The Splitting Extrapolation Method: A New Technique in Numerical Solution of Multidimensional Problems

THE SPLITTING EXTRAPOUTION METHOD

SERIES ON APPLIED MATHEMATICS Editor-in-Chief: Frank Hwang Associate Editors-in-Chief: Zhong-ci Shi and Kunio Tanabe

Vol. 1

International Conference on Scientific Computation eds. T. Chan and Z.-C. Shi

Vol. 2

Network Optimization Problems — Algorithms, Applications and Complexity eds. D.-Z. Du and P. M. Pandalos

Vol. 3

Combinatorial Group Testing and Its Applications by D.-Z. Du and F. K. Hwang

Vol. 4

Computation of Differential Equations and Dynamical Systems eds. K. Feng and Z.-C. Shi

Vol. 5

Numerical Mathematics eds. Z.-C. Shi and T. Ushijima

Vol. 6

Machine Proofs in Geometry by S.-C. Chou, X.-S. Gao and J.-Z. Zhang

Vol. 7

The Splitting Extrapolation Method by C. B. Liem, T. Lu and T. M. Shih

Series on Applied Mathematics A Volume 1

THE SPLITTING EXTRAPOLATION METHOD A New Technique in Numerical Solution of Multidimensional Problems C. B. Liem The Hong Kong I'olylcc/inic

Univerai.ty

T. Lii Chengdu Inal.il.utv of Computer Application a

T. M. Shih The I long Kong Polytechnic

University

World Scientific Sinaapore •• New NewJersey Jersey • London L Singapore • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 9128 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Tao, Lu, 1940Splitting extrapolation method / by Lii Tao, T.M. Shih, C.B. Liem. p. cm. - (Series on applied mathematics: vol. 7) Includes bibliographical references and index. ISBN 9810222173 1. Splitting extrapolation method. I. Shih, T. M., 1937II. Liem, C. B., 1934. III. Title. IV. Series. QA297.5.T36 1995 519.4-dc20 95-18449 CIP

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Preface

Large scale computation in science and engineering is one of the most challenging areas in computational mathematics. Despite the significant progress m a d e in digital computers and computational m e t h o d s over t h e last two decades, to date the solution of problems with complicated domains and high dimension remains difficult. To solve a high dimensional problem, the computational work and computer storage requirement increase expo­ nentially with respect to the dimension. This phenomenon, known as "the dimensional effect", is the major obstacle in the application of computa­ tional methods to the solution of high dimensional problems. In order to overcome this obstacle, a computational m e t h o d would, ide­ ally, fulfil the following requirements: ( 1 ) it should be a parallel computational method which can be imple­ mented in multiprocessor computers requiring minimal communica­ tion among processors; ( 2 ) it should possess a high order of accuracy with a posteriori error estimation and adaptive grid refinement so t h a t it is able to evaluate the computational results in real time; ( 3 ) the complexity of the algorithm should be independent or almost independent of the dimension of the problems. This volume describes the splitting extrapolation m e t h o d t h a t contains all the above-mentioned characteristics. T h e idea of splitting extrapolation was first developed by two Chinese mathematicians Qun Lin and Tao Lii in 1983. T h e authors of this book conducted a detailed analysis of splitting extrapolation and its applications in 1990, and identified it to be an extension of the classical Richardson extrapolation, t h a t is, the multivariate Richardson extrapolation. In recent years, promising progress has been m a d e in parallel computa­ tional methods, multilevel m e t h o d s and sparse grid combination techniques. T h e combination of these m e t h o d s with splitting extrapolation has become an effective way of tackling "the curse of dimensionality". A comprehensive review of these m e t h o d s can be found in U. Rude (1991, 1993). v

vi

Preface

A lot of the recent research on splitting extrapolation methods has been published in Chinese journals but has not drawn much attention from over­ seas. I hope that with this publication, it will help to promote interactions among interested readers outside China.

Beijing, December 1994

Zhong-ci Shi Member of Academia Sinica

Acknowledgements

First and foremost, we would like to acknowledge Professors Zhong-ci Shi and Qun Lin for their valuable suggestions and unfailing encourage­ ments. We also express our sincere gratitude to Professor Zhong-ci Shi for writing the preface of this volume. We owe many thanks to our students at the Hong Kong Polytechnic University, Har-ming Chu, Yu-kin Yiu, Kin-fai Wong, Tung-kao Hung and Wai-lun Kwok, for performing many valuable numerical experiments, and Mr. Yi-xun Wang for his help in editing the manuscript. T h e authors would also like to thank World Scientific Publish­ ing Co P t e Ltd and Ms E.H. Chionh for their friendly cooperation. Finally, we thank the Hong Kong Polytechnic University Research Fund (340/284 and 340/181) and the National Science Foundation of China (19471075 and 863-306-05-01), without whose subsidies this work would not have been possible.

Hong Kong,

December 1994

Tao Lii Institute of Chengdu Computer Applications Academia Sinica

Tsi-min Shih,

Chin-bo Liem

Department of Applied Mathematics T h e Hong Kong Polytechnic University

vn

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Introduction

Most m a t h e m a t i c a l models of scientific and engineering problems are described in continuous forms, such as integrals, integral equations, differ­ ential equations and integro-differential equations. In order to numerically evaluate these models, it is necessary to choose a discrete parameter h and a discretization m e t h o d , so as to convert the continuous forms into alge­ braic equations, and then to obtain the numerical solution uh. Therefore, the accuracy of uh depends on h. If an exact solution is regarded as the limit of numerical solutions, con­ sider the dependence between the numerical solution and discrete p a r a m e ­ ters, one can derive naturally an extrapolation or an extrapolation to the limit. In essence, one can choose several different discrete parameters and evaluate the relevant approximate solutions. These solutions are then ex­ trapolated to obtain another approximate solution with higher order of accuracy. In fact, the idea of extrapolation was used in the ancient time for ac­ celerating the convergence. T h e Chinese mathematicians Liu Hui (A.D. 263) and Zhu Chongzhi (429-500) used the idea to evaluate the value of 7r, and Huygens (1654), a French mathematician, proposed an extrapolation 4 1 formula -Th - -T2h to evaluate TT. o o However, it was Richardson (1910) who first studied the extrapolation as common and effective algorithms, and established polynomial extrap­ olations. This was followed by the ^-extrapolation by W y n n (1956), the rational extrapolation by Bulirsch and Stoer (1964), and the general ex­ trapolation by C. Brezinski (1980). In practice, extrapolation methods have become effective techniques for accelerating the convergence in various branches of numerical m a t h e m a t ­ ics, including the Romberg algorithm (1955) in numerical integration, the Stetter Theorem (1965) and the Gragg-Bulirsch-Stoer algorithms (1964) in ordinary differential equations, the extrapolation in the finite difference m e t h o d by G. Marchuk and V. Shaidurov (1979), the extrapolation in the finite element m e t h o d by Q. Lin and T . Lii (1983), and by R. Rannacher (1988). IX

X

Introduction

T h e above-mentioned extrapolations, though varying in forms, are all based on the extrapolation with respect to a single discrete parameter. Nat­ urally, the extrapolation of such a kind will cause difficulties in solving high dimensional problems because of the dimensional effect. Take the evalua­ tion of an s-multiple integral by Richardson's extrapolation as an example, in order to obtain an algebraic precision of degree (2m -f 1), function val­ ues at 2 m 5 points are needed. This leads to an exponential increase in the computational complexity as the dimension increases. T h e dimension effect becomes more serious in solving partial differ­ ential equations: for an s-dimensional problem by means of one step of Richardson's extrapolation, apart from solving an algebraic equation with Ns unknowns, one has to solve an auxiliary problem with (2N)S unknowns. However, problems arising from science and engineering, such as the explo­ ration and exploitation of oil and gas, weather forecasting, pollution control, are often described by high dimensional m a t h e m a t i c a l models. It was on the grounds of such demands t h a t Q. Lin and T . Lii (1983) proposed the Splitting Extrapolation Method (SEM). T h e Richardson extrapolation is based on a single-variable asymptotic expansion of the error, while the SEM is based on a multivariate asymptotic expansion of the error. This simple change greatly improves the efficiency of the extrapolation. First, the computation complexity is almost optimal. For example, to evaluate an s-multiple integral by the SEM, in order to reach the algebraic precision of degree (2m + 1 ) , only the function values at ( m m ) P o m ^ s a r e needed. Second, SEM converts a large scale problem into several mutually independent subproblems which can be computed in parallel by a multiprocessor computer with a few communication among the processors. Here, the degree of parallelism depends on the numbers of independent discrete parameters, which do not need to coincide with the dimension of the problem, and can be set according to the scale of the problem and the available number of processors. Third, SEM provides a posteriori error estimations and self-adaptive algorithms. T h e latest developments of extrapolation m e t h o d s are related to the multilevel methods. Multilevel methods (such as multigrid methods) re­ quire the solution of several discrete equations with different grid sizes. These results can be used in extrapolation m e t h o d s to obtain economically a higher order of accuracy. There are two branches of this idea: A. B r a n d t (1983) and W . Hackbusch (1984) combined the Richardson extrapolation and the multigrid methods (called the r-extrapolation), and A. Schiiller and Q. Lin (1985) combined the SEM and multigrid m e t h o d s . Further, based on finite element multilevel subspace splitting, C. Zenger (1990) pro­ posed a sparse-grid m e t h o d for solving multidimensional problems, and

Introduction

XI

—1

—x

proved that s-dimensional problem needs only 0(/t 1 In /i|^ ) grid points and an equivalent number of base functions to obtain the accuracy of or­ der 0 ( / i 2 | l n / i | 5 _ 1 ) , whereas the standard finite element method requires 0(h~s) grid points for the accuracy of order 0 ( / i 2 ) . The former method therefore has a greatly reduced computation complexity. M. Griebel, M. Schueider and C. Zenger (1990) proposed the combina­ tion techniques based on the sparse-grid method. In principle, this method is similar to the SEM, in that both methods rely on a certain form of multivariate asymptotic expansion. However, the aim of the combination techniques is to combine the solutions of coarse grids to obtain an approxi­ mate solution which has the same order of accuracy as the solution of finer grids. A comprehensive report on the relationship between the SEM and the combination techniques can be seen in U. Rude (1991, 1993). Lastly, a new development of extrapolation, the implicit extrapolation, tries to obtain an approximate equation of higher order of accuracy by means of the idea of extrapolation. The developments of the implicit extrapolation methods are also described in U. Rude (1991, 1993). This volume concentrates on the theory and application of the splitting extrapolation methods. Some of the materials are extracted from the work by the authors and are presented in this volume for the first time. Chapter I introduces the Richardson extrapolation and its application in numerical integration. This may be useful for readers who are not familiar with these topics. Chapters II to V cover respectively the SEM and their applications in multidimensional numerical integration, integral equations, and differential equations. Chapter VI discusses the combination methods related to extrapolation. Finally, Chapter VII introduces the combination techniques and the sparse-grid methods by C. Zenger.

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List of Symbols

■z m |n

set of integers

m Jin

n is not divisible by m

R M'

set of real numbers

X

= ( x i , • • •, xs), a point in Ms

e

j

unit vector in the direction of Xj -coordinate

fi

open set of Ms

an

boundary of Q

Q

closure of Q,

0

empty set

Tfc.1

set of gridlines, see §5.2

Th

set of grid points, see §5.2

nh n*,.-

set of regular grid points, see §5.2

dtlh

d f i n l \ / , see §5.2

«*,«

&h U Qhti U d£lh

M{x)

discrete neighbourhood of gird point x, see §5.2

h

= (fei, • • •, hs)

ho

= m a x hi

a

= ( a i , •• •, a 5 ) , a multi-label

M

= Qfi H

a!

abbreviation of oc\\a2\ • • - a 5 !

/i

a

h

n is divisible by m

Euclidean space

s-dimensional

set of irregular grid points, see §5.2

multi-parameter mesh width

ha,

abbreviation of h^1 • • • ftf*, see §2.1.1 ii

• .•

w

^i

h

v

s 0rt abbreviation of ( — , • • • , —^-), see §2.1.1

xm

xiv

List of Symbols

1-1-a

abbreviation of (, • • •, - — - — ) , see §2.1.1 V l + c*i 1+ a/' *

/(#)

abbreviation of function / ( x i , • • •, xs)

f
f-y(xi, * • j xs) is a homogeneous function of degree 7, see §1.2

D{

abbreviation of the partial derivative operator -7;— OXi

a

1

D dx C m (ft) C m+<7 (ft) L p (ft) W™(ft) Co°(fi)

abbreviation of D" • • • D?' abbreviation of dx\ • • • dxs set of m order continuous differentiable functions on ft Holder space, 0 < a < 1 Space of p-integrable functions on ft Sobolev space set of infinitely differentiable functions with support in ft

W™ (ft) Hm (ft)

closure of C^(Q) in W™(Q) abbreviation of W? (ft)

tf^(ft) I • \\m,P,n ii| m>Pj n

abbreviation of W? (ft) norm of W^(ft), denoted briefly by || • || m)P , see §5.3.2 seminormof W ^ f t ) , denoted briefly by |-|m,p sometimes, see §5.3.2

I • || m ,n |.||oo

abbreviation of || • ||m,2,r2, denoted by || • ||m>2 sometimes, see §5.3.2 normofL°°(ft)

\u\\'^q

norm of ( fl W™(e)) n ^ ( f i ) , see §5.3.2

||u||'m A L Lh Sj K A"* 3t(K) k(x, y)

abbreviation of ||u||^ | 2 Laplace operator elliptic partial differential operator discrete approximation of L, see §5.2.1 second order central difference operator, see §5.2.2 integral operator, see §4.1.1 conjugate operator of K, see §4.1.1 the range of operator K kernel of K, see §4.1.1

eeJh

List of Symbols * ( n ) ( * , y)

■£ K

K

* „ -

K

XV

n iterated kernel of K, see §4.1.1 {Kn}

point wise converges to K, see §4.1.1

{Kn}

collectively compact converges to K, see §4.1.1

P(K)

the resolvent set of K

Jh

partition with grid parameter h

h

5 (fi)

test function space, see §5.3.1

5j(fi)

Sh(n)nH%(Q),

n"

interpolation operator, see §4.7

Pn

projection operator, see §4.7

see §5.3.1

Eh

orthogonal projection, see §4.7

h

interpolation operator mapping C(Q.) on SQ (Q), see §5.3.1

e

element of Jh,

P!(e)

set of linear functions in e, see §5.3.1

QiOO

s-linear function on e, where s is the dimension of e

«(•,•)

energy inner product

Rh

Ritz projection

see §5.3.1

h

finite element approximate solution of u

u

1

interpolation function of u

Jk

fc-th refinement of the initial partition Jo, see §7.1

M*

set of all nodes of Jk, see §7.1

Sk

piecewise s-linear function corresponding to Jk, see §7.1

hu

interpolation of function u on Sk, see §7.1

Tk

hierarchical basis subspace, see §7.1

*5n,m

bilinear element space on rectangular grids, see §7.1.2

9°)m °n

sn>mr\H^)

u

sparse grids space, see §7.1.1 u1

interpolation function of u in S% m, see §7.1.2

K,n

combination approximation of u, see §7.2.1

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Contents Preface

v vii

Acknowledgements

ix

Introduction

xiii

List of symbols I. Generalization and application of Richardson's extrapola­ tion 1. Polynomial extrapolation 1.1. Interpolation polynomials and extrapolation 1.2. Polynomial extrapolation algorithms and their generalization 1.3. Stability and convergence 1.4. A posteriori error estimation 2. Application in numerical integration 2.1. Euler-Maclaurin summation formulae 2.2. Euler-Maclaurin formulae for integrands with singularities at end points 2.3. Multidimensional Euler-Maclaurin expansion 2.4. Asymptotic expansion of multidimensional numerical quad­ rature of functions with singularities 2.5. Numerical results II. Splitting Extrapolation Methods 1. Multivariate asymptotic expansion 2. Recurrence algorithm of splitting extrapolation 3. Coefficients and stability of SEM 4. A posteriori error estimation of splitting extrapolation 5. Fractional power expansion and successive elimination methods.

III. Application of SEM to multidimensional numerical inte­ gration 1. Splitting extrapolation algorithms for smooth functions 2. Change of variables of multidimensional improper integrals 3. Duffy transform for multidimensional improper integrals 4. Multivariate asymptotic expansion of multidimensional improper integrals xvn

1 1 2 5 9 15 20 20 29 31 33 39 43 45 47 51 59 61 69 70 79 83 85

xviii

Contents

5. Integration on multidimensional simplexes 90 6. Integration on multidimensional domains with curved boundaries 96 IV. SEM for integral equations 99 1. General theory of integral operators 99 2. Approximate quadrature methods 105 3. SEM for multidimensional integral equations with smooth kernels 109 4. SEM for integral equations in polygonal domains 114 5. SEM for eigenvalues and eigenfunctions 125 6. SEM for integral equations with nonsmooth kernels 132 7. SEM for collocation-method solutions of one-dimensional weaklysingular integral equations 143 8. SEM for collocation-method solutions of multidimensional weakly-singular integral equations 153 V. SEM for differential equations 159 1. SEM for the collocation-method solutions of two-point boundary value problems 160 1.1. SEM for the collocation-method solutions of the quasilinear two-point boundary value problems 160 1.2. SEM for the collocation-method solutions of Sturm-Liouville type eigenvalue problems 163 1.3. SEM for singular two-point boundary value problems 168 1.4. SEM for two-point boundary value problems with discon­ tinuous coefficients 170 2. SEM for finite-difference approximations 2.1. Difference equations and the discrete maximum principle 2.2. Multivariate asymptotic expansion of the finite-difference approximation on a domain with a smooth boundary 2.3. Multivariate asymptotic expansion of the finite-difference approximation on a rectangular parallelepiped 2.4. Numerical examples

172 172

3. SEM for finite-element approximations 3.1. Finite-element approximations of second order elliptic equations 3.2. Basic expansion of bilinear rectangular elements 3.3. Expansion of three-dimensional problems 3.4. Piecewise uniform rectangular partition and multiparameter asymptotic expansion 3.5. Piecewise strongly-regular partition 3.6. Numerical examples

200

178 189 196

201 204 217 223 227 230

Contents

xix

V I . C o m b i n a t i o n m e t h o d s for accelerating t h e convergence .. 232 1. Combination methods 232 1.1. Combination principle 232 1.2. Combination methods for integral equations 234 1.3. Combination methods for difference equations 236 2. Combination methods for collocation-method solutions of quadratic and cubic splines 241 3. Combination methods for the Nystrom solution of boundary integral equations of the second kind 247 3.1. Boundary integral equations of the second kind 247 3.2. Combination methods for the Nystrom solutions 248 3.3. The nonsmooth cases 253 3.4. Neumann boundary value problems 255 3.5. Numerical examples 257 V I I . Sparse grid m e t h o d s a n d combination techniques 1. Sparse grids 1.1. Multilevel splitting of finite element spaces 1.2. Two-dimensional sparse grids 1.3. Higher dimensional sparse grids 1.4. Finite element equations on a sparse grid 2. Combination techniques 2.1. Two-dimensional combination techniques 2.2. Three-dimensional combination techniques 2.3. Numerical examples 2.4. Comparison of combination techniques, SEM and sparse grid methods 3. Outline of the implicit extrapolation

260 261 261 265 268 271 273 274 276 280

Appendices

293

Bibliography

303

Index

313

284 290

Chapter I GENERALIZATION AND APPLICATION OF RICHARDSON'S EXTRAPOLATION

In this Chapter, we discuss only the Richardson extrapolation and its application in numerical integration. As the application of Richardson's extrapolation in partial differential equations is concerned, see the mono­ graphs by G. Marchuk and V. Shaidurovt 90 !, and by Q. Lin and Q.D. The contents of this Chapter are mainly based on the survey by D.C. Joycet 48 ! and P. Rabinowitzt 100 ^ 1. Polynomial extrapolation One of the basic topics of numerical mathematics is that: given a con­ tinuous problem (such as an integral, an integral equation, a differential equation), discretize it with mesh width h into an algebraic problem, which is solved with the help of computers to obtain an approximate solution T(/i). The precision of T(h) depends on the mesh width h. Usually, the smaller the /i, the finer the grid partition, the higher the precision of T(/i), and the larger the amount of work. In many cases, the exact solution ao depends continuously on h > 0, i.e., a 0 = limfc-o T(h) = T(0),

(1.1.1)

and also there is an asymptotic expansion of the following form: T(h) = a0 + ai/i^ 1 + a2h?* + • • • + amhPm + 0(fe*"»+l),

(1.1.2)

where ai, a2) • • •, am; Pi,P2r '' >Pm+i are constants and 0 < pi < p2 < • • • < Pm+i- In practice, some asymptotic expansions may have terms of the form hPi lnfe. It will be proved later that such terms can be eliminated by means of the polynomial extrapolation. In general, for smooth problems, the expansion (1.1.2) contains only even powers of h. An extrapolation with the expansion (1.1.2) is called the polynomial ex­ trapolation, (also called Richardson's extrapolation or Romberg's extrap­ olation). In 1910, Richardson combined two approximate solutions T(h) 1

2

I. Generalization and application of Richardson's

extrapolation

h A h I and T ( —) to get - T ( — ) — ^ ( ^ ) j which is a solution with an order of accuZ o Z o racy 0 ( / i 4 ) , called the /i 2 -extrapolation by Richardson. In 1955, Romberg used the Euler-Maclaurin asymptotic expansion to successively eliminate /i 4 , /t 6 , • • •, and obtained approximate solutions with higher and higher or­ der of accuracy. It is the famous Romberg's m e t h o d for numerical integra­ tion. In fact, in the 3rd century, the Chinese m a t h e m a t i c i a n Liu Hui (A.D. 263) already possessed a primitive thought of extrapolation. In Europe, Huygens (1654) used (4S2n — Sn)/3 to evaluate TT. At present, applications of the extrapolation in improving the accuracy of approximations can be found almost everywhere in mathematical computation.

1.1. Interpolation

polynomials

and

extrapolation

If the approximate solution has an asymptotic expansion of the form (1.1.2), and there are already m + 1 approximate solutions, namely T(h{), i = 0, • • •, m, where ho > hi > • • • > hm > 0, then the m - t h extrapolation value Tm is determined by the following linear system: T(hi) = 7 l 0 ) + ai hPl + • • • +

amhpr,

i = 0,---,m,

(1.1.3)

where Tm , a i , • • •, a m are unknowns. T h e value Tm ' is of interest because the expansion (1.1.2) shows t h a t the order of accuracy is 0 ( / i Q m + 1 ) . In fact, instead of solving the linear system (1.1.3) to find Tm\

one can

m

derive an interpolation polynomial f(x)

= Y ^ 6,-xPi satisfying the interpo-

lation conditions: / ( / i , ) = T(hi), i = 0 , 1 , • • •, m. Once the interpolation polynomial is obtained, / ( 0 ) = 60 = T^ will be the required extrapolation value. This relationship between the extrapolation and interpolation poly­ nomial makes it possible to use the properties of interpolation polynomials to study extrapolation algorithms. For the sake of simplicity, the following asymptotic expansions with even powers are discussed: T(h) = aQ + aih2

+ • • • + amh2m

+ 0(/i2(m+1)).

(1.1.4)

Let x = /i 2 , and the corresponding interpolation problem becomes: find a polynomial P m ( x ) = a 0 + aix H h a m x m , satisfying Pm(xi)

= T(hi),

i = 0, • • •, m,

(1.1.5)

where X{ — hf and Pm(xi) = / ( x x ) . This problem is equivalent to finding coefficients ao, • • •, am, satisfying the linear system a0 + axxi + • • • + amxm

= /(x t -), i = 0, • • •, m.

(1.1.6)

1. Polynomial

extrapolation

3

Since the coefficient determinant of (1.1.6) is the Vandemonde determinant with value TT(xfc — Xj) ^ 0, the interpolation polynomial exists uniquely. k>j

It is well known that there are different expressions for the interpolation polynomial and each has advantages and disadvantages of its own. A) Lagrange's interpolation formula m

Pm(x) = Y,H*)f(*i),

(1-1-7)

i=0 ™

X — Xj

where Li(x) — \ \ ( jj=o = o Xi

) are called the interpolation basis functions

x

j

satisfying Li(xj) = 8iji i , j = 0,---,m,

(1.1.8a)

£ > ( * ) = 1.

(1.1.8b)

and

The Lagrange interpolation polynomials are expressed explicitly, they are convenient in theoretical analysis, but are not suitable in practical compu­ tation. For example, when the number of nodes increases, one has to start the computation from the beginning. B) Newton's interpolation formula Pm(x) = f(xo) + (x - x0)f[x0ixi] + (x- x0)(x - xi)f[xQl xi, x2] + • • • + (x - x0) • • • (x - ar m -i)/[xo, • • •, xm],

(1.1.9)

where /[xo, • • •, X{] is the i-th Newton's divided difference, which can be expressed as

/[*„,•••,».•] = E i f{Xi) =0 ' II(**-*J)

( L L 1 °)

A:=0

Newton's interpolation formula has many advantages: one of which is that when the number of nodes increases by one, only one more term is added, while the previous terms remain unchanged; another is that this formula

4

I. Generalization and application of Richardson's

extrapolation

can be generalized easily to multivariate interpolations. This property is very helpful in the discussion of the splitting extrapolation algorithm in Chapter II. C) Neville's interpolation formula

P
i=

0,..,m,

(x - *,-)J?-r(«) - (* - 'j+i)p?M*) j = 1, • • •, m; t = 0, • • •, m - j .

(1.1.11)

Neville's interpolation method is a recurrence algorithm in essence. From the interpolation polynomials of degree j — 1, one can easily construct interpolation polynomials of degree j . The following lemma justifies the Neville recurrence algorithm. Lemma 1.1. Pj'(x)

constructed by (1.1.11) is a polynomial of degree

j satisfying the interpolation condition: Pj(xk)

= /(#*;)> k = i, • • •, i + j .

Proof. By induction, when j = 0, PQ(X) obviously satisfies the inter­ polation condition. Suppose that PyL* \x) and PjJ^x) satisfy respectively the interpolation conditions: = /(**)> * = i + 1, ■ • •, i + J,

PJ?i\*k)

(1.1.12a)

and = /(**),

P&ixk) then for Pj(x) tion, we have

(1.1.12b)

determined by (1.1.11), according to the inductive assump­

x

n(i),

Pj

t = i, • • •, i + i - 1,

(xk)

(^k-Xi)f(xk)-(xk-xj+i)f(xk) =

= /(**)> Xj+i - Xi

(.)/ x

T («o =

i = t + l,---,i + j - l . 0-(xi-a? i:+ 0/(a:,-) x

j+i

= /(*.)

x

i

and (0

(*i+.--«0/(«i+j)-o x

j+i

-

x

>

1. Polynomial extrapolation

5

which completes the proof.

D

The Neville interpolation algorithm is summarized in Table 1.1. When­ ever a new interpolation point is added, only polynomials for obtaining the elements in the last row are evaluated. Table 1.1. List of Neville's interpolation f(x0) =

p(°\x) P(01\*)

f(x1) =

f(x2) = p(2\x) 3)

P[°\x) p[X\x)

P(°)(x)

p 2)

r^h*) P?\x)

/ ( * 3 ) = P
H0)(*) P(°)(x) P£\X)

Applying the mean value theorem, it can be proved that the error of a Neville interpolation polynomial satisfies

/(m+1)(£)'+m

Pk\x)

- f(x) = —

where min(a;, xiy xi+i, • • •, xi+m) 1.2. Polynomial lization

±± H (x - xk), (™ + l)! k=i

(1.1.13)

< f < max(x, x,-, xi+1, • •, x i + m ) .

extrapolation

algorithms

and their

genera­

Since the extrapolation value is equal to the value of the interpolation polynomial at h = 0, then substituting xt- = hf and using Neville's in­ terpolation formula (1.1.11), the Romberg extrapolation algorithm follows immediately: T 0 (0 = T(hi) = P o (O (0), i = 0 , . . . , m , T(0 = j

pCO/nN = j

L2

A

T(i)

i

_ i 2'7-0'+1)

«+i i-i V J - I h2 . -h2 n

i = l,---,m;

(1.1.14)

n

i+j

i

i=

0,-,m-j.

The error estimate of Romberg's extrapolation can be obtained from (1.1.13). Noting that / ( * ) = T(h) = a0 + alX + • • • + a m + 1 x m + 1 + • • •, / m + 1 ) ( 0 ) = (m + l)!a m + 1 >

I. Generalization and application of Richardson's

6

extrapolation

then 2*> - ao = P) 0 (0) - T(0) »'+;'

« - a i + i n ( - * t ) = (-l) i «i+i^---A. ?+i-

(1.1.15)

fc=t

Similarly, if the expansion of T(h) is T(h) = a 0 4- ax/i r + a 2 /i 2r + • • • + a m fc mr + 0 ( / i ( m + 1 ) r ) , (1.1.16) let x = hr,f(x) = T(/i), then substitute x = 0 in the corresponding inter­ polation polynomial P&'(x), and T^/ = Pm (0) is an approximation of ao. Therefore, the polynomial extrapolation algorithm is as follows: T0(0 = T ( M , i = 0,...,m, Lr

nr.(»)

LrT(*+l)

hUj-h

(1.1.17)

n(0

n(»>l)

(hi/hi+j)r j = l,---,m;

- 1 i = 0,

>™-j.

The error can be estimated by:

lj 0 -a 0 «(-iya i + 1 &f.

*»+i-

(1.1.18)

If T(/i) has a more general asymptotic expansion as follows: T(h) = a 0 + ai V + a 2 /^ 2 + • • • + a m W~ + O ^ " * 1 ) ,

(1.1.19)

with 0 < pi < • • • < pm < Pm+ii then the recursive formula (1.1.17) can only be used once. Here T1W = T(/i i + 1 ) +

Tjhi+J-Tihi) (hi/hw)Pl

- 1

(1.1.20)

and the error is of order 0(hP2). In this general case, (1.1.17) cannot be used to obtain further extrapolations. However, for a special choice of mesh parameters, i.e., there exist ho and 6, where 0 < b < 1, and Hi)

hi = h0b\

i = l , 2 , --jm,

Bulirsch-Stoer (cf. [48]) proved that there is a similar recurrence algorithm. Algorithm 1.1

(Bulirsch-Stoer) i?(6, ZioiPi, • • • ,Pm)

1. Polynomial extrapolation

7

S t e p 1. Let T 0 (0 = r(/i 0 6'), t = 0, • • •, m, i := 1. S t e p 2. Set r («) J

_

=

J-1

^zi 1 - &w r (»'+i) _ T ( 0 i 1 " _w_ii~1, i = 0,.--,m-i. 3 ffi) +

(1.1.21)

S t e p 3. If j = m, output the extrapolation value Tm , otherwise set j := j + 1 and go to Step 2. Algorithm 1.1 has to be verified inductively. For j = 0, it follows from (1.1.19) that T0(0) = T(fto) has the following asymptotic expansion T0(0) = T(fco) = <*o + a i ^ 1 + • • • + amhp0" + 0(/i£ m + 1 ). Assume that for Tj_\ = 7}_i(/io), there is an asymptotic expansion T^

= Tj-tiho) + ...

= a0 + afhft

+

a^hp0i+l

p +i + aU)hP™+O(h 0" ),

(1.1.22)

where en , af2i, • * • are constants independent of ho. Hence, substituting Tfl,

= 75-i(M') = a 0 + a^htfb'n

+ ■■■ + a«)fcg-6*- + 0(^ m + 1 6 , p ™+')

into (1.1.21), we have T(i)

'

J =

y , J

j-1

J-l

1 - Vi = a0 + ayL J+

+a£>

l-6« 1 — V>

(6'Ao)" i+1 + • • •

(6l7.o)Pm + 0 ( ( 6 % ) P m + 1 ) -

Set x
- 6Pi)/(i - 6*), t = j + 1, • • •,m,

8

I. Generalization and application of Richardson's

extrapolation

which are constants independent of ft0 and i. Thus Algorithm 1.1 eliminates successively the terms in the asymptotic expansion, and the error satisfies ri0)-ao=0(^m+1)-

(1-1-23)

Now let us consider the case that the expansion has terms including In ft, e.g., T(h) = a0 + axh^ In ft 4- a2ftP2 + • • • + a m ft^- + 0(hP**+l).(1.1.24) In order to eliminate the second and the third terms, what we need is to call the program £"(&,fto;Pi, Pi) twice. The proof is as follows: Since T0(0) = To(fto) = a 0 + aiftg1 In ft0 + a2hPl + • • •, T0(1) = 7o(6ft0) = a 0 + a ^ f t g 1 ln(6ft0) + a2^hPl

+ • • •,

then i f > = T^fto) 6PMn(6ft 0 )-^x 1 lnft 0 = a 0 -f ai ft£ = ao +

H

a(?)hp0l+0{hp0>),

where a ^ z z a ^ i In 6/(1-fe^ 1 ) is a constant independent of fto and i. Hence, through the first hPl-extra­ polation, one can only eliminate the c^fto1 term but also convert the term aiftg1 In fto into a\ hPl, which will be eliminated by another /^-extrapo­ lation. If the asymptotic expansion contains ftPl(lnft)2 and hPl In ft, i.e., T(ft) = a0 + aiftP^lnft) 2 + a2ftPl In ft + a3ftPl + • • •,

(1.1.25)

then we can call the program £(&, fto,Pi,Pi,Pi) repeatedly to eliminate the 2-nd, 3-rd and 4-th terms. An analogue is valid for asymptotic expansions with logarithmic terms of higher degree. Such terms often occur in the evaluation of singular integrals, boundary value problems of ordinary dif­ ferential equations with singular points, partial differential equations on the concave domains and etc. The computational process of Romberg's extrapolation algorithm, the polynomial extrapolation algorithm, or the Bulirsch-Stoer extrapolation algorithm can be implemented according to Table 1.2.

1. Polynomial

extrapolation

9

Table 1.2. Recurrence table for extrapolation T(h0) = r0(°> T(h!) = T 0 ( 1 ) -

TJ
T(h2) = 7*( 2 ) .

n.(i) n n

T(h3) = T 0 (3) ; T(ft4) = T0(4)-

1.3. Stability

and

T(0) J

2

(2)

n(l)

-(0)

(3)

^(2)

n(l)

n(0)

convergence

From (1.1.3), there exist constants {Cmtj}JL0 tion value satisfies

such that the extrapola­

(1.1.26) j=Q

The constants {Cmj^-Q are called extrapolation coefficients. Obviously Cm j is the solution of the following system of equations: — 1,

(1.1.27a)

EOnj/if =0, 1=1,-

(1.1.27b)

/_^Cmj j=o

It is easy to see from (1.1.27) that for fixed m, C m j have alternating signs, and the sum of their absolute values,

M(m) = ^2\CmJ\

(1.1.28)

j=o

is an important quantity in studying the numerical stability of the extrap­ olation, and is called the stability factor of the extrapolation algorithm. In fact, due to the rounding error in the evaluation of T(/ij), the result T(hj) is merely an approximation of T(hj)} and the corresponding extrapolation

10

I. Generalization and application of Richardson's

extrapolation

value Tik is also an approximation of Tm • Hence, according to (1.1.26), the error between Tm and Tm satisfies

0)

m

W - ^ l = I E <w r ( & *) - ?(*i))| m

< M^maxlT^O-f^)!,

(1.1.29)

which implies that the smaller the M(ra), the more stable the extrapolation algorithm. For large M{m)^ the extrapolation may not reach the desired accuracy, hence it is necessary to use a double or even higher precision. Now let us consider the stability of the polynomial extrapolation algo­ rithm. T h e o r e m 1.1. The coefficients of the 'polynomial extrapolation algo­ rithm (1.1.17) have the following properties: m a

)

/ _j Cmj

— 1,

b) ( - l ) m - ' ' C m i i > 0, m

c) M(m) = y^ |Cmj| increases with the number of extrapolation, and d) The sufficient condition for that there exists a positive number M independent of m such that M{m) < M, Vm > 0 is that the mesh parameters satisfy hi+i/hi < p< 1, Vi > 0,

(1.1.30)

where p is a positive number. Proof. Setting x = 0 in the Lagrange interpolation formula (1.1.7), one obtains the extrapolation value. Setting x = 0 in the identity (1.1.8b) and a) is proved. Taking X{ = \i\ in the computation of coefficients Cm j , and

1. Polynomial

extrapolation

11

noting h0 > hi > • • • > /i m ,

(-hr3- )

m

Cm,i = L,(0) = TT

-

•■£■1 fti — - v f ^it" *1 TT 1 h\ - h* )

(1.1.31)

1-

1=0

which proves that b) is valid. It follows from (1.1.31) that m+l

|Q m +1 (-l)| = X) l^+l-'l > \Qm(-l)\ = £ l^.'l' t=0

(L1-32)

i=0

where Qm(x) — ^ C m > t x * , hence c) holds. i=0

Finally, estimate the stability factor M(m). By (1.1.31), m

m

m

M

ib=Oj=o \hrj — hrk\

fc=o

j?k m

=Y: fc=0

m

UT"

n -r-v—

J=o

l^i "" ^m-Jfel j^m — k m m—k — 1 1

m

- E n r^^i; n Jb=0 j = 0

1 "" Kn-k/h]

m m—k

1

*

fc=0 J = l * oo ^

P j = lP oo A;

j=l A —P

jk = 0 j = l

oo < TT

l2

°° y

j=m-k+l

1

T; Kn-kl' ] 1

*

1 l

rk(k+l)/2 (1.1.33)

< M, which proves d).

D

For Bulirsch-Stoer's algorithm, the following theorem is valid.

12

I. Generalization

and application

of Richardson's

extrapolation

T h e o r e m 1.2. If the mesh parameters satisfy the condition Hi), then the coefficients of the Bulirsch-Stoer extrapolation algorithm has the same properties as those stated in Theorem 1.1. Proof. Construct an auxiliary polynomial m

Qm(x) = ^2cm)kxk1

(1.1.34)

Jb=0

with {Cmyk} satisfying equation (1.1.27a), then m

Qm(l) = £ c m , f c = l,

(1.1.35)

fc=0

i.e., property a) is valid. By using (1.1.27b), m

m

o = ]rc m ,*/# = £cm,fc(/io&*r m

i=l,-,m,

(1.1.36)

Pj

i.e., b (j = l , - , m ) are roots of the polynomial (1.1.34). Applying (1.1.35), there is m

X-bPj

Qm(x) = U—-—. i=l

(1.1.37)

1 bpj

~

Expanding (1.1.37) and comparing it with (1.1.34), it can be shown that the signs of coefficients satisfy {-\)m->Cmj

> 0, j = 0,...,m,

(1.1.38) m

i.e., property b) is valid. This property leads to \Qm(—1)| = J ^ |Cm,fc|, fc=0

hence, m

m 1 + bPj

M(m) = 53|Cmii| = I ] T^TT' which implies that M(m) increases with m, and ~ 1 + IP*

M(m) < f| -f-Jil-bPi-

—

< M.

( 1L39 )

1. Polynomial

extrapolation

13

T h u s properties c) and d) hold. If the asymptotic expansion (1.1.16) holds, (1.1.18) is an error estimate of the polynomial extrapolation algorithm; if the asymptotic expansion (1.1.19) holds, then (1.1.23) is an error estimate for the Bulirsch-Stoer extrapolation algorithm. These facts show t h a t if there exists an asymptotic expansion, the extrapolation m e t h o d can accelerate the convergence rate of {T(hi)} and improve the order of accuracy. If the asymptotic expansion is absent, the extrapolation may not improve the result. However, if the choice of mesh parameters enables the extrapolation coefficients satisfy the conditions of Theorem 1.1 or Theorem 1.2, then the Toeplitz Theorem will still guarantee the convergence. It must be pointed out t h a t when the asymptotic expansion is absent, the Toeplitz Theorem only guarantees the convergence, and no conclu­ sion about the acceleration has been drawn. Besides, even if there is an asymptotic expansion, how to choose the mesh parameters {hj} to save the amount of work and to attain the required precision is still a noteworthy problem. As previously mentioned, the numerical stability of extrapolation depends on M ( r a ) , the sum of absolute values of extrapolation coefficients. T h e smaller the M{m)) the more stable the extrapolation algorithm. How­ ever, smaller M(m) means, in general, t h a t grid points will increase rapidly as the number of extrapolation increases, which will lead to rapid increase of the amount of work. As examples, consider the following refinements: Hi)

A,-=2-*'fc 0 , i = 0 , 1 , - -,

H2)

hi=h0/(l

H3)

hi

H4)

hi=h0/Fi,

+ i), f = 0 , 1 , - •-, 2-(«+i)/2 / l 0 )

if i is odd,

3_i2_(,_2)/2/l0)

if j is even.

i =

0,l,2,--,

where Fi =

(lW5)<+2

_ ( 1-V5 ) t -

V5

is the i-th term of the Fibonacci H i ) is Romberg's refinement tion. It has a good stability. In point of a small interval obtained

sequence: {1, 2 , 3 , 5, 8,13, • • •}. used frequently in the numerical integra­ addition, each new node is just the mid­ in the previous partition. This refinement

14

I. Generalization and application of Richardson's

extrapolation

will simplify general Romberg's extrapolation (1.1.14) to

itf^TQio/T),

2 = 0,...,m,

i f = T^ + (T^ - iflj/tf

- 1),

j = 1, • • •, ra, i = 0, • • •, m - j .

(1.1.40)

Bauer et a/J 4 ^ proved that the stability factor of Romberg's extrapo­ lation of Hi) is Mi(m) < 1.97, Vm > 0.

(1.1.41)

Compared with Hi), the stability of H 2 ) is poor. M 2 (12) = 5730,M 2 (14) = 27670, and as m increases, M^^m) increases rapidly and tends to infinity. But the number of nodes of the H 2 ) refinement increases most slowly, and so H 2 ) partition is still effective for those problems, e.g. numerical solution of partial differential equations, in which the amount of work increases rapidly with the number of nodes. For these problems, in general, small number of extrapolation is needed and double precision can be used to reduce the rounding error. As a compromise proposal suggested by Bulirsch, H3) has a good stability as well as a slow increase in the number of nodes. But its disadvantage is that the computation scheme is not uniform. H4) was proposed by Z.F. Qin[yy^. Since the Fibonacci numbers satisfy

Fn+i/Fn

- 1.618, (n - 00),

(1.1.42)

the number of nodes increases slower than that of Hi), and the stability factor MA{m) < 3.59, Vm > 0 is still very small.

(1.1.43)

1. Polynomial

extrapolation

1.4. A posteriori

error

15

estimation

There are two kinds of error estimates: a priori error estimate and a posteriori error estimate. By a priori error estimate we m e a n t h a t the or­ der of accuracy of an approximation is obtained from the original d a t a and the computational m e t h o d used, for example, in numerical integration, the order of error of an approximation is judged by the smoothness, singular­ ity of the integrand and the quadrature m e t h o d used; in the root-finding problem of an algebraic equation the position of a root is judged according to the alternation of signs of the function / ( x ) ; and in the solution of a differential equation by the finite element method, the order of error is de­ termined according to the smoothness of the domain and coefficients, the properties (such as positive definiteness, energy inequality) of the differ­ ential operator, and the choice of the test function space. A priori error estimate is i m p o r t a n t , b u t it cannot obtain an accurate range of the error because the estimate factor often depends on the exact solution, which is unknown. For a posteriori error estimate, the error is estimated according to the d a t a obtained in the computational process. Moreover, a posteriori estimate is sharper t h a n a priori estimate. Results from a priori estimate is usually enlarged, up to hundred, even thousand times to the real one. A posteriori error estimate can give an estimate with the same order as the real one. It can also give an evaluation to the approximation in each stage of computation. T h u s one can judge if the computation should be stopped or continued. Therefore, a posteriori error estimate is a necessary p a r t of an adaptive software, and plays an important role in modern computational methods. Archimedes (3rd century B.C.) got a two-sided inequality for 7r by com­ puting the perimeters of an inscribed polygon and a circumscribed polygon. He estimated the computational error by a two-sided inequality, which is the earliest example of a posteriori error estimate. Archimedes' method has m a n y developments. We denote by T(h) the 1 semi-perimeter of the inscribed n-polygon (h = —) in the unit circle, then n T(h)

1 = - sin(7r/i) h 7T 3

= -K

IT5

h2 + — hA 3! 5!

X2''+1

h2j + • • •, (1.1.44)

+ (-1V (2j + l)

which shows t h a t T(h) has an asymptotic expansion with even powers. Sim­ ilarly, denote by U(h) the semi-perimeter of the circumscribed n-polygon,

16

I. Generalization and application of Richardson's

extrapolation

then 1 U(h) = - tan(/i7r) h TT3

27T 5

177T 7

= TT + —ft2 + A4 + ft6 3! 15 315

+ • • •.

(1.1.45)

Huygens (1654) improved the Archimedes' inequality T(h) < TT < U(h)

(1.1.46)

to (4T(/i) + T(2/i))/3 = S(h) < TT < V(/i) = (2U(h) + T(2/i))/3.

