JbJ

jlnjJxJZc0jlkj

jlnj

JbJ

Jb1J: ð4:20Þ

Since JbJZJb1J and jlkj=jl1jZgk40 with gk � Oð1Þ. Eq. (4.20)leads to

Cond_effZc0jlkj

jlnjZc0gk

jl1j

jlnj¼ c0gkCond¼ c0Cond,

where c0 ¼ c0gk. This is the desired result (4.17), and completes theproof of Theorem 4.2. &

Remark 4.1. Theorems 4.1 and 4.2 can be applied to manynumerical methods for BIEs of the first kind in [13] and the secondkind in Atkinson and Han [1], where the pesudodifferentialoperator is of the boundary integral type, and other boundaryintegration equation methods (BIEM) [18,25]. The stability analysisin this paper may also be extended to the BIEs with dual boundaryintegral equations (DBIEs) in Chen et al. [6–8]. Moreover, theconclusions in this subsection are also valid for the boundaryelement method (BEM).

4.2. Effective condition number for numerical PDEs

Now, we turn to study Cond_eff for numerical PDEs. The discretealgebraic equations of (4.1) from the finite difference method(FDM) or the finite element method (FEM), and are denoted by thematrix form

Khx¼ b, ð4:21Þ

where the matrix Kh is symmetric and positive definite. Also denoteðli,uiÞ the eigenpairs of matrix Kh, and the eigenvalues are alsogiven in a descent order [26,27]

c1h�p ¼ l14l2Z � � �Zln ¼ c040, ð4:22Þ

where pZ2,c1 and c0 are two constants independent of h. For thehomogeneous boundary conditions (i.e., g¼0), the following rela-tions are satisfied (see [26,27,30,32–34]):

JxJ � JbJ � Oðh�1Þ: ð4:23Þ

We can derive the following theorems by following the argumentsof Theorems 4.1 and 4.2.

Theorem 4.3. Let (4.22) and (4.23) be given. Suppose that x has a

portioned distribution on a low frequency eigenvector uk. Then

Eq. (4.4) holds.

From Theorems 4.1–4.3, the effective condition number may besignificantly smaller than Cond only for numerical PDEs, but not fornumerical BIEs whose pseudodifferential operator is of boundaryintegral type as given. The intrinsic characteristics result from thefollowing facts. The differential operator �D is unbounded, so that foreigenvalues of the discrete matrix have the bounds in (4.22)

ln � Oð1Þ, l1 � Oðh�pÞ, pZ2: ð4:24Þ

On the other hand, the integral compact operator is bounded, so thateigenvalues of the discrete matrix have the following different boundsin (4.6):

jl1j � Oð1Þ, jlnj � OðhpÞ, pZ1: ð4:25Þ

Hence, we conclude that the effective condition number is significantfor numerical PDEs, but not for numerical BIEs. For numerical BIEs, thetraditional Cond ¼ O(h�1) is not large, and the Cond_eff is insignificantfor stability. This fact also displays that the Cond_eff is really significant

when the traditional Cond is large for numerical PDEs.For numerical PDE by the Trefftz method (TM) and the colloca-

tion Trefftz method (CTM), Eq. (4.21) is obtained with matrixKhARm�nðmZnÞ, the above analysis and conclusions are also valid(see [28,29,31,35–37]).

5. Numerical experiments

We carry out three experiments by AQMs and h3-Richardson’sextrapolation or splitting extrapolation methods (SEM), and verifythe stability and the error analysis made in the above sections.

Example 1. Let G be a circle with radius e�1=2. Based on [1,44,47],CG ¼ e�1=2a1. Consider the typical BIEs of the first kind

�

Z p

�pln 2e�1=2sin

t�t2

��������wðtÞ dt¼ p

2cos2t, ð5:1Þ

where the true solution w(t)¼cos 2t. In Table 1, we list Cond,Cond_eff and the errors

en ¼ max1r irn

jwðtiÞ�whðtiÞj, eEn ¼ max

1r irnjwðtiÞ�wEðtiÞj,

where Richardson’s interpolation

wEðtiÞ ¼ ð8wh=2ðtiÞ�whðtiÞÞ=7:

Table 2 lists Zk and yk, to indicate v(t) having a proportioneddistribution on uk, discussed later.