(1.1.47)

The left-hand side is just an /^-extrapolation. Huygens proved (1.1.47) by a geometric method. By the asymptotic expansions, 7T5

S(h) = *

h4+

30

7T 7

h6

,

252

and 2TT5

V(h)

= 7r-h

2TT7 4

/i +

/i 6 + - - - ,

15 63 b o t h S(h) and ^ ( / i ) have an order of accuracy 0 ( / i 4 ) and the signs of the second terms are different, this is the feature of a two-sided inequality. By using this feature, one can construct a two-sided approximation and a posteriori error estimate of the polynomial extrapolation. According to (1.1.18), the error of the polynomial extrapolation is

jf) _ ao = (_i)iftrftr+i.. .ftr+.{oi+1 + O(ftr)}.

(i.i. 48 )

This implies that for fixed j , as i increases, {l^ } monotonically ap­ proaches ao = T(0). If one can construct a sequence {U^} which con­ verges to T(0) from the direction opposite to {^- }, then the two-sided approximation of T(0) is obtained. For this purpose, construct a sequence

Uf> = (1 + a)TJi+1) - <*Tf\

(1.1.49)

where a is an undetermined constant, a has to be chosen such that Uj — T(0) and

T)°-T(0)

have opposite signs. Substituting (1.1.48) into (1.1.49),

1. Polynomial

extrapolation

17

we have

UJ,)-T(0) = (-iyhr...hr+j .|fl i + 1 [(l + a)-^ti-a]+0(*T)},

(1.1.50)

which implies that we may choose a satisfying (1 + a)

a= -1,

hr{

or 2 Q

1-1-

(1.1.51)

(Vfc*+i+i) r -i'

With such a choice of a, {t/j } and {T^ } approach T(0) from two oppo­ site sides, hence its average has a posteriori estimate

| i ( ^ + ^)-T(0)|
(1.1.52)

But it follows from (1.1.49), (1.1.51) and (1.1.17) that

\{TfHuf) T(t

j

~

+ 1)

7T7h

T(t)

v

7~

i+1

(1.1.53)

'

and 2V

j

j

;

hr

T (»+i)

T(0

j

= h

h

( i+j+l/ iY

j

-hr

j

= T^ - TV

j

(1.1.54)

- *

Substituting into (1.1.52), we have a posteriori estimate of the ext;rapolation value T / ; \

|^;\-T(O)|<|^-T;«|. For the sake of simplicity, it is very effective to choose i+1)

uf = 2TJ

- i f>

(1.1.55)

(1.1.56)

18

I. Generalization and application of Richardson's

extrapolation

as the opposite side approximation of T(0). This is equivalent to taking a = 1 in (1.1.49). Substituting into (1.1.50), we have

v? -r(o) = (-iyk; •••*;„

It can be seen that if the mesh parameters satisfy 2(&i+i+i/fc,-)r-K0>

(1.1.57)

then Tj, Uj are still two-sided approximations of T(0), hence, (1.1.52) is still valid. In this case,

+1 i(:zf+ E/f)=7f \ 2'

I(^>-7f)=7f +1 >-lf, hence, by (1.1.52) we get a posteriori estimate |T/*' +1) - T(0)| < |7f + 1 ) - lf\

(1.1.58)

However, this estimate is rather conservative. By the asymptotic expansion (1.1.48) and

lji+V-1<»

(-iyhri+1...h'i+J+1

=

.L+1(l--p—)+0(h:)\,

(1.1.59)

n

i+j+l

(1.1.58) can be improved to |nn(« + l )

|^*'+1) " T(0)| « J L L

{hi/hi+j+x)

_rp(i)\

- U - = 1 ^ - 2? +1 >|. (1.1.60) - 1

A posteriori estimate (1.1.55), (1.1.58) and (1.1.60) show that the error of any element in Table 1.2 can be estimated by the difference between this element and the previous element in the same column or the previous (or the next) element in the same row. That is, in implementing the extrapo­ lation program, one can stop as soon as he finds that the difference of the adjacent elements in the same column (or row) satisfies the required preci­ sion. However, this analysis is based on that for fixed j , as i is sufficiently large, {?j } converges monotonically to T(0). How can one judge whether

19 1 9

1. Polynomial extrapolation 1. Polynomial extrapolation r

1 or not is sufficiently large so t h a{Tt {T' } monotonic is monotonic or 0(h (1.1 .50) {) in or not i isi sufficiently large so that ' } is or OQi -) in (1.1.50) can be ignored? T h e most reliable way is to judge whether the following can be ignored? The most reliable way is to judge whether the following expression holds: expression holds:

_ JVQ) rAi rAi +3l) +_l) JVQ)

~ i -*-ra ) —_« 1. if T(0) i f > _ T(0)

(1.1 .61) (1.1.61)

is so, regard t {Tm } converges monotonically. (1.1 .61) If itIfisit so, oneone cancan regard thatt h a{im } converges monotonically. ButBut (1.1.61) cannot be used directly, since T(0) is unknown. Instead of (1.1 .61),wewe cannot be used directly, since T(0) is unknown. Instead of (1.1.61), consider consider 7

'

rp(t)

" V • '

"W

m(» + l )

T(*-l) T(«)

*?+i+i - *r' if-if- 1 *

• • -fc? +i {aj +i+11 (^ (fcr - />r) ( - i(y- l^)+^1 +■1 ■■^■{a - ftp + p(ftr)} + o(fc rj > 1+i+1

j

"(-ly-AJ •^ . . +i • ^. 1+ {a { a i(/ir i r +-j hr_ - fcr_ i . 1 i+1 + 1 ( /+j i} + oQi^)} "(-ly-AJ.. x) + " ( ^ - 1 ) :\ (1.1 .62) « 1. « 1. (1.1.62) obtained during computational process, it can used 2}j Dj cancan be be obtained during thethe computational process, it can be be used to to judge ^} converges monotonically. W h eDj n Dj« 1, « it 1, means it means ; that ; i is judge if {if i {T. f } converges monotonically. When that i is sufficiently large. sufficiently large. e posterior estimate polynomial extrapolation mentioned above TheT hposterior estimate of of polynomial extrapolation mentioned above can be applied to Bulirsch-Stoer's extrapolation, provided t h a t the following can be applied to Bulirsch-Stoer's extrapolation, provided that the following modifications m a dfor e for (1.1.51), (1.1.55), (1.1.61), (1 .1.62): modifications areare made (1.1.51), (1.1.55), (1.1.61), (1.1.62): — IXl — a =ru +

2 2

1 1

>

(1.1. .51)' (1.1.51)'

11,

6-Pi+i 6-Pi+i _ i_ 1

+1)

1 1 i f >|, l ^ - ^ ^ r r ^ llf ^ "- ^ '

(1.1 .55)' (1.1.55)'

+1)+1)

(1.1 .61)' (i.i.6iy

\TE ~ T(0)| < ^

1

9W i+ v.
1 -f-

J

1 _ bPj+i

jj,(»'+l)

J,(t)l

j

j - T(0)| ' _— |7} ( ;\- if - +1) if +|,1 ) |, p L A =' = |7f|lf - T(O)| » i«^ — |T/;\

6-Pi+i _ l («' +_l ) T_( 0rp{i)

T l) r(i+

^

= ^iT-Z7i^b~Pl+1 i i

* 1-

(1.1 .62)'

(1-1-62)'

20

I. Generalization and application of Richardson's

extrapolation

(1.1.62)' indicates that if it is hard to determine pi,P2, • • • in the asymptotic expansion of T(/i), then for sufficiently large i, we can use T(t+i)

#>;+i « _ i rp(i) _ 3

_

T(.)

(1.1.63)

L_ rp(i-l) 3

to determine the exponents. 2. Application in numerical integration Numerical integration is an important field of computational mathemat­ ics. Many mathematical areas, such as function approximation, functional analysis, mathematical statistics, analytical number theory, even topol­ ogy and algebraic group theory, find their applications in the computation of numerical integration. In this section, we discuss the applications of Richardson's extrapolation in numerical integration, and those of splitting extrapolation in multidimensional integration will be studied in chapter II. A comprehensive survey of applications of extrapolation in numerical integration can be found in [100] and that of recent developments in [15]. 2.1. Euler-Maclaurin

summation

formulae

Consider the integration on a finite interval [a, b] f f(x)dx = 1(f). Ja The simplest quadrature formula is the trapezoidal rule R"(f) = h \\f(a)

+ f(a + h) + -.. + f(b-h)

+ i/(6)],

(1.2.1)

(1.2.2)

with b— a , n >0. (1.2.3) n If the integrand is a smooth periodic function (with period b — a), then the trapezoidal rule has a very high order of error. If the integrand is an analytic periodic function, then the error is of exponential order (e.g., 0(e=^~) as h —► 0). If the integrand is an aperiodic function, then the order of error is only 0(h2). But by means of the Euler-Maclaurin expansion of the trapezoidal rule, Romberg combined the trapezoidal rule with the Richardson extrapolation and got Romberg's extrapolation for numerical integration. Since the Bernoulli polynomial plays an important role in the Euler-Maclaurin formula, we shall now introduce it briefly. h=

2. Application

in numerical

integration

21

A polynomial of degree k, denoted by Bk(x), is called a Bernoulli poly­ nomial, if it is obtained from the following recurrence relations: a) B0(x) = 1, b) B'k(x) = k-Bk-1(x), c) / Bk(x)dx

i > l , and

= Q, i > l .

JO

According to the recurrence relations above, it is easy to find the Bernoulli polynomial of any degree, e.g., 1

B2(x) = x2-

B1(x) = x - - , 3 1 B3(x) = x3- -x2 + -2 , 2

6

1

BA(x) = x4 - 2x3 + x2-

30'

It follows immediately from b) and c) that Bk(0) = Bk(l),

(1.2.4)

k>2,

which implies that a Bernoulli polynomial can b e expanded periodically to the whole real axis. For example, P\(x) = x

- W - « is

simply the

extension of B\(x) with period 1, where [x] stands for the largest integer which is less than or equal to x. The Fourier expansion of Pi(x) is °° 2sin(27rnz)

(1.2.5a)

2?rn

n=l

By integrating (1.2.5a) repeatedly, we obtain

p 2j (x) =

J2L 2 cos(27rnx)

(

- i y - ^ - - —2 - , i = l,2,---, „ = 1 (2*n) (2Ttn)^' °° 2sm(27rnx)

p2j+i(x) = (-ly- 1 y

Obviously, by b), Pj(x) is an extension of (1.2.5), we can prove that

, i = o, i, • -•.

(1.2.5c)

Bj(x)/j\ with period ]L. Applying

PUI(*) = ft(«). P 2 ;+i(0) = P 2 i+i(l) = 0, i = 1,2 2 1 ^2,

t ' ' '

p2j(o) = p2,(i) = c-iy- y K) '

(1.2.5b)

(1.2.6a) (1.2.6b)

>

B2j

r= —, 2

„^i(2™) '

(2j)\

(1.2.6c)

22

I. Generalization and application of Richardson's

extrapolation

where B2j = ^2;(0) is called the Bernoulli number. Applying -Pfc(#), ^ is easy to prove the following Euler-Maclaurin expansion: T h e o r e m 2 . 1 . If f e C 2 * +1 [0, n], /Aen

i/(0) + /(l) + ... + /(n - 1) + i/(n) = jT /(*)<** + ^ [/'(») - /'(0)] + • • • + ^ - [f^\n) 2!

(2fc)!

+ ./of P 2 H i W /

( 2 H 1 )

- /^(O)]

(^-

(1-2-7)

Proof. Using integration by parts it is easy to verify the identity

^ [/(*) +/(* + !)] = J

f(x)dx + J

=

f(x)dx+

(x-[z]--)f(x)dz P1(x)f'(x)dx.

(1.2.8)

Summing up for k = 0,1, • • •, n — 1, it follows that

^/(0)+ /(1)+ ••• + / ( * - l ) + i / ( n ) = /

/(*)<**+ /

P1(x)f'(x)dx.

(1.2.9)

Applying integration by parts to the last integral of (1.2.9), we obtain

f P1(x)f'(x)dx

Jo

n

=p2(x)f(x)\ - r p2(x)f(x) 10

Jo

= -^[f'(n) ~ /'(0)] - f P2(x)f"(x)dx. (1.2.10) 2! »/o Applying again integration by parts to the last integral of (1.2.10) and by using (1.2.6b), it follows that

r Jo

p2(x)f((x)d2

2. Application

in numerical integration

= P3(x)f"(x)\n

23

10

- f1 Jo

P3(x)f'"(x)dx

= - f P3(x)f'"(x)dx. (1.2.11) Jo Continue in the same way, we obtain the Euler-Maclaurin summation for­ mula (1.2.7). □ Theorem 2.1 has been used in summations of finite series for a long time. Not until 1955 did Romberg apply this asymptotic expansion to Richardson's extrapolation and obtained the famous Romberg's algorithm. T h e o r e m 2 . 2 . Let g(x) G C 2 f c + 1 [a, 6], take h = (b - a)/n, following asymptotic expansion of the trapezoidal rule is valid:

then the

Tn(9) = h [-g(a) + g(a + ft) + .. • + g(a + (n - l)ft) + -g(b)] = / g(x)dx + Jo. 2!

^-h2[g'(b)-g'(a)}+.>

—h>k[g&-1Xb)-g
+ (2k). +h2k+1

[" P2k+1(n—)gVk+1\x)dx. Ja b— a

P r o o f . Take f(x) = g(a+hx). immediately. □

(1.2.12)

Substituting into (1.2.7), (1.2.12) follows

C o r o l l a r y 1. If g(x) = C 2 f c + 1 [ M ] , and g'{b) = g'(a),g"\a) = g'»(b), • • •, 0, such that | ^ ( 2 f c + 1 ) ( x ) | < Af, a<x
(1.2.13)

then I / g{x)dx - Tn(g)\ Ja

< C/n2k+1,

(1.2.14)

where C = M(b-

a) 2 f c + 2 2- 2 f c 7r- 2 *- 1 C(2ib + 1)

(1.2.15)

oo

is independent

of n, and £(&) = / ^ i - * is the Riemann

zeta

function.

24

I. Generalization and application of Richardson's

extrapolation

Proof. Under the assumptions and by applying (1.2.12), we have / g(x)dx-Tn(g) Ja = - Z ^ + i / ' p2k+1 (/_Zl)g(2k+x){x)dx

{1 2 16)

b—a

Ja

By using (1.2.5c), for any t there is

l i W O I < £ ^r-^TT = 2-2^"2fc-1C(2A: + 1). j = l (27Tj) 2 f c + 1

Substituting into (1.2.16), it follows that (1.2.15) holds.

D

Corollary 2. If g(x) £ C 2fc+1 (—oo,oo), and g(x) is a periodic function with period 2w, then | / Jo

g{x)dx - Tn(g)\ < 4irMC(2k + l)/n2k+\

Proof. The periodicity implies g'(0) = 0'(2TT), • • •, g^'^O) (2TT). It follows from Corollary 1 that (1.2.17) holds. D

(1.2.17) = 5f(2fc+1)

Definition 2.1. A quadrature rule is said to have an algebraic precision of degree m, if this rule is exact for any polynomial of degree not higher than m, but is not exact for at least one polynomial of degree (m + 1). Theorem 2.2 shows that if no extrapolation is implemented, the alge­ braic precision for the trapezoidal rule is of degree 1. But as far as a periodic function with period p = b — a is concerned, the trapezoidal rule is very sharp. Let

ETM) = - £ fi-p) - f f(*)dx

(1218)

n fc=0 n Jo be the error of the trapezoidal rule, one can easily prove that ( #Tnexp(

2irijx P

) =

P,

3 ± 0, n | j ,

0,

otherwise,

(1.2.19)

where i = >/^T. (1.2.19) implies that the trapezoidal rule is sharp for triangular polynomials of order n.

2. Application

in numerical integration

25

Theorem 2.2 can be extended further in two directions. The first one is to establish the general offset trapezoidal rule for 0 < 0 < 1, = - £ / ( — ) •

rt\f)

n

t=o

(1.2.20)

n

The other is to reduce the condition /C*- 1 ) e C[0,1] to that /C*- 1 ) is a function of bounded deviation. Therefore, we have the following: Theorem 2.3. Suppose that there exists an integer k > 1 such that /( f c _ 1 )(x) is a function of bounded variation on the interval [0,1], and if f(x) is continuous at points Xi = (i-\-Q)/n (i = 0 , - - , n — 1), then the following generalized Euler-Maclaurin expansions hold:

I

o

f(x)dx =-J2 / ( — ) + 4 0 ) (/. 0), k = 1, n

(1.2.21)

n

i=Q

and

i

i

1 "zi

I f(x)dx = -J2f(

i +0

)

.=0

fc / ( i - i ) ( i ) _ /O'-i)(0) - E — Bj(6) + R^(f,e), i=i i!";

* > 1,(1.2.22)

1 f1 ; # > ( / , 0) = — f Pk(9 - nx)df^k-1\x). n* Jo

(1.2.23)

with

Proof. We prove by induction, when k = 1, Pi(0 - nx) = 6 - nx - - - [6 - nx] is continuous for x ^ (i + 0)/n, and jumps at x = (i + 0)/n. Hence the integral on the right of (1.2.23) exists, and for 0 < x < 1, 1 ,i Px{6nx)df(x) n Jo 1 ,1 1 ri = -P1(0-nx)f(x)\ - - / f(x)dPi(0 n '° n Jo /(l)-/(0) 1 r1 = -II —BiV) - - / f(x)d(6 n n Jo

- nx) 1 -nx--) z

26

I. Generalization and application of Richardson's

extrapolation

r1

1

+ - / f(x)d[0 n Jo /(l)-/(0)

- nx] l""1

A

n

.

/

o

n

i+ 0 n

t=0

}

= i# (/,0),

(1.2.24)

which proves that (1.2.22) holds for i = 1. If (1.2.22) holds for k := AT—1, we shall prove that it also holds for k. By the absolute continuity of Pk{0 — nx), we obtain

Pk(9-nx)dfV°-1\x)

— I k n

Jo

= —Pk(0

- nz)/*-1)^)!1 - — / l0

71*

/ ( * - i ) ( l ) _ /(*-i)(o) n n~k /(*-!)(!) _ /(*-i)(0)

fV-x\x)dPk{0

nx)

1 Pk(0) + -— - 1 f /^-^(aOPjb-iC* n * Jo Bk{6) + — --1 I Pk-i(6 n * Jo

k\nk

-

71* JO

nx)dx

nx)df(k-2\x),

-

by inductive assumptions we prove that (1.2.22) holds.

D

C o r o l l a r y 1. If / ^ " ^ ( x ) is a function of bounded variation ^ e n under the assumptions of Theorem 2.3, we have

on [a, 6],

rb

" "iX I f{x)di

Ja n-1

*

fO*-l)(J)_ / 0 " - l ) ( a )

i=o

j=i

i!

= & 2 /(a +ft(i+ *)) - £ rb

+hk f Pk(6+ Ja

x —a h

— WBj (0)

X

—^)df^k-1){x))

(1.2.25)

with h = (6 — a ) / n . P r o o f . Let y(x) = (b - a)f(a + (6 - a)x). By / J6

/(x)dx = /

g(x)dx,

(1.2.26)

JO

replacing f(x) in (1.2.22) with g(x), (1.2.25) holds.

D

2. Application

in numerical

integration

27

An important case of Theorem 2.3 is for 0 = - , it is called the midrectangular rule: n-l

j

M„(/) = ^ A / ( a + A ( i + - ) ) .

(1.2.27)

z

t=o

Noting that at x = - the Bernoulli polynomials of odd degrees ssatisfy (1.2.28)

B 2 i + i ( - ) = 0 J i = 0,l,...,

while that of even degrees satisfy (1.2.5b), then by Theorem 2.3 and its corollaries, we have the following: T h e o r e m 2.4. If g G C 2 * + 1 [a,t], then

Mn(f)-J

9(x)dx

= -^h2[9'(b)

- g'(a)) + -^h4[g'"(b)

- g'"(a)} + • • •

+—h1>h[gVk-V(b)-gVk-i\a)] (1.2.29) la

*

h

with C 2 i = B2j ( i ) = - ( 1 - 2 1 - 2 '")B 2i , j = 1, •' •, *• Proof. It follows from (1.2.5), (1.2.25) and (1.2.28).

□1

Obviously, according to Theorem 2.3, the condition g^ G C^faji] stated in Theorems 2.2 and 2.4 can be reduced to a weaker one, i.e., g(k) is a function of bounded variation. As a periodic function is concerned, the trapezoidal rule Tn(f) and mid-rectangular rule Mn(f) are identical, i.e., both are equal to the simple average of function values at the nodes. For an aperiodic function, the midrectangular rule belongs to quadrature rules of open type, i.e., the function values at two end points a and b are not included in the formula; while the trapezoidal rule is of closed type, i.e., the function values at two end points a and b are needed. In section 2.3, we shall discuss integrals where the

28

I. Generalization and application of Richardson's

extrapolation

integrands have singularities at the end points. Certainly, the open type formulas are sharper. Combining expansions (1.2.12) and (1.2.29) one can obtain the following Simpson's rule with higher order of accuracy: 2 1 Sn(g) = -Mn(g) + -Tn(g) 3 3 = -T2n(g) - -Tn(g), (1.2.30) 3 3 which can be regarded as a combination of the mid-rectangular rule and trapezoidal rule, or performing one step of extrapolation to the trapezoidal rule. The error of the Simpson's rule also has an expansion Sn(g) - f 9{x)dx = -^h4[g'"(b) - g'"(a)} + ■■■ Ja 4! D2k + _ p(2*-i)( a )] + 0(/> 2 * + 1 ), (1.2.31) fc2*k(2*-i)(6) (2k). 2 1 1 with constants D2j = -C2j + -B2j = -(2 2 " 2 ^ - l)B2j} j = 2,3, • • •, k. O

o

O

In the following, we consider the error of Tm obtained from implement­ ing Romberg's extrapolation to the trapezoidal rule. T h e o r e m 2.5. If f(x) G C 2 m + 2 [0,1], then there exists £ e [0,1], such that T (fc)

4-*( m+1 )fl 2 m+2 ( 2 m + 2 ) ri = / /(*)<** + _ / (V{ J) . J0 2™(™+1)(2m + 2)!

(1.2.32) '

V

Proof. By means of the extrapolation coefficients, from (1.1.26), we have m

7#> = 5 ] C m i i r ( / i H i ) , hk+j = 1/2*+'". i=o

(1.2.33)

Substituting into the asymptotic expansion (1.2.12), and using (1.1.27), we have

r w - f f{x)dx JO rl rn

r<

I„ g i S S ^ J J ^ ' C ^ ) ^ " ^ 2 (2,

- ^

fo

S2m+2(x)fVm+*\x)dx,

(1.2.34)

2. Application

in numerical

integration

29

with S2m+2(z)

= (-1>m E ^ ^ ( i W ° ) " P^+2(2^x)).

(1.2.35)

Applying properties of the Bernoulli polynomial of even degrees, it can be proved that 5 m +2(x) does not change sign on [0,1]. By the mean value theorem of integration, we have r1

Tm

r1

1

(t)JoS>™+i

Jofi*)dx-^m+2)kf (2m + 2)!2<2m+2)fc

(x)dx (1.2.36)

m,3/

~

Using (1.1.34) and (1.1.37), it follows that m

c

4 (m+i)j

(-ir E ->> "

1

1

=(- )" n — —

i=o

m

_ JT4-J — 2

m 4~('"+l) _ 4--i j=i

_m m

(1.2.37)

( + ).

Substituting into (1.2.36), (1.2.32) is obtained. 2.2. Euler-Maclaurin larities at end

1 — 4 ■*

1

formulae points

D with ,singu-

for integrands

Consider the following integral whose integrand has singularities at end points /(/) = f Jo

fi x

(l - x)"h(x)dx,

p,u>

-1,

(1.2.38)

with h{x) £ C ^ - ^ O , ! ] . J.N. Lyness and Ninham (cf. [25]) used Fourier transform to prove that the error of the trapezoidal rule and mid-rectangular rule have the following asymptotic expansions

ETn(f) =

Tn(f)-I(f)

30

I. Generalization and application of Richardson's extrapolation

"zM-i^aXH'-j) . + /Z : + 0(n-N),

(1.2.39)

= Mn{f) - 1(f) ^#(0)(2-"-'-i)C(-/?-i)

EMM)

»-i (-iy ^ ' ) ( i ) ( 2 — i - i)c(-u, - i> N + > +0(n~ ), (1-2.40) Jn

J=}1

where V-o(x) = (1 - x)uh(x),

= *"&(*)>

fa(z)

( 1, 13 = 0, Jo =< { 0, 0 # 0,

(1.2.41)

f 1, forw = 0, (1.2.42) h=\ [ 0, for w # 0,

OO

and £(s) = / ^ l/ 71 *'

( ^ e 5 > 1) stands for the Riemann zeta function,

n=l

which has an analytic extension on the complex plane 2(5-l)! « C(l - s) = cos (—)C(*)> s £ 1. V

J

V

(2TT)*

2 ^

V

^

r

(1.2.43)

Thus C(—2p) = 0 for positive integers p. Therefore, the result in the case of (3 = w = 0 without singular point is identical with that of the usual Euler-Maclaurin's expansion. Comparison between (1.2.39) and (1.2.40) shows that when the end point is an algebraic branch point, the mid-rectangular rule behaves better than the trapezoidal rule. Now consider the integral / xf3h(x)dxJ

0 < f3 < 1. Substituting into

(1.2.39) and (1.2.40), due to u> = 0, Vo(*) = h(x), there are

h(0)(2-e-i)c(-P)

W/) =

—

+- ,

h(0)C(-f3) The ratio of them is (1.2.44) ET.{{)

2. Application

in numerical integration

31

which implies that as /? —* 0, this ratio also tends to zero. For an integral with logarithmic singularities of the following form, 1(f) = /

x*(l-*)wlnxln(l-*)
(1.2.45)

Jo

one can also prove that it has an asymptotic expansion, Tn(f) ~ 1(f) = a i / n ^ 1 + a 2 I n n / n ^ 1 + a 3 / n / J + 2 + a 4 In n/n^2

+ • • • + fcx/n"*1 + b2 In n / n " * 1

+& 3 /n u; + 2 + 64 In n / n " + 2 + . • ..

(1.2.46)

We have proved that the logarithmic terms can be eliminated term by term by the polynomial extrapolation, provided that we choose the mesh parameters /i, = hob\ where 0 < b < 1. 2.3. Multidimensional

Euler-Maclaurin

expansion

Consider an integral on the s-dimensional cube

1(f) = I •••I /(*)<**.

(1-2.47)

Jo Jo where x = (xi, • • •, xs), dx = dxi • • • dxs. Construct the i/-point quadrature rule V

V

Q(/) = £«i/(*i). X >

= 1

-

(1-2-48)

The quadrature rule <3(/) is said to be symmetric if the nodal set {ZJ} is symmetrically distributed with respect to the plane X{ = - , i = 1, • • •, s. Q(f) is said to have an algebraic precision of degree d, if 1(f) = <2(/)> V/ G 7ra, but there exists / G 7Td+i such that / ( / ) ^ Q ( / ) , where Wd denotes the set of all polynomials of degree not higher than d. Q(f) is said to have a triangular polynomial precision of degree d, if 1(f) = Q(/)> V/ G Tj, where Td denotes the set of all triangular polynomials with period 1. The most popular one is the mid-rectangular rule: Q(/) = /(z), z = ( i , - - - , i ) ,

(1-2.49)

which is a symmetric quadrature rule with algebraic precision of degree one. One can also construct a non-symmetric quadrature rule with algebraic precision of degree zero. Take z = (zi, •••,*«) as an arbitrary interior point Q(f) = f(zu ■■-,*,), 0 < * , - < ! , l
(1.2.50)

32

I. Generalization and application of Richardson's

extrapolation

which is an offset rectangular rule. Applying (1.2.48), one can construct the quadrature rule with m replications ,ZU

z.

o w ( / ) = E - E E ^ / ( - ■■•■■— )• Jb1=0

Jb.=0j=0

m

m

(12-51)

m

The repeated quadrature rule can be understood as partitioning [0, l] 5 into 1 ms hypercubes with length —, then using the quadrature rule (1.2.48) for m each cube, and summing them up. Therefore if Q(f) is symmetric, so is Q(m\f), and they have the same degree of algebraic precision. J.N. Lyness^l generalized the Euler-Maclaurin expansion to the mul­ tidimensional quadrature (1.2.51). He obtained the following theorem: Theorem 2.6. If f(x) and its partial derivatives f^OL\x)i integrable functions on [0, l] 5 , a. = (c*i, •••<*,), then

|ot| < p are

J-i 5.

Q(m)/-/(/) = £

— + W ( m ) ; / ) , *,

(1.2.52)

where bj are constants, which depend only on Q and f, and are independent of m. And there exists a function /*« (Q( m ); rax), such that for s < I < p,

MQ(m);f) = — £ I ■■ I hcc{Q{m);mx)f(aXx)dx. m J Jo '\a\=i o

(1.2.53)

In addition, bj have explicit expressions (1.2.54) where Ca = Ca(Q)

= Q(<pa)

= 1 ■[ ha(Qim);mx)dx, Jo

(1.2.55)

Jo

with function Bai(Xl)
=

■

ai!

Ba2(x2)

Ba,(xs)

■

a^!

where B,,(x) are the Bernoulli polynomials.

—

as\

(1.2.56)

2. Application

in numerical integration

33

Corollary. If Q(f) is a symmetric quadrature rule, then 6t=0,

ifi

(1.2.57)

is odd.

Proof. It will be sufficient to prove that when i is odd, the coefficients of (1.2.54) are Co, = 0, | a | = i. N

By using (1.2.55) and (1.2.56), since Yjofj = % is odd, then there exists at i=i

least an odd component otj. Noting that the Bernoulli polynomials with odd degrees satisfy: BQj(t) = — Baj(l — t)y 0 < t < 1, which imply rl

Jo /o

Ba.(t)dt

(1.2.58)

= 0,

one can easily derive that when |ot| = i is odd, Co. = Q(<pa) ~ 0. 2.4. Asymptotic expansion quadrature of functions

of multidimensional with singularities

□

numerical

J.N. Lyness gave an asymptotic expansion for the error of numerical quadrature formulas for multivariate functions of the form (1.2.59)

f{x) = r«
where r = (x\ + h x^) 2 and 6 = (#2, * • *, 0«) are spherical coordinates, ip{Q),h{r) and g(x) are smooth. Hence, the origin is the only singular point. In application, the potential integral belongs to this type. Besides, this type of integral often occurs in boundary element methods where sin­ gularity comes from the homogeneous function ra. In order to consider the asymptotic expansion of the quadrature formulae of / ( x ) , it is necessary to study the properties of the homogeneous functions. Definition 2.2. A function /(a?i, • • •, xs) is said to be homogeneous of degree yf if for any real number X there is the identity

f(\x1,--,\xs)

= \y(x1,---,x,)

= \y(r,e).

(1.2.60)

In spherical coordinates, (1.2.60) is equivalent to f{\r,9)

= X>f(r,0).

In the following, we denote by fy(x)

(1.2.61)

a homogeneous function of degree

7, and by /y (x) the partial derivatives of order /3 = (/?i, • • •, /3S). The following properties are obvious:

34

I. Generalization and application of Richardson's

extrapolation

(i) The function f(x) = 0 is homogeneous of any degree. (ii) The function
is homogeneous of degree (7a + 6b).

(iv) The function |/ 7 (aj)| is homogeneous of degree 7. (v) The function /y

(x) is homogeneous of degree (7 — \/3\).

T h e o r e m 2 . 7 . ^ ' If fy(x) is homogeneous function of degree y, and is analytic on [0, l] 5 \ {0}, Q^m\f) is the quadrature formula defined in (1.2.51), then there exists an asymptotic expansion

m5+7 +

Ds+y\nrn — m5+7

j

= 1

,. +i4m)(/7),

mJ (1.2.62)

1>S + J,

where As+y,Byfj,Ds+y are constants independent of m and Rj remainder with an explicit expression as follows: n

i

Ui)—

m

1 r n .,,, 2^i / OL\y,x)Jj m |a|=/^[ '°°)

\x)ax,

(/ 7 ) is a

(1.2.63)

where ha{0\ x) are defined respectively by (1.2.55) and (1.2.53), and L[m, 00] = {x G Ms : m < max(z,), 0 < min(a:,)}. * * Corollary 1. The remainder R)m\fy)

= 0{m~l),

for I > 7 -f s.

Corollary 2. 7 / / 7 ( x ) is a polynomial, then (1.2.64)

A y + 5 = Dy + S = 0.

Corollary 3. If Q is a quadrature formula of symmetric type, then Byj = 0,

if j is odd,

Dy+S = 0,

ify + sis odd.

(1.2.65a) (1.2.65b)

In addition, if Q has an algebraic precision of degree d, then ByJ = 0, L>7+5 = 0 ,

j < d, 7 + s < d.

(1.2.66a) (1.2.66b)

2. Application

in numerical

35

integration

In the above argument, we assume that the function value /(0) is not included in the quadrature rule Q(f)) because / 7 (0) = 0 for 7 > 0; while the value of / 7 (0) is undefined for 7 < 0. However, this assumption is unessential. In fact, if the quadrature formula Q{f) needs the value of /(0), it is possible to define the following quadrature rules: Q(f)=Q(f)-™of(0), &m\f)

= Q(m)(f)

(1.2.67) ~ wof(0)/m',

(1.2.68)

where u>0 is the weight in Q{f) with respect to /(0). It can be proved that Q

(rn\

(/) has an asymptotic expansion as (1.2.62).

In the following we shall consider a function of the form (1.2.59). Definition 2.3. Consider a function of the form f(x) = r° —s. For example, it is easy to prove r ~ 4 x 3 G H_£, because r ~ 4 x 3 = r " 1 cos3 0 is obviously integrable. Conditions A2) and A4) cannot ensure that a function with singularity on [0, l] 5 is included in Ha* • For example, if / ( x , y) = (Ax + fiy)P, ft is not a non-negative integer, then for X/i < 0, / ( x , y) does not belong to this kind, while for /? > — 2, A/i > 0, / ( x , y) belongs to this kind. In both cases, the function is of the form r^(Acos0 + fism0)P and condition A2)' leads to the same conditions on A and //. Theorem 2.8.t°^ If f(x) G Ha* , then the asymptotic expansions

Q^\f) - i(f) = £ — — + £ — +°(™"'). t=0

m" T ' T *

s < / < a + s+p,

7Z] m"

a ^ integer,

(1.2.69a)

36

I. Generalization and application of Richardson's <extrapolation

and Z-l J±. _j_

ft.

/-I

]J.

Q ( m ) (/)-/(/) = E ^ L — ~ + £ — lnm + 0(m-J l n m ) , s < /, a = integer

(1.2.69b)

AoW. Corollary. / / Q is a quadrature formula of symmetric type,, then Bj = Dj = 0,

(1.2.70)

ifj is odd;

if Q has an algebraic precision of degree d, then Bj = Dj = 0,

if j is an integer less than d.

(1.2.71)

Theorem 2.8 can be further generalized to functions with logarithmic singularity, for example, F(x) = lnr • ra
(1.2.72)

where f(x) = ra(p(0)h(r)g(x) € Has\ For the case that a is not an integer, it is easy to get the following results: T h e o r e m 2.9. If a is not an integer, then the function F(x) defined by (1.2.72) satisfies Q<m)(F)-I(F) PllA'+i t=0

lnm

Pi*

m


m

»=0

j=0

1 rn

*-'). (1.2.73)

S < / < Qf + S + P ,

with _ 8Aa+s+i a+ +i

'

~

da

_ dBj '

j

~ da '

where Aa+s+i and Bj have the same meaning as those in Theorem 2.8. Proof. It follows immediately by differentiating both sides of (1.2.69a) with respect to a. D For the case that a is an integer, similar argument cannot be applied to (1.2.69b). However, J.N. Lyness^85! used the more detailed analysis to get the following theorem:

2. Application in numerical

37 37

integration integration

T h e o r e m 2.10. Let F(x) have an expression of the form (1.2.73) and a be an integer, then

m m

t=0 t=0

j =l

m m

^£l3_ lnm + TT~ + Lr n^<*+»+* ^ ^^Zi Zi_ (lnm) 2 + ++ -££«+,-+, — + 0 ( m - <( °o + '' ''>) ( l n m ) 2 ) . i=o *=o m Qf+,+5

(1.2.74)

It must be pointed out that, whether the coefficients of an asymptotic expansion are zero or not depends not only on the degree of algebraic pre­ cision and the symmetry of the quadrature rule but also on the integrand. For example, consider a homogeneous decomposition of the function f(x) p-i p-i

/(*) = £ /«+•'(*) + 9«+P(X)> t=0

(L2-75)

if it is known that one of the components fa+i is zero, then by the argu­ Daa+i+ +i+8s ment of Theorem 2.8, the corresponding coefficients Aa+i+s and D are also zero. As another example: let / ( x , y) = r~4x3 = r _ 1 sin# be a ho­ mogeneous function of degree — —1, 1, then then all all coefficients, coefficients, except except f-i(x, f-i(x, y), y), are are zero in the homogeneous decomposition. In general, it is possible to con­ clude that some coefficients of the expansion are zero by judging whether h{r)g(x) is an odd (or even) function.

T hh ee oo rr ee m m 2.11. 2.11. If/ / T h(r)g(x) = = h{-r)g(-x) h{-r)g(-x) h(r)g(x)

(h(r)g(x) (h(r)g(x)

= -h(-r)g(-x)), -h(-r)g(-x)), =

(1.2.76) (1.2.76)

then the the coefficients coefficients of of (1.2.69a) (1.2.69a) and and (1.2.74) (1.2.74) satisfy satisfy then ^a+i+« = A*+t+5 =

dd

Aa+i+8 = 0,

da for ii odd odd (correspondingly, (correspondingly, even). even). for

(1.2.77) (1.2.77)

Proof. If h(r)g(x) = h(—r)g{—x), then the term fa+i(x) = 0 (i is an odd number) of the homogeneous decomposition (1.2.75) follows immedi­ = ——h(—r)g(—x), ately, and so one obtains (1.2.77). Similarly, if h(r)g(x) = h(—r)g(—x), one obtains fa+i(x) = 0 (i is an even number). The proof of Theorem 2.11 is thus complete. □

38

I. Generalization and application of Richardson's

extrapolation

Consider an important case: a = 0, and / ( * ) = h(r)g(x),

(1.2.78)

with the expansion (1.2.69b). By (1.2.64), Ai+8 = A+* = 0 (i is an even number), hence, 7-1 J±.

(m)

1-1

ft.

Q (/W(/) = £ - 7 + £ - 7 >-\Dilnm + 22 m ,=, '

+ 0(m~l I n m ) , i-i

A.

for s even,

(1.2.79a)

for s odd.

(1.2.79b)

1-1 ft.

(m)


m

2|«

>-\Dilnm + 22

+ 0 ( m - ' I n m),

2|t

If Q is a symmetric quadrature formula, we have the following: T h e o r e m 2.12. If f(x) = h(r)g(x) and Q is a symmetric quadrature formula with an algebraic precision of degree d, then J-i A-


— + 0(m~'),

for seven,

(1.2.80a)

t=d+l 2|t'

g(-)(/)-/(/) = 53 _ + J2 i=s 2|i

+ 5^ i=s

m%

i=
™l

+ 0 ( m - ' I n m), m%

for s odd.

(1.2.80b)

2. Application

in numerical integration

39

P r o o f . If Q is a symmetric quadrature formula with an algebraic preci­ sion of degree d, then applying (1.2.70) and (1.2.71) into (1.2.79), the proof of Theorem 2.12 is complete. □

2.5. Numerical

results

Numerical integration of multidimensional singular functions is very dif­ ficult. T h e weighted Gaussian quadrature rule can be regarded as a good m e t h o d , b u t the choice of the weights depends on the singularities. For some functions with mixed singularities, such as / ( x , y) = r~1g1(x,

f(x,y)

= r

1

y) + r~1l2g2{x,

gi(x,y)

+ r

y),

1

\nrg2(x,y),

there are troubles, and need different rules to tackle different singularities. In addition, most of the functions with mixed singularities cannot be simply decomposed. For example, the function

/(*> V) = (ax + h)/(cx + dy) is regular in [0, l ] 2 except at the origin, it has different limits when ( x , y ) approaches (0, 0) from different directions. T h e Gaussian quadrature rule is invalid. In this case, under the help of the asymptotic expansion introduced in the previous section, the m e t h o d of extrapolation will be very effective. T h e following examples are quoted from the work of J.N. L y n e s s t 8 6 l Example 2.1. rl JO

/•!

r . - 3 / 2eP"- rr e~x

dxdyn

2.5255135399.

JO

Let Q be a mid-rectangular quadrature rule which is of symmetric type. By (1.2.69a) and (1.2.70), there is an expansion ( a = - 3 / 2 ) : Q(-)(/)_7(/) Ai/2

Mil

B2

A5/2

A7/2

m3/2

m

2

m5/2

m7/2

B4

= —— + —— + — + —— + —— + — + • • •, (1-2.81) m

r

b u t e~ e~ we have

x

l/2

m

4

satisfies the conditions in Theorem 2.11, then, by using (1.2.77),

Aa+s+i

= Ai+i

= 0,

i = 1,3,5, • • •,

40

I. Generalization and application of Richardson's

extrapolation

and the expansion becomes Q(m)(/) - /(/) A1/2 B2 A5/2 B* A9/2 BQ = —— + —2 + _ 5L 2 . + —4 + _ L9 _2 + —6 + . . . . 1 2 m ' m m / m m / m

(1.2.82)

If Q is taken as the Gaussian quadrature rule G7 with an algebraic precision of degree 7, then the expansion is

A\I2

Ml2

^9/2

#8

^17/2

= - — + — - + —— + — + + • •.. (1.2.83) m1/2 m5/2 m9/2 m8 m17/2 If the trapezoidal rule R^m\f) under the modification given in (1.2.67) is used, the expansion is the same as (1.2.82). Example 2.2. r^2e-r\-x2dx JO

« 0.3386576711.

JO

Applying the symmetric quadrature rule with algebraic precision of degree 2, the expansion becomes Q(m) ( / ) _ / ( / ) B2 A5/2 B4 A9/2 B6 ^13/2 = —2 + 5 2 + —4 + —— + —6 + 1 3— + • • ■. 9 2 m m / m m / m m /2

(1.2.84)

Example 2.3. 7o »/o

rxdxdy * 0.4393173207.

Here a = 1 is an integer. By (1.2.69b), (1.2.70) and (1.2.77), the asymptotic expansion is Q ( m ) ( / ) - i(f) B2 D4\nm A4 B6 Bs B10 = -— + — + — + — + — + + •••. (1.2.85) v y ™2 ^ 4 ^4 ^,6 ^8 ™10 rrr The comparison between the numerical results of the trapezoidal rule and the 12-point Gaussian rule with an algebraic precision of degree 7 is illus­ trated in Table 2.1. For Example 2.1, G(732), G^128) have little improvement when compared with i?(32) and ij( 128 ), but the number of points to be evalu­ ated by the former are 11 times as large as those by the latter, because both

2. Application

in numerical

integration

41

have order of error 0 ( m - 1 / 2 ) . The improvement in Example 2.2 is better than that in Example 2.1, because the order of error of the trapezoidal rule is 0(m~2) and that of the Gaussian rule is 0 ( r a ~ 5 / 2 ) . The best improve­ ment appears in Example 2.3, because the order of error of the trapezoidal rule is 0(m~2) and that of the Gaussian rule is O(m~4 In m). Table 2.2 lists the results of the extrapolation method with different ex­ pansions from three examples, in which the mesh parameter is chosen to be H3) type, i.e., {m} = {1,2,3,6,8,12,16,24,32, • • •}, and the extrapolation is based on the modified trapezoidal rule with the origin omitted. These results show that the efficiency of Example 2.1 with the expansion (1.2.81) is not as good as that with the expansion (1.2.82). A survey on Tables 2.1 and 2.2 shows that the efficiency of the extrapolation is obvious. When the number of points used are v — 1633, the errors in three examples are all of degree 10" 1 0 . Table 2.1. Numerical comparison between the trapezoidal rule and Gaussian rule q u a d r a t u r e n u m b e r of rule Q

points v

R< 6 )

49

R(32)

R(

128

)

G?> GW G(ia.)

E x a m p l e 2.1 1.496

1089

E x a m p l e 2.2

E x a m p l e 2.3

QU) \QU) - /(/)!//(/) | Q ( / ) - / ( / ) ! / / ( / ) \QU) - /(/)!//(/) 0.4 0.17

2.080

9.3xl0~3

4.5X10"3

2.4X10"

4

1.6X10-4

5

16641

2.303

0.08

1.3X10"

12

1.977

0.08

2.5X10"3

4.0X10"6

12288

2.428

0.04

4.0X10"7

3.7X10"11

196608

2.477

0.02

1.2X10"8

i^(n) : n-repeated trapezoidal rule, G7 : n-repeated Gaussian quadrature rule with algebraic precision of degree 7. Table 2.2. Examples

Results of extrapolation Example 2.1

asymptotic expansion (1.2.82) (1.2.81) T (fc) V relative error error T(3) 37 1.7xl0"3 1.2X10-2 T(5) 121 4.5X10-5 2.3xl0-2 T(9)

1633

1.3X10"10

2.4xl0~7

Example 2.2 E x a m p l e 2.3 (1.2.83)

(1.2.85)

error

error

4.0xl0"5

2.2xl0~6

4.4X10"6

5.3X10"9

2.3X10"11

6.2X10"14

Error = | T 0 W - / ( / ) | / / ( / ) . Note that there are logarithmic terms in the asymptotic expansion of Ex­ ample 2.3, which has to be tackled by the method (1.1.24). J.N. Lyness^"^ gave the following example:

42

I. Generalization and application of Richardson's

extrapolation

E x a m p l e 2.4. / ( / ) = / / c o s M x d y = - [ l n ( V 2 + l) + V 2 - l ] . Jo Jo 2 Applying the symmetric rectangular rule of Q(f) = /(«>«)> / is a homo­ geneous function of order 0. By Theorem 2.7, the expansion is Q(m) ( / ) _ J ( / ) B2

D2

B4

BQ

= —z + —z l n m + — + — + . . . . m m m m

(1.2.86)

Table 2.3 lists the results obtained from Algorithm 1.1, in which b = 1 - , h0 - 1 and (pi,P2, •• • ,Pm) are respectively (2,4,6,8,10), (2,2,4,6,8) and (2,2,4,4,6). These results show that the precision with (pi, P2, • * *,Pm) = (2,2,4,6,8) is the best, which is identical to the result of expansion (1.2.86). Table 2.3 Errors of numerical results of three kinds of extrapolation number of extrapolation (2,4,6,8,10) 5.931X10" 2 T(l) 9.540X10 -3 9 T (2) 1.922X10" 3 9 T (3) 1.114X10" 3 T (4) 2.823X10" 5

Pi

(2,2,4,6,8) 5.931X10" 2 5.931X10 -2 9.960xl0~ 7 1.579X10"8 5.428X10 - 1 1

(2,2,4,4,6) 5.931X10" 2 5.931X10" 2 2.354X10" 6 3.187xl0~ 8 1.045X10 -9

Chapter II SPLITTING EXTRAPOLATION METHODS

T h e classical extrapolation methods, e.g. the Richardson's extrapo­ lation, polynomial extrapolation and ^-extrapolation, have been applied to various fields of computational mathematics. In the past two decades, Richardson's extrapolation m e t h o d has been widely used in solving partial differential equations. T h e works of the Russian mathematicians were sum­ marized in the monograph by G.I. Marchuk and V.V. Shaidurov^O] and the works of other countries can be found in the papers by V. Pereyrat 9 7 ! and T h e finite element extrapolation developed in the '80s is a sub area of the finite element method. Since the first paper by Q. Lin, T . Lii and S.M. Shent 5 8 ) , published in 1983, which stimulated a series of works done by Q. Lin and his cooper at ors T . Lii, C M . Chen, Q.D. Zhu, R. R a n nacher, H. Blum, J . C . Xu and J . P . Wang, the finite element extrapolation becomes not only a reliable method for improving the accuracy of the finite element solution b u t also a basis for a posteriori error estimation and for discovering the super convergence points. This new development is the socalled interpolation finite element theory which simplifies t h e complicated finite element theory and therefore greatly expands its range of applications. For details, see [77], [101]. However, there are some difficulties in applying Richardson's extrapolation to higher dimensional problems. Many scientific and engineering problems can be reduced to multidimensional m a t h e m a t ­ ical problems. Examples include exploration and exploitation for oil and gas, design of aircraft and other large-sized frameworks, weather forecast­ ing, pollution control and computation for nuclear reactors. From these problems, multidimensional integrals, integral equations or linear and non­ linear systems of partial differential equations can be extracted. T h e main difficulty involved in multidimensional numerical m a t h e m a t i c s is t h a t the number of discretized equations increases exponentially with the dimension. Take the solution of an s-dimensional second order elliptic equation as an example, in order to obtain a solution with error of order 0 ( / i 2 ) , discretized by the usual finite element or finite difference methods, a linear system with 0(ji~s) unknowns has to be solved. This implies t h a t

43

44

II. Splitting extrapolation

methods

for a time-dependent problem, a large scale linear system will have to be solved at each time step, which is difficult even using a supercomputer. No wonder the numerical analysts call the above-mentioned problem "the dimensional worry". T h e splitting extrapolation m e t h o d (SEM) studied in this chapter is one of the most efficient m e t h o d s for tackling the "worry". This m e t h o d is essentially based on a multi-parameter asymptotic ex­ pansion of the approximate solution. For instance, in the numerical solution of an s-dimensional problem, the s mesh parameters are mutually indepen­ dent and the error of the approximation has a multivariate asymptotic expansion with respect to these parameters. Instead of taking a global re­ finement in all directions as suggested by the classical Richardson extrapola­ tion, the splitting extrapolation needs only to carry out some unidirectional refinements. T h e SEM is specially suitable for large scope multidimensional problems due to the following properties:

( a ) In order to get an approximate solution with a certain order of er­ ror, the complexity of SEM increases according to a polynomial of the dimension, while t h a t of Richardson's extrapolation increases ex­ ponentially with the dimension. ( b ) Problems can be solved in parallel, and the higher the dimension, the higher the degree of parallelism. In fact, SEM consists of two steps: solving first several coarse grid problems independently and then combining the solutions obtained to get an approximate solu­ tion with higher order of accuracy. SEM is ideal for implementing in multiprocessor computers and almost no communication among processors is required. (c) SEM is self-adaptive, and the stopping criterion can be obtained through a posteriori estimate.

In 1983, based on the multivariate asymptotic expansion, Q. Lin and proposed the m e t h o d of splitting extrapolation. Later, P. Neit95 t a a n m a k i a n d Q . Lint ] and A. Schiiller and Q. Lin[H3]More general appli­ with one split to solve partial differential equations. applied the m e t h o d cations of SEM appeared in 1990, T . Lii, T . M . Shih and C.B. Liemt 8 1 ' derived a recurrence algorithm and proposed two types of refinements of SEM. Their work makes the splitting extrapolation m e t h o d more practi­ cal and powerful. T h e extrapolation process can be carried out in the recurrence form as many times as required and the computational cost is almost minimal. U. carried out numerical experiments of SEM

1. Multivariate

asymptotic

45

expansion

on partial differential equations and obtained the combination coefficients of Type I methods for two-dimensional problems. T.M. Shih et oiy^^ gave the coefficients of Type I and Type II methods for multidimensional problems. The numerical experiments show that the SEM is very efficient for computation of multidimensional integrals, integral equations and par­ tial differential equations, and the higher the dimension, the more efficient the SEM. In this chapter, we shall study the multivariate asymptotic expansion of approximate solutions and introduce the Type I and Type II splitting extrapolation methods.

1. Multivariate asymptotic expansion Consider an s-dimensional problem whose numerical solution u(h) = u(hi, • • •, hs) depends on the mesh width h = (/ii, • • •, hs), where hi is the mesh width in the x,-axis. In the following, we shall prove that, in many multidimensional problems, linking the approximate solution u(h) and the exact solution u there is a multivariate asymptotic expansion with respect to hi, • • •, hs:

u(h) = u+

Yl

Cah2« + 0{hlm+2),

(2.1.1)

l<|a|<m

where Ca are unknown coefficients with a = (c*i,••-,«,), M = a i + --- + a,, h° = h%x-~h°', h2a = hlai.-h2sa% h0 = max hi. l
If hi = • • • = /i 5 , the expansion (2.1.1) is the Richardson expansion. The difference between (2.1.1) and the Richardson expansion is that hi are mu­ tually independent. And so it is possible to replace the global refinement of the Richardson extrapolation by refinements in different directions in­ dependently, and reach the same order of accuracy as that of Richardson's extrapolation. Furthermore, the amount of computation and computer storage are reduced and the degree of parallelism is greatly increased. We highly recommend the following two types of splitting extrapolation refinements which are analogs of the one-parameter grid refinements Hi) and H2) in the case of multi-parameter grid. T y p e 1. Given the initial mesh width h = (hi, • • •, hs), refine succes­ sively by the mesh width A/2" = (Ai/2^ 1 , • • •, A,/2"')> 0 < \/3\ < m, the corresponding approximate solution is denoted by u(h/2^).

II. Splitting extrapolation

46

methods

T y p e 2. Given the initial mesh width A, refine successively by the mesh width fc/(l + P) = (fti/(l + /?i), • • •, fc,/(l + /?.)), 0 < |/?| < m, the corresponding approximate solution is denoted by u(h/(l + /?)). In the Type 1 refinement, replacing & of (2.1.1) by ft/2^ and omitting the higher order term 0(Ao m + 2 ), one gets ti(fc/2") = u+

Cah2a/22^fi\0<

£

\fi\ < m,

(2.1.2)

l<|cr|<m s

where (a,/?) = Y ^ a « A - Similarly, for Type 2 refinement, »=i

«(A/(1 + /?)) = S+

£ l<|a|<m

Caft2V(l + /?)2°,

0 < |/?| < m,

(2.1.3)

where

fc2«/(i + /?)2a = ( ^ 7 ( 1 + A)2"1, • • •, A?°7(i + A)aa") • The basic idea of SEM is that by Type 1 or Type 2 refinement of the initial grid, we find out the corresponding approximate solution u{h/2^) or u(h/(l + /?)), 0 < \P\ < m, and then solve the equation (2.1.2) or (2.1.3) to obtain the splitting extrapolation value u with error 0(h2m+2). Since the approximate solution u{h/2^) (or u(h/(l+/?))) can be obtained in parallel, SEM is very suitable for implementing in multiprocessor computers, and there is a one-to-one correspondence between the number of independent solutions and the number of terms in the multivariate expansion. For sim­ plicity, only the two-dimensional case is considered. We arrange all terms of two-dimensional multivariate expansion in the following triangular pattern

1 h^y

hi

K

hex hi ...

hlh] h%h\ hlhl ■■■

h* hlh* h6y

hi

h*h*

hlhl ■■■

...

2. Recurrence algorithm of splitting extrapolation

47

and the corresponding approximate solution can be arranged in the same form u(hXihy)

ti(-f,A y )

u(-£,hy)

u(-£-,hy) 8

•••

u{hx, y ) «(*-,^)

«(y,£)

"(ftx.y)

2. Recurrence algorithm of splitting extrapolation In order to obtain the splitting extrapolation value u for Type 1 meth­ ods or Type 2 methods, it is inconvenient and expensive to solve the linear equations (2.1.2) or (2.1.3) directly. The relation between splitting extrap­ olation and the multivariate Lagrange's interpolation makes it possible to construct a recurrence algorithm. Let uJ(x) be the interpolation polynomial of the function u(x), and the basic points for Type 1 interpolation are taken as: h*/4P = ( A J / ^ V - A V ^ H O < |/?| < m,

(2.2.1a)

and for Type 2, they are ft2/(i + /?)2 = (h\/(i + A ) 2 , . . . , / i 2 / ( i + & ) 2 ) , 0 < |/?| < m.

(2.2.1b)

Set uI(x)=

£

Caxa,

x = (xlr-.,x,),

(2.2.2)

0<|a|<m

then for Type 1 methods, (2.1.2) is equivalent to the interpolation condition u^tf/tf)

= u(h/2P), 0 < |/?| < m,

(2.2.3)

and for Type 2 methods, (2.1.3) is equivalent to « V / ( 1 + /?)*) = « ( * / ( ! + /?))•

(2.2.4)

II. Splitting extrapolation methods extrapolation methods

48

Since under the interpolation condition (2.2.4), the Lagrange's interpolation polynomial (2.2.2) exists uniquely, which implies that the equation (2.1.2) or (2.1.3) has aa unique solution, and that once the interpolation polynomial x ) is constructed, there must be ui /7((x) (2.2.5) (2.2.5)

11 = tZ = 1/(0). t/(0).

the interpolation condition of of the form (2.2.3), the the multivariate For the Newton's interpolation formula can be generalized easily from the univariate Newton's interpolation formula (1.1.10). We denote the multivariate Newton's divided difference by U

u u

— i*i/i,•••,»* 1*1/1,•••,»* — — UL vv — L 1 i ''

'' ii

''

'' * * ''

'' * * J' J'

> x^Xj > >• • • > > Xj Xj33\ \ 1 < <j < < s, s, and where xXj^ >

{x{Vl\...,x^), 0<M (a^,-..,*<"•>), H<m, are basic points of the interpolation. Again, let X« = Xflizi) = (*,- - x^). • .(*,- - z^-% 1

1 < i < s, ,,

s

(2.2.6)

Xv =Xu(x) = Xi i>---Xi \

Making use of the univariate interpolation formula (1.1.10) and by induc­ it is easy to prove the multivariate Newton's interpolation formula tion, it

* ( x ) ~ 5Z X"u>"

u

(2.2.7)

0<|i/|<m 0<\v\<m

and we have the splitting extrapolation value

u = t/(0) = ti = t/(0)=

Y, ^ ( 00)K ^ Y, - M 0<|i/|<m 0<|i/|<m

j 1 = (-i)wHnnnnN v = E (-i) *p)«,. 0<|i/|<m 0<|i/|<m

(2.2.8)

tt == ll ; ;==00

Applying the error estimate of multivariate interpolation, we get the inter­ polation error I I u(x)-u (x)= e(x) = u(x)-u (x)=

J2 J2

XvD"u(Z XvD"u{Zv)lv\> v)lv\,

|i/|=m+l

with

D> = B? •••£>;-, Di = A , i/! = 1/1! •••!>,!, 6 = ( & , , • • ■ , & , . ) ,

^

€[0,ft,-], €[0,fc,-], l < i < « .

i
(2.2.9)

2. Recurrence algorithm of splitting extrapolation

49

By Newton's interpolation formula (2.2.7), it is easy to derive a recur­ rence algorithm. Let Pm(x) = P m ( x i , • • •, xs) be an interpolation polyno­ mial of degree m with the basic points zp = (x j *', • •, x[ ) , 0 < |/?| < m. It follows from (2.2.7) that

Pm(x) " Pm-l(*) = J2 *"M*)>

( 2 ' 21 °)

|i/|=m

where dv are undetermined coefficients. In order to find dVj note that the interpolation condition implies Pm(zfi) - Pm-i(zp)

= 0, 0 < \/3\ < m - 1,

(2.2.11)

and Xy(zp) = 0, |/?| = M = m, /? ^ v.

(2.2.12)

In fact, if |/?| = |i/| = m, but /? ^ i/, then one can always find /?,• < i/,-, 1 < i < 5. By the definition (2.2.6), x £ } ( x ^ 0 ) = 0, and (2.2.12) holds. Now let x — zv and \v\ — m in the identity (2.2.10). By using (2.2.12), we have Pm(Zv) - Pm-l(Zu)

= dvXv(zv),

\v\ = m,

or d„ = (Pm(zv)

- Pm-i(zy))/Xy(zv)y

|i/| = m.

(2.2.13)

Substituting into (2.2.10), we obtain Pm(x) =

Pm-l(x) + ]T

{Pm{Zu)-Pm-l{zv))Xv{x)/Xv{zv).

(2.2.14)

|i/|=m

Therefore the recurrence algorithm for evaluating the splitting extrapola­ tion value P m (0) is derived. A l g o r i t h m 2 . 1 . (Splitting e x t r a p o l a t i o n r e c u r r e n c e a l g o r i t h m ) SPE(s, m, ft, Type, um) Begin P 0 (x) = u(h) For i = 1 t o m If Type = 1, t h e n x{p = h2k/4j,

0 < j < i/fc, 0 < |i/| < i, 1 < k < s

compute u(ft/217),

|i/| = i in parallel.

50

II- Splitting extrapolation

methods

Set Pi(x) = Pi-!(X)

+ £

(u(h/2^-PM(hyAv))X¥(x)/X¥(h2/4r)

else

xlj) = h2k/(i + j)2, o < i < i / f c > o < H < « ,

i<*<*

compute u(h/\ + ^ ) , W\ = * in parallel. Set Pi(x) = Pi.1(x)+y£(u(h/(l + v)) - P * - i ( f t 2 / ( l + vf))Xv{x)/Xv e n d If

(/> 2 /(l + i/)2)

t£,-(0) = Pi(0) e n d For Um = P m (0) end

R e m a r k 1. Similar to the case of one-parameter extrapolation, the numerical stability of Type 1 methods is superior to that of Type 2 methods. However, the former needs greater amount of computation and storage than the latter. As the numerical solution of partial differential equations is concerned, the number of split is usually less than or equal to 5, then the cost of Type 2 methods will be much less than that of Type 1 methods while the order of accuracy of both types' methods are comparable. A Type 3 refinement similar to H4 originated from the Fibonacci numbers is given in the following: T y p e 3. Given n and the initial mesh width h, refine step by step the mesh width to h/Fa = (fti/F a i > • • •, / i , / F « J , 0 < |a| < m, where Fa is the a t -th Fibonacci number. It can be proved that the numerical stability of Type 3 methods is superior to that of Type 2 methods while inferior to that of Type 1 methods, and the amount of computation and storage needed by Type 3 methods are greater than those of Type 2 methods and less than those of Type 1 methods. Remark 2. Suppose that the approximate solution has the following

3. Coefficients and stability of SEM

51

more general asymptotic expansion: ti(ft) = ti +

C a / i r a + 0(/io ( m + 1 ) ),

J2

(2.2.15)

l<|a|<m

where r is a positive real number, hra = /i™1 • • • hr3a'. In this case, the grid refinements of Type 1, Type 2 and Type 3 can also be used, and the splitting extrapolation value will be recursively computed by Algorithm 2.1, while the basic points are correspondingly Type = l,

[ (hk/V)\ _

XU) x

k

(hk/(l

—

+ j))\

[ (hk/Fj)rJ

Type = 2, Type = 3,

0 < j < vk, 0 < \v\ < m, 1 < k < s.

3. Coefficients a n d stability of S E M Equation (2.1.2) can also be solved by the elimination method provided that one obtains in advance the extrapolation coefficients a^, 0 < |/?| < m, satisfying the equations 0<|/?|<m

]T

(2.3.1)

apj^^

= 0, 1 < M < m,

0<\p\<m s

where (/?, a) = y^/%a t -. Once ap, 0 < \/3\ < m are found, then multiplying i=i

both sides of (2.1.2) by ap and summing up with respect to /?, and one immediately gets um=

Y,

«fiu{h/&)'

(2-3.2)

0<\(3\<m

Similarly, it follows from (2.1.3) that the extrapolation coefficients ap of Type 2 methods satisfy the equations o<|/?|
V

(2

"'

= 0, H)

1 < |a| < ro,

3 3)

II. Splitting extrapolation

52

methods

and the extrapolation value of Type 2 is

\J-"r^/

0<|/9|<m

In computing the extrapolation coefficients, it is inconvenient t o solve directly the equations (2.3.1) or (2.3.3). An easier way is to apply the recurrence algorithm 2.1. As an example, for fixed /?, 0 < |/?| < ra, in order to obtain the ap by a T y p e 1 m e t h o d , in Algorithm 2.1, we set hi; = 1, 1 < i < s,

v*P,

«(£) = | 0,

0 < \v\ < m,

1,

u = (3,

and then let ap = um. Similarly, in the case of T y p e 2, in Algorithm 2.1 we set hi = 1, 1 < i < 5,

1 0, v?/3,

«(r5-) = < V

0 < \i/\ < m,

1, v = p,

l + l/

and let the extrapolation coefficients ap = um, 0 < \/3\ < m. Similar to the theory in Section 1.3, Chapter I, the s u m of absolute values of splitting extrapolation coefficients

Mj(s]m)=

M , i=Type

Yl

(2.3.5)

0<\P\<™

is called the stability factor of splitting extrapolation m e t h o d s . In fact, e.g. in T y p e 1, due to the rounding error, an approximation of u ( — j ) , say ~,h \ . u[ — ) , is obtained, then the difference between the numerical and exact

? .

splitting extrapolation values is \um -Um\

= 0<|/J|<m

l

l

h

< £ M u(-)-u(-) V 0<\P\<™

< Mx(s

m)

maIX 0<\(3\ <m

h u{—)

h

2^ - u(

i) IP'

3. Coefficients and stability of SEM

53

Similar discussion can be applied to Type 2 and Type 3 methods. The sta­ bility factor Mj = Mj(s,m) increases rapidly as s, the dimension and m, the number of split increase, and lim m _ 0 0 Mj(s, m) = oo, j = 1,2,3. This is totally different from the Richardson extrapolation and shows that the stability of the splitting extrapolation is not as good as that of the Richard­ son extrapolation. In order to estimate the stability factors of the splitting extrapolation methods, we shall prove the following theorem concerning the signs of splitting extrapolation coefficients in Type 1 methods: T h e o r e m 3.1. If dp, 0 < \/3\ < m, are the extrapolation coefficients of an s-dimensional problem under the grid refinement of the Type 1, then they satisfy (-l)M+map>0.

(2.3.6)

Proof. We use the extrapolation coefficients ap to construct an svariate polynomial (2.3.7) 0<\p\<m

By using the equation (2.3.1), f(s) possesses the property (2.3.8a)

/ ( i , ■ - - , i ) = i, and / ( x ^

■■■,*)=/(*<'>)

=

(>,

(2.3.8b)

where

*"=(•&>->■&)>

1<M<™,

(2.3.8c)

hence f(x) is also an interpolation polynomial satisfying the condition (2.3.8), and can be expressed by Newton's interpolation formula as

/(«)=

Jf„(*)/[*i 0 ) ,-,«W;... ;«?>,■ •■,«<•">]

£ 0<|i/|<m

By the divided difference expression (1.1.10) and the interpolation condition (2.3.8b), J l*i

i

>* i

)

>*»

>

i*»

J

54 54

II. Splitting Splitting extrapolation extrapolation II. _ V~>

J \xl

^^

> x*

)

methods methods

)

u=o ..=o "Q ( g (,(fc> _ , 0 . ) ) ) *.-o ,-.=o -Q ( JJ (,(fc> _ ,00)) —

5

I/fc

*=1

jk=0 =0 jk&k:tfc

— f \iA < m = /*, 0(\ << |i/|< m,

1

niK - *«* °) °)

(2.3.9) (2.3.9)

**r == l jl jf cf c==l l

hence hence

;

£

/(«) =

UXy{x).

0<|i/|<m 0<|z/|<m 0<|z/|<m fc Jfc) = l/4*fc By x^ x[Jk) < 1, 1, it it follows follows that that /„ /„ > > 1, 1, and and By = l/4* <

/* > fu> / „ , for for z/> i/ > i/, i/, fv> where vv > > vv where Consider Consider

(2.3.10) (2.3.10)

implies i/* v^ > > i/jt, i/jt, 11 < < A A:: << 5,5, and v. implies and vv ^ ^v. the function the function

/(-*)= j ; (-l)°a a x a = 0<|a|<m 0<|a|<m

£

h x

- °-

00<\a\<m <|a|<m

Obviously, if if one one can can prove prove Obviously, m

> 0, 0, >

(-l)mbbaa (-l) then (2.3.6) (2.3.6) holds. holds. By By (2.3.9) (2.3.9) then / ( - .«) =

£

(2.3.11) (2.3.11)

/„**(-- X )

000<\v\<m << || ii // || << m m

= J ] (-l)V^n(*), 00<|i/|<m <|i/|<m

(2.3.12) (2.3.12)

with with ( 1)

n(*) = YS?( n 1 (xXl1))...y n(«) ■ ■lc;)(«.), -YS:\xt),

(2.3.13a) (2.3.13a)

and and

Y^(Xk) = (Xk + x^y ■ • ( « * + * ( ; - 1 ) ) , l
(2.3.13b) (2.3.13b)

3. Coefficients and stability of SEM

55

Expand Y„k (x*), and let Y£H**) = E E ^ x i V k - j \

l
(2.3.14a)

j=o

Yv{x) = £

2 # ) * « , 4 " ) = Ei[lai

■ ■ ■ E%-2a.,

(2.3.14b)

where E^ are the coefficients of the product in (2.3.13b). Substituting into (2.3.12), it becomes

/(-*)= E (-ir/.E^"^ 0<|i/|<m

= E

1

a
Et- )"-^*"'

(2-3.15)

0<|i/|<ma
hence the coefficients of the term ara, denoted by 6 a , have the form ba = ^ ( - i r / , 4 ^ -

(2-3.16)

In order to prove (2.3.11), let

9k = E /"£« }> i«i < ^ < "»>

(2-3.17)

i/>a

and obviously, &a = 53(-l)*flf fc , 0 < | a | < m .

(2.3.18)

Now we shall prove 0<9k

<9k+i <--
/?>i/

and

(2.3.19)

Take |/?| = |i/| + l.

It follows from (2.3.10) that fp > f„. On the other hand, by using *fc° > 0, 0 < j < I/* - 1, 1 < i < 5, we derive E^

> 0, and also the following recurrence relation: £J**+1> = E^l + x ^ E ^

> E^_kl

(2.3.20)

II. Splitting extrapolation

56

It follows from (2.3.17) that gk < gk+li Now by (2.3.18) and (2.3.19), ( " l ) m 6 a =9m~

i.e., (2.3.19) holds.

+ ■ ' • + (-l)m'M9\a\

9m-l

methods

which completes the proof of (2.3.9).

> 0,

□

Corollary 1. M±(s, m) = | / ( - l , • • •, - 1 ) | . Proof. It follows immediately from Theorem 3.1 that

£

\aa\ = (-irf(-l,...,-l).

D

(2.3.21)

0<|a|<m

Corollary 2.

Let

"<*")=£ n (7-3=7) n \a\=mk

l

=l

~

X

h

Jb=l

+

n >

(2.3.22)

1 - * (i*)

■=,

J*=I


(2.3.23)

Proof. From

/(-i,... ( -i)= £ (-i)-n n'(i+.if'V. 1 0<|or|<m

Jk = l j f c = 0

and

/« =

-, X(fc0) = 1, 1 < * < 5,

niK1-*?0)

we obtain /(-l,

••,-!) 5

Qffc-1

/][_!_

,(ifc) \

1

n 5

2 . (- )* 11 11 7—fc 11 7-0=) • (2-3-24) 0<|a|<m

1

lk=l\l-xk

ib=lifc=l V — ^fc

Now denote

&=nnB,nr S Offc-1 / l

,

/r0'fc)\

5

/

.(«o

)

3. Coefficients and stability of SEM

57

and

Nk = £ Stt.

(2.3.25)

I«|=fc Then by using (2.3.24), ^(-l)kNk.

/(-1,..,-1) =

(2.3.26)

*=o

Noting that 0 < x, 0 j p0; < 1, 1 < j k < aki 1 < k < s, one obtains 0<7V 0 < Ni <

< Nm.

It follows from (2.3.21) and (2.3.26) that

0*^™) = Yl KI = (-i) m /(-i,••-,-!) 0<|a|<m

(-l)mNm)

= (-l)™(No + --- +

= Nm - iV m _! + Nm.2 - ••• + (-l) m i\T 0
(2.3.27)

= N(s)m)}

which gives the proof for the inequality on the right-hand side of (2.3.23). The inequality on the left follows immediately from the fact that {N{} is a monotonically increasing sequence. □ An upper bound of Mi(s, m) is given in the following theorem: T h e o r e m 3.2. For Type 1 methods, the stability factor Mi(s,m) the following estimate: Afi(«, m) < ( m +„* " 1) 3*W 3 .

has

(2.3.28)

Proof. By (2.3.25) and x[jk) = 4 " ' \ 1 < j k < ak, 1 < k < s, we have

n (i+4_ifc) Offc

\a\=l

|a|=/

*=1

(2.3.29)

11(1-4-")

By Bernoulli's inequality: If, for 1 < i < n, cti > — 1 and have the same sign, then (l + a i ) . - - ( l + a „ ) > l + ai + --- + an,

(2.3.30)

II. Splitting extrapolation

58

methods

and the following inequalities can be derived: crfc-i

oo

JJ (l + 4-'*)<exp(£4-'") = 1[[ ( 1 - 4 - ' * ) >

(2.3.31)

l-^4--'fc 4"1

2

(2.3.32)

Substituting (2.3.31) into (2.3.29),

w = E 5« ^ E =

3V/3

(' + « - l ) 3 V / 3 .

Taking / = m, and using (2.3.23) the proof of (2.3.28) is complete.

(2.3.33) D

(2.3.28) shows t h a t M i ( s , m ) increases exponentially with dimension 5, and as fast as a polynomial of the number of split, m. T h e stability factor M 2 ( s , m ) or Ms(s,m) of Type 2 or T y p e 3 m e t h o d s increases more rapidly t h a n t h a t for Type 1 methods, this indicates t h a t the number of split should not be too big, and t h a t the use of double-precision word length in computation is necessary. Table 3.1 lists the stability factors of T y p e 1 and T y p e 2 m e t h o d s . Obviously, the stability factor of T y p e 1 m e t h o d s is less t h a n t h a t of Type 2 methods, and b o t h increase rapidly with the dimension. If the number of split is small, e.g., m < 5, then the stability factors of T y p e 2 methods are comparable with those of T y p e 1 m e t h o d s . More tables of the splitting extrapolation coefficients are given for ref­ erence in Appendix I. Note t h a t the extrapolation coefficients possess a symmetric property, i.e., if /? = (/?i, • • •, /38) is only a different arrangement of the components from a = (c*i, • • •, a 5 ) , then aa = ap.

4. A posteriori

error estimation

of splitting

extrapolation

59

T a b l e 3 . 1 . T h e s t a b i l i t y f a c t o r s o f s p l i t t i n g e x t r a p o l a t i o n for T y p e 1 a n d T y p e 2 D i m : 5 Split:m M i ( * , m ) A f 2 ( a , m ) 2

9.89

12.38|

3

16.75

31.9l|

4

24.19

78.781

5

31.84

190.04|

6

39.56

452.061

7

47.31

1065.48|

8

55.06

2495.27|

9

62.82

5816.491

10

70.57

13510.491

11

78.33 31296.28|

1 ^

1

2

*■

1 *" 1 ^

2

25.00

3

62.37

95.07|

4

124.56

282.711

5

214.73

790.061

6

334.41

2121.14|

7

484.25

5538.621

2

47.22

52.20

3

157.76

214.64

4

410.53

758.47

5

895.89

2440.31

28.731

2

76.56

82.78

3

321.89

409.58

2"

113.00

120.47

3

573.73

698.87

4. A posteriori error estimation of splitting extrapolation Let um(h) be the splitting extrapolation value of u with m split, and suppose that the asymptotic expansion

u-um(h)=

£

Caft2« + 0(ft2(m+2))

(2.4.1)

|a|=m+l

holds, and h^ is the refinement of the initial mesh width h = (hi, /12, • • •, h5) with only one split in the i-th component /i,-, i.e., ft(0 = (/i!, • • •, /!,_!, hi/2, hi+ly • • •, hs) 1 < i < s.

(2.4.2)

II. Splitting Splitting extrapolation e x t r a p o l a t i o n methods methods II.

60 60

Choosing h^ h^ as as the the initial initial mesh mesh width, width, and and denote denote by by uumm (h^), (h^), 11 < < ii < < s, s, Choosing the m-th m-th splitting splitting extrapolation extrapolation value. value. Taking Taking the the average average of of these these splitting splitting the extrapolation values, values, by by (2.4.1), (2.4.1), one one obtains obtains the the asymptotic asymptotic expansion expansion extrapolation

ft(,) «-7E«-( ) *=1 *=i

C°(-st^)h2a

= £ |c*|=m+l \a\=m+l

+ °(hl{m+2))-

243 ((2.4.3) )

»=1 «=1

Set Set

iimm(h)=-J2u (fc)="E« mm(fc«) u (h^),> sS » = 1 «=i

m 1

b= max max {{ iIyy"-- iia-- }} == iI(5-l-f4b= ( 5 - l + 4 - m-- 1), )' )' | c r | = m + l l* 8S^^ 44° * ' SV l«l=m+l »=1 and and Mm(h) * ) = (1 + 7)Mm(fc) U j)um(h) ~- JUmih), yum(h), where 77 is is an an undetermined undetermined constant. constant. By By (2.4.2) (2.4.2) and and (2.4.3), where (2.4.3) u- m(h) Um(h) u-U 4 E ~°'-T 1= 1

|cr|=m+l |cr|=m+l

2a (m+2) + O(h 0 ( / i20o(m+2) hft2a + ). ) .

(2.4.4)

t=i

Choose 77 such such that that Choose 5s

II ±± ll ££44--<<""- -7 7< (<(1+7)6-7 l + 7 ) 6 - 7 <0, <0, t*== 1 l

V|a| = = m m+ + 11 ,, (2.4.5) V|a| (2.4.5) which implies implies that that for for sufficiently sufficiently small small /i /i00,, uumm(h) (h) and and U Umm(h) (h) approach approach uu which from two two sides sides and and therefore therefore the the following following estimate estimate holds: holds: from \\(Umm(h) (h) \Uu

+ uumm(h)) (h)) --u\< \\Umm{h) (h)\. + „| < i|l7 (fc) -- uumm(h)\.

(2.4.6) (2.4.6)

But But

l(U (h) + uumm(/i)) (h)) mm -(C/ (ft) + 1

= 22 KK1 ++ 77)«m 7«m(fc) ++ U m{h)} = ) « m ( fW t ) -" 7«m(fc) «m(ft)] — 7 I— 1 = « m ( ft) + 2 ^ ( « m W ■ - «m(fc)), = «m(ft) + — ( « m C l ) ~ «m (>»))>

(2.4.7) (2.4.7)

5. Fractional power expansion and successive elimination

methods

61

hence Um(h) - um(h) = (1 + 7 )(tJ m (>i) -

(2.4.8)

um(h)).

Substituting into (2.4.6), we have l^Umih)

- ^±um(h)

- u \ <

l±±\um(h)

-

Um(h)\.

(2.4.9)

Now by (2.4.5), taking

s - l + 4-m-1

b 1

~ 1-6 ~

l-4-m"1

'

(2.4.10)

and substituting into (2.4.9), the a posteriori estimate of splitting extrap­ olation is obtained and this is valid for both Type 1 and Type 2 methods. 5. Fractional power expansion a n d successive elimination methods The asymptotic expansion (2.1.1) with even powers exists for smooth problems only. The singular problems, such as multidimensional improper integration, integral equations with weakly-singular kernels, and partial differential equations with nonsmooth coefficients or concave polygonal do­ mains, may have the following asymptotic expansion: m

u(h) = u + Y,T,

C

<*hPk0 + 0(^ m+l(m+1) ),

(2.5.1)

* = l|cr|=Jb

where pi, • • •, pk are positive real numbers satisfying 0 < px < 2p2 < • • • < (m +

l)pm+1.

(2.5.2)

We are going to eliminate the term hPka by successive splitting extrapola­ tion. Denote by ui(h) the /-th splitting extrapolation value, then ui(h) (\a\ = ib, 1 < k < m) possesses a higher order asymptotic expansion m

Ul(h) = u+ £

£<^l>ft'>-+0(ftS-+l(m+1)),

k=l+l\a\=k

1 < / < m,

(2.5.3)

where Ci, are constants independent of the mesh width h = (hi, ■■■>h.). In case of only one split, denote (2.5.1) simply as u(h) = u + dih*1 + ■■■ + dshP> + O(h20p3).

(2.5.4)

II. Splitting extrapolation

62

methods

hi

Let ftW = (fei, • • •, ft,*-i, -«r-, ftt+i, • • •, /*«) be the initial mesh width with only one refinement in the f-th direction, and replace h in (2.5.4) by h(*\ it becomes

U(ft<0) = U + dxh? + • • • + di-lfcfi! + ^/if 1 + di+1h^+1 + • • • + rf.ftj1 + 0(fc?a)» 1 < « < m.

(2.5.5)

Summing both sides of (2.5.5) with respect to z, it follows that

X>(/»(,)) = «« + ( « - ! + 2-"1) E **?' + °(ftoP2)«=i

(2-5-6)

»=i

Let

«i(fc) = j ^ d l X * ' 0 ) - ( « - ! + 2"Pl)«W]-

(2-5.7)

«'=i

From (2.5.6) and (2.5.4), we have

m(&) = u + 0(A2Pa), or in more detail, Ul (/i)

= «+ £

C<2>ft*« + 0 ( ^ - + l ( m + 1 ) ) .

^

(2.5.8)

fc=2 |cr|=ib

Once ui(h) can be expressed as a multivariate asymptotic expansion (2.5.3), u/+i(/i) is then the combination of u/(ft) and u/(r^-), V|/3| = / + 1. There are altogether 1 +

(ni)

values with / split have to be evaluated and

ti/+i(fc) =

^2

apu\{—) + a0uj(h).

(2.5.9)

For Type 1 methods, the extrapolation coefficients ao and {a/?}|/?|=/+i sat­ isfy the equations a0 +

^

a/j = 1,

|/?|=f+i

a0 +

^

a/ ,2-"+

1

W» a ) = 0, V|a| = / + 1,

(2.5.10)

5 Fractional power expansion and successive elimination methods

63

while for Type 2 methods, the coefficients satisfy the equations qp + < 1

ap = l,

y] m

^

(2-5.11)

aa

|/J|=|+1

V

^ ^

If /?' is obtained from /? by reordering the components, then ap = apt. Hence the distinct coefficients in (2.5.10) and (2.5.11) are not too many, e.g., for ui(/i), there are only two distinct coefficients ao and a± satisfying the equations

0/»2" Pl(/,,o) = a 0 + (s - 1 + 2 - * ) « i = 0,

< a0+Yl

(2.5.12)

l/9| = l

V|a| = l, where a\ = ap for \(3\ = 1, (a, a) = 1 for |a| = 1, and (0,a) = O, for |/?| = |a| = 1,

P±a.

Solving the equations (2.5.12), it shows that a\ — (l — 2~ P l ) , ao = -(l-2-pi)~1(sl + 2~ P l ), and again (2.5.7) is derived. In order to prove that the solution of (2.5.10) or (2.5.11) exists uniquely, we make use of the interpolation theory of multivariate homogeneous poly­ nomials. For this purpose, denote by /C/ := ICj(Ms) the set of the homoge­ neous polynomials of degree / with s variables. Obviously, the dimension of K\ is 1 i 1 . Given a set of points P a , |a| = /, the interpolation prob­ lem by homogeneous polynomials of degree / means to find a polynomial Hi{x) £ K\ satisfying the following interpolation condition: H,(Pa) = f(Pa),

V|a| = /.

(2.5.13)

S. L. Lee and G. M. Phillips discussed the existence, uniqueness and recurrence algorithm of the homogeneous interpolation polynomial. Define a hyperplane S$, £ G Ms satisfying the following linear equation s

(*,*) = 5 > x ' = °-

(2.5.14)

1= 1

Lem m a 5.1.V**] The homogeneous interpolation polynomials of degree is unique if and only if the set of interpolation points P =

II. Splitting Splitting extrapolation extrapolation II.

64 64

{Paiai \a\ \a\ == /} /} satisfies satisfies the the following following Geometric Geometric {P tions tion: Aii)) For For any any -point Pp ££ P, P, one one can can find find A point Pp does not not lie lie in in the the II superplanes superplanes but but the the others others does superplanes. superplanes.

methods methods

Characteristic (GC) (GC) condi­ condi­ Characteristic superplanes such such that that Pp Pp II superplanes lie in in at at least least one one of of these these II lie

Proof. If If Ai) Ai) is is satisfied, satisfied, then then for for any any P Paa G GP P there there exist exist superplanes superplanes Proof. Sa i ' (£<*>#) = 0, i = 1, •••,/, such that P does not lie in any SSatati> i> Saffi ' (£<*>#) = 0, i = 1, •••,/, such that Paa does not lie in any but Pp(f3 / a, |/3| = |a| = /) lies in one of S /s. One can construct the but Pp(f3 / a, |/3| = |a| = /) lies in one of Saa/s. One can construct the homogeneous polynomial polynomial of of degree degree // homogeneous /

s

v a

i

M « ) = *=i n (E*?^-)' i i= ' i=i «=i i = i and by the assumption Ai), L ^ 0. Now define a(Pa) and by the assumption Ai), La(Pa) i1 0. Now define La(x) = La(x)/La(Pa), V|a| = /. V|a| = /. Obviously, by Ai), La(x) = La(x)/La(Pa), Obviously, by Ai),

' 1,1, 0/3== a,a, Laa{Pp) (Pfi) == <{ L

0,

^2-5-15) (2.5.16) (2.5.16)

(2.5.17) (2.5.17)

P£a, p?a,

i.e., {{LLaa(x) /} constitute constitute the the interpolating interpolating basis basis functions functions of of the the i.e., ( x}) , |\a\ a | == /} homogeneous polynomial polynomial of of degree degree /,/, hence, hence, homogeneous Hi(x) == Y, Y, f(Pa)L f(Pa)Laa(x), (x), Hi(x) which completes completes the the proof. proof. which

(2.5.18) (2.5.18)

□ □

S. Lee Lee and and G. G. PP hh ii ll ll ii pp ss ^^ have have proved proved that that the the Type Type 11 grid grid refinement refinement S. satisfies the the GC GC condition, condition, and and the the solvability solvability of of system system (2.5.10) (2.5.10) is is therefore therefore satisfies guaranteed. guaranteed. Theorem 5.1. 5.1. For For Type Type 11 methods, methods, the the system system (2.5.10) (2.5.10) has has aa unique unique Theorem solution. solution. Proof. Since Since the the existence existence and and Proof. finding u/ and coefficients Ca finding u/ ++11 and coefficients Ca the equation equation the

uniqueness of of (2.5.10) (2.5.10) are are equivalent equivalent to to uniqueness ,, V|a| V|a| == // ++ 1, 1, which which satisfy satisfy uniquely uniquely

u U **(7p) ( £ ) = =« .W + ! ++ ££

M=i+i |a|=l+l

+ 11) )

^^ +

+1 a (^r dz r i *> . Z

(2-5.19) (2-5.19)

5. Fractional power expansion and successive elimination methods

65

where /? = 0 and \/3\ = I + 1. Let

Hw(x)

Ci/+1)^

= Yl

(2-5.20)

be the homogeneous polynomial of degree /, satisfying the interpolation condition # + i ( ( ^ ) P , + l ) = « « ( ^ ) - « i + i . V|/3| = / + l,

(2.5.21)

where uj+i is the undetermined splitting extrapolation value with (/ + 1) split. By (2.5.18), one can express ft (U*(-7)-UI+I)LP(X)

Ht+i(x)= J2

2

I0l=i+i

= £

U|(—)M*) -tl, + 1 £ 2

101=1+1

M*)«

(2-5'22)

101=1+1

Let x = ft, and tij(ft) = wj. By (2.5.19) and (2.5.22),

w/

ft

- Z , "<(—Wco 2

l/?l=J+i

= ti, + 1 (l- X) M*))-

<2-5-23)

l/?l=i+i Since Lp(h) is a homogeneous polynomial of degree / -+ 1, then

i^

5 ; Mft), 101=1+1

hence u/+i = w/ + i(ft) tii_

Z) 101=1+1

1-

£

u 2

i(~;)Lp(h)

L,(ft)

(2.5.24)

101=1+1 Therefore by the uniqueness of the interpolation polynomial one derives the uniqueness of the solution of (2.5.19). Now if we set ft = (1, • • •, 1) and ui(^)

= 6»,f>, ^

any i/,/J = 0 or |i/| = \/3\ = I + 1

66

II. Splitting extrapolation

methods

in the equation (2.5.19), then the extrapolation coefficients av = t*/+i, \/\v\ = Oor|i/| = / + 1 . □ T h e interpolation formula and recurrence algorithm for homogeneous polynomials are given in [53], and can be used to find the extrapolation coefficients, ao and av, V|i/| = / + 1 can also be obtained by solving directly the equations (2.5.10) and (2.5.11). In order to avoid rounding error, one may use the symbolic computation. T h e extrapolation coefficients of multi­ dimensional successive extrapolation are listed in Appendix 2 for reference. For Type 2 methods, it is not clear whether or not the interpolation points satisfy the G C condition. T h e solution of equation (2.5.11) may exist uniquely. Since the extrapolation coefficients of (/ + 1) split exist, one can successively eliminate the terms Ca (0 < | a | = / + 1 ) in the multivariate asymptotic expansion. T h e homogeneous recurrence algorithm can also be used for the case t h a t {pk} are fractional, while Algorithm 2.1, which is based on the multivariate Newton's interpolation formula, is not available. A l g o r i t h m 5 . 1 . ( R e c u r r e n c e a l g o r i t h m o f h o m o g e n e o u s split­ ting extrapolation) S t e p 1. h = (hlr-.,ha), /:=0, uo(h) := u(h). S t e p 2.

Evaluate in parallel ui(—),

V|i/| = / + 1.

Solve equation (2.5.10) to obtain ao and a„, \v\ = / + 1. Set ti/ + i(ft) = a0ui(h) +

^

avui(—).