Compared the kernel in (5.1) with that in (2.2), there is a factor2p of differences. We have from (2.14) and (2.23)

jl1j ¼maxijlij ¼ p, ð5:2Þ

jlnj ¼minijlij ¼

pn: ð5:3Þ

When n¼210¼1024, the theoretical value of jlnj is given by

jlnj ¼p

1024¼ 3:068ð�3Þ: ð5:4Þ

On the other hand, from Table 1 we can see the numerical data for n

¼ 210,

jl1j ¼ 3:141, ð5:5Þ

jlnj ¼ 8:506ð�3Þ: ð5:6Þ

Eqs. (5.2) and (5.5) coincide with each other perfectly. The value ofjlnj in (5.4) is smaller but close to that in (5.6), to verify our lowerestimation in Theorem 2.1 very well.

Table 5Errors, Cond and Cond_eff for (5.8).

(n1,n2) juh�uj jlnj jl1j Cond Cond_eff

(4,4) 4.413(�2) 0.104 4.287 40.865 12.042

(8,4) 2.166(�2) 0.113 4.147 36.420 11.979

(4,8) 2.166(�2) 0.113 4.147 36.420 12.049

eE 7.229(�3)

(8,8) 1.738(�3) 0.055 4.374 78.474 22.395

(16,8) 9.452(�4) 0.056 4.312 76.007 23.820

(8,16) 9.452(�4) 0.056 4.312 76.007 23.856

eE 7.495(�5)

(16,16) 1.383(�4) 0.028 4.378 154.828 44.046

(32,16) 7.805(�5) 0.028 4.357 153.680 47.609

(16,32) 7.805(�5) 0.028 4.357 153.680 47.618

eE 5.184(�7)

(32,32) 1.725(�5) 0.014 4.375 308.348 87.754

(64,32) 9.703(�6) 0.014 4.369 308.234 95.208

(32,64) 9.703(�6) 0.014 4.369 308.234 95.210

eE 1.350(�9)

Table 6

The values of lk , bk and cosyk at (n1, n2) ¼(8,8).

k lk bk jcosykj k lk bk jcosykj

1 4.374 3.006 0.752 9 0.280 �3.608(�16) 6.123(�17)

2 2.309 1.110(�16) 6.123(�17) 10 0.225 �0.021 5.334(�3)

3 2.076 2.542 0.636 11 0.218 8.326(�17) 6.123(�17)

4 0.941 0.653 0.163 12 0.187 0.014 3.496(�3)

5 0.551 1.110(�16) 6.123(�17) 13 0.184 4.718(�16) 6.123(�17)

6 0.532 0.258 0.065 14 0.162 5.153(�3) 1.288(�3)

7 0.396 2.775(�16) 6.123(�17) 15 0.090 �2.775(�16) 6.123(�17)

8 0.308 0.078 0.019 16 0.055 3.665(�17) 6.123(�17)

J. Huang et al. / Engineering Analysis with Boundary Elements 35 (2011) 667–677 675

Example 2 (Lu and Huang [38], Sidi and Israrli [43]). Let G bexðtÞ ¼ c0ðe

ffiffiffiffiffi�1p

tþc1e�ffiffiffiffiffi�1p

tÞ,tA ½0,2p�, which is an elliptic curve,where c0¼50 and c1¼0.5. Since CGa1, the boundary integralequation:Z 2p

0lnjxðtÞ�xðtÞjwðtÞ dt¼ 2plnjxðtÞj, ð5:7Þ

has the unique solution

wðtÞ ¼ 1þ4X1k ¼ 1

ð�1Þkck

1

1þc2k1

cosð2ktÞ:

The computed results are listed in Tables 3 and 4. A comparison ofTable 3 with Table 1 is given below.