M=i+i S t e p 3 . If / + 1 < m, set / := / + 1 and go to S t e p 2, otherwise o u t p u t the extrapolation value um(h). In the case of Type 2 grid refinement, Step 2 in Algorithm 5.1 is replaced by computing in parallel t / / ( -

) , and solving the equation (2.5.11) to

obtain the coefficients ao and a 7 , \y\ = / + 1 (see Appendix II). For asymptotic expansions with logarithm terms m

c &Pfc ° + d«hPka U(h) = u+Y^ 5Z ( « k=

* )

l\a\=k

+O(/i^+l(m+1)|ln/i0|5), where a = (c*i, • • •, as),

ln h

and 1,

for oti ^ 0,

0,

for oti = 0,

oli = \

(2.5.25)

methods 5. Fractional power expansion and successive elimination methods

67

l

ln*fc == lln* ln*fc n * l ffti...ln*'ft,. ti...ln*'ft,. In order order to to eliminate eliminate the the terms terms with with the the logarithm logarithm in in (2.5.25) (2.5.25) by by SEM, SEM, one In can make make use use of of the the expression expression (1.1.24), (1.1.24), i.e., i.e., applying applying the the extrapolation extrapolation co­ can in (2.5.10) (2.5.10) repeatedly, repeatedly, so so as as to to eliminate eliminate all all terms terms with with coefficients efficients in Caa and and ddaa (\a\ (\a\ = = k) k) at at the the same same time. C For example, example, u(h) u(h) has has the the asymptotic asymptotic expansion For 5

J2cihPi1

u(h) = u +

»=i »=i

s p1

(2.5.26) (2.5.26)

P

1 + ]^dTihdih* + O O(hl + (h2QP2 °lnh In h00),), i lnhIn i hi +

tt==ii

then then

PlPl pi P h +d +O(hl 'lnh0), Infto), +di(^) + YX/djh )W i +0(h*» i(^) \n^ In h- + j \nhj^

(2.5.27) (2.5.27)

where AW AW = = (Ai, (Ai, •• •• •,•, A,-_i, A,-_i, hi/2, A,-/2,AAf-f+-i,+i,••• ••,•,AA5),5), i t== 1,1, • •••,*. where • •, s. Summing up both sides sides of of (2.5.27) (2.5.27) with with respect respect to to i, it becomes both s5

s*

XX Xffcc(( 00 )) == ™ ™ ++ ] C£ c,<* (s(s -- 11 ++ 2-* 2 -1^) f c)i 1^ X tt ==li

;j ==1i s

s

s pi d 2 Pl +X + J2 d<*j i ( (s - - 11++22- p i ))fcf / i f InInhjfy++J^J^ di2i'Pl'

i=l

l nl n2 2/ l/ l

ff

i=l

(2.5.28) (2-5.28)

2 p

++O(h 0 ( C 20 l*n /Inl oft)0.).

By (2.5.7), By 5

Pl «, =- i T *^ l [EV T" J f^/ >) ^ -- ( («s -" !1 ++22~ «i - p i )WAYI «(^)],

(2.5.29) (2-5.29)

»»=1 = 1

by (2.5.26) (2.5.26) and and (2.5.28), (2.5.28), ui ui can can be be expressed expressed as by a

U! ? 11 ++ 00(/^ ( ^ PP22 lnfto), «! = = uu ++ ^5 c3i5h A? In Ao). i= l *=i

where where 1 1 ,l c, == (l-2-P 2-P'ln2rf,Ci (l-2-Pl)~)"12-i \n2di, . . •,« .,* ) t1== 1,1•, -

(2.5.30) (2.5.30)

68

II. Splitting extrapolation

methods

are independent of ft. Therefore the splitting extrapolation of the form (2.5.29) can be used once more: «i(A) = x ^ = p r [ I > ( A ( 0 ) - ( « - ! + 2 - p i ) 5 i W ] ,

(2.5.31)

1= 1

and we have Ul(h)

= u + O(h20P2 ln/io).

(2.5.32)

For general cases, suppose u,(ft) = u +

Y,

(c<*hPl+1<x + dahPl+ia In 5 ft)

|a|=/+l

+0(ftS,+2(/+2)|lnfton,

(2.5.33)

then it is easy to prove inductively that all the terms with coefficients ca and dQ (\a\ = / + 1) can be eliminated by the homogeneous splitting extrapolation.

Chapter III APPLICATION OF SEM TO MULTIDIMENSIONAL NUMERICAL INTEGRATION

Multidimensional numerical integration is one of the most important aspects in multidimensional problems. There are numerous papers related to this area published in the leading journals of numerical analysis over the world, and many monographs^']' ^^» ^^ dealing with multidimensional integration at present. P. J. Davis and P. Rabinowitz^^ divided multidi­ mensional integrals according to the dimension into three classes: (I) those with dimension 2-5; (II) those with dimension 6-12; and (III) those with dimension more than 12. For Class I, the product Gaussian rule may still be useful, and the ^-algorithm can accelerate the computation of some difficult integrals, but the dimensional effect becomes very strong for Classes II and III, even the product Gaussian rule may become invalid. The Monte-Carlo method and the number-theoretical method developed by N. M. Korobov, N. S. Bahvalov, L.G. Hua and Y. Wang! 47 ! remain effective, because these methods need only a simple summation, and the sample nodes are indepen­ dent of the dimension. The error of Monte Carlo method is probabilistic, and this is not a high accuracy method (with error

0(—T=),

where N is

the number of sample nodes). The number-theoretical method not only has the same advantage as the Monte Carlo method, but also is simple in programming and has a sharp error estimate. The number-theoretical method has a high order accuracy for smooth multivariate periodic func­ tions. Although any aperiodic function can be converted to the periodic one in principle, the complex structure of the converted functions restrict their application. Another weak point is that the Monte Carlo method and the number-theoretical method do not have algebraic precision, and their error will not decrease monotonically when the number of sample nodes increases. One of the new developments of number-theoretical method is the lattice rule approached by I. H. Sloan. 69

70

HI. Application

of SEM to multidimensional

numerical

integration

The algebraic methods try to use fewer points to achieve as high an alge­ braic precision as possible. A.H. Strod has constructed optimal quadrature formulae with an algebraic precision of degree five and seven. But there is no general optimal quadrature formula of higher degree. In Chapter I, we discussed the generalization of the Euler-Maclaurin for­ mula (1.2.52) to multidimensional integration, and by means of Romberg's extrapolation one can obtain higher order approximation with an algebraic precision of arbitrary degree. But the cost of Romberg's extrapolation will increase exponentially with the dimension, e.g., in order to obtain a so­ lution of an s- dimensional integral with an algebraic precision of degree 2m -f 1, function values at 2ms nodes should be computed by Romberg's extrapolation method, and at (ra + l ) 5 nodes by the product Gaussian rule. Thus, both methods should not be used for problems with dimensions higher than 5. The splitting extrapolation methods introduced in this sec­ tion not only have the same advantage as Romberg's extrapolation, but also need less amount of work, e.g., in order to get an algebraic precision of degree 2m + 1, at most ( ^ m ) function evaluations for Type 1, and ^2 2''■ ls + \ *=0

^

1

jforType 2 methods are needed. With such an amount

'

of work, the splitting extrapolation method has indisputable superiority for Class I problems, and is comparable with the number-theoretical method for problems in Class II and even Class III. 1. Splitting extrapolation algorithms for smooth functions Consider an s-dimensional integral 1= [ f(x)dx, Jv

(3.1.1)

V=[-l,l]'.

Divide V into n disjoint cuboids Vj with mesh width h = (h1} • • •, hs), i.e., hi

hi

)

*=1

where Mj = ( M ^ , • • •, Mja) is the center of Vj. Define the rectangular rule n

IR(h) = J2 meas(t$)/(My),

(3.1.2)

j=l

and the trapezoidal rule J

n

i

s

^) = E r E j=i

Zs

i=i

m

eas(^)(/(^) + / ( ^ r ) ) ,

(3.1.3)

1. Splitting extrapolation

algorithms for smooth functions

71 71

are where N* = {Mjxx, ■ • •, Mj. ± ~ i * * * > ^ i . } a r e centers of the two surfaces of Vj perpendicular to the x,-direction. We shall prove that the formulae (3.1.2) and (3.1.3) possess multivariate asymptotic expansion with respect to the mesh width h. 2m 2 T h e o r e m 1.1. If f e C 2 m + 2 ( F ) , then

I-IRR(h)=

c«/i22 " + 0 ( / i 22 mm + 22 ) ,

J2

(3.1.4)

l<|a|<m

where the the coefficients coefficients ccaa = = ccaii ...>at aii... >at where

are independent independent of of h. h. are

Proof. By By (3.1.2), (3.1.2), Proof.

(h) == J2f J2f 77 -- IIRR(h)

(/(*) "" f{Mj))dx. f{Mj))dx. (/(*)

(3.1.5) (3.1.5)

By the Taylor expansion,

f(x) - f(Mj) f(Mj) == f(x)

£ £

L^)(MjMj)(x )(x L^)(

l<|cr|<2m+l

a - Mj) Mj)a

aa-

-

m+2 +0(hlm+2),Vx£Vj.

(3.1.6)

Substituting (3.1.6) into (3.1.5), and noting that a {x-Mj)adx

/ 0, = < < == <

if a has odd components a,-,

na+2A) na+2/%)

^ »=i »=1

we immediately obtain we immediately obtain

I-In(h) I-In(h) -_£i = H<|«|<m ^ V<*h ' 7J ]« l E (l + ' 2a i )(^0(^.(3,,) i = H<|«|<m l**J" tJ= Jl ( l + 2 a { ) ^ t= l

(3.1.4) can b<s proved inductively. For k = 1, it is obvious. Suppose (3.1.4) can be proved inductively. For k = 1, it is obvious. Suppose expansion (3.1.4) holds for k < m. We shall prove that the asymptotic asymptot ic expansion the (3.1.4) holds for k < m. We shall prove that

that that it is it is

72

III. Application

of SEM to multidimensional

valid for k = m + 1. In fact, suppose / G C that

I-In(h)

„±

E

i=n<|«|<m+i

2m+4

numerical

integration

( F ) , it follows from (3.1.7)

^ » - W ) (|)- + oo.^) W-

JJ(i + 2aO «=i

=E

E

//<*■>(«)*—^

i=ii
-E

(V

( 2 a ) ! j J ( l + 2a.)

/ (/ (2a) w-/ {2a) w))^—J

E

i=ll<|a|<m+l" ,l 'i

(V

( 2 a ) ! j J ( l + 2«i)

+o(hlm+4)

=

2

//(«)*r

i
^

(V

( 2 a ) ! j J ( l + 2a,-) «=i

-E

E

/ (/^W-Z^CM,))^

i=ii
yi

Aa«

( 2 a ) ! f J ( l + 2ai) := 1

+O(h20m+4).

(3.1.8)

Now replace / ( * ) by / tion, the sum

(2or)

E/

(x), (1 < |a| < m-f 1), by the inductive assump­

(/^(zW^CM,-))^

has an asymptotic expansion. Therefore by (3.1.8) the asymptotic expan­ sion (3.1.4) holds for k = m + 1. D For the trapezoidal rule (3.1.3), there is an estimate similar to (3.1.4). Theorem 1.2. If f e C* 2m+a >(V), then there is an asymptotic expan­ sion I-IT(h)=

£ l<|a|<m

dah2a + O(h20m+2),

(3.1.9)

1. 1. Splitting Splitting extrapolation extrapolation

algorithms functions algorithms for for smooth smooth functions

73 73

where ddaa are the coefficients independent of h. h. Proof. By (3.1.2),

I-ITT(h) j = lJVS

ZS

izzl

(3.1.10) (3.1.10)

= Ji Ji + JJi, 2, with J

i =E /

(3.1.11) (3.1.11)

{f{*)-Wi))dx,

(/(Mi)- -i 7-E (wt)+W7)))*» 2*==EE/ / (/(^) £ (wt)+WT)))^

J

j=iJVJvj

j-l 3 n

meas (VM y* meas (V*)

*s s t = i

5* J

t=l

= -i =Ei —^ ~ ,=1 £ (/TO - 2W)+/TO) = - E - ^ r ^ E W ) - Wi)+/TO) 2

" meas(V;) meas(^)

'

™ ™ 11

^, , . .

AA; \ 22 *i

+2 +o(hlm+2)) +o(C 1 n a m 1 . 1 * 1 t hk) 2k k) = E E E — / D? f(x)dx(-) = - - E E E — / D? f(x)dx(-)2k n

L.

m

+ - E E E ^ / uT'/oo-aT/M))* s j=i ,. = i t = i ( 2 * ) ! * i=i 2 m+2 +O(h +O(h 00 )

= =

Jv Jv

i>

2

2

2k __ Il yV; E g J—_ [f D^ f(x)dXX(-) D^f(x)d (-)2k - i s m

1

n

-

2t ) ) +EE T^T EX E / ((^» /(.)-^" +7 E E^ ^/W - ^ //(M W )i))4 ) ^K") J)* 2 + o0((/ i^2m, m++22)) . Therefore by Theorem 1.1 the proof of (3.1.9) is complete.

(3.1.12) (3.1.12) D □

Corollary. 7/5 = = 3, 3, Men then ^ ( ^ 1 , ^ 2 , ^ 3 ) Aas an an algebraic algebraic precision precision of of degree three, and (3.1.13) i-h(h = 0{h%). xMM) I-lT(hi,h (3.1.13) 3,h3) = 0(ht).

74

III. Application

of SEM to multidimensional

numerical

integration

Proof. The comparison between (3.1.4) and (3.1.9) shows da = ca-lcai

V|a| = l,

hence for s = 3, da = 0, V|a| = 1 in (3.1.9).

(3.1.14) □

In the proofs of both theorems, only Taylor's expansion is used, and the proofs are simple. But the coefficients ca and da remain undetermined. One can also make use of the Lyness method to generalize the Euler-Maclaurin formula to the multidimensional case. This was shown in Theorem 2.6 of Chapter I. Now the coefficients of expansion (3.1.4) can be expressed explicitly: s

B2ai(-)

n—-

/ f^2a\x)dx, Jv

L\ (2*.-)!

V

1 < M < m,

(3.1.15)

/

where Bj(x) is the Bernoulli polynomial of degree j . Similarly, the explicit expression of da can also be obtained. Using the multivariate asymptotic expressions shown in Theorems 1.1 and 1.2, one can compute multidimensional integrals with the help of re­ finements of Types 1, 2 or 3. In the following, we give an estimation of the amount of work in finding the splitting extrapolation values by the three types of refinements. The amount of work for numerical integration depends mainly on the number of function evaluations. T h e o r e m 1.3. For Type 2 methods, at most ( ^ m ) function eval­ uations are needed in order to obtain an approximation with an algebraic precision of degree 2m -f 1. Proof. For the Type 2 refinement, the number of grid points with m splits is £

(1+ & ) . . . ( ! + &).

(3.1.16)

0<|/?|<m

We shall now prove

5 > + /?1)---(l + \p\=k

ft)=(2S+fc"1). v

J

(3.1.17)

75

1. Splitting extrapolation algorithms for smooth functions

In fact, set a,- = 1 + /?,-, i = 1, • •, s. Noting that oo

Jb = l

\a\=k

x'/(l-x)2',

=

(3.1.18)

5+

and comparing the coefficients of x * on both sides, (3.1.17) follows im­ mediately. By the following combinatorial identity

f:( 2s+ r i )=( 2 ^ m )' the proof is complete.

(3.1.19)

D

R e m a r k . In practice, the number of function evaluations is far less than that estimated by Theorem 1.3. This is due to the overlapped fine grid points and coarse grid points which were counted separately in Theorem 1.3. T h e o r e m 1.4.

m For the Type 1 methods, at most / ^ 2 *• ( (s

i=0

^

+

r')

function evaluations is needed in order to obtain a solution with an algebraic precision of degree 2m + 1. Proof. For Type 1 refinement, the number of grid points is obviously not more than m

m

m

/

\.

t=0

V

'

(3.1.20) t=0 |«|=»

t=0

|a|=i

and thus Theorem 1.4 holds.

□

For Type 3 methods, since Fibonacci numbers satisfy the asymptotic property: Fn^.\JFn ~ 1.618, the number of function evaluations is approx­ imately equal to \ J 1.618* I

j

1.

»=o Theorems 1.3 and 1.4 show that the SEM is very efficient. As an exam­ ple, take s = 10, m = 5, and the degree of algebraic precision is 11, at most 77505 function evaluations for Type 1 methods and 53130 function evalu­ ations for Type 2 methods will be computed. In contrast, 6 10 = 60466176 function evaluations will be needed for a product Gaussian rule, which is about one thousand times more!

76

III. Application

of SEM to multidimensional

numerical

integration

In the following some numerical experiments are given. E x a m p l e 1.1. Consider an s-tuple integral / • • • / exp(xi + ••• + x8)dxi--dxs = (e - l ) 5 . Jo Jo The comparison of SEM of Types 1 and 2 with the product Gaussian quadrature rule shows that the product Gaussian rule failed when the di­ mension s > 8, while both the SEM of types 1 and 2 work well and the relative errors are less than one percent for s = 20. Table 1.1. Comparison of SEM and product Gaussian quadrature rule DQAND Type 2 Type 1 dimen­ number of CPU(sec) sion S splits m rel. error CPU(sec) rel. error CPU(sec) rel. error 5 8 9

15 20

7

8.6E-13

335

5.0E-9

336

1E-8

0:42

8

1.9E-13

974

1.2E-8

656

1E-9

9 hours, fail

5

3.7E-7

618

3.7E-7

502

4.1E-8

11283

6

4056

9.4E-9

3092

4

9.7E-9 2.1E-5

176

1E-3

8 hours, fail

8^5E-7

1197

6

2.7E-8

1544 11391

2.1E-5 8.6E-7

144

5

2.7E-8

8915 same as above

3

2.9E-3

158

2.9E-3

146

1E-3

4

3.0E-4

4275

3.0E-4

3738

1

2

4.7E-2 8.5E-3

37

4.7E-2 8.5E-3

37

1E-3

627

594

1.2E-3 1.6E-2

33756 1659

1.6E-2

1568

3 4 24

3

same as above

These results are obtained on a VAX-750 computer. The subroutine DQAND is called from the IMSL library based on the Gaussian quadrature rule. Table 1.1 shows that DQAND is valid for s < 5 and the accuracy is limited, e.g., for s = 5, it is invalid if the tolerance is set to be IE — 9, while the Type 1 methods work even when the tolerance is set to be IE — 13. The Type 2 methods have worse stability for m large, e.g., when s = 8, the result of m = 8 is not as good as that of m = 7. But for the cases m < 5, the accuracy of Type 2 methods is comparable to that of Type 1 methods and needs less CPU time. E x a m p l e 1.2. Consider the triple integral l

/ / / Jo Jo Jo

(1 + ^xlX2X3 -f x\x22xl)eXlX^dx1dx2dx3

= e - 1.

1. Splitting

extrapolation

algorithms

for smooth

77

functions

This example is quoted from the monograph^4*! by L.G. Hua and Y. Wang. For the results evaluated by the number-theoretical method and the split­ ting extrapolation methods of Types 1 and 2, see Table 1.2.

T a b l e 1.2 C o m p a r i s o n o f n u m e r i c a l r e s u l t s b y SEM and number-theoretical method SEM n u m b e r of

Type 1

Number-Theoretical M e t h o d Type 2

point

error

CPU

splits

error

CPU(sec)

error

4

8.5E-5

0.46

8.6E-5

0.35

3237

9.5E-4

5

5.7E-6

1.24

6.3E-6

0.83

8197

9.2E-4

2.45

6

1.9E-7

3.16

3.4E-7

1.95

39027

6.2E-5

11.56

CPU(sec) number

(sec) 0.97

7

6.2E-9

7.5

2.2E-8

4.26

57901

3.2E-4

17.0

8

1.4E-10

16.7

3.2E-9

8.68

82001

6.7E-5

24.1

9

1.7E-12

35.1

1.7E-9

16.66

140052 6.9E-5

40.7

10

6.7E-14

70.1

9.5E-9

30.32

314694 6.3E-5

91.8

Table 1.2 shows that SEM converges very fast. Type 1 methods need greater amount of work but have better stability than that of Type 2 meth­ ods. The accumulated error of Type 2 methods increase when the number of split is higher than 9, while the error of Type 1 methods decrease mono­ tonically. The errors in the number-theoretical method do not decrease monotonically as the number of nodes increases. However, the number of nodes is independent of dimension, this leads to prominent superiority when the dimension s is large, e.g., s > 10.

E x a m p l e 1.3. Consider the following integral of a rational function

m

l

.

.

dx\dx2xz

. (x? + 0.1)(x^ + 0.1)(x§ + 0.1)

= 89.39853274047529.

The results of Romberg's extrapolation and the splitting extrapolation are listed in Table 1.3.

III. Application

78

of SEM to multidimensional

numerical

integration

Table 1.3 Comparison between numerical results of Romberg's extrapolation and the splitting extrapolations number of splits

Romberg error CPU(sec)

Type 2 Type 1 error CPU(sec) error CPU(sec) 0.01 1.23 0.01 1.23

Type 3 error CPU(sec) 1.23

0.02

1

1.31

0.02

2

7.7E-2

0.10

3.0E-2

0.03

9.1E-2

0.02

9.1E-2

0.04

3

5.8E-4

0.76

8.7E-3

0.09

1.0E-2

0.05

1.0E-2

0.09

4

7.9E-6

5.96

6.7E-4

0.27

2.5E-3

0.16

1.8E-3

0.20

5

4.7E-8

47.24

1.3E-6

0.67

1.4E-4

0.39

4.5E-5

0.43

6

4.7E-11

318.71

1.82

1.9E-5

0.77

6.9E-6

0.94

7

2.3E-6 1.4E-7

4.78

4.2E-6

1.50

7.4E-7

2.09

8

3.9E-9

12.05

2.1E-7

2.64

1.8E-8

4.39

9

3.5E-11

29.31

3.2E-8

4.59

1.4E-9

8.82

10

1.8E-11

70.08

6.9E-9

7.66

9.9E-11

17.40

Table 1.3 shows that the classical Romberg's extrapolation converges faster than SEM, but the CPU increases rapidly with the number of extrap­ olation. The convergence rate of SEM is not as good as that of Romberg's extrapolation, but the CPU increases slowly with the number of splits, and the CPU time is far less than that of Romberg's extrapolation for a given accuracy. In the comparison among the three types of splitting extrapola­ tions, Type 1 converges fastest and spends more CPU, Type 2 converges slowest and uses least CPU, while Type 3 is in between and is worth to be recommended. The above comparison is merely for a triple integral. For higher-dimensional integrals, the number of extrapolation of Romberg's method would be further limited. In the following, we give an example of an eight-dimensional integral, after performing Romberg's extrapolation twice, the CPU has over 1800 seconds and no result is obtained. E x a m p l e 1.4. Consider the following eight-dimensional integral / (4 - X!) 4 *$(l.l - x 3 ) 6 (5 - x 4 ) 3 (2 - x6fx5(9 Jv V = [0,1]*.

-

x8)7dx,

For the numerical results of Romberg's extrapolation and Types 1 and 2 SEM, see Table 1.4. Obviously, for higher dimensional integrals, the performance of Rom­ berg's extrapolation is very bad. On the other hand, when the number of splits is greater than 5, the convergence rate of SEM also slows down because of the accumulated error.

2. Change of variables of multidimensional

improper

integrals

79

Table 1.4 Comparison between numerical results of Romberg's extrapolation and splitting extrapolation for an eight-dimensional integral number Romberg Type 2 Type 1 of splits error CPU(sec) error CPU(sec) error CPU(sec) 12.78

1.02

36.70

0.17

36.70

>1800

3.34

1.55

3.35

1.37

3

0.14

10.62

0.14

8.83

4

2.9E-3

60.84

3.0E-3

45.73

7.6E-5 4.9E-5

310.17

8.1E-5 4.9E-5

210.91

1 2

5 6

1462.49

0.18

898.9

2. Change of variables of multidimensional improper integrals The singularities of multidimensional improper integrals behave so complicatedly that one has to treat different cases in different ways. In this section we try to eliminate the singularities and obtain a proper integral through transformation, and then compute it by the splitting extrapolation. For example, consider a multidimensional improper integral J1=

/ .../

Jo

l/

3*

«...z'/''-f(zu---,x,)dx1...dzl,

Jo

(3.2. 1)

where - 1 < pi/qi < 1, (i = 1, ••-,«), and pi, tfi,P2, 52, • • • ,P«, Qs are all integers. Without loss of generality, assume that the denominators qi > 0, (i — 1, • • •, s), and j{x\, • • •, xs) is a smooth function. By change of variables: X{ = yjl, (i = 1, • • •, s) t " 1+tl 1

fi

Jo

Jo 1 1

- ---ti'+ '- f{vll,---,lf)dvi---dyt.

(3.2.2)

Since —1 < Pi/qi < 1, then pi -f qi — 1 are nonnegative integers, and (3.2.2) is a proper integral with a smooth integrand and hence can be computed by SEM. It is well known that an improper integral of the form

h=

irL

-dxi■■-dx,

II(1-*. )

cosy,-, « = 1,

, s, into the

cost/x, •••, cos y9)dy1 •• dys

(3.2.4)

can be converted, by the transformation x* proper integral Jo

Jo

(3.2.3)

? 1/2

80

HI. Application

of SEM to multidimensional

numerical

integration

Similarly, 1 "

/

■

■

Jo

■

fjr. ... X I

.

/

Jo

' 1

'——dxx-dx,,

ni-'C -**)]

(3.2.5)

178

1= 1

can be converted by the transformation x,- = sin 2 t/,-, i = 1, • • •, s, into the proper integral h = 2° f •••/ /(sin 2 yu • • •,sin 2 ys)dVl Jo JO When the singular points of the improper integral Is = J

^dys.

. . . y f(xlr..iz,)dx1-..dx8

(3.2.6)

(3.2.7)

lie on the boundary, and the value of the integral over the neighborhoods of the singular points contributes a very small portion to the whole in­ tegral, the integral can be computed directly by the formula (3.1.2), and accelerated by the splitting extrapolation. In this case, in order to further weaken the influence of singularity, (3.2.7) can be converted, by the cosine transform xt- = cos y,-, i = 1, • •, 5, into h=

/ • • • / /(cos2/i,---,cos2/ 5 )TTsint/ l d?/i Jo Jo £_i

-dys.

(3.2.8)

5

Here, the factor TTsint/,- can reduce the singularity of the function *=i

/(cosj/i, •••,cosy,). For the case that more complicated singularities occur at end points, or the case that the integration interval is semi-infinite, one can generalize the following Double Exponential (DE) transform, proposed by H. Takahasi and M. Mori, to multidimensional integrals. For the one-dimensional integral f f(x)dx, (3.2.9) Ja construct a transform: x = (oo), then J=

oo

/

/(^(t))^(t)dt,

(3.2.10)

-oo


(3.2.11)

2. Change

of variables

of multidimensional

improper

integrals

81

The property (3.2.11) allows us to omit the part of integral over the interval far from the origin. Usually, we approximate J by

j=

r f(
»/_

f(
(3.2.12)

and the corresponding trapezoidal rule is N J N)

h

=

h

Yl

f(

(3-2.13)

i=-N

with h = 4/N. When t tends to infinity, all the derivatives of the integrand tend to zero as fast as a double exponential function does. This ensures that the trapezoidal rule has a better accuracy. Merit 94 ! proved that J — Jl ~

exp(-CN/\ogN). There are many double exponential transforms, such as x = y>i(i) 1

A"

= - [ t a n h ( - sinh(t)) + l] : (0,1) <- (-co, oo), ^ 2 7T

x —
/

f(x)dx,

(3.2.14)

-oo

if f(x) is a function which converges very slowly to zero as x —► ±oo, it can also be converted to a function, which varies to zero at a double exponential speed as x —► ±oo by the double exponential transform x = sinh(Jsinh(*)).

(3.2.15)

For details see [94]. The double exponential transform can be used in multidimensional sin­ gular integrals as well, e.g., for 75 =

/

'"/

/(Xl»"'"'x«)dxi'""rfx»'

(3.2.16)

III. Application

82

1

let Xi = MU) = o [

tanh

of SEM to multidimensional

IT sinh(

(2

numerical

integration

-TTT T 7 * Ccosh(£,)

^) + ^ A(U)

=

1

cosh2(-7T sinh(t,-))

i = 1, • • •, s. One can consider r4

/5« /

• /

r4

f(Mh)r''^s(ts))t[
(3.2.17)

which can be computed, for example, by the trapezoidal rule. When the double exponential transform is applied to multidimensional integrals, in order to prevent the computer from overflowing or underflowing, it is neces­ sary to choose appropriate ai,a2 and as in (3.2.11). In fact, our experience shows that this method is not suitable for high dimensional cases, but is an effective method for lower-dimensional cases. E x a m p l e 1.5. Consider a double integral -dxdy. Jo Jo

y/xy

Firstly we convert it into the form (3.2.17) by a double exponential trans­ form, then evaluate by the trapezoidal rule with splitting extrapolation. The results are listed in Table 1.5. T a b l e 1.5 N u m e r i c a l r e s u l t s o f a p p l y i n g S E M and double exponential transform in evaluating a singular integral number

Type 1

Type 2

Type 3

of splits

error

CPU(sec)

error

CPU(sec)

error

CPU(sec)

1

7.40E-1

0.17

7.40E-1

0.18

7.40E-1

0.14

2

2.60E-1

0.54

2.59E-1

0.49

2.59E-1

0.47

3

2.20E-1

1.52

2.24E-1

1.11

2.31E-1

1.15

4

3.23E-2

4.11

3.51E-2

2.21

3.52E-1

2.54

5

1.28E-3

10.26

3.11E-2

4.07

7.34E-4

5.24

6

6.18E-5

24.33

1.98E-3

6.76

7.71E-4

10.61

7

4.41E-6

57.58

2.51E-4

10.55

6.77E-5

20.51

8

7.24E-9

131.75

1.68E-5

15.90

1.40E-6

38.24

The table shows that the SEM acceleration is almost invalid for the small number of splits, but performs better where the number of splits becomes larger. The reason is that under the double exponential trans­ form, the terms with lower order powers in the asymptotic expansion of the trapezoidal rule are close to zero.

3. Duffy transform for multidimensional

improper integrals

83

Remark. M. Beckers and A. Haegemans compared numerical values of various transformations and found that the TMT transform is better than the DE transform for multiple integral (cf. [29], pp. 329-340). 3. Duffy transform for multidimensional improper integrals Consider the multidimensional improper integral Hf) = I • • • / f(xu...,xs)dxx...dx„ Jo

(3.3.1)

Jo

with integrand / ( * i , • • •, xs) = gfa, • •, xs)h(xly g(xu • '• •, xs) = x*1, • • •, x*'gp(xu

• • •, x,), • • •, x 5 ),

(3.3.2) (3.3.3)

where h(x\, • • •, xs) is a smooth function on [0, l] 5 , <7/i(xi, • • •, xs) is a ho­ mogeneous function of degree /i. Suppose A,- > - 1 , (i = 1, • • • , * ) , and /i + ^ A , - > - s . As an example, one can consider
2

5

, or gn(xir-,xs)

= (^Pa.xf)",

with

h

«i > 0,/? > 0 and a/3 = /i.

*=i

Many practical problems can be brought into the framework of improper integral of the type (3.3.1)-(3.3.3). For this type of improper integrals, Duffy^'! proposed the so-called Duffy transform which can eliminate the singularity at the origin caused by the homogeneous function #(xi, • • •, xs) of the form (3.3.3). Further, by some suitable transforms, it is often con­ verted into the proper integral which can be computed by the product-type Gauss-Jacobi formula. The Duffy transform considers integrals over an s-dimensional pyramid Vi = { ( s i , - - - , * , ) : 0 < * i < l , x i > X i , V i ^ l } ,

h(f)=

I I*1---!'1 Jo Jo

f{xlr..,x9)d*i~'d**

Jo

Jo Jo Jo •h(xi, - - • ,xs)dxi

• • dxs.

(3.3.4)

84

HI. Application

of SEM to multidimensional

numerical

integration

Taking the transform: x\ = 2/1, x 2 = t/it/2? * • •, z* = 2/i2/*> we have h{f)

= / • • • / 2/i 1 (yiy2) A 2 ---(2/i2/5) A '^(2/i,yi2/2, Jo Jo h(yliy1y2l • • •,yiVsWi' 1 dyi • • • dy,

= I

-,2/iy*)

M2-~v}'gr(i,V2,---,y*)

"1

Jo Jo •^(yi, 2/12/2, • • •, 2/12/5)^2/1 "dy5,

(3.3.5) s a

where 77 = Ai H h A, + /1 + s - 1, ^ ( 1 , 2 / 2 , ••, 2/«) * regular function, and h(yi, 2/12/2, • • •, 2/i2/«) is a smooth function of the variables 2/1,2/2, * • •, 2/«It follows from (3.3.4) t h a t 77 > — 1 , A,- > — 1 for 1 = 2, • • •, s, and hence the integral (3.3.5) is meaningful. If A, = Pi/qi, i = 1, • • •, s are fractions, then by a transform tA; = z\{, i = 1, • • •, s, the integral can be converted into a proper integral of type (3.2.3). If the integration domain is V — [0, l ] 5 , then V has the decomposition (3.3.6)

V=(jVi, »= 1

where V% is the closure of V*, and Vi is an s-dimensional pyramid Vi = { ( x i , . . . , x , ) :0<Xi

<

l,xi>xj)Vj^i}1

i = l , ••-,*.

(3.3.7)

It is easy to verify t h a t Vi fl Vj = 0, for i ^ j . In fact, if x £ Vi D Vj, then x £ Vi implies x,- > Xj. Again x £ Vj implies Xj > x,-, and this is a contradiction. On the other hand, Vi C V, and the volume of every Vi satisfies measV; = - , which leads to t h a t {Vi} is a decomposition of V, and (3.3.6) holds. Therefore, the integral (3.3.1) can be described as

1(f) =

/ • • • /

Jo

/(xii-

",x8)dxi-"dx8

Jo

= f f(x)dx Jv

,■=1

= /

- ^

[ ■■[ F(xi,-;x,)dxv-dx„

Jo Jo

Jo

(3.3.8)

4. Multivariate

asymptotic

expansion

of multidimensional

improper

integrals

85

with s

F{xu - . . , * , ) = £ / , ( * ! , . . . , *,),

(3.3.9)

and • -,x 5 ) = /(x,-,x t + 1 , • • • , x 5 , x i , - - - , x t _ i ) ,

fi(xlr

i = l, •••,*.

(3.3.10)

Since the singularity of F(x\, • • •, xs) comes from the function f(x\, • • •, xs) when the Duffy transform is applied to (3.3.8), the singularity can be soft­ ened, or even eliminated. If g(x\) • • •, xs) in (3.3.3) has logarithmic structure: ra lnr, then Duffy transform remains effective. The singularity is transferred to the form 1 /9

Xj*(l + x\ + • • • + xj) l n x i . In this case, (3.3.1) cannot be converted into a proper integral through transforms, but using the product form of the Euler-Maclaurin expansion (1.2.45) with singularity at end points, the splitting extrapolation method described in (2.5.25) still works. 4. M u l t i v a r i a t e a s y m p t o t i c expansion of multidimensional im­ p r o p e r integrals Consider the improper integral

Q(/)= f ■■■[ n(*rinA-xo^(«i.---.«.)d«i•••<'«.. Jo

Jo

i=1

- 1 < m < 1, A, = 0 or 1.

(3.4.1)

This type of integral not only occurs in practice, but is also obtained from applying the Duffy transform to the integral of the form (3.3.1), where H(xlr--,xs) e C 2 m + 1 ([0, l] 5 ). If m is an irrational number or A,- ^ 0, one cannot convert (3.4.1) into a proper integral, but can apply splitting extrapolation methods by means of multivariate Euler-Maclaurin expansion Qh(f)

where

Ni-l

JV.-l

»1=0

t,=0

hi = l/Ni,

0<9i < 1, i = l, • • . , « .

86

III. Application

of SEM to multidimensional

iintegration

numerical

If Oi = - , this is the rectangular rule. In (1.2.38) and (1.2.45), the onedimensional asymptotic expansion with singularities at end points is given. For the general case 0 < Q\ < 1, Navot (cf. [25]) proposed the following L e m m a 4 . 1 . If G(x) - xpg(x)

with - 1 < p < 0, g(x) € C 2 m [0,1],

and h = —, then the following asymptotic expansion holds [1

E(h)=

=

n-l

G(x)dx-h^2G((i

+ e1)h)

+P+I _ ^ 1 W ) G 0 . 1 ) ( 1 ) f c f .-E'""'?'"."')')"^ j=l

;=0

i=l i! m ++ 00 (( // ii 22 m )) ,, 3'

i=0

-7*

^*

(3.4.3) (3.4.3)

where (.(t^e) is the generalized Riemann zeta function defined as oo OO

C(M) = £ > + *)"'•

(3.4.4) (3.4.4)

Re * > 1,

fc=0 k=0

anrf (3.4.5)

C(-i,e) = - 5 i + 1 ( £ ) / ( i + 1), j = 0 , 1 , - . . , where Bj(x) are the Bernoulli polynomials. L e m m a 4.2. Under the assumptions of Lemma Ji.l} g(x)xp In x, — 1 < p < 0, then 2m-l

G{x) =

2m-l

Ajhj +lnh^2

£(/*) = £

if

Bjhj+P+1 j=0

j=l 2m-l

+ 5 3 C'i&i+|,+1+0(ft2m)>

-Kp<0.

(3.4.6)

£j^'+1+0(/i2m),

(3.4.7)

/ / G(x) = p ( x ) l n x , Men 2m-l

£(A) = ^

2m-l

AjW+logfc Yl

where Aj,Bj and Cj are constants independent of h, and depend on the derivative of g(x) at the end points.

4. Multivariate

asymptotic

expansion of multidimensional

improper integrals

87

Theorem 4.1. Suppose that Q(f) is an integral defined in (3.4.1), Qh(f) is the quadrature rule defined in (3.4.2), and H(xi, • • • , x 5 ) G C 2 m + 1 ([0, l] 5 ), then there exist constants Aa)Ba and Ca which are inde­ pendent of h = (fti, • • • }hs), and depend on H(x\y • • • , x 5 ) and its deriva­ tives such that the following multivariate asymptotic expansion holds

Q(f)-Qh(f)=

A ha

X

°

+

E

Ca^

+/i+1

l<|a|<2m

+

0<|a|<2m

Yl

5a/»a+"+i(in/lr

+ 0 ( ^ m + 1 ( l n / i ) A ) , (3.4.8)

0<\a\<2m

here (ln/i) A = (ln/ii) A l • • (ln/i 5 ) A », A = (Ai, • • •, A,),/i = (/ii, • - , / / , ) , ot = (ai, • • •, a,), and fc«+"+i = ft«i+^+1 . . . ft«.+<*.+i. Proof. We use induction with respect to the dimension. For s = 1, (3.4.8) follows from Lemmas 4.1 and 4.2. Now suppose that for (s — 1)dimensional case the integral asymptotic expansion (3.4.8) holds, we shall prove that it also holds for the s-dimensional case. For simplicity, let x = (x 2 , • • •, x 5 ), A = (A2, • • •, XS),~P= (/*2, • • • ,/**), dx = dx2 - - - dxs, and consider F(x1)=

I • • • / x^{\nx)XH{xux)dx. Jo

(3.4.9)

Jo

This is an (s — l)-dimensional integral with parametric variable x\. By the inductive assumption, there exist functions AQT(XI), B-a(x\) and C-a{x\), which are independent of h = (/12, • • •, hs), but depend on xi, such that F(Xl) N3-l

N.-l

5

» 2 =0

:,=0 j = 2

=h2...h, Y: ••• E n{[(ii+^)^]"i[in(^+^)^)] 0 •^(*i,(»'a + tfa)A2,---,(t. + ^ ) M

+

E

^ * i ) ^+

l<|«|<2m

+

E

E

^(xx^^'Onft) 1

0<|cF|<2m

^o(*i)A5+5r+1+0(^m+1(lnA))X.

(3.4.10)

0<|a|<2m

By the asymptotic expansion of one-dimensional integral and (3.4.10), Q(f) 1

Al

x5' (lnx 1 ) ^(x 1 )rfxi / Jo

88 88

III. Application Application of of SEM SEM to to multidimensional multidimensional III. N2-l

N.-l

s

numerical numerical

integration integration

A

l

= h2...h. ..-h, £ £ . .. ... ££ n{ife+«i)*ir[»«(( >[ln((ij+ej)hj)]!Ai'} =h } 2 t =0 ».=0 j = 2 t 22 =0

».=0 j = 2

•• // Jo Jo

x^(lnx1)1Xl )XlH(x H{x1,(i +e2)h2,---,(i, 1,(i z?(l*z + 02)h2r..,(i9 22

+ +

£ J2

IT h°

ll<|«l<2m <|o|<2m

+

Xl ff11x^(lnx )dx11 x^(\nx11))XlA^{x Ao{x11)dx

J

°°

J

^+Jr+10nA)X

J2 0<|o|<2m |«l<2m

+ +

/'x^(\nx1)XlB^{x1)dx1

■'O "/o +1I+1

Xl f1x^(lnx 1) C^{x1)dx1 fx^ilnx^C^xJdx!

h"+1I+1 T

££

+ 009a)h,)dx )hs)dx11 +

0<|a|<2m ^° 0< 2 m+1 A +O{h2 , m0 + 1 (ln/l) (\nhf) +0(A ) m+1 2 m+1 J X ), = /
where Ki Ki ,K ,K22,Kz,K± ,Kz,K± where of (3.4.11). (3.4.11). Since Since of

(3.4.11) (3.4.11)

stand for for the the four four summations summations on on the the right-hand right-hand side side stand

Xl Xl (\nXl H(xltlt Xl)) H(x // ** jj "" (\n Jo Jo

(ta + + 0022)h )h22,, •■■-,((,+ (ta •,(*, +

)htt)dxt )dxt eett)h

JV,-I JV,-I

= &i E

[(''i + ^i)/«i]'"[Jn((n + ^ 1 )/, 1 )] Al

*i=0

•//((«! + «!)&!, •••,(«, + 0.)h.) 2m 2m

2m 2m +/il+1 A1 +E E 66 ii (( (( ii 22 + + ^)/i2,--,(i, ^)/i2,--,(i, + + ^0MM ^0MM+/il+1 (lnAi) + (lnAi)Al

i=o i=o 2m 2m

1+1 E Cc>> ((*'» *.)ft.)M ++""1+1 +E (('2 ++ '»)*»• ^ 2 , •••,(*,+ •••,(!•.+ ff.)*.)M 7=0 J'=0

+0(fc*mm++11 lnA liiA11), ), +0(ftg

(3.4.12) (3.4.12)

where aj-(x a,(x 22,, •• •• ••,,xxa5),bj{x ), 6j(x • • •, xs) and • • •,>*») xs) are where andCj(x c ; (x2,2 ,-are coefficients coefficients of of the the 2,2> ■ ■• ,x,) expansion (3.4.6) (3.4.6) with with parameters parameters xx22,, ■ • •■•,■,x sx. sLet expansion . Let Q(g) = JJo ■■[ 9(x2,--,xa)dx2-dx, Jo

(3.4.13) (3.4.13)

4. Multivariate

asymptotic

expansion of multidimensional

improper integrals

89

be an (s — l)-tuple integral, and N2-l

AT.-l

Qh(g) = h2-.-hs 53

• 53 g((i2 +

» 2 =0

02)h2r~,(i.+0.)ha)

»t=0

(3.4.14) be the corresponding quadrature rule. Let , *,) = J J [XV(InXj)x'].

w(x) = w(x2,

(3.4.15)

i=2

It follows from (3.4.12) that in the expansion (3.4.11) 2m

2m

Ki = Qh(f) + ^ZQh(waj)h{

+

j=l

^Q^wb^hi+^ilnh^ j=0

2m

+E^(^)M + M l + 1 2m

2m

= Qh(f) + 53Q(^)/i{+53Q(^)M + / i i + 1 an/ii) A i 2m

2m

+ li+1

+ £ o(^i)M ' i=o 2m

+ £ {^(^i) - «(^i)}M j=i

+ £ {Q»M>;) - Q(^;)}ft{+/il+1(in fti)^ 2m

+ 53{Q,(^c i )-Q( U ;c i )}/i{ + ^ +1 .

(3.4.16)

i=o Applying the inductive assumption to the terms with braces, and noting that ahbj and Cj G C 2 m _ i ( [ 0 , l] 5 " 1 ), (3.4.16) becomes

Ki = Qh(f) +

53

*ahx

l<|a|<2m

+

53

^a^a+/i+1(ln/i)A+

0<jor|<2m

+0(ft^ m + 1 (lnft) A ),

53

5«^+Ai+10n/i)A

0<|cr|<2m

(3.4.17)

where AajBa and C a are constants independent of ft. Combining (3.4.17) with the terms K2lKs and K±y the proof of (3.4.8) is complete. □

90

HI. Application

of SEM to multidimensional

numerical

integration

Corollary 1. J/0, = - , i = 1, • • •, s, then the asymptotic expansion is

Q(f)-Qh(f)=

A2ah2°+

£

J2 Baha+»+\\nh)x

l<ja|<2m

+

0
Ca/i

]T

a+

^ + 1 - f O ( / i ' m + 1 ( l n / i ) A ) . (3.4.18)

0<|a|<2m

Corollary 2. If the integrand

= fl K't

1

- ^rOn^)^(ln(l -

Xj))^}H(xu

• • •, * , ) , (3.4.19)

where \j,r)j take value 0 or 1, AajBajCa and Da such that Q(f)-Qk(f)=

HJ^UJJ

£

> —1, then there exist constants

Aah"+^

0<\a\<2m

+

Yl

Baha+^l+

Y,

0<|a|<2m

+

Y

Caha^+1(lnh)x

0<|a|<2m Q

Dah +»+\\nhy

+ 0(hlm+1(\rih)x+ri).

(3.4.20)

0<|or|<2m

5. Integration on multidimensional simplexes The multidimensional simplex is the simplest one among various multi­ dimensional polyhedrons. Many complicated domains can be divided into simplexes. In order to apply the SEM, we suggest a method, similar to the Duffy transform mentioned above, to map the simplex onto a cube. Such a mapping has two advantages: first, a multidimensional simplex is mapped onto a multidimensional cube, and remains the smoothness of the integrand, which facilitates the use of the SEM; second, if the integrand has a singularity at the origin or on the surface of simplex, then the mapping can "soften" or even eliminate the singularity. Consider an integral I = / f(zir-,2s)dxi JT

= / / Jo Jo

••/ Jo

"dx3 / ( * i , - - - , * f ) d * i - -dx,,

(3.5.1)

5. Integration on multidimensional

simplexes

91

where T = {On, • • •, xs) : 0 < X! < 1, xx > x 2 > • • • > xs}

(3.5.2)

is a simplex in Ms. Applying the integral transform *i = i/i, xi = xi_1yi,

i = 2, ••-,*,

(3.5.3a)

xi = 2/i, Xi = yi---y,-, i = 2, •••,*,

(3.5.3b)

or equivalently

it is easy to prove that the transform (3.5.3a) maps T onto [0, l] 5 , and the Jacobian of the transform is J = ys1ys2-1--ys-i.

(3.5.4)

Hence,

= J f(x)dx vly^1-'y*-if(yi>yiy2r-,yi'-ys)dyi'-'dys.

= / • • / Jo

Jo

(3.5.5) Obviously, if / ( x i , • • •, x 5 ) is a smooth function on T, then the new inte­ grand is smooth on [0, l] 5 . If / is a singular function of the form /(xi,---,x5) = x f • • • x?'gi(xi, • • •, x,)/i(xi, • •, x 5 ),

(3.5.6)

where - 1 ,

/it- + s + 1 - i > - 1 (i = 2, • • •, s),

(3.5.7)

Men Me following improper integral on an s-dimensional simplex T I=[

x^gi(x)h(x)dx,

(3.5.8)

JT

converges, where x** = x^1 • • -x%9, and

Vo ./o -9i(l, *2, z 2 x 3 , • • •, x 2 • • • x5)/i(xi, x x x 2 , • • •, xi •..x5)dx. (3.5.9)

HI. Application Application of of SEM SEM to to multidimensional multidimensional HI.

92 92

numerical numerical

integration integration

Theorem 5.1 5.1 shows shows that that for for an an improper improper integral integral of of the the form form (3.5.8) (3.5.8) Theorem denned on a simplex, if /ii, • • •,//, and / are fractional numbers, then this denned on a simplex, if /ii, • • •,//, and / are fractional numbers, then this integral is of type (3.2.1), and can be converted into a proper integral of integral is of type (3.2.1), and can be converted into a proper integral of type (3.2.2) and computed by SEM. If/ has a logarithmic singularity, e.g., type (3.2.2) and computed by SEM. If/ has a logarithmic singularity, e.g., //(a?i, ( a ? i , •• •• •• ,, ** ,, ))

p/(xi • • • i xs)Kxli.. •, xB), (3.5.10) 3 5 10 == nn [*ron*o [*r(inx0AiAi]]#(*!» ' • • •' *• w*i. • • • * *•)■ ( - - ) s

»=1 »=i

with At- = 1 or 0, then through the transform (3.5.1), with At- = 1 or 0, then through the transform (3.5.1),

I = / f(x)dx

I = /

J J°

°

f(x)dx J J°

j=2 ° j=2 jj == ll (3.5.11) •^/(Ij xX2, " ' ' , *2 ' ' ' Xs)h(xi,XiX • • Xs)dx.(3.5.11) •0/(1, • •'■' ,•*, !Xi • • •• x,)dx. 2 , • • •, x 2 • • • xs)h(x llx1x2, 2j

Thus, under under the the assumptions assumptions of of Theorem Theorem 5.1, 5.1, the the improper improper integral integral of of aa Thus, singular function defined in (3.5.10) on the simplex T converges. Applying singular function defined in (3.5.10) on the simplex T converges. Applying the quadrature quadrature formula, formula, the the error error expansions expansions of of integrals integrals of of this this type type are are the described by by (3.4.8) (3.4.8) and and (3.4.15). (3.4.15). One One can can obtain obtain the the approximation approximation by by described the successive successive recurrence recurrence algorithm algorithm 1.2 1.2 of of SEM. SEM. the Consider the the following following multiple multiple integral integral over over aa simplex simplex Consider

h = = /f f(*)d* f(x)dx

M where where

== / / Jo Jo Jo JO

••••/ ••/ Jo Jo

(3.5.12) f(xi r"i1x..8)dxi'"dx / ( * i , - ",x dxS)8 , (3.5.12) s)dx

T11 == J| (( xx 11 ,, .. .. .. ,, xx O O :: ^^ xx ,, << ll ,, xx ll >> 00 ,, ii == l ,l ., ,. 5, 5l .l . T

(3.5.13) (3.5.13)

Obviously, the the transform transform Obviously, tt -- ii

X\ = 2/i, 2/1, s.Xi = y .i -((ll - ]^^3Xj j) ), ,(( ** = 2,---,s), 2,-.-,s), *i i=i i=i maps Ti Ti onto onto [0,1]*. [0,1]*. Besides, Besides, (3.5.14) (3.5.14) has has an an explicit explicit expression expression maps «-i

*i = l/i, Xi = yiY[(1-"yj)>(i i=i

= 2

>'-,s),

(3.5.14) (3.5.14)

(3.5.15) (3.5.15)

multidimensional 5. Integration on multidimensional

simplexes simplexes

93 93

and one obtains »—1

» — l

= U(1-VJU*

i-Y,*

(3.5.16) (3-5.16)

= 2>---,').

it follows immediately from (3.5.14) that for ti = = 22 In fact, by induction, it (3.5.15) and (3.5.16) obviously hold. Now assume that for ii — 1, (3.5.16) holds, then by using »-i

»

i=i

i=i

=n( -w)-wii( -w)=ii (!-»)' (3-5-i?) tt - -l l

i

xx --l 1

j=l

i

i

ii ==l l

JJ== l

i, (3.5.16) holds. Again, by (3.5.14) and (3.5.17), we obtain that for i, t

t

= w+ill( 1 -w)'

< 3 - 5 - 18 )

it is proved that for i + 1 , (3.5.15) holds. Denote by * the contents not used in the evaluation of the determinant. Since ' 0, 0, dxj

i i>j, > h

(( ll -- t»/ ii )) -- -- -- ((ll--t M / ; -_ il )) ,,

dyi

, *> which means that the Jacobian is

(3.5.19) (35.19)

ti = J, = i,

* i
JJ = = (1 - 0i)*(l 2/i)*(l - 0i fae))"- 1 • -• •((1I - y ., - i ) . (3.5.20) (3-5.20) 5 TAe transform (3.5.15) maps 7\ T\ on [0,l] [0,1]*,, and the T h e o r e m 5.2. Tfce s integral (3.5.12) is converted into the following integral defined on [0, l ] 5 .

h = f f{x)dx ix

5 >+i * f(yi>y2(i-yi),---,y f[(i-yi))dy ---f ll^-yiY~ == /1--Y1n([i-%) " /^'^(i-yi)r--,y/na 1

1

i 1

t

*/o

Jo

i=1

j=1

1

= Jo Jo

Jo j ==11 Jo J —•*■

yj))^y

[ -rf[z'j-

i+1

f(l-z1,z1(l-z2),...,(l-zs)Sf[zj)dz. , i== 11 (3.5.21)

94

HI. Application

of SEM to multidimensional

numerical

integration

P r o o f . Under the transform (3.5.15), it follows from (3.5 .20) t h a t

Jo

Jo

=1

5-1

(3.5.22)

•/(2/i, 2/2(1 - y i ) , •• • , y , [ | ( l --yj))dyi--dy.. Again, applying the following transform i = !>••• >*,

Z

J = (i-Vj)'

(3.5.22) becomes

«/o

./o

=1

• / ( l - z i , z i ( l - z 2 ), • • - , ( ! - 2,)*i .■ • • z , _ i ) d z i - - • ^ 5 , which completes the proof of (3.5.21).

D

C o r o l l a r y 1. If gi(x\} • • • , x 5 ) is an homogeneous with a unique singular point at the origin, and / ( x i , • • •, xs) = x^1 • • • x*'gi(l

functuon of degree I

- a?i, x2} • ••,**)M*i>--

where ft(a?i, • • •, ar5) is a smooth function,

and

s

fij > - 1 , / + * + ^ ^ .

•>**)i

s

> —1,^ —i-hl-h

i=2

^

/i, > - l ,

t=j+i

(3.5.23) //ten Me integral of f(x)

on T\ converges,

f f(x)dx= / 1 .../ 1 *j+'+*»+

+

and

" ' { f l ( l - -'i)"'}

•{n*;-' + 1 + w + i + - + "-}„(i >

1 - * 2 | . - - , ( 1 - -2*)*2 • • • * * )

i=2 • A ( l - * i , • • • , ( ! - - * * ) / i i •••/**-_i)cfc.

(3.5.24)

C o r o l l a r y 2 . Tf/ii, • • •, JI 5 and / are fractions satisfying the condition (3.5.23) ; Men Mis integral is of type (3.2.2) and can be computed by SEM.

5. Integration on multidimensional

simplexes

95

Moreover, consider the integral over T\ with the integrand

/(x 1 ,...,x 5 )=nK j ( i n ^) A j i •0/(1 - a?!, x 2 , • • •, xs)h{xu

• • •, x a ) ,

which has a logarithmic singularity with Aj = 0 or 1. W h e n the integral is converted into the type (3.4.13) on [0, l ] a , the asymptotic expansion (3.4.20) shows t h a t it can be computed by SEM. We have discussed the simplexes T defined by (3.5.2) and T\ defined by (3.5.13). By a suitable affine transform one can m a p an arbitrary sdimensional simplex onto one of these two standard forms. Let T3 be an 5-dimensional simplex with vertexes Ai, • • •, As+i, the base corresponds to the vertex A\ satisfies the plane equation Lx(x)

= a i * i + ■ • • + asxs - 6 = 0.

(3.5.25)

Consider the following integral /

w(x)g(x)dx,

(3.5.26)

JT3

where g(x) is a smooth function on T3, and s

wx=

s

aiXi

()

\Yl

-H^(M]C a * x , '~H)

»=i

1

»A«I

> -iiAi = o,i,

i=i

is a function with singularity on the base corresponding to A\. Map Z3 onto the standard simplex T\ by an affine transform, and suppose ^^(1,0,--.,0),-.-, A , - . ( 0 , 0 , ■■•',!), i 4 , + i - ( 0 , . . . , 0 ) . Under this mapping, the new base corresponding to A\ satisfies 2/1 = 0. Thus, / JT3

w(x)g(x)dx=

I

y^(lnyi)^g(y)dy.

(3.5.27)

JT2

Applying the transform (3.5.15), it is converted into the form (3.5.21). If the integral on a cube has a singularity on the diagonal plane, it can also be decomposed so t h a t the singularity will be eliminated. For example, the integral \x-y\fil(\Ti\x-y\)Xlg(xiy)dxdy,fi 1 > - 1

h= Jo Jo

III. Application

96

of SEM to multidimensional

numerical

integration

has a singularity on the diagonal line x = y, then [0, l ] 2 can be decomposed into two triangles with this diagonal line as hypotenuse. For details, see A. Sidi's w o r k t 1 1 ' } . This article discusses t h e Euler-Maclaurin expansion for an integral over the triangle a n d t h e integrand h a s an algebraic a n d logarithmic singularity. T h e transforms discussed in this section n o t only can m a p s-dimensional simplexes onto [0,1]*, b u t also reduce t h e m t o [0, l ] 5 so t h a t SEM can b e used. 6. I n t e g r a t i o n o n m u l t i d i m e n s i o n a l d o m a i n s w i t h c u r v e d b o u n d ­ aries Many integrals on the domains with curved boundaries can be converted into integrals on [0, l ] 5 by using suitable transforms. For example, let

ft = {(*!,-••,*,) : 0 < z i < 1,0 <ar,- ^ f l i - i f a i , . . . ,*,•-!), 2 < t < * } ,

(3.6.1)

and 1=

I I Jo Jo

■■ Jo

f(x1,---,x,)dx1---dxs.

(3.6.2)

Consider the transform * i = 2/i, *2 = 2/2#i(zi), • • •, x9 = ys0s-i{xly

• • •, x , _ i ) .

(3.6.3)

Obviously, such a transform m a p s ft onto [0, l ] 5 . In order t o express ex­ plicitly this transform, determine recursively * i = 2/1,

*2 = 2 / 2 # i ( z i ) = y20i(2/i) = 2^2(2/1,2/2),

and * 3 = 2/302(*i, x2) = 2/302(2/1,2/201 (2/1)) = £3(2/1,2/2,2/3).

In general, if x, = £,(2/1, • • •, y : ), i < s, have been defined, then x,+i = yi+16i(xu

••-,x t -)

= 2/t>i0.- (yi, ^2(2/1, y2), • • •, * . ( y i , • • •, y,)) = £»>i(yi,---,y«+i).

(3.6.4)

Now define 0»(yi,---,y»)

= 0i(yi,x2(yi,y2),-;*i-i(yw-,yi)),

i = i,-••,*-!,

(3.6.5)

6. Integration on multidimensional

domains with curved boundaries

97

the transform can hence be expressed explicitly as xi =yi,x.-+i = y,+i0*(yi,---,y,), t = 1, • • • , * - 1.

(3.6.6)

Denote by * the contents not used in the evaluation of the determinant. Since the elements of the Jacobi matrix are 0, dxj — =^ <9y«-

j
i = t,

the Jacobian is .(3.6.7)

J = e1{y1)02(yuy2)---6s-1{ylr..,ys-1).

For the integral on a more complicated domain with curved boundary,

h = [ f(x)dx rl

rV?i(ari)

/»V.-i(a?i,---,a?.-i)

••/

= »/0 Jxpi{xx)

/(xi,---,x5)dxi---dx5,

^I-I(*I,-,*I-I)

(3.6.8) where fii = { ( x i , - - . , * , ) : 0 < a ? i < l , ^.•-i(xi,---,x.--i) < xt- < ^•_i(a?i,--- > a:i_i),

i = 2, •.,*},

(3.6.9)

one can use the transform x i = yi,

xi = yi - ^ t - i ( x i , - - - , X i _ i ) , i = 2 , - - - , «

(3.6.10)

to convert it into one of the types (3.6.2). Express recursively the transform (3.6.10) by xi = x i ( y i ) = yi,

Xi = yi - tl>i-i(xi(yi)r = Xi(yi,---,yi),

• - ,Xi-i(yir

-,yi-i))

i = 2, • • • , « .

Let 0i{yi, • • •, yt) =
i = ! , • • • , * - 1,

(3.6.11)

98

III. Application

of SEM to multidimensional

numerical

integration

(3.6.8) becomes ri

rOi(yi)

r*i(yi,—,y.-i)

/

h= Jo Jo

g(yi,-~,ys)dyi~-dy„

(3.6.12)

Jo

with g(yi,

• • •, y*) = / ( ^ i ( 2 / i ) , • • •, * . ( y i , • • •, 2/5)).

Integral (3.6.12) can now be converted by the transform (3.6.6) into an integral on [0, l] 5 . E x a m p l e 6.1. Let

fi2 = | (*ir • •, x,) : J2 ( ^ ) m i ^ L XL- • •>«» > 0 > ,

(3-6.13)

with ai,rrii > 0 (i = 1, • • •, s). The transform yi = &)mi,i=l,-,'

(3.6.14)

a,-

maps Q2 into the standard simplex T\ and / f(x)dx= Jn2

m\rri2 • • rns

• / ny;1+1/mV(*iy^^

(3-6-15)

This can be converted into an integral on [0, l] 5 by using (3.5.15), its sin­ gularity may then be eliminated by the Duffy transform, and hence may be evaluated by SEM.

Chapter IV SEM FOR INTEGRAL EQUATIONS

1. General theory of integral operators Suppose that X is a Banach space and K : X —* X is a bounded linear operator. Consider the approximate solution of the operator equation u - Ku = / ,

(4.1.1)

where / G X, 1 G p(K), and p(K) is the resolvent set of K. The operator K is called an integral operator, if Ku = / k(x,y)u(y)dy,

V* G

ft,

(4.1.2)

where ft is a bounded domain in Ms. Equations of the form (4.1.1) are called the Fredholm integral equations of the second kind. k(x,y) is called the kernel of the integral operator K. If k(x, y) G C(ft x ft), then Ar(x, y) is a continuous kernel. If fc(x, y) is continuous and fc(z, y) G C r (ft x ft), r > 0, then k{x) y) is a smooth kernel. If there exists a(x, y) G C(ft x ft) such that a discontinuous kernel is of one of the following forms, k{xyy) = a(x,y)\x-y\-vJ

0
(4.1.3)

or k(x, y) = a(x, y) In |x - y|,

(4.1.4)

then it is called a weakly-singular kernel. It is noteworthy to study the condition under which an integral operator K is compact. Graham and I.H. Sloan (cf. [20]) proved the following L e m m a 1.1. Suppose that for 1 < p, q < oo, p~x + q"1 = 1, K is a compact operator mapping i/(ft) (q < r < oo) onto C(ft) if and only if the function kx{y) = k(x,y) satisfies (i) sup \\kx\\p < oo, sen

and 99

100

IV. SEM for integral

equations

(it) lim ||i c . - i,||p = 0, Vx',z G fi. x'—*x

Due to the following inclusion relation C(Q) C L°°(n) C Lr(Q) C L^Cl),

(4.1.5)

the conditions in Lemma 1.1 imply that K : 27 (ft) —* I/(£2) and L°°{Q) —► C(Q). The latter is more important in application. Similar to Lemma 1.1, the condition that K is a compact operator mapping C(Q) onto itself can be given as follows: Lemma 1.2. If the kernel k{x,y) (i) sup / \k(x, y)\dy
satisfies

and

(ii) lim / \k(x'iy)-k(x,y)\dy

= 0, V * ' , * G O ,

/Aen K is a compact operator mapping C(Q) onto C(Q). Proof. Setting p = 1 in Lemma 1.1, it follows that K is a compact operator mapping L°°(Ct) onto C(ft). Due to the fact that C(Q) C £°°(Q), it is clear that K is a compact operator mapping C(Q) onto C(Q). Lemma 1.2 can also be proved directly by the Ascoli-Arzela Theorem. Corollary 1. If the kernel k(x,y) operator mapping C(Q) onto C(Q.).

£ C(Q, x Q), then K is a compact

The condition under which an integral operator K : L2(£l) —► L2(Q) is a compact operator is as follows: Lemma 1.3. If the kernel k(x1y) M2=

satisfies

[ f \k(x,y)\2dxdy<

oo,

(4.1.6)

JVLJil

then K is a compact operator mapping L2(Q) onto L2(Q) satisfying \\K\\ = M. Proof. First, for any u 6 X 2 (ft),

PHI 2 < / \K*,y)?dy I \u{y)\2dy,

(4.1.7)

hence under the L 2 (fi) norm, \\Ku\\2 < M 2 ||«|| 2 , V« € £ 2 (fi).

(4.1.8)

1. General theory of integral operators

101

Hence it follows immediately that \\K\\ = M. Second, since k(x, y) G X 2 (fix Q), and {
*(*>y) = £ ^wWw(»).

( 4L9 )

•j=i

where {^>»(x)} is a complete orthonormal system of £ 2 (Q), and ilii= /

I k(xiy)(pi(x)(Pj(y)dxdy.

(4.1.10)

Let n

M * i y ) = ^2 AijWWwiy)

(4.1.11)

be a degenerate kernel, i.e., the range of the corresponding integral operator Kn is finite-dimensional and hence is a compact operator. Since \\K-Kn\\2

= /

\k{x,y)-kn{x,y)\2dxdy

I oo

= £

4

- 0 (n - oo),

(4.1.12)

i,j=n

then if, as the limit of the uniformly convergent sequence {-Kn}, is a com­ pact operator. □ In the following, we shall prove that the integral operator corresponding to a weakly-singular kernel is also a compact operator. Define *(2)(*,y)= / k(x,z)k(z,y)dz

(4.1.13)

as the iterated kernel of k(x, y), and fc(n)(x,y)=

/

fc(x,2:)fc(n_i)(z,y)(f2:

(4.1.14)

Jet as the n-th iterated kernel of k(x, y). Letfc(,-)(ar,y) be the kernel of If*. The weakly-singular kernel has the following well-known properties: L e m m a 1 . 4 . ^ Suppose that P{x,y) and Q(x,y) are two weaklysingular kernels, and there exist constants A\ and A2, such that |P(z,y)|<% ra

\Q(x,y)\<^,

(4.1.15) rp

102

IV. SEM for integral

equations

s

with 0 < a < s, 0 < /? < s, r = | x - y | = ( ^ ( * t - Vi)2)1 *, ihen for the t=i

kernel R(x, y) = / P(x, z)Q(z, y)dz,

(4.1.16)

the following estimate holds:

\R(x,y)\<{

C,

for a + $ < s,

C | l n r | + Ci,

for a + /? = 5,

[ C/ra+?—,

(4.1.17)

/ora + /?>s,

where C and C\ are positive constants. L e m m a 1.5. Ifk(x,y) is a weakly-singular kernel, then the correspond­ ing operator K is a compact operator mapping L2(Q) onto L2(Q) and C(Q) onto C(Q). Proof. If Jb(x,r/) is weakly-singular, so is k(y1x). kernel

By Lemma 1.4, the

t(x, y) = / t(x, z)k(y, z)dz

(4.1.18)

Jfi

is also weakly-singular, which is the kernel of the integral operator T = K*K. Repeated applications of Lemma 1.4 show that there exists a suf­ ficiently large integer n such that t( 2 n)(x,y), the 2 n -th iterated kernel of t(x,y), is a bounded function. By Lemma 1.3, the corresponding operator T 2 is a compact operator. It is easy to prove that if A*A is a compact operator, so is A. It follows that T = K* K is a compact operator, hence K is a compact operator mapping L 2 (Q) onto L 2 (Q). The fact that K is a compact operator mapping C(ft) onto C(Q) can be easily proved by the Ascoli-Arzela Theorem. D Consider the convergence of the operator sequence {l^n}Definition 1.1. a) {Kn} is said to be pointwise convergent to K, denoted by Kn -^> K, if and only if for any u G X, Knu —> Ku. b) {Kn} is said to be uniformly convergent to K, if and only if \\K — #„||->0(n-oo). c) \I^n} is said to be collectively compact convergent to K denoted by Kn —► K, if and only if

1. General theory of integral operators

(%) Kn A K,

103

and oo

(ii) the set S := (J (K — Kn)B

is a relatively compact set in X, where

n= l

B is the unit sphere in X. Using the fact that Kn approaches K, construct the following approxi­ mate equation of (4.1.1): un-Knun

(4.1.19)

= fny

where fn is an approximation of / . The following problems are of great importance: Pi) The existence, uniqueness, convergence and stability of the approx­ imate equation (4.1.19). P2) The complexity in solving the approximate equation (4.1.19). and P3) The existence of methods for accelerating the convergence of the approximate solution un. For the problem Pi), we have the following theorem: T h e o r e m 1.1. If (I — K)'1

exists and is bounded, and Kn satisfies

\\K-Kn\\<\\(I-K)-l\\-\

(4.1.20)

then (4.1.19) has a unique solution un and the following error estimate:

llu

u

II < IKJ-^)- 1 H(l|Jg-^n|||H| + ||/-/nH)

, 4 ^

Proof. By I-Kn

= (I= (I-

K) - (Kn - K) K)[I - (I - K)~\Kn

- Kj\,

let Q = (I - K)-\Kn

- K).

It follows from the assumption (4.1.20) that

WQWKWil-K^UK-KnWKl, hence, (I — Q)"

1

exists and oo

IKJ-Qm^EllGH^i/U-IIQII)n=0

(4.1.22)

104

IV. SEM for integral

Substituting into (4.1.22), (7 - Kn)'1

equations

exists and satisfies

(J-iQ-i = (I-Q)-i(/-K)-\ and

Ha-**)- 1 !! <

i-IIQII

,

iie-^ni

, 4123)

which proves the existence, uniqueness and stability of the solution of (4.1.19). For the error estimate, since (I-Kn)(u-un) = u - Knu - u„ + Knun = Ku + f - Knu - Knun - /„ + Knun

=

(K-Kn)u+(f-fn),

then u - un = (I - Kn)'1 [(K - Kn)u + ( / - / „ ) ] .

(4.1.24)

By the estimate (4.1.23), < | | ( I - Kn)-l\\]\\K

- Kn\\ || U || + | | / - fn\

1

^IIU-^)- ll[ll«-^n|||HI + ||/-/„||; l-\\(I-K)-i\\\\K-Kn\) which completes the proof of (4.1.21). Corollary. If (I — Kn)~l

(4.1.25)

D

exists and

(I - Kn)-1-^ Kn±K,

(I - K)~\ fn->f,

(4.1.26a) (4.1.26b)

then un —+ u (n —► oo). Proof. By the Banach-Steinhaus Theorem, (4.1.26a) implies that the operator sequence {(/ - Kn)'1} is uniformly bounded. Therefore by the condition (4.1.26b) and the identity (4.1.24), the proof is complete. D

2. Approximate

quadrature

methods

105

As far as the problem P2) is concerned, the complexity in solving the approximate equation depends on the construction of the approximate op­ erator Kn. Usually, Kn is constructed as a finite dimensional operator, i.e., the range of Kn is in a finite dimensional subspace. There are several methods to construct approximate operators of an integral operator K: a) b) c) and d)

the approximate quadrature method, the collocation method, the Galerkin method, the degenerate kernel method.

With the exception of a), all these methods can be brought into the framework of projection methods. From the computational point of view, a) is the simplest, followed by b). In method a), no integral evaluation is needed, while in method b), each element in the discrete matrix requires the evaluation of an integral on Cl. The most complicated one is c), in which each element in the discrete matrix requires the evaluation of an integral on Q x Q. Since most of the CPU time is used in the integral evaluation, the Galerkin method has little practical value. However, different methods require different conditions on the smoothness of the integral kernel: the condition required by the approximate quadrature method is stronger, by the collocation method is weaker, and the Galerkin method requires a much weaker condition. For the problem P3), in this book, we concentrate on the method of splitting extrapolation. For details of other methods, please refer to the work of ChatelW20! and I.H. Sloan and V. Thomeet 122 '. 2. Approximate quadrature methods Consider the integral on fi / ( / ) = / f(x)dx

(4.2.1)

/„(/) = 5>i B) /(»0-

(4-2.2)

and the n-point quadrature

*=1

Here, the basis points {y{\ and the weights {w\n'} conditions: Ai) V/ € C(J1), / „ ( / ) - / ( / ) (n - 00);

and

satisfy the following

106

IV. SEM for integral

equations

A2) there exists a constant C independent of n, such that

t=i

Using the quadrature rule / n (/)> one can construct n

Knu = ^ u ; { n ) i ( x > W ) t i ( W ) ,

(4.2.3)

i=i

which is an approximation of the integral operator

Ku= I k{x,y)u(y)dy,

(4.2.4)

and Kn is called the Nystrom approximation of K. Concerning the rela­ tionship between Kn and i£, AnseloneW proved the following: T h e o r e m 2.1. / / the integral operator K has a continuous kernel fc(x, y) on £2 x fi, anrf /Ae quadrature rule satisfies the conditions K\) and A 2 j, then inC{Q), Knc-$K. Proof. First, by the condition Ai), one can easily obtain (4.2.5)

Kn £ K. OO

Hence it is sufficient to prove that the set 5 = |J (K — Kn)B is relatively n= l

compact, where B is the unit sphere in C(Q). By the Ascoli-Arzela Theo­ rem, in order to prove that S is relatively compact, we need only to prove that the functions in S are uniformly bounded and equicontinuous. In fact, VuE5,

|A>|<^K( n) ||^,2/i)IK2/OI t=l n

< maxli^.yJIIHIoo^KI t=l

< Cmax|Jb(z,y)|,

(4.2.6)

x,y£Cl

i.e., the sequence {/^n^} is uniformly bounded. Furthermore, it is equicon­ tinuous, since for arbitrarily x, x' £ fl, \Knv(x) -

Knv(x')\

< C\\v\\oo max \K(x, y) - K(x', y)\.

(4.2.7)

2. Approximate

quadrature methods

107

When \x — x'\ —► 0, the right-hand side tends to zero independently of n. Thus, {/
(J KnB is a relatively compact set. Since K is a compact operator, then n= l oo

S = \J (K — Kn)B is relatively compact. According to Definition 3.1, n=l

Kn " K.

D

In general, the Nystrom approximate method is different from other ap­ proximate methods (e.g., the collocation method, the Galerkin method and etc.), Kn cannot converge uniformly to K. Theorem 2.1, which guarantees that Kn —+ K, is a theoretical foundation of the approximate quadrature method, because Kn —► K guarantees the following theorem: T h e o r e m 2.2. If K is a compact operator on the Banach space X, 1 is not an eigenvalue of K, and suppose Kn -+ K, then \\K2 - KnK\\ -► 0, / o r n - 4 o o , \\K2-KKn\\-+^ /orn-^oo, and when n is sufficiently large, (I—Kn)~l

(4.2.8) (4.2.9)

exists and is uniformly bounded. oo

Proof. We shall now prove (4.2.9). Since Kn -$ K, then Si = (J KnB n= l

is a relatively compact set, and for arbitrarily small e > 0, there exists a finite e-system: {<7i, • • •,flfm},such that for any Knf £ Si, one can always find <7i, such that (4.2.10)

\\9i ~ Knf\\ < e. However, Kn A K, then, for sufficiently large n, there are \\(Kn-K)gj\\
i = l,--.,m,

(4.2.11)

which imply

\\(Kl-KKn)f\\ =

\\{Kn-K)Knf\\

< \\(Kn - K)(Knf

- 9i)\\ + \\(Kn -

K)9i\\

<(||iC„|| + ||A:||)c + £, V / € B . (4.2.12) Since e > 0 can be arbitrarily small, (4.2.9) is proved. Similarly, we can prove (4.2.8). Finally, by the identity (I-K

+ Kn){I -Kn) = I-K

= (I - K) [/ + ( / - K)-\KKn = ( / - K)[I -(I-

KY^Kl

+ (KKn - Kl) - Kl)) - KKn)},

(4.2.13)

108

IV. SEM for integral

equations

and (4.2.9), for sufficiently large n, there is

WV-Kr^KZ-KKJWKl.

(4.2.14)

This implies that the right-hand side of (4.2.13) is invertible, and so is (J — Kn). Furthermore, (I-Kn)-1 = [l-{I-K)-x{Kl-KKn)}-\l-K)-\l-K The uniform boundedness of the operator {(I—Kn)'1}

+ Kn). (4.2.15) follows from (4.2.14).

D

In the following theorem, by using Theorem 2.2, we prove the conver­ gence of un to u} where un is the solution of the Nystrom approximation (4.2.16)

(I-Kn)un=f, and u is the solution of the integral equation (Z - K)u = f.

(4.2.17)

T h e o r e m 2.3. Under the assumptions of Theorem 2.1, the Nystrom approximation u„ converges to u. Proof. By (4.1.24), ll« " «„|| < ||(/ - Kn)-^\\

\\{K - Kn)u\\.

(4.2.18)

By Kn —► K and the uniform boundedness of ( / — J £ n ) _ 1 , the proof is complete. □ In practice, the numerical solution of (4.2.16) is divided into two parts. Firstly, solve the linear system n u

n(yi) - ^w$n)k(yi>yj)un(yj)

= f(yt),

i = l,- -,m,

(4.2.19)

to obtain the values of un(x) at the nodes {t/i} t _ 1 ; secondly, set n

un(x) = f(x) + 2 u ; 5 n ) t ( * > y i K ( y i ) ,

(4.2.20)

i=i

which is the required numerical solution. {un(yi)}^_1 of (4.2.19) is also called the Fredholm approximation of u, and (4.2.20) is called the Nystrom interpolation.

3. SEM for multidimensional

integral equations with smooth kernels

109

3. SEM for multidimensional integral equations with smooth ker­ nels Consider the 5-dimensional integral equation u(x) - / jb(x, y)u{y)dy = g(x), V = [0,1]*. (4.3.1) Jv Suppose that g(x) and k(xyy) are smooth functions on V and V x V re­ spectively. Divide V by the mesh width h = (/ii, • •, h$) into the union of n cuboids, the widths of its sides are /ii, • • •, /i, respectively: U Vj.

V=

(4.3.2)

7= 1

Let Mj be the center of V}, using the rectangular rule (3.1.2) we construct the Nystrom approximation n h

K u = ] T meas(^)ib(a:,M i )t/(M i )

(4.3.3)

i=i

of the operator i f t i = / k(x,y)u(y)dy,

(4.3.4)

and the corresponding approximate equation uh-Khuh = g.

(4.3.5)

The following theorem shows that the accuracy of the numerical solution of (4.3.5) can be improved by the SEM. Theorem 3.1. If I is not an eigenvalue of K, g E C 2 m + 2 (V r ), k e C ( V x V), then there is a multivariate asymptotic expansion 2m+2

u(x)-uh(x)=

£

h2°wa(x)

+ 0{hlm+2),

(4.3.6)

l<|a|<m

where wa(x) are functions independent

ofh.

Proof. By the assumptions, u G C2m+2(V). It follows from the asymp­ totic expansion (3.1.4) of numerical integration that there exist functions wa E C 2 m + 2 - 2 l a '(V r ), such that Ku - Khu =

^ l<|or|<m

h2awa(x)

+ O(h20m+2).

(4.3.7)

HO

IV. SEM for integral

equations

By (4.2.18), for m = 0, u-uh

=

(4.3.8)

0(hl).

For m = 1, we shall prove that there exists the expansion u(x)-uh(x)

= J2 h2awa(x)

+

0(ht).

(4.:J.9)

|a| = l

In fact, let eh(x) = u(x) - uh{x). By using (4.3.7), we have (I-Kh)eh

= (K-Kh)u=

^

h2awa{x)^-0{ht),

|a| = l

with wa(x) e C2(V) and efc(x) =

u(x)-uh(x)

= ^^(/-/^r^^+o^) |a| = l

|a| = l

+E

fc2a[(/«^fcr1-(/-^)-i]t5tt(x)+ o(hi).

Letting u; a (x) = (I - K^w^x), and by (4.2.18), (4.3.9) holds. Now suppose that for /, 1 < / < m, there already exist the functions wa(x) independent of A, such that u(x)-uh(x)=

£

h2awa(x)

+

0(h2l+2)

(4.3 .10)

i<M<J

holds. We shall prove that (4.3.10) holds also for replacing / by / + 1. In fact, by (4.3.7), (I-Kh)eh

=(K-Kh)u

=

£

h*°wa(z) + 0{hll+*).

(4.3 .11)

i<M<'+i Construct an auxiliary integral equation (7 - K)va = wa, 1 < |or| < / + 1

(4.3 .12)

and the approximate equation {I-Kh)vha

= wa, 1 < H < / + 1 .

(4.3 .13)

3. SEM for multidimensional

integral equations with smooth

kernels

ill

By the inductive assumption, there exist functions wap, such that

vvaa-v -v\= = a

h^wa, + 0(hll+'-2^),

Y,

il<'+i-M for 1 < | t * | < / + l .

(4.3 .14)

Substituting (4.3.13) into (4.3.11), one obtains

(I-Kh)(eh-

h2av

J2

°)

i<M
= (I-Kh)(ek-

£

h

*"v«

l<|a|
+

£

h*°+Vw2a+o(hi<+<))

£

l<|a|<7+ll<|/?|
= (I-Kh)(u-uh-

J2

h2a™a + 0(hll+*))

l<|a|
(4.3.15)

= 0(hl<+*). Using Theorem 2.2, the proof of (4.3.6) is complete.

□

Example 3.1. Consider the solution of the two-dimensional integral equation u{x1,x2)

= exp(-xi — x2) -

Jo Jo

x1x2exv(y1+y2)u(yl,y2)dy1dy2.

Its exact solution is u = exp(—x\ —x2)—-xy.

Let the initial mesh width be

h = ( - , - ) , using the Richardson extrapolation method and the Type 1 and Type 2 methods of splitting extrapolation, the numerical results obtained are illustrated in Table 3.1. For h = ( —, — ) , without acceleration by extrapolation methods, the maximum error of the Nystrom approximate solution is 0.00128, the con­ vergence is very slow. Although the Richardson extrapolation converges faster than the SEM, the CPU time and storage increase rapidly with the number of extrapolations. The SEM not only reduces the CPU time and storage but also has higher order of error.

IV. SEM for integral

112

equations

Table 3.1. The comparison between numerical results of the Richardson and SEM for Example 3.1. number of

Richardson

Type 1

Type 2

extrapolations max error CPU(sec) max error CPU(sec) max error CPU(sec) 2.0E-2

0

2.0E-2

1

1.2E-4

0.18

2.0E-2 1.2E-4

0.11

1.3E-4

0.14

2

5.3E-8

6.67

5.3E-7

0.65

1.6E-6

0.46

2.1E-8

1.64

3

Fig. 3 . 1 .

P l o t of log ( m a x i m u m a b s o l u t e error)

-*— Richardson's extrapolation -*- Type 2 Splitting extrapolation TTl = ?7l-step/split extrapolations 771 = Nystrom method with step size /i/2* 1

Type 1 Splitting extrapolation Nystrom method

E x a m p l e 3.2. Consider the solution of the two-dimensional integral equation u(xx, x2) = exp(xi

+x2)

-[(exp(xi + 1) - l)(exp(x 2 - h i ) - l ) ] / [ ( * i + 1)(* 2 + 1)] + / / Jo Jo

z*v{xiyi+x2y2)u(yl,y2)dyldy2)

with the exact solution u(xly x2) = exp(xi + x 2 ). The equation is solved by the Nystrom approximate method, with an initial mesh width h = ( - , - ) ,

3. SEM for multidimensional

integral equations with smooth

kernels

113

and then is accelerated by the Richardson extrapolation method and the SEM. The numerical results obtained are illustrated in Table 3.2: Table 3.2. The comparison between numerical results of the Richardson and SEM for Example 3.2. number of Richardson Type 1 Type 2 extrapolations max error CPU(sec) max error CPU(sec) max error CPU(sec) 3.6E-1

0

3.6E-1

1

3.0E-3

0.2

4.3E-3

0.14

4.3E-3

0.12

2

1.9E-5

6.56

7.4E-5

0.69

8.6E-5

0.49

6.5E-7

5.36

1.2E-6

1.80

3

3.6E-1

Fig. 3.2. Plot of the number of kernel evaluations

For h =

(T^>7^)>

the maximum error of the Nystrom approximate

solution without extrapolation is 0.02278. The SEM not only improves the accuracy but also reduces the amount of function evaluation and storage. Figure 3.1 gives the accuracy of the four methods, Fig. 3.2 compares the amount of work of the three extrapolation methods, and the amount of storage of these methods are given in Fig. 3.3.

114

IV. SEM for integral Fig. 3.3.

equations

P l o t of s t o r a g e

4. SEM for integral equations in polygonal domains The SEM can also be applied to the numerical solution of integral equa­ tions on complex domains such as plane polygonal domains. Since a polygon can always be divided into finite number of triangles, we shall discuss firstly the extrapolation of numerical integration on triangular domains. Suppose that A is a given triangle. Consider the integral on this trian­ gular domain J w i t h y = (2/1,2/2), f(y)

I f(v)dy,

= /(2/1,2/2), dy =

(4.4.1)

dyidy2.

In order to establish the quadrature formula for (4.4.1), one can divide A by successive refinements, that is, the k-ih refinement is obtained by joining the midpoints of the sides of each triangle of the (k — l)-th refinement. After / refinements, A is divided into 4' congruent triangles: A = (J A*i=i hl

Let A

= — measA. One can construct two kinds of numerical integral

formulae: the first one is the rectangular rule 7*1

J

R

=

Ah^f(Mj),

(4.4.2)

4.

SEMforinitegral equations in polygonal

domains

115

where Mj is the barycenter of A j , the other one is the trapezoidal rule 1

*'

(4.4.3) i=l

where Aj, Bj and C, are the three vertices of Ay. For both (4.4 .2) and (4.4.3), we have the following theorem: T h e o r e m 4.1.( 65 ) Iffe

C 2 m + 2 ( A ) , then m

ER(f) = J-JhR>= 5>,-(/)A?< + 0(fcf"+2),

(4.4.4)

t= l

and m

Er(f) = J-Jfr=

£>,(/)/*?* + 0(hlm+2),

(4.4.5)

1= 1

where Wi(f) and Wi(f) are constants which are independent of hi = 1/2' and depend on f. This theorem will be proved later. Consider the evaluation of the following integral equation of the second

kind

u(x)=

/ k(x,y)u(y)dy Jn

+ /(x),

(4.4.6)

where Q is a polygonal domain. According to the Nystrom method , (4.4.6) can be computed by the following steps: S t e p 1. Subdivide the polygonal domain Cl into an initial triangulation such that

ft = U Qt. S t e p 2. For each triangle, join the midpoints of the sides and thus obtain the next refinement, after /, refinements, ft,- = (J A , j , n, = <*'',(»• = 1

\

1, •••,*•)• S t e p 3. Solve the linear system r

^(Mij) -J2Y,

nk

&hkHMij,Mkt)uh(Mki) = /(Ma),

i= 1, ••,»*, j = ! , - • • , «i

IV. SEM for integral

116

equations

and thus obtain {uh (My), i = 1, •••, r, j = 1, • •, n , } , where A^ = meas(ft,-)/4*S hi = 2 -'*, i = l , - - , r , fc = (fci,---,ftr)> ^o == max/i,-. S t e p 4. Compute the Nystrom approximate solution uh(x) z Jb = l

* = 1

For simplicity, (4.4.6) can be expressed in the operator form u — Ku = / ,

(4.4 •7)

and the corresponding Nystrom approximate equation is u h - J^t/*1 = / ,

(4.4 .8)

with Rhv

= E E Aji(x,Mfct)t;(Mfc0.

Now we prove the following expansion theorem of the Nystrom approx­ imate solution: T h e o r e m 4.2. If f E C 2 m + 2 (ft), *Ae ifcerne/ k(x,y) G C 2 m + 2 (ft x Q), and 1 is no/ an eigenvalue of K, then there exist functions wa(x) G C 2 m + 2 ~ 2 l a l(Q), which are independent of h = (/i1} • ••, hr), such that u(x)-uh(x)=

h2awa(x)

J2

+ 0(hlm+2).

(4.4.9)

l<|a|<m

Proof. Under the assumptions, u G C 2 m + 2 (ft). By Theorem 2.1, Kh ^ J£. Let e,l(ar) = tx(ar) — wh(x) be the error of the Nystrom approximate solution. Hence by Theorem 4.1, it follows that

(I-Kh)eh

(K-Kh)u

= = f r

h(x9y)u{y)dy^^f^A^k(xiMij)u(Mij) m

'EE^w+o^.

(4-41°)

»=ii=i

with fity G C 2 m + 2 - 2 i ( f i ) . Similar to that in Theorem 2.3, we shall prove inductively the asymptotic expansion (4.4.9). For the case m — 1, it is easy

4. SEM for integral equations in polygonal

domains

117

to prove that there exist functions Wi G C 2 (Q), which are independent of ft, such that r

«(*) - uh(x) = £&?«;,•(*) + 0(ht). »= 1

Suppose that for / < m, the asymptotic expansion u(x)-uh(x)=

h*awa(x) +

]T

0(hl'+2)

(4.4.11)

l<\a\
holds. Now we shall prove that it also holds for replacing / by / + 1. Con­ struct an auxiliary problem (I - K)vij =Wij, t = 1, • • •, r; j = 1, • • •, / + 1,

(4.4.12)

and the corresponding Nystrom approximation is

(I - K ^

= wih i = l,...,r; i = l , - " , / + l .

(4.4.13)

By the inductive assumption, we have aa +4 w +4 aa +4- w w-)- )-v^= J2 J2 **aa *"y-+o(*? "y-+o(*? -v^= -v^= J2 "y-+o(*? )1<|<*|<M-1-J 1<|<*|<M-1-J

Vij Vij Vij --

(4.4.14)

l<M
Substituting (4.4.13) and (4.4.14) into (4.4.10), we obtain

(I-Kh)(eh

rrr

-r

rl+l (+1 r/+1 /+1J+l l+l

t : = 1 ::=1 t : = 1 =1

ii

hVv\ hVv\

= (I-Kh)(u-u»-J2!£h?vij 1= 1 >= 1

/+1 rr r/+1 /+1

hfh w , +, o(hl'+*)) + o(hl'+*)) hfh £ hfh w w , + o(hl'+*)) £ «=i i = n < | o | < / + i - i

l 1< |o|
= 0(ft*+ ).