Example 3 (Sloan and Spence [45]). Let G be an open contour oflength 2, in the form of a right-angled wedge:

G¼ fðx1,0Þ : 0rx1r1g [ fð0,x2Þ : 0rx2r1g :¼ G1 [ G2:

The integral equation is chosen as

�

ZG

lnjy�xjvðxÞ dsx ¼ 1 for ðy1,y2ÞAG: ð5:8Þ

We compute the numerical solution of

uðyÞ ¼ �

ZG

lnjy�xjvðxÞ dsx

at (0.5,0.5), whose true value u(0.5,0.5) ¼ 0.621455343.

From [45], although the true solution vðxÞ ¼ Oðjx�x0j�1=3Þ singu-

larity at the right-angled corner, the dominant singularities in v(x)occur at two ends, as u¼Oðjx�x0j

�1=2Þ. Based on [23,24], usingj6ðtÞ inthe periodic transformation (3.7), we obtain the numerical results atQ¼(0.5,0.5) by AQMs and list Cond and Cond_eff in Table 5. Let n1 andn2 be the numbers of uniform partition on [0,1] for G1 and G2,respectively. Based on (3.16), we can obtain the splitting extrapolationerrors eEðQ Þ ¼ juEðQ Þ�uðQ Þj, where the splitting extrapolation values

uEðQ Þ ¼8

7

Xd

m ¼ 1

uhðmÞ ðQ Þ� d�7

8

� �uhð0Þ ðQ Þ

" #, d¼ 2:

The errors juhðQ Þ�uðQ Þj and the splitting extrapolation errors eE(Q) arealso listed in Table 5.

Now, let us examine more the results in Tables 1 and 3. We cansee numerically,

e¼ Oðh3Þ, eE ¼ Oðh5Þ,

and

jl1j � c, jlnj ¼Oðh�1Þ: ð5:9Þ

Table 4

The values of lk , bk and yk at n¼25.

k lk bk yk k lk bk yk

1 24.58 1.534(+2) 0.000 17 0.406 �2.365(�14) �1.570

2 4.712 �2.931(�14) �1.570 18 0.369 �8.881(�15) �1.570

3 1.964 �1.964(�14) �1.570 19 0.368 1.159(�14) 1.570

4 1.571 �2.086(�14) �1.570 20 0.339 8.881(�15) 1.570

5 1.180 4.440(�15) 1.570 21 0.339 2.471(�14) 1.570

6 1.179 �9.014(�15) �1.570 22 0.316 �8.881(�15) �1.570

7 0.918 �1.536(�14) �1.570 23 0.316 1.366(�14) 1.570

8 0.838 8.881(�14) �1.570 24 0.299 �1.332(�14) �1.570

9 0.740 1.831(�14) 1.570 25 0.299 1.233(�15) 1.570

10 0.653 6.217(�15) 1.570 26 0.287 3.908(�14) 1.570

11 0.614 1.222(�14) 1.570 27 0.287 �1.411(�14) �1.570

12 0.540 0.000 1.570 28 0.278 0.000 1.570

13 0.523 �2.735(�14) �1.570 29 0.278 6.920(�15) 1.570

14 0.464 �3.730(�14) �1.570 30 0.273 �8.881(�15) �1.570

15 0.457 �3.193(�14) �1.570 31 0.273 2.418(�14) 1.570

16 0.409 �1.687(�14) �1.570 32 0.272 �5.329(�15) �1.570

Eqs. (5.9) coincide with the theoretical estimates of jl1j and jlnj

given in Theorem 2.1. Next, from Tables 1 and 3 we find

Cond¼Oðh�1Þ, Cond_eff ¼Oðh�1Þ, ð5:10Þ

which are also consistent with Theorems 2.1, 2.2 and 3.1. Moreover,from Table 5 we have

juh�ujð16,16Þ

juh�ujð32,32Þ

¼1:383ð�4Þ

1:725ð�5Þ¼ 8:01, ð5:11Þ

and

eEð16,16Þ

eEð32,32Þ

¼5:184ð�7Þ

1:350ð�9Þ¼ 384: ð5:12Þ

Eq. (5.11) displays the O(h3) convergence rate, and Eq. (5.12) alsodisplays the O(h6) convergence rate since 384Z26

¼ 64. Hence, theSEMs can provide more accurate solutions. Note that from Table 5with the total number n ¼ n1 + n2 ¼32 and 64, the errors of SEMsare 5.184(�7) and 1.350(�9), respectively. In contrast, whenn¼256 the solution uh

¼0.62125 is given in [45] by Galerkinmethods. This fact displays the efficiency of AQMs and SEMs.