2a 2a 2a ijc ijc ijc

(4.4.15)

Because tty-, u>tja are independent of ft, and Kh —» K guarantees that (I—Kh)~l exists and is uniformly bounded, the proof of (4.4.9) is completed by grouping the terms with the same power. □ Note that the number of the mesh parameters ftt- (i = 1, • • •, r) men­ tioned above is not the dimension but the number of triangles in the initial triangulation. This shows that lower dimensional problems can also have many independent mesh parameters. Because only one mesh parameter is

IV. SEM for integral

118

equations

refined at each time, the more the independent mesh parameters, the less the amount of work and storage needed by the SEM, and the higher the degree of parallelism. For example, let ti§ = u(hi} • • •, hr) be the solution with the initial triangulation, and u1* = ti(Ai, • • •, hi/2, • •, hr) be the solu­ tion with one refinement in ft,-, then the error with only one split of SEM is

* " ^ 4 X > ? - (*" - *)<] = °C*o),

(4.4.16)

where tig, uf (i = 1 , - -,r) can be evaluated in parallel, which are more economical than the Richardson extrapolation. We now go back to the proof of Theorem 4.1. The asymptotic expansions (4.4.4) and (4.4.5) are very useful not only in the SEM but also in the evaluation of three-dimensional boundary integral equations. Suppose that a triangle A is divided into 4' elements after / refinements. Let $ = {1, • • •, 4'}. A,-, i G $ is called an element with positive direction if it can be obtained by a parallel and similarity transformation from A. The numbering set of these A,- is denoted by ^"^. Let 3>~ = $ \ 3>+, then each A,- (i G $ " ) is called an element with negative direction. Consider a fixed side of A, say AB, for each element Aj (j G $ " ) , there exists a unique consecutive A,-, i G $ + , such that Aj and A,- form a parallelogram /^7j and their common side is parallel to AB. Let 0 be the set of all numberings of the elements with positive direction without pairing element. These elements have a side on AB. In the following, denote by rrij (j G $"") the center of /Z7j and denote by my (j G $°) the midpoint of the side on AB of A ; (j G $°). In order to prove Theorem 4.1, the following lemmas are needed. L e m m a 4 . 1 . There exist constants Ca and Da independent of hi = 1/2', satisfying f (y-Mi)ady

= Cah[alAh\

ie$+,

[ (y-Mi)ady

= (-l)°Cah[alAh\

j (y-mi)ady

= DahiflAh*,ie90,

(4.4.17a) i e r ,

(4.4.17b) (4.4.17c)

'Ai

ft

/

Jcyi

VM J °' a

(y - mi) dy = i I 2Dah[a]Ah>,

2 W

f '

2||a|.

i€$~.(4.4.17d)

4. SEM for integral

equations

in polygonal

domains

119

Proof. If |a| = 1, it is trivial that Ca — 0. Suppose that i G $ + , i.e., A,- is an element with positive direction. Take a transform y — Mi = hi(z — M) : At- —» A, where M is the barycenter of A, then [ (y- MiYdy

= h[al+2 I (z - M)adz,

\*\ > 1.

JA

JAi

Set h\ Ca = ^ A"

1

r

(z- M)adz,

(4.4.18)

JA

which is a constant independent of hi. Therefore, (4.4.17a) holds. Further, for i E $ ~ , take the transform y—M{ = —hi(z-M) : A,- —> A, f (y- Mi)°dy = (-irh[a{+2

f (z-

M)°dz.

JA

JAi

It follows from (4.4.18) that (4.4.17b) holds. (4.4.17c) can be proved in a similar way. Because rrij is the symmetric center of CJj, (4.4.17d) also holds. D L e m m a 4.2. If f E C 2 m + 2 ( A ) , then there exist constants &,(/) (i = l,---,ra) independent of hi, such that

Ei(f) = f f(y)dy-2

£

/(m^A** - £

/(m^A^

m

= 5>(/)^+0(A?m+2).

(4-4-19)

*=1

Proof. It can be proved by induction. For the case m = 0, by a simple estimate, (4.4.19) holds. Suppose that (4.4.19) holds for m < k - 1, we shall prove that it also holds for m = k. By Lemma 4.1, £

i(/)= E / (/(»)-/K)) d w+E / (/(y)-/^))^

jLL{y-mj)2adyf{2a){mi)

=£ E +E

E

-i / (»- -i) a ^/ (a) K)+o(Af +2 )

120 120

IV. SEM SEM for for integral integral equations equations IV.

= EE EE(2a)! ^ ' •gzt^W^) W'-W = je*~

i<\<*\
je*~ i
——I v ; *

+ EE EE ^ | a^hr^f^imj) W 2 «>(m,-) + jG*°l<|a|
att*

'

a| +a + EE EE ^^ AV->(wj)+o(*? )) + ^/w^j+o^ je$°\<x\eik je$°\<x\eik

= -1<M<* E ^^l''MMf™)+ + E

E 1<M<* ^

2a) f \y)dy ^*l/ wy {2a

''

A

^ l A ^ / ( « > ( m i ) + 0(/if+2),

E

(4.4.20) (4.4.20)

jj€^°|a|€/fc €*°|o|€/fc

where I* I* — — {1, {1, 3, 3, •• •• •, •, 2k 2k 44- 1} 1} stands stands for for aa subset subset with with odd odd numbers. numbers. By By where 2a the inductive inductive assumptions, assumptions, Ei(f( Ei(f(2a ^) (1 (1 < < ||a| < k) k) possesses possesses the the following following the ^) a| < expansion with with even even powers powers only only expansion

£?!(/('«>) = J2 b2aJh? + 0(hfk-l«'+1>),

(4.4.21) (4.4.21)

where &2a,j &2a,j are are constants constants independent independent of of hi. hi. On On the the other other hand, hand, by by the where the Euler-Maclaurin expansion, Euler-Maclaurin expansion, ^—' a! a!

= =

x

£

E E Ai/^i) >*i/(a)(™;)

a!

j€$°

ftW-^^/Ja ' , ••'

,

a

/

f(a)(vi(t),

V2(t))dt - d2hl

, 2 i b - | a j - l , ^ / , 2Jb-|<*|-l\l

, rf jk-|a|-lfti , 2 i b - | a j - l +, 0^( f/f tt, !2 J b - | a | - l)\ lj , 2 <*2fc-|a|-lfci + 0( l )j, VMEJfc, VHG4,

(4.4.22) (4.4.22)

where cf, cf, are are constants constants which which are are independent independent of of hi hi but but depend depend on on // and and where a, and (yi(t) y (t)) is a point on the line AB. Substituting (4.4.21) and 1 2 a, and (yi(t)1y2(t)) is a point on the line AB. Substituting (4.4.21) and (4.4.22) into (4.4.20), the proof of (4.4.19) is complete. D (4.4.22) into (4.4.20), the proof of (4.4.19) is complete. D Proof Proof expansion expansion

of Theorem T h e o r e m 4.1. 4 . 1 . First, First, we we shall shall prove prove that that the the asymptotic of asymptotic (4.4.4) of the rectangular rule holds. By induction, for m m = = 00 (4.4.4) of the rectangular rule holds. By induction, for

4. SEM SEM for for integral integral equations equations in in polygonal polygonal domains domains 4.

121

(4.4.4) obviously holds. Suppose that (4.4.4) holds for m = k -— 1, then for

£ / (f) == J2f

E E(f)

(f(y)-f(Mj))dy (m-f(Mj))dy

E == E

E E

2 (» -- M^dyf^Mj) + 0{h?») i3 // (y M^dyf^Mj)+o(^*+ ) JA > ' k+2 "^i

i £ * l<|«l<2Jb+l je* l<|a|<2fc+l

where where

(4.4,23) (4.4.23)

= Ji + J2 + 0(hl ), = / i + J 2 + 0(/i?*+ 2 ),

= EE EE h = ^

j€*l<\a\
7£y/jLfiy-M^dyf^iM,) (y-Mj^dyf^Mj)

2o) = EE EE % ^ ^ / ((2o) w = %^^/ w))

2 ! l [/(2^)-/(2a)W)]^ = - E E %^T/

~ k s f e , < °> ^ + i <£M < *^*l' M [ 1 1

; ;

''

fi^wv-

A A

4 4L24) ((4.4 - - 24 )

i<M<* ' By the inductive assumptions, Ji can be expanded in even powers; of h\. By the inductive assumptions, J2 can be expanded in even powers of h\. Second, Second, r

/■'"'A**

0) ** = = EE EE ^ i ^ / (^-^m (^) ii€€$$||aa||G G//ffcc

E E E == E

L {a) ^!^(M ^!^-[fL[f{a) (M j)-&HMp} j)-&HMp}

je*-\a\eik je*-\a\eik

+ ^E ^E

+

je*o\<*\ei je*o\<*\eikk = J3 J3 + + J4, ^4, =

^ f^w) ^^-^m

a!

(4.4.25) (4.4.25) 1. Hence, Mf is the bary barycenter of the the pairing pairing element element of of Aj Aj (.; (j G G $~). $~) Hence, where Mj" center of there exists a vector a, which is independent of /ii, such that Mj = = rrij m;- + ftia, Mj M;* = = rrij raj — — /iia. hia. Mj Since

^a\Mj)-^(M;)

122

IV. SEM for integral

equations

= E s^/C^KO + o^r2"1^), Me/* We/*

|/?|<2fc-a \p\<2k-a

then

* - E EE

j € * - | ao r | € / *

E

|/?|€/fc W\<2k-\a\ |/?|<2*-|«l

^ -b^r■^/*"'♦- ^ " o" -^. )

+0(/i + 0 ( / l2f*+ + 22))..

(4.4.26)

On the other hand, Mj = m, + /ii
*= E E ^ A ^ / W ^ + M)

= r ^ / 1( /^'^) "+^) =EEEE EE ^^V^ \6\<2k-\<*\ |/?|<2fc-|a|

+E E je*°\a\6ik

E \p\eik |/?|<2*-|a|+l \(3\<2k-\a\+l

r

/i|a+/?la/J

„,,, ^f^\rnj) 'p'

= JJ5 ++ /6 ( / ik^++2).2 ) . = J6 ++ 00(hl

+ 0(hl^) (4.4.27)

By Lemma 4.2, JZ + JQ can be expanded in even powers of h\. By the EulerMaclaurin expansion of integral on AB, J$ has an asymptotic expansion with even powers of h\. In summary, J\ = J 3 4- J$ + «/6 has an asymptotic expansion with even powers of h\. Therefore, the proof of (4.4.4) is complete. Concerning the proof of (4.4.5), we note that

£ETr{f) (/)

= £ / (/M ./(A) + W)+/(a))dv^ =£/ im.m±m±m j-l^A, 3 j=iJ±i 4'1 4' , 4

3

4'1 , 4'

4 (f(v)-f(Mj))dy-,,£\[

= =E E //

(f(v)-f(Mj))dy- £\[

4 ! , 4' ' i r ~ E (f(Bj)-f(M j))dy - E ^W/ (f(Bj)-f(Mj))dy i=i

3

^ i

,

(/(A^-fiMjfidy (/(A^-fiMjfidy

4. SEM for integral equations in polygonal domains 4

' 1 /

' 1 r -- £EW W 3 3

j=l i=i

j

123

(m)-m))dy. (fm-fMDdy.

A] ±i

J

(4.4.28)

(4.4.28)

By (4.4.4), the first term possesses an asymptotic expansion with even powers of hi. For the second term, by the Taylor expansion,

//

JA,

hl = =A Ahl

( /(/(A^-fiM^dy (^-/(M,))^ 1 m 2 m+2 /f(a^\(MM ).). (4.4.29) j)(A, J) ^ - -- M,-)*/" Mj)a/a\ ++ 0(hl 0(&J+ (4.4.29)

5Yl3

l<|a|<2m+l l<|«|<2m+l

There exists a vector a independent of j and hi, such that

Aj-Mj=ahu

VjE*+, Vje*+,

A,- - M,- = -—afti, a/ii,

Vj E $ ~ ,

and when |a| is an even number, we have 4'

i=i 41

a , , aa,, hhll ((a) = 53a ^ a a / /i i 11 A ( M ii )/a! )/a! = A // a ) (M

= ^

E

/

[/(O)(M,)-/<«>(
f_^_ y/ /W( + +—h f(a)(y))rfdy-

(4.4.30) ( 4A3 °)

By (4.4.4), in this case there exists an asymptotic expansion with even powers of hi. When \a\ is an odd number, 4'

Y^A^f^iMjKAj-Mjr/al , a,

-= 5J2 3 a^ , 11a , A^(/W(M jj)~/W(M;))/a! + 53 aafc|a|AfcV(a)(JW;)/a! = J7 J7 + + // 88 ,, =

(4.4.31)

where J7 and Jg have asymptotic expansions with even powers of hi, the proofs are the same as those for J3 and J4. Using the similar argument,

124

IV. SEM for integral

equations

one can show that the asymptotic expansion (4.4.5) holds for the third and fourth terms on the right-hand side of (4.4.28). D Corollary. Define the Simpson rule of numerical integration on trian­ gular domains as follows: J^ = ljRl

+ \ 4 \

(4-4.32)

then the error satisfies the asymptotic expansion Es{f) = w2h\ + w3hl + ... + wmhlm

+ 0(hlm+2).

(4.4.33)

Proof. By direct verification, J^1 (/) has an algebraic precision of de­ gree 2, hence the h\ term in the asymptotic expansion of Es(f) vanishes, that is, there exist constants Wi (i = 2, • • •, m) independent of Ai, such that (4.4.33) holds. D The following asymptotic expansion was given in the monograph by H. Engelst28^ ET(f)

= C2h2 + C3h3 + CAhA + • • •.

(4.4.34)

In our proof, this result is improved to one which includes even powers only, and so the extrapolation results will be more accurate. E x a m p l e 4 . 1 . Evaluate the following integral: ex+ydxdy,

/ JA

where A is a triangle with vertices (0,0), (0,1) and (1,0), and its exact value is 1. The extrapolation results obtained from the rectangular rule, trapezoidal rule and the Simpson rule are as follows: 1° The rectangular rule 0.97386027 0.97316288

0.99959403

0.99827062 0.99997340

0.99999869

5. SEM for eigenvalues and eigen functions

125

2° The trapezoidal rule 1.07276061 1.02015570

1.00262078

1.00516571

1.00016905

1.00000560

3° The Simpson rule 0.99859042 0.99991064

0.99999865

0.99999439

0.99999998

0.99999999

According to (4.4.34), /^-extrapolation error of Ex{f) will be 2 x 10~ 4 , but the error obtained in this example is just 5 x 10~ 6 , which is only about l/40 of the former. In addition, the rectangular rule and the trapezoidal rule have the same order of error, but the latter needs only (2l + 1)(2* + 2)/2 nodes, while the former needs 4l nodes. Hence, the latter is superior to the former, and this superiority will be more obvious in the evaluation of integral equations. 5. SEM for eigenvalues and eigenfunctions Consider the eigenvalue problem of the integral equation / k(x,y)u(y)dy = Au(x), (4.5.1) Jn where Q is a bounded domain in Ms, k(x, y) is a smooth function o n f i x f i , and so the integral operator I k{x,y)u{y)dy (4.5.2) Jn is a compact operator mapping C(Q,) onto C(fi). Let In(f) be the quadra­ ture formula defined by (4.2.2), satisfying the conditions Ai) and A2), Kn be the Nystrom approximation of K, which is defined by (4.2.3). Construct the approximation eigenvalue problem Ku=

(4.5.3) and consider the evaluation of the first eigenvalue (i.e., the one with the largest absolute value) of K.

126

IV. SEM for integral

equations

Suppose the first eigenvalue A of K is simple, since Kn -^ K, the first eigenvalue An of Kn is also simple, and An —* A (cf. [20]). It is impossible to compare the approximation of the eigenfunctions without standardization. Let ij) be the eigenfunction of K*, i.e., (\-K*)i/>

(4.5.4)

= 0,

where A is the conjugate complex of A and K* is_the conjugate operator of K. By the properties of the compact operators, A is the simple eigenvalue of K*, hence ip is unique except for a constant factor. Take a fixed %j>y the standardization conditions for u and un are chosen as <«,V> = 1, («n,^) = l,

(4.5.5)

where (•, •) stands for the pairing of elements in X — C(Q) and its conjugate space. We now estimate |A — A n | and \\u — un\\ under the condition (4.5.5). By using (A - K){u - «„) = -Xu„ + Kun = (A„ - A)«„ + (Kun - Knun),

(4.5.6)

one obtains 0 =

{(X-K)(u-un),i,)

= (A„ - A)(u„, ip) + {Kun - Knun,rl>) =

\n-\+j{K(K-Kn)un,4>),

hence |A„ - A| = 0(\\K(K

- Kn)\\ I K H ) .

(4.5.7)

We shall now prove that {||txn||} is bounded. Note that the restriction of A — K in the subspace V={v:{v^)

(4.5.8)

= 0}

is a bounded reversible operator, and u — un G V, hence, by using (4.5.6),

II«-«„II < C(|A - A„| ||u n || + ±-\\{K - Kn)Kn\\

IKH),

where C is a positive constant independent of n. Due to |A-A„|^0,

\\{K-Kn)Kn\\-+Q,

(4.5.9)

5. SEM for eigenvalues and eigen functions

127

and

ll«»ll
- Kn)\\), - Kn)\\).

(4.5.10a) (4.5.10b)

Theorem 2.3 guarantees the convergence of the Nystrom approximate solu­ tion of the eigenvalue problem. If the kernel is symmetric: Ar(x, y) = k(y, x), then the corresponding integral operator K is self-conjugate, which implies that the eigenvalues of K are real numbers, and the set of eigenfunctions forms a complete orthogonal system in L2(Q). In order to apply the SEM to the eigenvalue problems, suppose that Cl = [0, l] 5 , and the Nystrom approximation Kh of K is defined by (4.3.3), then the approximation eigenvalue problem is Khuh

= Xhuh.

(4.5.11)

The following theorem gives SEM for eigenvalue problems of integral oper­ ator with symmetric smooth kernel. T h e o r e m 5.1. Suppose that ft = [0,1] 5 , the kernel k(x,y) of the in­ tegral operator defined by (4-5.2) is symmetric, and Ar(x,t/) = k{y,x) G C 2 m + 2 (fl x ft). Again, suppose that X is a simple eigenvalue of K, and u, uh are eigenfunctions corresponding respectively to X and Xh, then there exist constants aa and functions va(x)y which are independent of h — (/ii, • • •, hs), such that the following asymptotic expansions hold: Xh = X+

Yl

a«h2a + O(h20m+2)

(4.5.12a)

l<|a|<m

and uh = u+

£

a

2 a

+ 0(C+2).

(4.5.12b)

l<|a|<m

Proof. First, when / = 0, the estimate (4.5.10) implies that \Xh - X\ = 0{hl) and \\u - uh\\ = 0(hl). Second, suppose that for / < m, there exists an asymptotic expansion up to h2a (\a\ — I). Now we shall prove

128

IV. SEM for integral

equations

that for the case / + 1, there also exists an asymptotic expansion up to h2a (|a| = / + l ) . Consider (A* - Kh)(u - uh) = Xhu - Kh u

\u + {\h = (K-Kh)u

-X)u-Khu + (\h-\)u

=

=

wa(u)h2°+

£

£

a*fta«ti + 0(ftg , + 4 ).

(4.5.13)

Here we use the asymptotic expansion (4.3.7), and assume

\h-X=

«aft2or + 0(/»o'+4).

J2

(4-5!4)

1<|«|<J+1

where {a£} are undetermined constants. Construct an operator equation tofind{5*,o*}, (Ah - Kh)v* = wa(u)+Z*u,

(4.5.15a)

(wa(u) + ahau, uh) = 0.

(4.5.15b)

{Ah,w'1} is the simple eigenpair of Kh, which guarantees that the system (4.5.15) has a unique solution. It follows that a* = - ( « ? « ( « ) , « ) .

(4.5.16)

Now construct another operator equation for finding { v a , a a } , (A - K)va = wa(u) + aau,

(4.5.17a)

(iZa(u) + aau,uh)

(4.5.17b)

= 0.

The condition (4.5.17b) implies a a = -(ffi a (ti),ti),

(4.5.18)

and the solution va exists uniquely. By the inductive assumption and wa(u) £ C 2 m + 2 ~ 2 ' a '(fi), there exist constants bap, which are independent of h, such that a*-aa=

^ E l<|0|
- J 2/l . 2 /+ ? _0(fc? i . n ^ +4 + --2|tt| H ). »«/ift

h

2/

4

2

(4.5.19)

5. SEM for eigenvalues and eigen functions

129

By (4.5.19) and Theorem 3.1, there exist functions wap{x) independent of h such that

?*-*« =

^ f t w + o(fc?+4-aw).

£ l<|/?|<J+l-<*

Substituting these results into (4.5.13), it becomes

(\h-Kh)(u-uh-

vhah2a)

]T l<|a|
h

h

h

= (\ -K )(u-u -

Yl

^h2a

l<|a|
-

E

E

l<|a|
»<*Aa(fl,+')+o(Ay+4))

= 0(fc* + 4 ).

(4.5.20)

Noting that = 0 ,

which implies that the left-hand side of (4.5.20) belongs to Vh — {vh : (vh,uh) = 0}, so that the restriction of (Xh — Kh) on Vh is invertible, and the inverse operator is uniformly bounded. Hence, by (4.5.20),

u - uh -

v°h2a

Yl 1<|<*|<J+1

- E

E ».**<*»

l<|a|
Grouping the terms with the same power offt,we obtain

uh - u -

Yl

l<|a|
v h2a

°

= °(hll+4)-

(4.5.21a)

In addition, substituting (4.5.19) into (4.5.14), it follows that \

h

- \ -

]T

aah2a = 0(hll+4).

(4.5.21b)

1<M
Therefore, the asymptotic expansion holds for / + 1.

□

The symmetry assumption on k(x,y) in Theorem 5.1 is unessential. It can be proved that the asymptotic expansion (4.5.14) also holds for

130

IV. SEM for integral

equations

nonsymmetrical kernels. But the standardization condition (4.5.5) makes the proof more complicated. In the following, only the expansion containing the h2 terms is proved. Theorem 5.2. If ft = [0,1]* and the nonsymmetric kernel k(xiy) £ C 4 (Q x Q), then there exist constants a,- and functions Vj(x) (i = 1, • • •, s), which are independent of h, such that 5

XAh - X T a ; / i ? + O(ftJ), A A = ]53««*? O(^),

(4.5.22a)

ti — = i1

and 5s

uh-u- u = Y^vihl

+ °(ht)-

(4.5.22b)

t=i

Proof. Apply the method of undetermined coefficients. First assume that (4.5.22) holds. Evaluate 55

5 5

5 5

K\u + J2 viht) vih}) ~(\ + J2aih?)(u + J 3 vihf) Vihf) K\U+53 - (A+53 aih?)(u+53 «= 1 «=i

1= 1 »=i

5

1= 1 »=i

5

5

2 iKhVi - ^ Y,hfcLiU X]Th v{ ++ O(Aj) = Khuu - Xu + JY^h T &?#**,ft?a,-ti - A ] T2ft?t;,O(Aj)

i=i

*=i

t=i

55

fc 2 = ti -- Ku) (KhhVi iU = ifti Ku -- Ati Xu + + (# (Khu Ku) + + Y Y hh1(K Vi -- a am - Xv Xv{{)) + + O(Aj) 0(h*) 55

5 5

i=i

»=i

ii = = ii

h h = Yh ViVi -- aiauiU- - Xvi) X )2iWi{u) h2iWi{u) ++ ]Thf{K ] T h2(K At;,-)++ 0(ht) OfaS) 5*

5 5

i=i '=1

»=i

= 53 ] T A? (Xi;,. -- A Xvi T fc?(lif*«,iU ++ffii(ti)) = *?(#«< w<- -aa,« £,(«))++] 53 *?(#*«<- - lift;,-) #»i) +0(ht). +o(K).

(4.5.23)

Choose Vi and at- (i = 1, •• •, s), satisfying the operator equations Kvi — Xvi=—a,u aiU Wi(u), - Xvi -— Wi{u), (atu Wi(u), V > ) = 0, i ==l ,1,. .• -•, •, (aiu u?i(ti), t/>) « .s.

(4.5.24a) (4.5.24b)

Since A is the simple eigenvalue of Ky then (4.5.24b) guarantees that the solution Vi satisfying (viy V>) = 0 exists uniquely, and a, (wi(u),i>). ai = {wi(u),xl>).

(4.5.25)

5. SEM for eigenvalues and eigeniunctions

131

Due to u e C 4 (ft), it follows that Wi(u) £ C 2 (Q), and so v.- G C 2 (fi), hence \\Kvi-Khvi\\

(4.15.26)

t = l , . " s,s.

= 0{hl),

It follows from (4.5.24) and (4.5.26) that 5

5

h

K (u + J2 vihf) - (A + j ; o,-fc?)(u 1= 1

t=l

1= 1

(4- 5.27)

= o(^). 1

1

Note that A' and tx' satisfy Khuh -\huh

= 0.

(4- 5.28)

Subtracting (4.5.27) from (4.5.28), we have s

Kh(uh -u-J2

vih?) - \h(uh - u »=i

_(A*_A-X)a,-fc?)(« + E i«^?) 1=1

1= 1

(4- 5.29)

= 0(h*). Let V'/l he the standardized eigenfunction of the eigenproblem

(&-(Khy)*ph = o. Acting the operation (-,il>h) on (4.5.29), one obtains s

s

1=1

1=1

O(fcJ).

(4-.5.30)

Since V,/l —► Vs then 3

(u + 5 3 Vih2i^H)

"" (w' ^) = 1, as ^

-o,

i=i

which gives 5

A " - A - £ a , / i ? = 0(/i4). 1= 1

Substituting into (4.5.27), it leads to s

{Kh - \h)(uh - ti - 5>?«.-) = C ( ^ ) . 1= 1

(4 .5.31)

IV. SEM for integral equations

132

Note that Kh - Xh in the invariant subspace Vh = {vh : (vh, ^h) = 0} is invertible as well as uniformly bounded. Although s

uh-u-y52h?vi

= g<£Vh,

«=i

the spectral project of g on Vh satisfies 9-

,

h

lhu

€Vh.

And it follows from (4.5.31) that

9-r^

u h

=0 ^ -

(4 5 32)

--

By using the standardization condition (4.5.5) and (vt, V>) — 0, one obtains <<7,V>h) (uh,i/>h)

=

i + ^ i (u\i>») n

-I {uhAh) = o(ht).

Substituting into (4.5.32), it becomes s

g=

Uh-U-Y,hiVi=0(hi). 1=1

The proof of (4.5.22) is complete.

D

6. SEM for integral equations with nonsmooth kernels Theoretically, the extrapolation methods and the SEM for integral equa­ tions with nonsmooth kernels are far from perfect. Q. Lin and J.Q. Liu''"' have discussed the extrapolation algorithms of the one-dimensional Green's kernel, which is now introduced as follows: Consider the one-dimensional integral equation «(:<)- ■ f k(t,T)u(r)dT Jo

= g(t).

(4.6.1)

6. SEM for integral equations with nonsmooth kernels

133

Suppose that 1 is not an eigenvalue of the integral operator, &(£, r ) is a Green's kernel, i.e., (4.6.2)

*(t,r) = { k2(tir))

r>t,

where fci(*,r) and k2{t,T) are smooth functions, and ki(t,t) = k2(t,t). These facts show that fc(£,r) is a continuous function on [0,1] 2 , but its derivatives are discontinuous at t = r. Rewrite (4.6.1) as u(t)-

I i i ( t , r ) t i ( r ) d r - I k2(t,T)u(T)dr

= g(t).

(4.6.3)

Take the mesh width h\ = 1/n and the nodes tj — Tj = jhi. Construct, by using the trapezoidal rule, the Nystrom approximation of the integral operator K

»" *»/(*) = - £ ^ . ^ / ( T J ) ,

(4.6.4)

with //

£ *(*.*i-)/fo) j=0

+k(i, r i ) / ( n ) + • • • + *(t, r n . i ) / ( r n _ i ) + - i ( t , r n ) / ( r n ) and the approximate equation ti„ - i £ n u n =flf.

(4.6.5)

The values {u n (£,)} of un(x) at the nodes are obtained from the linear equations >

Un(U)

hfc,

Tj)un(Tj)

Tj
Yl Tj>ti

k2(ti,Tj)un(Tj)

= g(ti), i = Or--,n.

(4.6.6)

134

IV. SEM for integral equations

Now we shall prove that there exists a function w(x) independent of n, such that «»(*<) = «(**) + HU)/n ™(ti)/n2 2 + 0(n-4),

(4.6.7)

i.e., at the nodes <,• (i = 0, • • •, n), the approximate solution can reach the order of error 0(n~4) by extrapolation. Note that 1 n" " "(*«)- ~nrrE HU,Tj)u(Tj) nrr n r,11 j = 0 I! n ""

T T = 9(ti) = // tk(ti,T)u(r)dT--J2 ( « i , r ) « ( r ) d r - - V tHti, f t . }M ^ u fi)o ) ++ $(*,•)

= { //

fci(^,r)u(r)drfcl(ti,T)«(T)dT-

+{ /

- ]T Tj )U(T,-)| T fc^tj, fci(tj,Tj)«(Tj))

*k22(U,T)u(T)d,T (*i,r)«(T)dT

- r V^ M*« h(ti,Tj)u(Tj)\ Y) > i)«(rj)i

+g(ti), •••,!», +$(*•)> «'t = 0, ••-,«,

(4.6.8)

and the Euler-Maclaurin expansion, U f/"*' 11 vv-^" ^" r fci(ti,r)w(r) *i(*i> i)"( T i) // *i(f,-,r)u(T)dTti^.-.Tj-H^-)

=

T -ii^^^'^))|;:%o =-is?^^' ^))i;ro+o((Bn-), -4), Z"1 / k2(ti,T)u(T)dT /

1 v-^ " >

fc2(t,-,r)u(r)dr--

=

1

d

-Y^TM^

> T)U{T))\

k2{U,Tj)u(Tj) k (ti,TJ)U(TJ) |T =2 1

+ 0(n-- 4 ) .

let let let w(t) = - ^ [ s : ( M ' * , r M r ) ) | ; % | : ( * a ( « * , r ) t , ( r ) ) | ^ 1 ] >

5(0=

~iyi^ i( '*' r)u(r) C + ^ (fc2(
substituting into (4.6 .8), one obtains

1 "" substituting into«(4.6.8), ( * * ) - one - £ obtains *:(<»> T;)u(r.z) nrrn 1 w(tt) " =
(4.6.9)

= *(*<) + - ^ 2 i + 0 ( n - 4 ) , t = 0 , - - . , n .

(4.6.9)

j=o

6. SEM for integral equations with nonsmooth kernels

135

Since w(x) is a smooth function, the solution w of the integral equation (I-K)w = w

(4.6.10)

(K-Kn)w = 0(n-2).

(4.6.11)

is also smooth, and so

Hence, (I-Kn)(un = 9-(I= 9-(I= g-(I-

-u + n~2w) Kn)u + n-\l - Kn)w Kn)u + n~2(I - K)w + n-2(Kn Kn)u + n~2w + 0 ( n " 4 ) .

- K)w (4.6.12)

Let t — ti (i = 0, • • •, n) in (4.6.12), which forms a linear system with un­ knowns v(U) = un(ti)-u(ti)-n~2w(ti) (i = 0, • • •, n). Kn ^ /^ guarantees that the inverse of the coefficient matrix of (4.6.12) exists and is uniformly bounded and (4.6.9) shows that the right-hand side of the system is of 0(ji~4)y hence, un{U) - u(U) + n~2w(ti) = 0{n~A),

i = 0, • • •, n.

(4.6.13)

T h e o r e m 6.1. If the kernel of the integral equation (4.6.1) is the Green kernel defined by (4.6.2), and one is not an eigenvalue, then un(ti) (i = 0 , - - , n ) , the solution of the linear system (4.6.6), have the asymptotic ex­ pansion (4.6.13). Theorem 6.1 guarantees that the nodal values can be extrapolated, while it may not be true for the non-nodal values determined by the Nystrom interpolation (4.2.20). A detailed analysis can further prove that (4.6.13) has a higher order provided ki(x}y) and £2(2,2/) are smooth enough. The proof is omitted here. The disadvantage of the global extrapolation mentioned above is the large amount of work involved in obtaining the approximation u-in* The SEM reduces the amount of computation and storage, and it can be im­ plemented on multiprocessor computers in parallel. For this aim, divide r

the interval [0,1] into r subintervals: [0,1] = (J /,-, U = [Tv_i,rt-], where »=i

0 = ro < Ti < • • < Tr = 1, i = 1, • • •, r, and I,- are not necessary isometric. In /,-, insert (nt- — 1) isometric nodes: TtJ- = T,_I -fj/i, (j = 0, • • •, n,), hi = (T,- — T,-i)/n t -, i = l , - - , r are r independent mesh widths. Define the

136

IV. SEM for integral

equations

Nystrom approximate integral operator as rr m ni "" Kh*f{t) * /« = E 5 3E 5 3 WKt,7*i)/(7y). *.-*(*, T ^ / t e ) .

(4-6.14) (4.6.14)

t=lj=0

tij = r , j , (i = 1, • • •, r; j = 0, • • •, n t ), and {ti {uhfc(ti(*j Let ty m)} m )} satisfy the linear system // -- ll ni ni 'I a

h U hik1(tim,Tij)uhhfaj) (Tij) Uh(Um)-^2J2 (*Jm)-53X3 Ml(*lm,^>

»=1 j=0

m n, m" " m "" t J 3 ^ iM m^, i^^) f^a^j ) - 5 3 ft'*2( Im,^j)lifc(77J-) -5 f a^m l*2(*lm,T>j)u*fa) j—0 j=0 j=m j=m r

ni n< a "

- 5 3 1 3 ftf*2(tIm,nj)wfc(rO')=^(t/m), =00 t = /J++lUj =

/ = 1, •■•,!•; m = 0,---,n/.

(4.6.15)

Noting that {u($jm)}> the values of the solution u(x) at the nodes, satisfy rr

ufam)

m ni II" fc

- 53X3 tt = = ll jj = = 00 JJ -- ll

,,

" " ii " "

fc fc i(^m,^)w(r)c?r-53

= = 5 53 3^ ^ // t=l

«'M*fm,T,j)utej)

i(^m,^)w(r)c?r-53

^

A A «'*i(*im,njMrij)}

«'*i(^m,ni)ti(rij)}

i=0

r)u(r)dr - 5 3 &i(
/hlkl l/ 1

T ^ (*/m' ^i V)} ^ ))UU((^J)}

i=o 0

**2(*Jm,r)ti(r)dr-53 2 ( * J m , r ) t i ( r ) d r - 5 3 ft/fcft2(^m,rzj)ix(r/j)} /fc2(^m,rzj)ix(r/j)}

+{ / r

.

m»

+ 5 3 { / M*Jm,r)ii(r)dr-53 i=J+l

^

A

»'*2(*Jm,njMr,-,-)}

j=0

+0(*/m), / = 1, • • •, r; m = 0, • • •, n,.

(4.6.16)

Applying the Euler-Maclaurin asymptotic expansion on every term on the right of (4.6.16), we have r w

ni a

(^m)~53X3 t = l j=0

^K^mjTij^Tij)

6. SEM for integral equations with nonsmooth kernels ,_1

=

137

1 d ft ?

E-i2 « ^N*'m'rH'-)]|

+

tim T)u

-^'{^^^^t., i^ ' ^0 r

1

fi

tt

- E ^ ^ V N ^ . ^ M ] ' +^/m) + 0(^), (4.6.17) •=/+i

i Z

a r

u<

~1

with ho = maxft,-.Let l
-j2^[klit'T)uiT)] l t i - i >

'

for t > t{,

~^Mt,rHr)} l t i - l > for t < U.lt

Wi(t) = <

(4-6.18)

-B0^ [ i b l ( t ' T ) u ( T ) C- 1 + | ^ < ' r ) u ( r C } for tj_i < £ < t{,

z = l , - -,r.

i

Obviously, W{(t) are functions independent of h. Therefore, by (4.6.17), I

ni "

»=1J=0 r

= ^«!.(<;m)ft, ? + ff(tim) + O(ftJ).

(4.6.19)

t= l

Using the argument similar to that used in proving Theorem 6.1, we can prove the following asymptotic expansion for splitting extrapolation. T h e o r e m 6.2. If the assumptions of Theorem 6.1 are satisfied, and [0,1] = U I,-, /,- = [r t _i,r t ], where 0 = r 0 < 7i < • • • < r r = 1

(4.6.20)

is a fixed partition, hi = (r,- — r,_i)/n,- (i = 1, • • • ,r) are isometric refine­ ment mesh widtf 15 of I{, and {uh(tim) : / = 1, • •, r; m = 0, • • •, n/} is the

138

IV. SEM for integral

equations

solution of the Nystrom equation (4.6.15), then the nodal values of uh(t) have the asymptotic expansion

uh(tlm) - u(tlm) = J2wi(tlm)h? + 0(h*),

(4.6.21)

»= 1

where Wi(t) (i = 1,• • •, r) are functions independent of h. Furthermore, if i i ( t , r ) G C 2 m + 2 ( Q i ) , k2(tyr) G C 2 m + 2 (ft 2 ), ^ i = {(*, T) : 0 < t < 1, 0 < r < t < 1} and ft2 = {(*, r ) : 0 < t < 1, 0 < * < r < 1}, one can then obtain the asymptotic expansion of higher order: Uh(tlm)-U(tlm)=

]T

^ ( ^ m ) f t 2 a + O(/i 2 0 m + 2 ),

l<|<*|<m

where wa are functions independent of h. If ii(-, •) and k2(-,-) are nonsmooth on ^ i and Q2l for instance, k(t, r) = \t- r|"tf(t, r ) , 1 > /i > 0,

(4.6.22)

where H(t, r) is a smooth function on [0,1] x [0,1], then *!(*, r ) = (t - r)"ff (t, r ) , 1 > * > r > 0,

(4.6.23a)

k2(t, T) = (T- t)»H(t, r ) , 1 > r > * > 0.

(4.6.23b)

and

Obviously, the derivatives of ki(t, r) and k2(t) r ) are not continuous. How­ ever by using the Euler-Maclaurin expansion (1.2.38) with the singularity at the end points, one can still obtain the following: T h e o r e m 6.3. If the kernel of the integral equation (4.6.1) is defined by (4.6.22), 1 is not an eigenvalue, and (u n (i,)} is the Nystrom approx­ imation determined by the equation (4.6.6), then there exist functions w^{t) and Wi(i), which are independent of n, such that un(U) - u(U) + w^n-v-1 i = 0,

-,n.

+ w^n-2

=

0(n"2^^), (4.6.24)

6. SEM for integral equations with nonsmooth

kernels

139

Proof. By (4.6.8), 1 " "

"(*'•)-~y) *(**> Ti)u(ri) = { /

(*,- - T)»H{h, r)u(r)dr

- V

n-^U

,i

- r^HiU, T3■Mrj)}

» "

+ { / ( r - < , - ) " f f ( * i , r ) u ( T ) d r - 5 3 n " 1 ^ - - *,)"#(*i > T j) u ( r i)} 7 —*

+9(U) (4.6.25)

t = 0, • -,n.

= Ji + J2+9{ti),

By using (1.2.39), we have Ji = n - " - 1 C ( - | i ) / r ( t i ,<,>(<,) - n - 2 C ( - l ) | - [ ( * i - r)"i/(<,-, r)ti(r)]| + 0 ( n "2 " " ) , at/ lr=o /2 = - n - " - 1 C ( - p ) f f (<<,«.■)«(«.•) + n - 2 C ( - l ) ^ - [ ( r - ti)"H(tit oy

T)U(T))\

lr=i

+ 0{n~ 2 - / A

Let fi^t)

=

-2<;(-f*)H(t,t)u(t),

wtf)

= -C(-l){|-[(*-ry'tf(*,r)U(r)]|T=o

_-![(<-,-)"#(*, r)U(r)]|

}.

Substituting into (4.6.25), we obtain «(*.•) -

=

1 1

n ^

nn

"

k(U, Tj)u(jj)

Wfx{U) +

1

i? m(U) + 9(U)

+ 0 ( n - 2 - ^ ) , i = 0.--.,n.

(4.6.26)

Let u;^ and wi satisfy respectively the following equation: (I - K)w^ - uip, (I - K)wi = w\.

(4.6.27)

IV. SEM for integral

140

equations

It is easy to prove that (K - Kn)wp = O f a - 1 - " ) , (K - Kn)w1 = C ^ n " 1 - " ) . Hence, (I - Kn)(un - u + n " 1 - ' 4 ^ + n" 2 tx;i) = < / - ( / - tfn)u + n - 1 - ^ / - X ) ^ + n " 2 ( / 1

=

g

K)w1

i

+n- ~> (K - X n ) ^ + rT\K - 2Cn)wi - (/ - 7
(4.6.28)

On both sides of (4.6.28), let t = U (i = 0 , - - , n ) . By (4.6.26), each component of the column vector on the right is 0 ( n ~ 2 - / i ) , but Kn —+ K guarantees that the inverse of the coefficient matrix on the left exists and is uniformly bounded, which implies that (4.6.24) holds. □ Corollary. If the kernel function is k(t, T) = \t - r|" In \t - r\H(t, r ) , 0 < \i < 1, then there exist functions w^^^w^t) n} such that

(4.6.29)

and w(t), which are independent of

un(ti) - u{U) + Wnitfin-"-1 Inn + wpL(ti)n'fA"1 + ^ i ( ^ ) n " 2 = 0 ( n - ^ 2 ) , i = 0, -,n.

(4.6.30)

Proof. It follows from the expansion (1.2.58) and the proof is similar to that of Theorem 6.3. D By means of induction which has been used repeatedly, the asymptotic expansion up to a higher order can also be obtained, provided that H{t,r) is sufficiently smooth. Further, Theorem 6.3 can be extended to higherdimensional integral equations, provided that the kernel can be expressed as

*(*.*) = n; = i {l*i - ViHlnte -% I)*'}#(*,
(4.6.31)

where x = (x x , • • •, x,), y = (y1} • • •, y 5 ), 0 < /xj < 1, Xj = 0 or 1. By using the asymptotic expansion (3.4.8) of the quadrature formulae, one can obtain the multivariate asymptotic expansion of the Nystrom approximation at nodes. Obviously, the SEM can improve the accuracy of the approximate solution. For the one-dimensional nonsmooth integral equations, Theorems 6.2 and 6.3 can be generalized to the following:

6. SEM for integral equations with nonsmooth

kernels

141

Theorem 6.4. Suppose that the kernel of the integral equation (4.6.1) has the form of (4.6.22), and one is not an eigenvalue of the integral op­ erator. Again, suppose that the nodes {tim : / = 1, ••-,?•; m = 0, — , n r } and the mesh width h = (/ii, • • •, hr) satisfy the assumptions of Theorem 6.2, then there exist functions W{(t) and w t/i (t), i = 1, • • •, r, such that the values of the Nystrom approximate solution uh(x) at the nodes have the asymptotic expansions uh(Um) - u{tlm) =

w^(t1m)h}+ti

•=1

for 1 < / < r, 0 < m < nu

(4.6.32a)

and uh(tj) - u(U) = wi-ltr(ti)ti±»

+

w^(U)h^

r

+ Y^Wi(tl)h? + 0(hl+>i)) / = l,..-,r. (4.6.32b) t=i

Proof. Since tim £ Ii (i / /), then fc(J/m,r) is smooth on Jt- (t ^ /), and nonsmooth on //. On the right-hand side of (4.6.16), using the EulerMaclaurin expansion for the error of the numerical integration on 7t- (i ^ /), while using the expansion with singularity at the end points for the error of the numerical integration on I\ = [2/,2/m] U[*'m>*i+i]j (4.6.32a) follows from the same argument used in proving Theorem 6.3. Similarly, since tj £ 1^ (i z£l — 1, /), k(ti, T) is smooth on 2, (i ^ I — 1, /) and nonsmooth at the end points of J/_i and //. Hence, on the right-hand side of (4.6.16), using the Euler-Maclaurin expansion for the error of the numerical integration on Ii (i z£ /, /— 1), while using the expansion with singularity at the end points of //_i and /,, (4.6.32b) holds. D The previous analysis shows that the Nystrom approximate solutions of the integral equations with nonsmooth kernel defined by (4.6.22) or (4.6.29) has a lower order of error, which is 0(/iJ + / i ) or 0(/iJ + / i | ln/i 0 |) respectively; and after extrapolation is 0{h2^^) or 0 ( / I Q + / 1 | ln/i 0 |) respectively. Kantorovich and Krylov^ proposed an algorithm with higher order of error, called the modified quadrature method. Its basic idea is to describe the integral equation (4.6.1) in the equivalent form [l - / Jb(t, r)dr\ u(t) + / .*(*, r) \u(t) - ti(r)] dr = g(t), (4.6.33)

142

IV. SEM for integral

equations

and to construct, by the quadrature rule, an approximate equation ,1

1

n »

[1 - / k(t, T)dT]u(t) + -J2 Kt, Tj)[un(t) Jo nfT'o

«B(TJ)]

=

9(t),

or equivalently, 1 " " [1 + - V k(t,TJ)-

f1

k(t,r)dr]«B(0

1 n " — Y,Kt,Ti)un(Tj)

= g{t).

(4.6 .34)

Take J = r,- (i = 0, • • •, n), then (4.6.34) is converted into linear equafrions of the nodal values {w n (r,), i = 0, - , n } . After getting {u n (r,)} and substituting into (4.6.34), one can obtain

^ (f\ —

1 n " 9(f) + -^2 Ht>Tj)Un(Tj) i5° n

j=o

(4 fi.35)

J

°

The modified quadrature method was studied by Baker'"!, Anselone' ' and Chatelint 20 !. Intuitively, the approximate equation based on (4.6.33) has higher order of error. In fact, if the solution u(t) is smooth, then the expansion of the quadrature formula is /

k(t, r)[u(t) - u(r)]dr - -J2 =

-iS?|:W''r)W«)-tt(T))]r;i + o ( ^ 7 ) ,

because —[k(t,T)(u(t) <9T

*(*, Tj)[u(t) - ufo)] (4-6 .36)

— u(r))] is not only continuous but is also of the

form \t-r\»H{t,T). In order to discuss the convergence and stability of the approximate equation (4.6.34), (4.6.35) can be expressed in the operator form (4.61.37)

(I-Kn)un=g, with Kn = Kn - (Knv - Kv)I,

v e C[0,1], v = 1.

(4.61.38)

7. SEM for ID weakly-singular integral

equations

143

Kn are denned by (4.6.4). Since fc(x, y) is continuous on [0,1] x [0,1], then Kn ^ K, which implies that Kn -^ K, and \\Kn - Kn\\ = \\Knv - Kv\\ - 0, (n -> oo).

(4.6.39)

Consequently, iCn —► I<\ Hence for sufficiently large n, (I — Kn)'1 exists and is uniformly bounded, which proves the existence and uniqueness of the solution of the modified quadrature formula (4.6.34). Repeating the argument used in the proof of Theorem 6.4, one can derive the SEM for the approximation obtained from the modified quadrature method. T h e o r e m 6.5. Under the assumptions of Theorem 6.J^, the solution of the modified quadrature method, at the node tjm has the following asymp­ totic expansion: r

h

u (tlm)

- u(tlm) = £ > ( t i m ) f t ? + O(h30+^).

(4.6.40)

»=i

/ / the kernel has a logarithmic singularity, described in (4.6.29), then the remainder of (4.6.40) is of 0(/io + / i |ln/io|)7. S E M for collocation-method solutions of one-dimensional weak­ ly-singular integral equations Consider the one-dimensional weakly-singular integral equation u(t) - / Jb(*, r)u(r)dr Jo

= g(t),

(4.7.1)

where the kernel k(t> r) is a function of the form jb(t, r ) = |i - r\aH{t, r ) , - 1 < a < 0,

(4.7.2a)

k(t, r) = \t - r\a In \t - r|, - 1 < a < 0.

(4.7.2b)

or

Since fc(t,r) is discontinuous on [0,1] x [0,1], the approximation operator {i£ n } derived from the quadrature method does not collectively compact converge to K. The numerical results show also that the approximate solutions obtained by the quadrature method diverge. At present, the main methods for solving the weakly-singular integral equation are as follows: a) the collocation method, b) the product integration method,

IV. SEM for integral

144

equations

c) the modified quadrature method, and d) the Galerkin method. The Galerkin method has a higher order of error. However, the gener­ ation of the elements in the coefficients matrix involves the evaluation of multiple integrals, it leads to a larger amount of work and so may make this method impractical. The principle of the product integration method is similar to that of the collocation method. In this section, only the collo­ cation method and its extrapolation are introduced. Both the collocation and Galerkin methods can be brought under the framework of the projection method. The difference between them is that the former bases on the interpolation projection, and the latter on the orthogonal projection. In order to construct the interpolation projection, construct the first initial partition of the interval [0,1]: 0 = r 0 < r i < . - . < r r = 1.

(4.7.3)

r

Let I{ = [ri-i, rj], I = |J /,-, and refine the fixed interval I,- (i = 1, • • •, r) »=i

by the mesh width hi = (r,- — r t _i)/n,-. Let r,j = r,_i + jh{ (i — 1, • • •, r) and regard ft,- as an independent mesh parameter. The set {T,J} forms a partition on [0,1] : 0 = T0 < Tn < . . . < r l n i = n < < r r _ i < rr>1 < • • • < rr = 1.

(4.7.4)

h

Let S [0,1] be the set of the piecewise linear continuous functions under the partition (4.7.4), and denote by u1 = nhu the interpolation function of the continuous function u on S^O, 1], i.e., Hhu G S^O, 1], and rtufaj)

= ii(r t j ), i = 1 , . . . , r; j = 0, • • •, n,-.

(4.7.5)

Obviously, the linear operator is a projection operator and is called the interpolation projection. Applying nh, one can obtain a collocation solution uh(x) G S^O, 1] from the following operator equation: uh - nhKuh

= nV

(4.7.6)

The following lemma guarantees the existence of uh and the stability of the equation (4.7.6). L e m m a 7.1. If (I — K)'1

exists and is bounded, and ho = max A,l<*
is sufficiently small, then (I -

C[M].

rfK)'1

exists and is uniformly bounded in

7. SEM for ID weakly-singular integral equations

145

Proof. By Lemma 1.5, K is a compact operator. Obviously nh A 7.

(4.7.7)

It will be proved that (4.7.7) implies || nh K - # | | — 0, (ft0 — 0).

(4.7.8)

In fact, suppose that B is the unit sphere in C[0,1], and KB is a compact set, then for any e > 0, there exists a finite ^-system {vi, • • •, v,-}. By (4.7.7), there exist 6 > 0 and when ho < 6,

I K - n ^ H ^ , i = i,...,jv.

(4.7.9)

Take arbitrarily w £ B, then there exists v,- such that ||/^-t;t||<e.

(4.7.10)

Hence,

|| n" Kw - Kw\\ < ||(i - n > , | | +1|(7 - r\h)(Kw -

Vi)\\

< ( 2 + ||n"||)£.

(4.7.11)

According to the Banach-Steinhaus Theorem, (4.7.7) implies the uniform boundedness of {||n/l | | } . Since e can be arbitrarily small, then (4.7.8) holds. The conclusion of Lemma 7.1 follows from Theorem 1.1 and its corollary.

□ We now convert the operator equation (4.7.6) into a linear system. Let {^fj } be the basis functions of 5^[0,1]. That is,
{

1,

for i = /, j — m,

0,

otherwise,

i = l , - - - , r ; i = 0,...,n,-.

(4.7.12)

Set Mh

r

r»i

W = EEa*iW(0-

(4-7.13)

Substitute into (4.7.6) and obtain the linear system with respect to {a,j} : dim ~ ]T) ILS aii /

k Tlm T (

(

' ) Pij(T)dT

7=l,...,r; m=l,..,n,-.

= 9{T\m)> (4.7.14)

IV. SEM for integral

146

equations

After solving {a t J } from (4.7.14), one obtains the collocation-method so­ lution uh(t). Further by using (4.7.1) and uh(t), one can also obtain an iterative solution uh(t) = g(t) + I uh(r)k(t, r)dr. (4.7.15) Jo Obviously, (4.7.14) shows that at the collocation points, uh(t) = uh(rim). In the Chatelin's book!20', (4.7.15) is called the Sloan iteration. For the Galerkin method, Sloan and Thomee^ 122 ' has expounded in detail that the Sloan iteration can improve the accuracy, but for the collocation-method solution, it is not the case. What we are interested in is whether the extrap­ olation method can improve the accuracy. In the case of smooth kernels or the piecewise smooth kernels (the Green kernel), Q. Lin and J.Q. Liu gave a positive answer. Concerning weakly-singular kernels, F. Chatelint 20 } said: "If the kernel is not smooth, the method is no longer founded theo­ retically." The following lemma states the fact that the difference between Ehv, the orthogonal projection from L2(Q) to 5^(0,1), and the interpola­ tion projection \lhv has an asymptotic expansion. For details, see the work of P. HelfricrJ44!. L e m m a 7.2. Assume that v G # 4 ( 0 , 1 ) , then the asymptotic expansion E»v

-nhv = -J2 ^ f ( f t V ' ) + q(v) t=i

(4.7.16)

iZ

holds, where E^ : L2(I{) —► Sh(I{) is an orthogonal projection, rji is the characteristic function on /,-, and the remainder q(v) has the estimate ll«(«)IU»(o,i) < Cl4\\v\\Ht(0il).

(4.7.17)

Proof. By the Euler-Maclaurin expansion (1.2.12), the integral on the interval J,j = [Ti,j-iiTij] (* = 1> *, r ; j = 1, •••,»!,•) has the asymptotic expansion

/ w(i)dt = —'- f w"(t)dt +/i 4 /

wW{t)PA((t

- Ttj-i)/hi)dt,

Vw G # 4 ( 0 , 1 ) . (4.7.18)

Let w(t) = (v- vT(t))iP(t), where v G # 4 ( 0 , 1 ) , v1 is the interpolation of i>, and ip G Sh[0,1]. Substituting into (4.7.18), we have

/

Jin

(vW-v'itfimdt

7. SEM for ID weakly-singular integral equations

147 147

= -ii/./ww* = -fnJj.v"imt)dt

™-jf k w - ^ w w * +hf [t;<44>(<)W) + 3t; 3»<(s)8(<)^(*)] >(<)^(t)] +hf f/ [»< )(<)^(0 + •fry ■Pi((t 7ij-i)/hi)dt. •P4((* - nj-i)/hi)dt.

(4.7.19)

Noting that the second term on the right-hand side vanishes, and by (1.2.6a), PP4((t nj-i)/hi) 4((< - TiJ-j/hi)

= hiPi((t hiPL((t -

TiJ^/hi). rij-i)/hi).

Substituting into (4.7.19), and integrating by parts,

/f („(*)-,/(<))(<)<«

•fry

== "-%l ^ / 12 12

Jin Jlij

v"(t)mdt + h} I[ v^\t)i>{t)dt v"(t)mdt+h} j*\twt)dt Jin Jlij

-h* -/if /[ 3v&(t)1>'(t)ft((t-nj-i)/hi)dt. 3«< >(iW(t)P5((t-7ij_i)/Ai)
(4.7.20) (4.7.20)

Hence,

/

Jo ./0

(v(t) -

v^t^^dt

= ("(O-^C))^*)* =E E EE/ / Wt)-At))*W t=l

7 j7== 1 l ^7»i

,=1

i2

-fr.

»=1

-- X > «? ? £E // !> «=1 »= 1

j =

JlJl

-fr*

4 4 7 21 3«< >(Otf'(t)A((*-n,i-i)A0
ll iJ'i

By the inverse estimate^**}, there exists a constant Co independent of /i,-, such that 1 k WIU»(i y ) < Coh^WnvVv), (4.7.22) II^HL»(J«) < Gofer II^IIL»(/«), VV VV- €€ s5*[0,1]. p>, i]. (4.7.22) It follows from (4.7.21) that for any i/> 6 Sh[0,1], 7

h

2

2 7 (v (O,I) = C ^ f - vw'. , V0L (O,I) (t; - vt>7,, V0L V>)i»(o,i) = (^ ^)i 2(o,i)

148 r

V h2

v

12

IV. SEM for integral , ^ ) £ a (A)

rr h? A? = - Yl TZ^id'y

i=l

iZ

equations

+ «(»)

^)L>(0,1)

+ «(f)

c l = = --Et^|f ^ ( ^( v^ ) ^") )L '» ^ ( ow , i ))+ «+ ( v?W. ).

4 7 23 ((4.7.23) -- )

where rji(t) are characteristic functions on I,-, i.e.,

{

1, l,

0,

Ti_i < < < Ti, T»-I << n,

otherwise.

(4.7.24)

Since

l«(«)l

(4) (4) (4) ft <<^Eii« ^ E n« iu»eoiMM'o+ah* E Ew(4) ii«ii^('«)ii^ii^«) iu»(i«)ii^iU'(/y) iU'('oii^iU'(io+^ oEEii *=i r

«=ij=i

r

< ^ ( I*=i > ( 4 ) I I L xi/nizunh ■(/«>)* tt==ii *=i

^hu ± f; n^)|ii,(/y)) * ( E E I iwiWo-)) wiWo-))** t = i ii == i t'=i

<

»=ij=i

(4.7.25)

C^II^H^O.DIIVIU^O.I),

where Ci and C are constants independent of h{ (i = 1, • •, r), then (4.7.25) and (4.7.23) imply

£*„ _ vi = £ _£*( V„'')^L Ehv -v'=J2 -^W ' ) ^ ++ ,(«), «(«), i=i

(4.7.26) (4.7.26)

^

and II«(«)HL»(O,I) II«(«)I|L»(O,I)

4 ^ Cfc CfeJHvllff^o,!). < IM|ir«(o,i)-

The proof of Lemma La 7.2 is complete.

(4.7.27)

D

Under the conditions that v = # 4 ( 0 , l ) n # o ( 0 , 1 ) and the mesh widths satisfy hi = ft2 = • • • = hr, Helfrich^ obtained h\ h A? h l , v = -—E v" Ehvh w-n - H g(v). f? w = - —1Zf 7 h w / / ++ «(»).

(4.7.28) (4.7.28)

7. SEM for ID weakly-singular integral equations

The argument here does not need the assumption v G

149 HQ(Q,1).

Since

r

2_] i]i(t) = 1, (4.7.28) is derived from (4.7.16) when the partition is isometi=i

ric. Lemma 7.3. Suppose that X is a reflexive Banach space, {Pn} is a sequence of projection operators, and Pn —► I for n —► oo, then the conjugate operator P* of Pn is also a projection operator of the conjugate Banach space X* f and P* -+ J,

for n -f oo.

(4.7.29)

Proof. Since Pn is a projection, then P% = P n , which leads to P*2 = P*, and P* is a projection operator of X*. In order to prove (4.7.29), we shall first prove that ~oo

U ft(Pn*) = X*.

(4.7.30)

n= l

where 5J(P,J) is the range of P*. Without loss of generality, suppose ^(P*) C &CPn+i), n = 1,2,--.. If (4.7.30) is not true, then there ex­ ists v, satisfying 0 / v G l = X**. For any tp G X* and any n, there is

(P^v)

=

(^Pnv)=0.

Let n —* oo, one obtains (V>, v) = 0, which implies i; = 0, this is contrary to the assumption v ^ 0. Second, by ||P*|| = ||P n ||, (4.7.29) implies that {||P*||} is uniformly oo

bounded. For every w G (J ^(P^*), there must be a sufficiently large integer i=i

m, such that tu G 3J(P^), hence, P , > = W, n>m.

(4.7.31)

oo

Applying the denseness of (J 5R(P^) in X* and the Banach-Steinhaus Theo»=i

, the proof is complete.

O

Corollary. Under the assumptions of Lemma 7.3, and ifK is a compact operator mapping X onto X, then \\K{I - Pn)\\x

- 0, f o r n ^ o o .

(4.7.32)

Proof. By P* — I (in X*) and p C ( I - P „ ) | | x = \\(I-PZ)K*\\X.,

(4.7.33)

IV. SEM for integral

150

equations

noting that K* is a compact operator of X*, using the argument that was used in the proof of Lemma 7.1, the proof can be completed. □ In the previous theorem and corollary, the reflexivity of X is an indis­ pensable condition. If Pn is the interpolation projection of C[0,1], (4.7.32) is invalid. In the following theorem, the norm || • || is taken in L 2 (0,1). Theorem 7.1. Suppose that u E # 4 ( 0 , 1 ) 25 a unique solution of (4.7.1), u is a unique solution of the collocation equation corresponding to the par­ tition (4.7.4), and u1 is an interpolation of u, then there exist functions Bi(u) (i = 1, • • •, r), which are independent of h — (/ii, • • •, hr) and depen­ dent on u, such that h

r

uh(t) - uT(t) = ^h?Bi(u)

+ 0(Afci),

(4.7.34)

»=i

with €l

= ||(/ - nh)K\\ + ||(I - Eh)K* || - 0, h -+ 0.

(4.7.35)

Proof. By (4.7.1) and (4.7.6), uh - u1 = nhK{uh h

- u) h

= n K(u - u1) + nhK(uJ - u), hence,

(J - nhK)(uh - u1) = nhK{u* - u) = tfKiu1 - Ehu) + nhK(Ehu

- u),

(4.7.36)

where Eh is an orthogonal projection mapping L2(0,1) onto Sh[0,1]. It follows from Lemma 7.1 that (I-^K)-1 exists and is uniformly bounded, hence, uh - u1 = (/ - n ^ ) " 1 n h Kiu1 - Ehu) -{I - nhK)-1

nh K(I - Eh)u

= Ji+J2-

(4.7.37)

First, by Lemma 7.1 and the error estimate of the interpolation of the linear finite element, we have

IWI < ve mf

||(7 - nVO" 1 n" K(i - Eh){u - v)\\


(4.7.38)

7. SEM for ID weakly-singular integral equations

151

Second, by Lemma 7.2, it follows that tfK)-1

Jx = (I -

Hh K{u* -

= (7 - K^Kiu1

-

1

Ehu)

Ehu)

+ [(/- n ^ ) - n* -(I - K)-1)^ = (7 - KYxK(v*

- Ehu)

- Ehu) - (7 - n * # ) - 1 ( J - nh)K(uT

- .Ehu)

- ( / - nh7s:)"1(7 - nh)7v:(7 - K)-1 nh Kiu1 - Ehu) = (7 - K)-1^1 - Ehu) + O(h20\\(I - nh)K\\) = - £ S ( J - Ky^KE^mu) + o(h20\\(i - nh)i<\\) t = l

1 Z

= - £ j | ( J - KY^iviu) + 0{hl\\K(I - Eh)\\) +0(hl\\(i-nh)K\\).

(4.7.39)

Let Bi(u) = — T^(I — K)"1K(rjiu),

i = 1, • • •, r, one obtains

r

(4.7.40) t= l

Corollary. For the nodal values of the piecewise linear collocationmethod solution uh(t), there is the following asymptotic expansion r

u\nj)

~ «(TJ>) = £

h?Bi(u(nj))

+

O(h2o£l).

(4.7.41)

»= 1

Further, the iterative solution uh defined by (4.7.15) can be extrapolated even at non-nodes. T h e o r e m 7.2. Under the assumptions of Theorem 7.1, there is r

uh(t) - u(t) = J2h2Bi(u(t))

+ O(Agei).

(4.7.42)

»= 1

Proof. By (4.7.15),

u-uh

=

K{u-uh)

= K[(I - KYl9 -{I- rfKY1 n" g] = K[(I - KY1 - (J - rtKY^g + K(i - n h 70 _1 ( J - nh)a = K(i - nhi
+ g) (4.7.43)

152

IV. SEM for integral equations

Substituting the identity K(I - nhK)-1

= (I -

Kn^K

into (4.7.43), u - uh = (I - KH^Kil

- n h )u

= (I - KH^KiE11 1

- nh)u + (7 - Knh)~lK(I

h

= (I - K)- K(E

+[(/ - # n*)

Eh)u

-

h

-f\ )u _1

- (7 - K)-x]K{Eh

-

nh)u

+(I-Kr\h)-1K(I-Ehu) = (I-K)-1K(Eh -nh)u -(I - Kn^Kil

- Uh)(I - K)-1K(Eh

fc

--n

)u

+(I-Knh)-1K(I-Eh)u = J1 + J2 + h-

(4.7.44)

By the identity

(7 - AXI*)-1 = I + K(i - nhK)-1nh, one obtains that {||(7— A'n A ) —1 ||} is uniformly bounded, hence there exists a constant M > 0, such that

l|/2||<M||(/-nh)^llll(^-n;'H| =

0(hl\\(i-nh)K\\)

|| Jsll < M\\K(I - Eh)u\\ < M\\K(I - Eh)(u -- « J ) i i <Mp:(j-.Efc)||||t«-t,'|| = 0(hl\\(I-Eh)K*\\). By Lemma 7.2, J

i = - E i|( z - *r1*£*(*•«)+O(AS)

= - E § i(7 - 'o-1* OH+(J - * r ** a-£*)(.»«)} ■ +0(ft$). Setting B,-(u) = - l ( I - J i r ) - 1 l f ( i K i i ) > t = ! , . . - ,r,

8. SEM for MD weakly-singular integral equations

153

and substituting into (4.7.44), it becomes r

«(*) - «*(*) = 5>?Bi(«) + O(fcfci). t= l

The proof is complete.

D

8. SEM for collocation-method solutions of multidimensional weakly-singular integral equations Consider the SEM for the weakly-singular integral equations on plane polygonal domains. Suppose that ft C M2 is a plane polygon. Consider the integral equation on ft u(x) = I k(x, y)u(y)dy + g(x),

(4.8.1)

where x = (a?i,x 2 ), y = (2/1,2/2), dy - dy^dy2, and k(x,y) is a weaklysingular kernel expressed by (4.1.3) or (4.1.4). For weakly-singular integral equations, the Nystrom method introduced in Section 4 cannot guarantee the convergence of the approximate solution, one has to use the modi­ fied quadrature formula method, the collocation method or the Galerkin method. Now consider the convergence and the splitting extrapolation of the collocation method. r

Let ft — (J ft, be the initial triangulation of ft. Each triangle ft, is t=i

divided into n,- = Ali smaller congruent triangles after /,- refinements, i.e., m ft, = (J Aij (i = 1, • • •, r). Let /i,- = 2~li (i — 1, • • •, r) be r independent i=i

mesh widths, h — (hi, • • •, /i r ), and ho — max hi. l<»
Denote by Shi(tt) the linear finite element space on ft,-, if v G Sfci(ftt-), then v G C(ft,), and v\ is a linear function. Again, let Sh(Q.) = {t)E Loo(ft) : v | n . G 5 h i (ft,), 1 < i < r } . Denote by AihBij and C^ (i = 1,. -,r; j = 1,- - , 4 n 0 the vertices of A,-,. n h : C(ft) -► S*(«) is an interpolation projection, i.e., Hhv G ^ ( f t ) , and nfct;(ZJ
A,B,C]

i = l , " - , r ; j = l,--,n,,

(4.8.2)

Thus construct the collocation equation ufc = HhKuh + n V (4.8.3) By means of the basis function on 5 h (ft), similar to (4.7.13) and (4.7.14), (4.8.3) can be converted into a linear system. The conclusion of Lemma

154

IV. SEM for integral

equations

7.1 remains valid, that is, if (I — K)~x exists and is bounded, and h is sufficiently small, then (I—rfK)'1 exists and is uniformly bounded. If uh has been found, one can then obtain the iterative solution uhh = Kuhh + g.

(4.8.4) h

h

The comparison of (4.8.3) and (4.8.4) shows that u and u take the same values at the nodes. Theorem 7.2 can be generalized as follows: Theorem 8.1. If I is not an eigenvalue of K and (4.8.1) has a unique solution u G H3(£l), then the collocation-method solution determined by (4.8.3) and the iterative collocation-method solution uh determined by (4.8.4) have the following asymptotic expansions: r

uh(x) {x) - «'(*) u\x) = Y,B Y,Bi{u{x))hl i(u(x))h?

+ 0{hlei), 0(hle1),

(4.8.5)

t= l

rr

uhh{x) ^2Bi(u(x))h] (x) - u(x) = Y /Bi(u(x))h?

+ O(fcfci), 0(hle1),

(4.8.6)

t= l

where functions Bi(u), i = 1, • •, r, are independent of h and depend on u, u1 = \lhu, h h e11 == \\(i-n \\(i-nhh)K\\ )K\\ + \\(i-E )K*\\, + \\(i-E )ir\\, h

and E

2

(4.8.7) h

is an orthogonal projection mapping L (Q) onto S (Q).

In order to prove Theorem 8.1, we shall first generalize Lemma 7.2 and then prove that Ehu — f]hu has an asymptotic expansion. Lemma 8.1. If u E H3(Q), then there exist functions Ui(u)) i = 1, • •, r, which are independent of h and depend on u, such that r

Ehu - Uhu = ^2hfE Y/h?EhUi{u) Ui(u) + + 0(hl). 0(hl).

(4.8.8)

»=i

Proof. Take any ip £ Sh(Q) and consider the inner product in L2(Q), (Ehu-nhu,xl>) r

=

ni

=

(u-uI,tl>)

-

«x)-uI(x))i>(x)dx

E E / /

t=ii=i- ^«i

= E E { / ("(*) - «'(*)MM0-)ds t= l j = l

+ / h + h, =h h,

JA

<3

(u(x) - «u71(x))(V>(x) (x)){4>{x) (u(x)

rP(Mij)dx\ i>{Mij)dx\ (4.8.9) (4.8.9)

8. SEM for MD weakly-singular integral equations

155

where M,j is the bary center of A, j . Using the argument in proving Theorem 4.1, it is known that there exist functions Ui(u),i = 1, • •, r, which are independent of h and depend on the second order derivative oft/, such that h = Y, hf ( Ui(u)xP(x)dx + qi(u, VO ,=i

J

n<

= E A.? / w(*)c'<(tt)^(*)d*+?i(«, v>) r

= 53fc?(^,^)+9i(ti,^).

(4.8.10)

»=i

Here, rji(x) are the characteristic functions of Q,-, and there exists a constant C, which is independent of ft, such that \qi(u,i>)\
/a = E E /

("(*)"w/(*)XV-(*)-i>(Mii))dx

=E E /

(«(*)-^("OXiK*) -v-(M,i))dx

,=1 i = l - ^ i i

t=l j =l J A O -

+ E E /

(«(*) - u/ (*))M*) - v>(M,v))«ix

= J 3 + 74.

(4-8.12)

/

Noting that (u(x) - u (x))(V'(x) - ip(M{j)) is a piecewise cubic function, which vanishes at three vertices and the barycenter, hence h = 0.

(4.8.13)

Obviously, by the inverse estimate, we have \h\ < C 1 /iS||u||„3 (n )||V'||jJHn) < C2fcSll«||*»(n)||^||. Substituting (4.8.10)-(4.8.14) into (4.8.9), we have

(4-8-14)

r

(Ehu - nhu, V>) = E hKmUi, VO + «(«, 1>) «=i

= j ^ h^Eh(ViUi), »= 1

i>) + q(u, V>), Vt/> € Sk(0).

(4.8.15)

156

IV. SEM for integral

equations

Letting U% = rjiUi and noting

l«(«,^)l
D

By Lemma 8.1, the proof of Theorem 8.1 is similar to that of Theorems 7.2 and 7.1, except in Lemma 7.2, for having a remainder with order 0 ( / I Q ) , v G # 4 ( 0 , 1 ) is required while for the extrapolation, only v G H3(0y 1) is required. Similar asymptotic expansions also hold for higher-dimensional weaklysingular integral equations. For example, let Q = [0, l] 5 be an s-dimensional cuboid, K be a weakly-singular integral operator, and 1 be not an eigenvalue of K. Consider the integral equation / k(xiy)u(y)dy

u(x)-

= g(x).

(4.8.16)

Jn Applying the mesh width h = (fci, • • •, /i 5 ), divide Q into N = (hi • "hg)""1 cuboid elements. We denote by Sh(Q) C C(Q) the corresponding piecewise s-linear finite element space. nh : C(Q) —► Sh(Cl) is an interpolation pro­ jection operator, Eh : L2(Q) —► Sh(Q.) is an orthogonal projection, uh is a collocation-method solution, satisfying uh - V\hKuh = Hhg,

(4.8.17)

and uh is an iterative solution, satisfying uh = Kuh +g. h

(4.8.18)

h

T h e o r e m 8.2. If u,u and u satisfy respectively (4.8.16), (4.8.17) and (4.8.18), and u G H3(Q), then there exist functions Bi(u) (i = 1, • • •, s), which are independent of h and depend on u, such that s

uh - u1 = £ft?£fcflf.(ti) + °(fto)>

(4-8-19)

»=i

and s

uh-u

= 53fc?S«(n) + 0{h3).

(4.8.20)

«=i

Proof. The proof is similar to those of Theorems 7.2 and 8.1. We need to prove that Ehu — F\hu possesses the following asymptotic expansion: s

Ehu - nhu = J2hiEhUi(u) «=i

+ O(h30),

(4.8.21)

8. SEM for MD weakly-singular integral equations

157 157

where Ui(u) are functions independent of h and depend on the second order derivatives of u. In order to prove (4.8.21), take \j) £ Sh(Q) and consider the inner product in £ 2 (ft), h h h h (E u u,,V>) ( « ---nnfcuti,A) {Ehuu --UU VO === (ti VO

= / (u(x) —- u11(x))t/;(x)dx (x))ip(x)dx Jci ./ft N

rt -u (z))^(z)
/e

» = 1 J*i N

f

= E/(«(*)-- « (x))di^(M.) /

»=1 » = 1 • ' «^'

+ E /(«(*)--u (z))(V'Oc)-V'(M;))
t=i

^

(4.8.22)

= Ii I1 + + IIa, 2,

where t/J = n'1!/, e,- is a cuboid element in Q, and Mt- is the barycenter of ti. Use the asymptotic expansion of the error of the trapezoidal rule JV N

Ii

» s

h2 f dd22uu j—2(Mi) + gi(« l z J ox

= EE-12 4 t=i i=i

*t

= EEEE -12 | = t=i i = i »=i i = i

,vo

j

^(Aft-)

^

W

i

,

h f d a2«u At = EE- * * * * * JV N

5 s

•=lj=l t=ij=i

2

)

x

meas(e meas(ett)) -f + 52(^ * 2 (,V0 ^)

2

iZ ^e 12 yieCFX , j

+ 93(1* ,V0

5

12 7ft ax? ^

< + ^(u,V>)>

(4.8.23)

with l«s(«, VOI < Ci||«||j lfl3(«,^)|
(4.8.24)

We now consider the estimate of I^. The function (u(x) — u1(x))(ip(x) — \p(Mi)) vanishes at vertices and the bary center of e», which shows that if u is a piecewise quadratic polynomial, \\) is an s-linear function, then I2, as a, bilinear functional, has zero value. By the Bramble-Hilbert Lemma^^ or the technique used in the proof of Lemma 8.1, \h\ < C 2 ||u||ir.(n)||*||fc8. C2\\u\\m((1)U\\hl

(4.8.25)

IV. SEM for integral

158

equations

Substituting into (4.8.22), it follows

(Ehu-nhuJip) = £-i^?( i=i

B

; ^ ) + «(«^)> ^eSh(Q).

(4.8.26)

U

d2u Here, let B{ = — - , and |<j(u,VOI < CftolMljf^njIMIj therefore the proof

dxf

of (4.8.21) is complete.

D

R e m a r k 1. Only the proof of the SEM for the collocation method in the space of piecewise linear functions has been given. For the collocation method in the space of piecewise constant functions, one can also prove, by the asymptotic expansion of the orthogonal projection and the interpolation projection, that the accuracy is of 0(JIQ). However, the existence of a corresponding asymptotic expansion cannot be guaranteed. R e m a r k 2. The accuracy of the SEM depends on the estimate of = ||(/-n f c )iif|| + ||(/-^ fc )-K'*||, which bases on the analysis of properties of the kernel k(x,y). For the details, see I.H. Sloan and V. Thomeet 122 !.

Sl

Chapter V SEM FOR DIFFERENTIAL EQUATIONS

One of the main purposes of numerical computation is to obtain numeri­ cal solutions of partial differential equations by solving large scale algebraic systems yielded from appropriate discretizations of mathematical models. Different discrete methods lead to different research directions. The main discretization methods include the finite difference, the finite element, the collocation and the boundary element methods. The finite difference and the finite element methods remain to be the leading methods in solving partial differential equations. The finite difference method can be used to solve nonstationary (parabolic and hyperbolic) equations, but not for problems with complicated curved boundaries. The finite element method was initially applied to solid mechanics, and it was subsequently developed to be applicable in fluid mechanics. It can deal flexibly with complicated boundaries and there are many available softwares. The boundary element method can reduce the dimension of the discrete problem to be solved, and so it is suitable for the outer boundary value problems in particular. However, it can only be used for a class of typical problems and is thus un­ suitable for solving partial differential equations with variable coefficients. The collocation-method is easy to operate, and has wide applications in twopoint boundary value problems and boundary integral equations. But so far the research on the application of this method to partial differential equa­ tions is still inadequate. The Richardson extrapolation acceleration of these four methods are all successful. For details about the extrapolation of the difference method, see the monograph by Marchuk and Shaidurovt 90 ! and those of the finite element method, see the monograph by Q. Lin and Q.D. Zhuy']. However, the cost of global refinement required by the Richardson extrapolation is very high for higher-dimensional problems, it is the SEM that can fill this gap. Based on the multivariate asymptotic expansion, the SEM refines only one mesh parameter at each step and achieves the same order of error as that of the Richardson extrapolation, while the scale of the problem is greatly reduced. Obviously, the idea of reducing a big problem into several smaller problems is suitable for evaluating the problem in par­ allel on multiprocessor computers. It must be pointed out that the number 159

V. SEM for differential

160

equations

of independent mesh parameters does not necessarily equal to the dimen­ sion of the problem. One can construct several independent parameters even for a one-dimensional problem. In this Chapter, we discuss the SEM for the collocation-method solution of the quasilinear two-point boundary value problems in §1, then the SEM for the solution of the difference methods in §2, and finally, the SEM for the finite element approximation in §3. Only stationary problems are discussed in this Chapter. The SEM can also be applied to nonstationary problems effectively, because the time step can be regarded as an independent parameter. For details of the asymptotic expansion of nonstationary problems, see Marchuk and Shaidurovt 90 ]. 1. SEM for the collocation-met hod solutions of two-point bound­ ary value problems 1.1. SEM for the collocation-method linear two-point boundary value

solutions problems

of the

quasi-

Consider the boundary value problem of the following quasilinear ordi­ nary differential equation •

u" = f(t,u,u'), < [ aju(j) + bju'U) = dh

0<*<1,

(5.1.1a)

j = 0,1.

(5.1.1b)

The difference method, the finite element method and the method can all be used for the evaluation of the numerical (5.1.1), among them the collocation-method is the simplest and sively. In order to apply the SEM, divide the interval [0,1] into i.e., first choose the initial partition as 0 = TO < ri < • • • < r r = 1,

collocationsolution of used exten­ r segments, (5.1.2)

and then refine the interval /,- = [T,_I,T,] (i = l , - - , r ) by a new mesh width hi = (T,- - T t _i)/n t -. Let Tti = r,_i + jht (i = 1, • • •, r; j = 0, • • •, n t ) be the nodes, and the corresponding partition becomes Jh : 0 = T0 < <

< n | 2 < • • • < TI < T 2 ,I < Tr_i < • • • < r r = 1.

TM

(5.1.3)

Let Sp3(Jh) be the cubic spline function space corresponding to Jh, i.e., Sp3(Jh) = span{ 1,M 2 , (* - n>)+» * = 1, • • •, r; j = 0, • • •, n,-}. The cubic spline approximate solution u^ 6 Sp$(Jh) satisfies the following collocation

1. SEM for two-point boundary value problems

161

equations <{nj)

= f(rij, t i f o ) , < ( r t i ) )

<

« = 1> • • • i r; j = 0, • • •, H,-,

, fljUfcO") + *i*fc(i) = <*>> i = 0,1.

(5.1.4a) (5.1.4b)

As we know that if / has 3-rd order partial derivatives and the exact solution u* e C 4 [0,1] is an isolated solution of (5.1.1), then there exists a unique approximate solution Uh in a sufficiently small neighborhood of u*, and the following error estimate holds \\Bi(u* - uh)\\oo < C f t ' l p V H o o , j = 0,1,2,

(5.1.5)

d where D = —, and C is a positive constant independent of h. dt For the sake of simplicity, introduce the following notations: f(U) = /(*,«,«'),

f0(U) =

at/ f(U(t))

-f(U),

ou

= f(t, u(t),«'(«)),

/*(!/(<)) = ^ / ( ^ ( 0 ) - (5-1.6)

In the following, we shall prove, under certain conditions, that there is a multivariate asymptotic expansion with respect to h = (fci, • • •, /i r ). T h e o r e m 1.1.1°™ Suppose that u* € C 2 [0,1] is an isolated solution, Uh is the approximate solution of u* determined by (5.1.4), then there exist functions Wi(t) G C 2 [0,1], i = 1, • • •, r, swcA Ma/ r

„•(«) - uh(t) = £ > ' ( * ) * ? + O(*o),

(5.1.7)

*=1

wiM /in = max /i,-. l<j
Proof. Let r\h be the piecewise linear interpolation projection operator determined by the partition Jh. Since u'^ is a piecewise linear function, then (5.1.4) is equivalent to the system

' u» = nhf(t,uhlu'h)

(5.1.8)

< <*jUh(j) + l>ju'h(j)=dj,

i = 0,1.

162

V. SEM for differential

equations

Let eh = u* - uhy from (5.1.1), (5.1.5), (5.1.8) and Taylor's expansion, eh satisfies = (I - n*)/(l/*) + 0{h%)

el - h{U*)eh - f2{U')e'h ajehU)

+ bje'h(j) = 0,

j = 0,l.

(5.1.9)

Let H(r,t) be the Green function of the boundary value problem (5.1.9), and set g(t) = f(V) = /(*,«*,(«•)')• Noting that g(t) G C 4 [0,1], one obtains eh(r) = C H{r,i){I Jo

- nh)g(t)dt + 0(h*)

= J2 [ H(T,t)(I-r\h)g(t)dt

+ 0(h$).

(5.1.10)

.=i Jit Hi

Since I{ = (J Iih JtJ- = [r t > j -i, r t -j], for j = 1, • • •, n t , then

y

ff(r,i)(i-nfcMt)A

= E / #(r> ')(' - n *M0*.

(5.1.H)

By using the properties of the Green function, if r ^ 7 t j, then H(r^i) is a smooth function on 7,j. It follows, from the estimate of the remainder of Simpson rule, that H(r,t)(I-nh)g(t)dt

/ Jin

= ^(r,m,,0[,(m,,)-^^±^l]

+0(ft»)

1 = - - i f ( r , m 0 y ' ( m t i ) / » ? + O{h50) 12" = -j2h^l where mtJ- = n , j - i H

H(T,t)gH(t)dt

+ 0{hl),TtIti,

(5.1.12)

is the midpoint of 7,;-. If r € Iij, then the derivative

of H(r, t) is discontinuous at r = t. Considering the error expression of linear interpolation, for t € Iij, we have (/ - r\h)g(t) = ^ ' ( T S J - O C * - 7ij_!)(t - r 0 ) + O(ftg).

1. SEM for two-point boundary value problems

163

By using the mean value theorem, there exists t G iij, such that

H(r,t)(I-r\h)g(t)dt

I

= \f =

H(T,ty'(nJ-I)(*

\H(T,*)y" 2

= ~

j

(T»J_I)

- nj-iX* - m)dt + 0(hj)

f (tJin

H(r,t)g"(t)dt

nj.i)(t

+ 0(h$),

- Ttj)dt +

0(hf)

r € /,;•

(5.1.13)

Combining (5.1.12) and (5.1.13), we obtain

H{T,t)(I-nh)g(t)dt

J

= ™*? j Letting w;(r) =

H(T,t)g"(t)dt

f H{r,t)g"{T)dt,

+ 0(ht),

i = I, •••,!•.

(5.1.14)

it follows from (5.1.10) that

r

eh(r) = 5 > , ( r ) f c ? + 0(h*),

(5.1.15)

»= 1

this completes the proof of Theorem 1.1.

D

Further analysis can lead to a higher order expansion of the error, which is omitted here. In practice, it is most convenient to choose the cubic £-spline function as the basis function of Sps(Jh). For details of the construction of nonisometric 5-spline, see Schumakerl 114 !. 1.2. SEM for the collocation-method solutions Liouville type eigenvalue problems

of

Sturm-

Consider th e Sturm-Liouville type eigenvalue problem ( <

du d , ^ Lu = - ( P ( 0 — ) - «(*)<* = - M * K t/(0) =atr/(l) = at0,

/ p(t)u2(t)dt= i Jo

1.

(5.1.16)

V. SEM for differential

164

equations

Here, suppose that p(t),q(t) and p(t) are smooth functions, and p(t) > d > 0, p(t) > C2 > 0, Vt E [0,1].

(5.1.17)

All the finite difference, the finite element and the collocation methods can be used for evaluating the numerical solution of (5.1.16), among them the collocation method is comparatively simple and effective. Let Jh be the partition of the interval [0,1] defined by (5.1.3). To find the cubic spline collocation-method solution {A/^ti/j} of the eigenvalue problem (5.1.16) is equivalent to the problem: find Uh G Sp$(Jh) satisfying the collocation equation /

Luh(Tij)

-\hp(Tij)uh{Tij),

i = l , - - , r ; i = 0,--,n t -,

< \

=

(5.1.18)

tifc(0) = tifc(l) = 0, / p(t)u2h(t)dt = 1. Jo

It has been proved in references^!' ^ ^ that the approximate error satisfies |A - \h\ = O(fcg), \\D°(u - uOlloo = O(ftg), i = 0,l,2.

(5.1.19)

In order to apply the SEM to improve the accuracy, it is necessary to verify that the errors of the approximate eigenvalue and the corresponding eigenfunction have multivariate asymptotic expansions. For convenience, introduce the Hilbert space H = L2(p; 0,1), and define the inner product as

Jo Suppose that p(t) is continuously differentiate, then problem (5.1.16) can be rewritten as the following equivalent form: * Tu = u" - ai(t)u' - a0(t)u = \g(t)u, < ( «(0) = «(1) = 0, («,«) = !. And the corresponding discrete problem is to find uh e Sp^(Jh), 1 Tuh{Tij) = \

\hg{Tij)uh{Tij),

1 «fc(0) = u k (l) = 0,

(«k,«Jk) = l,

(5.1.20) satisfying (5.1.21)

1. SEM for two-point boundary value problems

where g(t) =

165

-p(t)/p(t).

T h e o r e m 1.2.1 4 "' //{A, u) is a simple eigen element o/(5.1..16), {A&, Uh] is the corresponding approximate eigenelement determined by (5.1.18), then the error of the approximate eigenvalue has the following multivariate ex­ pansion: X

" A* + ^ X > ? / rt*M*)«(4)(0* - O(*o). 12 . = 1 Ju

(5.L22)

and there exist functions Wi{i) (i = l , . , r ) , which is independent of h, such that r

u(t) - uh(t) = Y,wi(t)h? + O{h30).

(5.1.23)

Proof. Let Hh be a linear interpolation operator with base points {r,j : i = 1, • • •, r; j = 0, • • •, n,}. (5.1.21) is equivalent to finding Uh E Sps(Jh) and Ah satisfying f

<

u'l = Vr(a\v!h + aoi/^ + ^h9Uh), tifr(O) = Ufc(l) = 0,

(uhyuh)

(5.1.24)

= 1.

Let en = u — Uh. By using (5.1.20) and (5.1.21), it follows that (T - \g)eh = (A - \h)gu + (A - \h)geh +(I<

nh)uh,

(5.1.25)

e*(0) = e h (l) = 0, where Uh = a\ufh + ao^/i + ^h9^h- By direct evaluation it is easy to verify that -T is a self-conjugate operator in H. Divide both sides of (5.1.25) by a if g, and take the inner product with u, one obtains

A-A„ = -(A-A f c ) / /#)«(<)«*(')*+ / Jo

= f

Jo

p(tHt)(I-nh)uh(t)dt

P(t)u(t)(l-n

h

)uh(t)dt + 0(ht)

Jo = / p(t)u(t)(l-nh)u"(t)dt Jo

+ 0(ht).

(5.1.26)

166

V. SEM for differential

equations

Here u'^ = V\hv>h and the error estimate (5.1.19) are used. Evaluate (5.1.26) by the Simpson rule, and assume that u G C 6 [0,1], then A-

\h h

= [

P(t)u(t)(l-r\

)u"(t)dt + o(h$)

Jo

= E E r rt*Mwj - n")«"(()^+°(ht) i=ii=rT».M r

ni

r2

= ^2 J2 { vhiP(mii)U(mij)[U"(mij)

-,1"(r'j)+2""(r"")]+oW)}

+

oW)

= - E E J|P(™.. )"("■., )«(4)(™li) + 0(14) (5.1.27) *=I12

^

the proof of (5.1.22) is complete. In order to prove (5.1.23), define Hi = {

(5.1.28)

The operator 5(A) = ( - ( T - A p ) ) " ^ / - E) is meaningful^ . The corresponding Green function G(T, t) is piecewise continuously differentiate. Rewrite the identity (5.1.25) as - ( T - A^e^ 9 = (A-Afc)ii + (A-A fc )c fc

+ ! ( / _ nfc)ti" + V - nfc)(5fc - u").

(5.1.29) 9 9 Let Wh = Uh — u". Operate 5(A) on both sides of (5.1.29), it becomes (/ " E)eh

1. SEM for two-point boundary value problems

=

( A - A h ) / G(r,t)eh(t)dt+ Jo 1

167

f G(r, t)g-\t)(I Jo h

-

nh)u "(<)<*<

+ f G(r,0<7- (0a-n KW^ Jo

(5.1 .30)

= J1 + J2 + J3.

Using (5 .1.19), it is easy to obtain the estimate ||Ji||oo = O(ho). For J2, there is J2=

/ Gfafy-^tXl Jo

-nh)u"(t)dt G(T,t)g-\t)(I-nh)u"(t)dt.