Finally, to scrutinize the spectral distribution of vector b in uk,we compute all bk ¼ uT

k b and yk ¼ arccosðbk=JbJÞ, and list them inTables 2 and 4 in the descending order of jlkj. We can see thatyk � 7p=2 except y4 ¼�0:102 and y1 ¼ 0:000 from Tables 2 and 4,respectively. This fact coincides with Theorem 4.2. In Table 6, welist all bk and cosyk ¼ bk=JbJ, to find the dominant distribution ofbk. We can see that cosy1 ¼ 0:752, cosy3 ¼ 0:636 and cosy4 ¼ 0:163.Since cos2y1þcos2y3þcos2y4 ¼0:996569� 1, the dominate vec-tors (i.e., the proportional distribution) of b happen just at the threelow frequency eigenvectors u1, u3 and u4, also to support theassumptions in Theorems 4.2 and 4.3.

J. Huang et al. / Engineering Analysis with Boundary Elements 35 (2011) 667–677676

6. Concluding remarks

1.

A new and systematic stability analysis is made for the advanced(i.e., mechanical) quadrature methods (AQMs) for the first kindBIEs in [20–24,38], based on Cond and Cond_eff. For continuous,singularity and discontinuous problems [20], there exist thebounds

Cond_eff � Cond � Oðh�1Þ: ð6:1Þ

Eq. (6.1) display an excellent stability of the AQMs. For discretematrix Ah in (2.5) from (2.3), the bound as (2.23) in Theorem 2.1is derived, by using the special cðzÞ function.

2.

Numerical experiments are carried out for the arbitrary boundaryG with CGa1 by AQMs and SEAs, and the computed results inExamples 1–3 and those in [20] agree with (6.1) perfectly. Also, thenumerical condition numbers show the optimal O(h�1).

3.

Let us compare the stability analysis in this paper with that inChristiansen and Saranen [13] in more detail. In [13, p. 48], thealgorithms (1.4) were also proposed, where the local conditionnumber was called. In this paper, we discuss three types of thefirst kind BIEs in Sections 2 and 3. Although the error analysishas been made in [20–24,38], the excellent stability as (6.1) isconfirmed for the AQMs, to enhance its application.

4.

In Section 4, based on matrix analysis, we explore the intrinsiccharacteristics, to give a theoretical justification to explain the factthat Cond and Cond_eff have the same growth rate for numericalBIEs. Note that the improvements of stability by the Cond_eff arenot as significant as those for numerical PDEs in [26–37,19]. Therationale is that the operators of BIEs and PDEs are bounded andunbounded, respectively. Hence, the eigenvalues of their discretematrices have different bounds in (4.24) and (4.25).

5.

The analysis in Section 4 also explains that why the effectivecondition number is not developed until its applications tonumerical PDEs in [26–37,19]. Because the effective conditionnumber is significantly smaller than Cond for numerical PDEs,but not for numerical BIE in this paper and numerical linearalgebra [27]. This provides an important and complete knowl-edge of effective condition number, as a concise review oneffective condition number in [26–37,19].

Acknowledgements

Authors are grateful to Prof. J.T. Chen and the reviewers for theirvaluable comments and helpful suggestions. This work is sup-ported by the National Natural Science Foundation of China underGrant 10871051, Doctoral Program of the Ministry of Educationunder Grant 20090071110003 and 973 Program Project underGrant 2010CB327900, and Shanghai Science & TechnologyCommittee under Grant 09DZ2272900 and Shanghai EducationCommittee (Dawn Project).

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