= E E /

(5.1 .31)

If r g i t f , by using the Simpson rule, it is easy to prove that G(r,0«7-1(0a-nh)«"(0^

/ = ~

l

G{T,t)g-\t)uW{t)dt

+ 0{h
(5.1 .32)

If r € lij , by using the interpolation expansion (I - n V ( 0 = « ( 4 ) (nj-i)(< - 7i,i-i)(* - n,-)/2 + 0 ("•?). Vt G ly, and the :mean value theorem for integration, there exists £ G /<;, such that

L

G(r,^- 1 (0a-n A K(0^ :i /

G(r ) 0«7- 1 (0(<-n,y_i)(<-r i j )d<«( 4 )(r i , j _ 1 ) + O ( ^ )

= iG(r,o.r1(0«(4)K;-i) / (*-^-i)(*-ni)«ft + o(fc?) (5.1 .33) Substitute (5.1.32) and (5.1.33) into (5.1.31), it follows that J 2 = - - V > ? [ G(T,t)g-\t)uW(t)dt 12 fz? 7/;

+ 0(h$).

(5.1 .34)

168

V. SEM for differential equations

For the estimate of J3, obviously, there exists a constant C > 0, which is independent of fc, such that (51-35)

H/slloo < C M K I l o o , where w'h = — [ai(u'h - u') + a 0 (« h - u) + (XhUh - Au)]

= ^ M«k - u ')] + ^ h ( « * -«)] + A f c K - t i " ) + (Ak-A)«'.

(5.1.36)

By using (5.1.19) and (5.1.22), one obtains the estimate (5.1.37)

UKWoo = 0(hl). It follows from (5.1.30) that

( I - £ ) c f c = - - V > ? / G ^ O ^ ' W ^ W ^ + O^).(5.1.38) 12 . = 1 Jit Noting Eeh = (c fc ,ti)w= --{u-uh,u-uh)u

= 0(/i^),

taking iy,-(r) =

/ Gfatfg^lfiuWtydt, 12 hi tuting into (5.1.38), we obtain

(i = 1,

- , r ) , and substi-

r

eh(t) = u(t) - uh(t) = ^2hfWi(t)

+ 0(fcg),

t=i

and the proof of (5.1.23) is complete. 1.3. SEM

for singular

two-point

□ boundary

value

problems

Consider the quasilinear two-point boundary value problem d , , .du,

, .du

-*<**)*)+*)#

=*>")> *<*<>>

(5 . L39)

u(a) - u(b) = 0. Suppose that p(t) > 0, W G (a, 6), p(a) = 0, and

fw)dt<°°-

(5L40)

1. SEM for two-point boundary value problems

169

For this class of singular boundary value problems, Babuska^' discussed the error of a finite difference approximation in the case p(t) =ta, 0 < a < 1, and proved that the error at the isometric nodes is O(ho). In the following we shall prove that it is possible to eliminate the singularity by using a simple transformation. (5.1.39) is converted first into a regular two-point boundary value problem, and then obtain an asymptotic expansion of the mesh parameters obtained from an isometric partition or a piecewise iso­ metric partition. Under this transformation, the isometric partition in the new equation is equivalent to the gradual partition in the original equation, that is, the closer to the origin, the denser the grid points. The gradual partition depends on the property of the zeros of p(x). Consider the transformation r = r(t) = I p'itydt.

(5.1.41)

Ja

Obviously, r : [a, 6] —» [0,T], where T = r(6). By the assumption p(t) > 0, W £ [a, 6], r(t) is a monotonically increasing function, then the inverse function t = t(r) exists uniquely. Applying the derivative relation du

du

dr

dt

(5.1.39) can be converted into ( 5"(r) = q(r)u'(r) - p(r)f(r, ^ I u(0) = u(T) = 0,

«(r)) t

(5.1.42)

where « ( T ) = u(*(r)),p(r) = p(t(T)),q(r) = « ( * ( T ) ) , / ( T , « ( T ) ) = f(t(r), u(t(r))). (5.1.42) is a special case of (5.1.1), and so applying Theorem 1.1 one can obtain the asymptotic expansion of the collocation-method solution r

«(r) - « k (r) = 5 > , ( r ) A ? +

0(h*),

t= l

0 < r < T.

(5.1.43)

Substituting r by r(t), 0 < t < 1, in (5.1.43), we get that r

u(t)-uh(t)

= ^wi(t)h?

+ 0(h$),

0
(5.1.44)

i=i

This transformation can also be used in the eigenvalue problems described in (5.1.16).

170

V. SEM for differential

equations

The relation between the grid points on [0, T] and that on the original interval [a, 6] can be explained as follows: Suppose that p(t) = ta, 0 < a < 1, and [a, 6] = [0,1], then r(t) = j t'adr t(r) = (1 -

= *1-tt/(l - a),

a f - M / f H

If the new interval [0,T] is partitioned isometrically with grid points lj— }? , then, in the original interval [0,1], the grid points are (1 — ot)n *—° i * { (—) I +" } , which is a gradual partition.

1.4. SEM for two-point boundary value problems discontinuous coefficients

with

Consider the two-point boundary value problem -(pit')' + qu = / ,

a < t < b. (5.1.45)

u(a)

= WQ,

u(b) = ti 1 }

with p(t) > Ci > 0, q(t) > 0. If any one of the coefficients p(t),q(t), f(t) has a finite number of dis­ continuous points in the interval [0,1], then (5.1.45) is called the Dirichlet problem with discontinuous coefficients. For simplicity, suppose that there is only one discontinuous point £ G (0,1), then (5.1.45) can be regarded as the couple of two differential equations defined on (0,f) and (£, 1), and at t = f, it satisfies the following conditions of consistency:

«tt + 0) = 11(^-0), (5.1.46) Ptf + OXtf + 0) - ptf - 0)ti'(£ - 0) = g, where g is a fixed constant, u'(£ + 0) = limix'(£ + 6), and u'(£ - 0) = 6>0

limti'(£ + «). a—o We denote by Q* the set of the functions which possess piecewise con­ tinuous derivatives up to order k on [a, 6], and the first order derivative has discontinuity of the first kind at f.

1. SEM for two-point boundary value problems

171

Marchuk and S h a i d u W 9 0 ) proved that if p £ Q^ + 1 , / , ? £ <2£, r > 2, then (5.1.45) has a unique solution u £ <3£+2 H C[0,1]. They also proved that there is an asymptotic expansion of the difference approximation uh at grid points (cf. Theorem 3.2 in [90]). (5.1.45) can also be solved by the collocation method, provided that the discontinuous points of the coefficients are taken as the initial partition points. The procedure is as follows: convert (5.1.45), by the transformation (5.1.41), into a simpler form u"(r) = q(r)u(r)-p(T)f(r\ (5.1.47) u(0) = a, u(T) = b. If £ is a discontinuous point of the coefficients, then r(£) is a discontinuous point of p,q/f. Take the initial partition as 0 = r 0 < n < • • • < r r = T.

(5.1.48)

All the discontinuous points of coefficients r(£) £ { r , } . . Refine the subintervals respectively by mesh widths hi = (rt- — r,-_i)/nt- (i = 1, • • •, r), and let Tij = r,_i + jh{ (i = 1, • • •, r; j = 0, • • •, n,). Furthermore, let tih be the cubic spline approximate solution with respect to the nodes {rt-j}, u be the solution of the equation (5.1.47) with the approximation u^ of the collocation method, and ?& = u — Uh be the error. Repeating the derivation used in (5.1.9), it can be proved that e>>-qeh=(I-n»)(qu-pf)-(I-n»)(qeh),

(5.1.49)

eh(0) = e,(r) = 0. Let G(T, S) be the Green function of the problem (5.1.49). It follows from the discontinuity of q(r) that G(r, s) is possibly discontinuous at r,-. How­ ever, G{r> S) is smooth in (r,_i, Tj), then we still have

= / Jo = J2f

G(r,s)[(I-nh)(qu-pf)-(I-nh)(qeh)]ds G(r, s) [(I -nh)(qu-p

/)] ds

-J2[ G(T,s)[(I-nh)(qeh)}ds = E - ^ f t « ? / G(r,s)(qu-pf)"ds + 0(hi).

(5.1.50)

V. SEM for differential

172

equations

Here the smoothness of the integrand on /,- and the inequality

||(i-n f c )(« cOHoo,/, < Cft?||ff cOloo,* < ChtWD^uWoo are used. Consider now the original equation (5.1.45), the error of the approximate solution Uh of (5.1.45) has an expansion r

u(t) - uh(t) = ^2Wi(t)h? + O(ftJ),

(5.1.51)

*=i

where Wi(t) are independent of h = (/ii, • • •, hr) and depend on the coef­ ficients and their discontinuous points. SEM for the collocation-method solution of a quasilinear two-point boundary value problem with discontin­ uous coefficients can be considered similarly. 2. S E M for finite-difference a p p r o x i m a t i o n s 2.1. Difference ciple

equations

and the discrete

maximum

prin­

Consider the Dirichlet problem of a linear elliptic equation Lu = > a » ( z ) - £ - 2 + l^hi(x)-^-

u{x) = g(x),

+ d(x)u

= f(x)

inficF, on dtl,

(5.2.1a) (5.2.1b)

where Q is a bounded open domain in Ms, a,-,6,,c/ and / £ C(Q),<7 £ C(dQ). Assume that a = min mina,(x) > 0, and the equation (5.2.1) has i<*<5 xen

a unique solution u. In order to construct a difference scheme of (5.2.1), define the grid lines I \ / and the grid Th on Ms as follows: First take the mesh width h = (/ii, • • -,/i,), and then let

i \ / = {x e Ms: XJ e m, x{ = mhif Hi e %>, i=l,-..,«, i ^ i } , Th = {xeMs : Xi = riihi, i = 1, • • •, *},

(5.2.2a) (5.2.2b)

where 2Z stands for the set of integers. Let e, be the unit vector in the i-th coordinate and let ft/* :={xe£l:x±hiei Qhti : = ( f l n r f c ) \ n f c ,

e f t , i = 1, • • •, s} H Th)

an fc :=flnnr fc| i, o M = nhunfc|l-udfifc.

(5.2.3a) (5.2.3b)

(5.2.3c)

2. SEM for finite-difference

approximations

173

Q/i is called the set of regular grid points, Clhti the set of irregular grid points, dClh the set of boundary grid points and Qh,t the union set of all three kinds of grid points. For a regular domain Qh,i = 0- Using the central divided difference, it is easy to construct the following difference equation of (5.2.1): s h h

Lu

= ]£
+ ^bi(x)[uh(x

+ hid) - uh(x -

hiei)]/(2hi)

t=i

+d(x)uh(x) = / ( * ) , uh(x) = g(x), Vx G dQh.

Vx€fifc,

(5.2.4a) (5.2.4b)

If ^lh,i ^ 0, and x G ftjif», then there exists at least a direction ej such that x -f hjej £ Q. There are many choices for setting the difference equation at the irregular point x G £lh,i- Suppose that x* = x + Sjhjej G d£lh, 0 < Sj < 1, is an intersection point of dCl with a segment between x and x -f hjej, we can use x* and x -f ^ e j , and replace the central difference in (5.2.4a) by a non-isometric difference. Another choice is called the simple trans­ port method, in which, let uh(x) :— uh(x*) and substitute into (5.2.4a). If a higher order scheme with an asymptotic expansion is required, then the Lagrange interpolation method should be employed. The detailed con­ struction will be introduced in the next section. Regardless of the methods of discretization, the difference equation can be described as

Vx€fi f c U «*,,-, where A(x, y) are coefficients which depend on x, y G fi/i U £lh,i and depends on a,, &», / , g and x.

(5.2.5) Fh(x)

Definition 2 . 1 . For any x G &h U Q^,,-, Me discrete set N(x) = {yenhU

Qh)i : A(x, y) ± 0},

is called a neighborhood of x. Definition 2.2. The set Qh U fi^,,- is said to be connected, if for any x,y £Qh U&h,i, there exists a finite sequence { x ( n ) } n = 1 , such that x^1' = x, x^m) = y, and x( n ) G ^ ( x ^ - 1 ) ) , n = 2,3, • • •, m.

(5.2.6)

174

V. SEM for differential

equations

By using Definition 2.1, the difference equation can be rewritten as

Lhuh(x) = £

A(z,y)u\y)=Fh(z), VsEflfcUfifc,,-.

(5.2.7)

The problems concerning stability and convergence of the difference equa­ tion (5.2.7) are of great importance. The discrete maximum principle, an analog of the maximum principle of differential equations, is the main tool for solving the problems. Before introducing the principle, the concept of operators of positive type should be defined. Definition 2.3. The difference operator Lh is said to be an operator of positive type, if a) b)

A(x, y) < 0, Y^

^(*>y)>0,

Vz G tth U Qfc|l-, y G M(x)y VxGftfcUQ*,,-.

x^y, (5.2.8a) (5.2.8b)

T h e o r e m 2 . l J 1 A ^ ( T h e discrete m a x i m u m principle) Suppose that Lh is an operator of positive type, and Q'h is a connected subset of Qh Utlh,i- If on the set

K = U ^(*),

(5.2.9)

uh(x) is not a constant, and Lhuh(x)

< 0, (or Lhuh > 0), Vx G

h

fi'fc,

then u (x) cannot reach the positive maximum (or the negative in Cl'h.

(5.2.10) minimum)

Proof. Suppose that Lhuh(x) < 0, Vx G fi^, and at the point x G Sl'hi u (x) reaches the positive maximum value h

uh(x) = maxu h (x) > 0, then Lhuh(x) = A(x,x)uh(x)

+

Y2 yeM'ix) y?x

A(x,y)uh(y)

(5.2.11)

2. SEM for finite-difference Approximations

=[ £

175

A(x,y)}u\x)

yeM'(x)

A(x,y)(uh(y)-uh(x)).

+ £

(5.2.12)

y£Jsf(x) y^x

It follows from the condition b) of Definition 2.3 that the first term on the right-hand side of (5.2.12) is non-negative, and from the condition a) and (5.2.9) that the second term is also non-negative, then Lhuh(x)

> 0.

Combining with the assumption Lhuh(x) Lhuh(x)

(5.2.13)

< 0, it follows that (5.2.14)

= 0.

According to (5.2.9), (5.2.12) holds only if £

(5.2.15)

A(xyy) = 0,

yeM'(x)

and uh(x) = uh(y), Vy E M(x).

(5.2.16)

By the assumption uh(x) ^ const., x E Qh, there exists a point x E &h, such that uh(x) < uh(lx). And by the connectivity of ftl, one can always find a sequence xW E fl'h (i = l , - - , m ) , such that x^1' E A / ^ ) , x^ E Af^-V), i = 2 . . - - , m l x € A/" (x< m )). By using (5.2.16), uh (x) = uh (x&). Without loss of generality, we may assume 0 < uh(x) = uh(xW)

= • • • = tx h (x( m )).

Otherwise, there exists x( m °), 1 < m 0 < m, then take x = We now evaluate L h ti h (x( m >),

Lfctifc(*<m)) = (

J2

(5.2.17) m +1

x^

° h

A(x(m\y))uh{xW)

y€JSf(x(m))

+ J2 ^(*(m),»)(«*(»)-«*(*(m))) yeJV(x<m>)

>

A(x^m\x)(uh(x)-uh(x^))

= i4(* ( m ) , *)(«*(*) - u*(3t)) > 0, which contradicts the assumption that Lhuh(x)

< 0, Vz € &'h-

(5.2.18)

176

V. SEM for differential

equations

Similarly, for Lhuh(x) > 0, replacing uh(x) by -uh(x) in the argument above, one can obtain that uh(x) cannot reach the negative minimum in Corollary 1. If in QhC\Clh ,-, Lhuh(x) < 0 (or Lhuh(x) exists at least one grid point x^°\ such that D(zW)=

> 0), and there

^(0)>2/)>0,

£ y€^(x(°))

x^enhunh)il h

(5.2.19)

h

then u (x) < 0 (or u (x) > 0) in Clh D Qhti. Proof. Take £l'h = fl/i O fij^-, if uh(x) ^const., then it follows from Theorem 2.1 that ^ ( x ) < 0 (or uh(x) > 0). If uh(x) = const., then

LV(*<°>) = D(*<°V(* (0) ) +

£

A(x(°\y)(uh(y)-uh(x0))

y€jV(x<°)) y**(0)

= D(x^)uh(x^)

< 0,

which leads to uh(x) = uh(x^) < 0, Vx G &h H fih,»- For the case Lhuh(x) > 0, a similar argument works. □ Corollary 2. i/* Me difference operator Lh satisfies the assumptions of Theorem 2.1 and Corollary 1, then the difference equation (5.2.5) has a unique solution. Proof. It is sufficient to prove that the homogeneous equation Lhuh(x) = 0, Vx G fi/i O Clhti has only a trivial solution. In fact, the condition Lhuh(x) = 0 of Corollary 1 guarantees that uh(x) < 0 and uh(x) > 0 hold simultaneously, which implies uh(x) = 0. □ Corollary 3. (Comparison Theorem) Suppose thatuh(x) is the solution of the difference equation (5.2.7) with the right-hand side F (x), and \Fh(x)\ < Fh(x),

Vx G Qh U n M >

(5.2.20)

then \uh(x)\ < uh(x)} V x G ^ U

fiM.

(5.2.21)

2. SEM for finite-difference

approximations

177

Proof. By using Corollary 1 and noting that -^h

Lh(uh ±uh) = F (x) ± Fh(x) > 0, one can derive uh(x) ± uh(x) > 0, therefore, ^ ( x ) ! < uh(x).

D

Theorem 2.2. Suppose that the coefficients of the difference equation

Lhuh(x) = Fh(x)y uh(x)

= gh(x),

xenhunh)i,

(5.2.22)

xetth

has the following properties: D{x) > 0, Vx G fi/» and D(x) > 0, Vx G Qh,i, then equation (5.2.22) has a unique solution, satisfying max\uh(x)\ < max\g h (x)\ + max\U h (x)\ + max\F h (x)/D(x)\,

(5.2.23)

where Uh(x) is a majorizing function, which is the solution of the following problem: -r;h,

Lhuh(x) = F"(X), Uh(x) T=T^/

where F'\x)

> \Fh(x)\,

xenhunh>i,

>0,

(5.2.24)

xedQhi , -rzh,

Vx G J2fc> an 0, Vx G fi*,,-.

Proof. The uniqueness of the solution (5.2.22) will be derived from Corollary 2. In order to prove (5.2.23), let ti fc (x) = tif(a:) + ti5(a:) + ti§(x) l

where uf(x), t = 1, 2, 3, satisfy respectively the equations Lhuh1{x) = 0, x€tlhUQhti;

h

Fh(x),

u$(x) = g(x), x e dQh, (5.2.25a) xeQh,

L u*(x) = { 0,

xEtth,i,

0,

a; e fih,

(5.2.25b) Ǥ(*) = 0, x G 5 f i A >

and

£*«£(*) = <

h

F (x),

(5.2.25c) x G fift,,-. «S(x) = 0, x E dQh.

V. SEM for differential

178

equations

By using Theorem 2.1 and its Corollaries, max|tij(*)| < max \g(x)\, flh,t

(5.2.26a)

s£dflh

and max\u$(x)\ < max\U(x)\.

(5.2.26b)

We shall now prove max \u$(x)\ < max \Fh(x)/D(x)\. nhunhti xerih,i

(5.2.26C)

For this purpose, replace Fh(x) in (5.2.25c) by F (x) = \Fh(x)\, the solution be 1X3(2;). By Corollary 3,

and let

h*3(*)l<*3(*)> Vx€SlfcUfi fc| «. Using Theorem 2.1, 1X3(2?) can attain its maximum value only at a point x(°) on £lh,i- However, at this point, \F(xW)\ = D(x°)uh3(x^)

A(x(°\y)(uh3(y)-uh3(xW))

£

+

t/CJV(*(°>) h

> D(x°)u 3(x(%

*<°> € n M >

which leads to <

max

!!*(*) = «*(*«»)

< |F(*<°>)/£>(s<0>)|. Thus the proof of (5.2.26c) is complete. The estimate (5.2.23) follows im­ mediately from (5.2.26). □ 2.2. Multivariate asymptotic expansion of the ence approximation on a domain with a smooth

finite-differ­ boundary

For simplicity, consider the Dirichlet problem of Poisson's equation -Aw = /, u =
inftCiR5,

(5.2.27a)

on <9ft.

(5.2.27b)

2. SEM for finite-difference approximations

179

Assume t h a t the boundary 9Q G C 4 + a . Take the mesh width h = (fti, • • •, hs), construct grid lines I \ / and grid points I \ , and denote, using (5.2.3), the set of regular grid points, irregular grid points and b o u n d a r y grid points by fth, Clh)i and dQh respectively. If x G Qh, then construct the difference equation 5

- ^ j V ( x ) = / ( x ) , Vx G J2fc>

(5.2.28)

7=1

where the central difference operator Sj is defined as Sjuh(x)

= [uh(x + hjej) - 2uh(x) + uh(x - hjej)]/h].

(5.2.29)

If x G £}&,*, and x + hjej £ Q, then (5.2.24) cannot be used to form (5.2.28). However, it is possible to stipulate first the meaning of uh(x + hjej) by the interpolation, and then to construct the difference equation by (5.2.29) in the following way: Suppose x -f hjej £ Q, b u t x — khjej G ^ / i U fi/i,i, k = 1, • • •, n, then the segment between x and x + hjej intersects dQ and let the intersection point be x -+- r\hjej, 0 < rj < 1. Construct a Lagrange interpolation polynomial p n ( 0 of degree ( n - f 1), such t h a t Pn(-khj)

= ^ ( x — khjej), fc = 0, • • •, n,

Pn(rjhj) = g(x + rjhjej).

(5.2.30)

Define Sffc(x + A i e i ) = p n ( f c i ) ,

(5.2.31)

and £ j V ( x ) = [ ^ ( x + fye,-) - 2ti / l (x) + u f c (z x G $}*,,-, x + fye,- £ Q.

hjej)]/h], (5.2.32)

Similarly, a difference equation for the case x G Slh,i but x — /ijej £ Q can be defined. T h u s , Vx G fi/i,», define a difference equation

- Yl *i"*(*)" X) ^ W = /W, VxGQ^, (5.2.33) with J x ( x ) = {; G { 1 , 2 , - - - , « } : x ± jhj (£ H, x G fifc|i}, J 2 (a?) = {1, 2, • • •, s} \ J i ( x ) . Combining (5.2.28) and (5.2.33), we obtain the differ­ ence equation of the problem (5.2.27). T h e convergence and stability of this equation depend on whether it satisfies the m a x i m u m principle. In other words, it is necessary to verify carefully whether (5.2.33) satisfies (5.2.8).

180

V. SEM SEM for for differential V.

equations

m+a ^(t) G G Cm+a tp(t) is is the the Lagrange L e m m a 2 . 1 . Suppose that \p(t) [—1,1], tp(t) of ip(t) with the the following set set of of interpolation base interpolation polynomial of points: {{^^jj,, 0, —/ij, •••,— nhj}yy where TJ G (0,1), then

p?m m

(o) == ^*iM _-r(ro) ) --^2(«Qo)) + ^»(-*,) i ) __ ^ ( 0 ) h

3

+,,

= Ep^)i* E(^^ (M+!, (»>+^. = (») + ^.

(5.2.34)

= [min{n — 1, m m — 2}]/2 loii/i wiiA remainder where rr = C55W, W, for for 7i ++ 22 << m , C

[

r C66/i™~ /if - 2 +2+o °, , [ C

(5.2.35)

l < m < n ++ l.

90 (cf. Marchuk and and Shaidurovt Shaidurovl90 By the the Lagrange Lagrange interpolaProof, (cf. ! .).) By tion formula, evaluate ip(h) ip(h)

n

(5.2.36)

k=0 fc=0

with aan,fc

"'*

aa

=

»." =

(-!)*+!(„ +1)!(1_,) (-l)*+i(„+ 1)!(1_,) (Jb+l)!(Jb + «,)(!»-*)!' ij)(n-t)!' (* + l)!(*+ " i +1 1 1 T~r~ 11

(5.2.37)

h be the the interpolation error. If If m m >> nn ++ 2, 2, then by Let ijRfc == ip(hj) — ip(hj) be the remainder estimating the

I**I

n +22)) ( 0 | / ( n + < (Ai fo - ^ •i ))Af ei (i 2( 2/ ih, -i ) .-.-.-((((nn ++ l ) hAi i))__™ ™x«i1| l^^»( + ( l / ( » + 2)! ( n\t)\/(n +2) h»+\l2 (l -- V,)) _max _maxii |V\4n+2 2)!. << A?+ (0l/(n ++ 2)!-

(5.2.38)

m< < nn + + 2, let let y>(i) be be an an arbitrarily polynomial of of degree m, obviously, If m y>(x) =~ (x) xl>(x) Taylor's theorem,

V<x) = W * - ^ + s m + 2 -i H < 1 , r~T tt=0 =0

t! t!

m!

(5.2.39)

2. SEM for finite-difference

with

approximations

181

max \0X\ < ||V,||o^-i-<*[—1,13- Take (p(x) = 22x%—M—'

^ f°llows

t=0

that \Rh\

=

Mhj)-$(hi)\


+

\Hhi)-lp{hi)\

< Mfy) - f{hj)\ + ^an
-
Jb=0

■fan ,r)\Hr)hj)i m+a

<

m!

-
-||V>||c~+«[--i.iKi+Ei"" M*" fc — 1 k=i

(5. 2.40) +an ,nn" ,vm+i + a" )\ • (5.:

l+a

Using (5.2.37), lan,fc| < (n + 1)!, 77an)l, < 77^0 < n + 1. Substituting into (5.2.40), we obtain |JJ*| < C 4 /i™ +a ,

(5. 2.41)

for m < n + 2,

where CA = ^ J I H I C - H . [ - I , I ] { 1 + (" + 1)[(" + l)!»

m+

°

+ 2m]}

Using (5.2.36), we have £2V-(0) ^.)-2(0) + V ( - M h

J

n

= 5 1 t>n,ki>(-khj) +

(5. 2.42)

bn^Tjhj),

Jt=0

with &n,0 = (O„,0 - 2 ) / / l | ,

6„,i = (On.1 +

l)/h],

K,T) = o.n,r,/h2j, bn>k = anik/hj,

k=

(5.2.43) 2,--,n.

Finally, by ?2V>(0) - V"(0) _ j,(hj) - 2V(0) + 4>(-hj) _

*J

+

fa)

- rP(hj)

*I

V. SEM for differential

182

equations

it follows from (5.2.38), (5.2.40) and Taylor's expansion that (5.2.34) and (5.2.35) hold. D L e m m a 2.2. For the coefficients 6n>jb, n = 1,2,3, k = 0,1, •••,?!, defined by (5.2.43), there is

( " *».o - J2 K*D/( 2 A; + E l6»>*0 > CT > 0, fc = l

(5.2.44)

Jb = l

where Ci is a constanty which is independent of hj and rj and is dependent on n. Proof. For n = 1, evaluate by (5.2.37), and the coefficients are 6l

"

2 7,(77 + 1 ) ^ '

=

2 -°= - ^ | ' ^

6l

2 (i+W

=

Hence, evaluate directly the left-hand side of (5.2.44), r 2

2

irjh]

1 r2 /

2 +

(l + f,)h]\ ih]

(l + fi)h]\

= 2 ^ 4 **,e{0,l). Thus, in the case n — 1, C7 = —. For n = 2, the coefficients satisfy 6

"•"- n{n + m

3-T,

+ 2)hy

h

-°-

4-2^

r,h]

)

1-7,

The left-hand side of (5.2.44) is 3-7,

\nh)

4-277

(l + t,)h]

i

r2

4-27,

4-2T?

6-

2r,2 + 7j3

^(13-T,2) > (6 - 5T? + 2 T I 2 ) / 1 3

> 3/13, 7,6(0,1).

/

l-m

r

,

4-2IJ

1

l

~* 1 (2 + r 7 ) f t ? J U | ( l T ^ | (2 + »7)ft?J

[3-«? 5TJ +

1-77

1-m

2. SEM for finite-difference approximations

183

3 Thus, in the case n = 2, C-j = —. For n = 3, the coefficients satisfy

24 , ( l + ,)(2 + ,)(3 + , ) / i f

h,rj = ■

7-577

32

3

2(2-,) ,fc? '

'°

-4(1-77)

k3=

^-(IT^J' * ' -"(2TW'

The left-hand side of (5.2.44) is

r2(2-,) L i,A?

7-5, (! + ,)/»?

4(1-,) (2 + ,)/.?

7-5,

4(1 - , )

■2 >

_

(^W

1-, i (3 + , ) / l ? J / 1- ,

I

(1 + ,)/.? + (2 + ,)/»? + (3 + ,)/>? J

+

ihj

1-T;

r2(2 - i|) _ 7 - 5 , _ 4(1 - , ) _ In

1

,

1 + 7? 2+ , 3-1' r 7 - 5 , 4(1-,) 1,

[2+ 1 + , +

2

+,

+

3J

> (24 - 3 2 , + 2 0 , 3 ) / ( 6 8 , + 12, 2 - 20, 3 ) > (6 - 8 , + 8 , 3 / 3 ) / ( 1 7 , + 3, 2 ) > ±, Thus, in the case n = 3, C7 = TJT. ou

, € (0,1).

D

L e m m a 2.3. //it*1 is a solution of the difference equations (5.2.28) and (5.2.33) satisfying the boundary condition uh(x) = 0 (x G <9ft/»), then there is an estimate max ^ ( x ) ! < £ max I/OOI + - i - A g max \f{x)\,

(5.2.45)

where b > 0 is a constant such that Q, C (—6,6)5, CV ts determined by (5.2.44), where n = 1,2,3, and h0 = max hj. Proof. Using Theorem 2.2, decompose uh = 1*2+1*3, where u* satisfies Lhu\{x)

= -J26]4(x)

= / ( * ) , Vx G Qh,

(5.2.46a)

1=1

1u5(*)

= 0, V x e f i M U 6 f i A ,

(5.2.46b)

V. SEM for differential

184

equations

and u% satisfies -J2t]u*(x)

= Q, Vx G Qfc,

(5.2.47a)

i=i

- J2 6H(z) - iZ SH(X) = /(*)■ V* € «M. (5.2.47b) ti§ = 0, V*€d«fc.

(5.2.47c)

Construct the following majorizing function: s

U(x) = (sb2 - Y

x2) max \f(x)\/2s.

(5.2.48)

»=i

Obviously, U(x) satisfies -AU(x)

= -y^6]U{x)

= max |/(x)|, Vx € n h , (5.2.49a)

;=1

cr(x)>o, Viefl^uatifc,

(5.2.49b)

which implies I^OOI < ^0*). V x G f t M -

(5.2.50)

On the other hand, by using (5.2.26c), it follows that max |*4(x)| < max \Fh{x)lD{x)\.

(5.2.51)

Therefore it is necessary to evaluate D(x), x G ftj»,i, by (5.2.47b). Since £ € ^M> ^ n e n ^i ^ $• Letting j G / i and using (5.2.42), we have 6Juh3(x) _ u%(x + hjej) - 2u%(x) + u%(x -

hje,)

n

= ^&n|*iifc(x-JbAiei),

l
(5.2.52)

Jb=0

where x G Slh.i, % + h ; e j G ft and ti h (x + rjhjej) = 0, the coefficients 6n)* are constants, which are dependent on x and j , and independent of hj. By

2. SEM for finite-difference Approximations

185

(5.2.44), the coefficients of (5.2.47b) satisfy

£>(*)> Mn.O-£>„,*|) fc=l

>CV(2//,? + X> n ,*|) *=1

> 2C 7 /IQ 2 , 1 < n < 3.

(5.2.53)

Since u%(x) = 0, (x G dft), then Fh(x) = f(x). By using (5.2.5), it follows that max|t*j|(*)| < max !/(*)/£>(*)I < -i-&»max|/(*)|,

(5.2.54)

and

i«kooi <\uh2(x)\ + \uh3(X)\
1 Vx G nh)t,

The proof is complete.

1 < n < 3.

□

T h e o r e m 2.3. If in the problem (5.2.27); <9Q G C 5+
+

0(h*+'),

i=i

x GJJfcUJJfc.i, where Wj(x) are independent of h.

(5.2.55)

X86

V. SEM for differential equations

Proof. Under the assumptions of Theorem 2.3, it implies u € C 5+<7 (fi) (cf. [52] ). Using (5.2.34), (5.2.28) and (5.2.33), one obtains

-LV*)-A«(x) = £ g ^

+ 0(/^).

xGfiftUftfc,,-. Let

I

d4u(x)

12 dxj problem

(5.2.56)

= Vj(x) G C 1+
- A W J ( X ) = VJ(X),

in Q,

u>j(x) = 0,

(5.2.57)

on <9fi, i = 1, •• - , s ,

with the solution Wj(x) G C 3+Qf . By using (5.2.5), we have -Lhwj(x)

- AWJ(X)

= 0(hl+a),

xetthV

fifc,,-.

Hence,

Lh(uh-u-J2h]™j) i=i

= - A w - Lfcu - ] £ A i w i " $ 3 ( A u ; i j=i

+

Lhw h

i) ]

i=i

= 0(/i^ +
(5.2.58a)

and u* - u - ^

^j WJ = 0, Vx G 9 0 .

(5.2.58b)

It follows from Theorem 2.1 and its Corollaries that (5.2.55) holds.

D

Theorem 2.3 shows that when n = 2, the boundary d£l and the func­ tions / and g are smooth enough, the extrapolation method can be applied once; when n = 3, the boundary and the functions are more smooth, the extrapolation method can be applied twice according to the following: T h e o r e m 2.4. If in the problem (5.2.27), the boundary dQ G C 7 +", the functions f G C 5 _ H T ( Q ) and g G C7+
2. SEM for finite-difference approximations

187

number of interpolation points n = 3, then the error of the finite difference solution uh(x) has a multivariate asymptotic expansion uh(x)-u(x)=

]T

hWwp(x) + 0(h\+*),

(5.2.59)

1
where wp(x) are independent of h. Proof. Under the assumptions of Theorem 2.4, it implies u G C 7+<7 (Q). By using (5.2.34), we obtain -Lhu-

Ahu

- l V / , 2J ^ 12f-f j=i

dxjJ

1 Af,t . 3 6 W

+ 36oE )^6+°(/lo+<7). i=i i Let v* = J

innkunh,,-.

(5.2.60)

1 d6"

, v,- G C 1 + a (Q) and let u>7- be the solution of the following 360 0s? '

equation: —AWJ = Sj,

in Q,

Wj — 0,

(5.2.61)

on 5Q, j = 1, • • •, s.

Obviously, Wj G C 3+<7 , and = 0(/iJ+a), j = l,..-,«.

-Lhwj-Awj

1 #4" „ , Vj G C 3+<7 (Q) and let Wj satisfy

Furthermore, let v; = 3

(5.2.62)

ndxj

-AWJ

J

= 5y,

{?j = 0,

in Q,

(5.2.63)

on <9ft, j = 1, • •, s.

Obviously, Wj G C 5 + a ( Q ) . Let iu^ be the finite difference approximation of Wj) satisfying LhS^ = VJ, J

wf = 0,

mQhunh}i, ond£lh.

(5.2.64)

188

V. SEM for differential

equations

By using Theorem 2.3, there exist functions Wjtk, which are independent of ft, such that the following multivariate asymptotic expansion holds 5

3 + 2.65) u nhM,,-. .(5.: wf(x) -- W WJ{X) xx 6€ ft/, Slh U (5.2.65) j ( x ) == Y, kl&jA*) 0 °), w* (*) " ,fc(x) + + O(h 0(A2+*).

fc=i

Using (5.2.61)-(5.2.64), we have 53

*J) E™J■*J)

Lh(uh -- u i=i

i=i

5

5

j=l

j=l

= -A« - Lhu - Y^^h] - ^vjhj =

+ O(h50+a °), O(^o )>

in

+ 0(hl+") (5.2.66a)

mQUJlfc,,-, «fc hunh,i,

and 5 S

h

Q h

u -u-Y^tftf-H2 3 t j=i ;=i

S3

= °>

i j=i =i

on5n on dfifc. *-

5 2 66b ((5.2.66b) -- )

By using Theorem 2.1 and its Corollaries, it follows that

2

^E^ = E^(

uh/l - - IX u — —

14

i3=1 =i

5s

i 3=1 =i 3 3

Sj

j=l

fc=l Jt=i

^l ++ o0(0 (^) 53

:hj .0 + JE"i ^ =l

in fi Q,hA.UQfc,,-. Ui]/, ( i. inQfcUQft,,-. in

;'=i

+f O (0(h?'), A^), (5.2.67)

In (5.2.67), grouping the same terms and letting wp(x) be the coefficient functions of h2f3 (1 < |/?| < 2), we obtain (5.2.59). D R e m a r k 1. When n > 4, Lemma 2.2 is invalid. However, Marchuk proved that the inequality (5.2.44) still holds if a larger ft = ph is used, where p is an appropriate positive integer. Obviously, the corresponding difference scheme will be more complex. R e m a r k 2. Theorems 2.3 and 2.4 are also valid for the elliptic differential equations of type (5.2.1). The crux of the proof is to verify the maximum principle. For details, see B o n W 1 0 ] .

2. SEM for finite-difference

approximations

2.3. Multivariate asymptotic difference approximation

189

expansion of the on a rectangular

finiteparallelepiped

Consider the quasilinear elliptic partial differential equation

Ati = /(*,ti),

inft = IKOAOi

(5.2.68a)

»=i

u = 0,

on dft.

(5.2.68b)

In order to guarantee that (5.2.68) has a unique solution, assume that / ( x , u) is a smooth function on ft x JR, and

(5.2.69)

fi(u):=K(x,u)>0.

Take the mesh width h = (h\, • • •, /i 5 ), where hi = 6,/iV,-, i = 1,•• •, s, and iV,- are positive integers. Let ft/i = {x = (xi, •••,*,) : Xi = jhi} 0<j<

dnh = Qhn 5ft, nh = nh\ 3ft.

Niy 1 < i < s},

Construct a difference operator s

Ahu(x)

= ] T [u(x + hkek) - 2u(x) + u(x -

hkek)]/h2ki

*=i

VxGft*,

(5.2.70)

and the corresponding difference equation Ahuh(x)

= / ( x , tifc(x)),

i t ^ x ) = 0,

in n fc , onfifift.

(5.2.71a) (5.2.71b)

Here, the scheme is simple since there is no irregular point. However, ft is a nonsmooth domain, it is hard to guarantee the smoothness of the solution of the auxiliary problem. The following lemma points out the conditions for the smoothness of the solution, and the relevant proof will be found in [90]. L e m m a 2.4. Consider the equation - A n + du = f,

u = 0,

in ft = (0,b x ) x (0,6 2 ),

(5.2.72)

on 3ft.

If the coefficients d and f G C 1 + a (ft), wh ere 0 < cr < 1, and / ( x ) = 0 at the corner points o/ft, then the solution u G C 3+
190

V. SEM for differential

equations

L e m m a 2.5. If the coefficients of the equation (5.2.72) d and f G C 3+<7 (Q) and satisfy the coherence condition at the corner points of Q d2f d2f = 0, 0, -— (Wx ) - - W r(*) = = 0, 0, //(x) (s) =

(5.2.73) (5.2.73)

then uGC 5 + < 7 (ft). It is easy to see that in Lemmas 2.4 and 2.5, the coherence condition at the corner points are not only sufficient but also necessary. In fact, if u G C 3+
+

8d2u(x) — = /(x). dx\

°=^r ^r= / w -

(5(5.2.74) 4)

"

In addition, if u £ C5+" at the corner points we also have the following: iz(x) d4u(x)

00 = =

= (

dx\ 5d2

2 dx dx\ d2f

3d44u(x) u(x) 5d2

dx\

dxl dx\ d2f

)Au{x) )Au(x)

== dxf — o(x)o —dx\ 0 y(x). 0.

(5.2.75) (5.2.75) K }

2

dx dx\ Lemmas 2.4 and 2.5 can also be generalized to multidimensional cases. Lemma 2.6. For the equation (

s3 6 in ft = n (°> j)> 5s >> 3>3, (0, &j), i=i on 3Q, on 3Q,

-Au + du = / ,

< II

uu = = 0, 0,

(5.2.76) (5.2.76)

if the coefficients d and_J G C 1+<7 (Q), then the necessary and sufficient condition for u G C 3+ *(fl) is /f(x) ( x ) = 0, Vx G G M.

(5.2.77)

3+or

Furthermore, if d and f G C (ft), then the necessary and sufficient conditions for u G C 5 +"(Q) are (5.2.77) and 02f

d2f

3xt3xf

<9x^ <9xj

w/iere M 15 the set of all k (< s — 2)-dimensional boundaries of£l.

(5.2.78)

2. SEM for finite-difference approximations

191

Proof. First, consider the case 5 = 3. It is easy to prove the necessity of the conditions. Consider an arbitrary point on one of the edges, e.g., A = (ai,0,0), 0 < ai < 6i, if ii G C 3+Qr (ft), then the vectors with A as a vertex, and rays parallel to any one of the three coordinate axes, are all on 9Q. Hence by the boundary condition, the following equality holds, 2

rd

d2u

u

d2u-i

LsrsrsjU"'™-

0 = 5+C

<5 9)

"

If u G C *(Q), then two inward normal vectors at point A lie on 80, and are parallel to 0 x 2 and Ox^ respectively. Hence, dAu

0

dAu =—(A)-—(A) ox\

ox\ K

dx\ 2

,d = ( K

dx2

d2f

dx\'^dx\ 2

d

dx\> '*=*

d d2 32N ) ( — + — + — In 2

w

dx\)K2 dx2

dx2j 1-=^

dx\

df

= — - (A) - —(A). dx\ dx\ Here the condition

92u

(5.2.80)

(A) = 0 is used. Therefore, the coherence conditions

dxi

on the vertices and edges for s = 3 are necessary. In order to prove that the conditions are sufficient, take an arbitrary point A = (ai, a 2 ,03) G ft. Let fii = ftf){x : xi = a i } , which is a cross sec­ tion perpendicular to the xi-axis and passing the point A. By tii(x 2 , X3) = u(ai,X2,X3), obviously, u\ is the solution of the two-dimensional problem

d2u\ ~

=

d2u\ r +

du

i

dx\ d2ux

dx\ d2ux d2Ul

dx\

dx\

dx2

= /(ai,x2,x3), in fii, i/! = 0, onffii,

+ dux (5.2.81)

and satisfies the coherence condition of Lemma 2.4 (or Lemma 2.5 respec­ tively), which implies that the 3-rd order (or 5-th order respectively) deriva­ tives of u(ai, X2, X3) with respect to x 2 and X3 are Holder continuous. Since

192

V. SEM for differential

equations

di may be any value within (0, fei), then for a fixed xi, u(xi, x 2 , X3) belongs to C 3+<7 (Qi) (or C 5+
+

d2v1 ^ +

dVl

= f

*>(ai>X2>X3)>

vi = 0, with fXl(x1)x2ix3)

inQl

>

(5.2.82)

on 3fii,

d

If / G C 1+ "(Q), then fXl G C"(0)

= —f(xuz2,x3). axi

and t;x G C 2+<7 (Qi). Since vi(x 2 ,x 3 ) = -r—(ai,x 2 ,x 3 ), thus - — — , o*\ dxidx^ d3u 33ti 3 2+a dx\dx\ and dxidx dx3 G C'(Q). If / G C +"(ft), then / P l G C (fi). 2

It is easy to prove that / r i ( a i , x 2 , X3) at four corner points of fii satisfies: - - = 0, which leads to v1 G C 4+Qf (Qi), or dx\ dx\ Dau G C^Q), |a| = 4, c*i = 1. Similarly, construct a cross section passing the point A and perpendicular to the x 2 -axis, one can also prove that the conclusions are valid for other mixed derivatives of u(x). For the case n > 3, the conclusion of Theorem 2.4 can be proved by induction, which is omitted here. D f(ai,x2}x3)

= 0,

Theorem 2.5. Let u be the solution of the problem (5.2.68). If u G C 5+
h

+ ^Tft'u/,- = 0(&o+")>

u -u

in Qh

>

(5.2.83)

t=i

where W{ G C 3+(7 (ft) (i = 1, • • •, s) are independent ofh. Proof. Since u G C5+"(Q), then v ^

A*II-AII = £ i=l

Let Vi

1 04« = ——e I2dxj

C1+°(n),

/i2

&

u

^ _

:0r?

+ o(fcj|+*)f

i = 1, • • •, s. Noting that

uh = u + O(h20),

innh,

infl*.

(5.2.84)

2. SEM for finite-difference approximations

193

there is (Ah - Mu))(uh

- u)

h

= f(x, u ) - f(x, u) - A h u + Au - /i(w)(« h - «) 8

= - X > , ? f . + 0(ft|!+<7),

infi„.

(5.2.85)

«= 1

Construct an auxiliary problem (A - f\(u))wi

= t;,-,

in Q,

<

(5.2.86) w;t- = 0,

on dfi, t = 1, • • •, s.

We shall now prove w{ G C 3 + a (ft). Since fx(u) and v{ G C 1 + *(fi), by using Lemma 2.6, it is sufficient to prove Vi(x)

= 0, Vx G M,

(5.2.87)

where M is the set of all k (< s — 2)-dimensional boundary points. In fact, take any x G Af, and draw a sufficient small segment /,- starting from the point x and parallel to the x,-axis, obviously, /,- C d£l. It follows from the boundary condition (5.2.68b) that ^ = vi(x) = 0, Vx e M, 12 5x?

(5.2.88)

which leads to Wi £ C3+
(5.2.89)

Substituting (5.2.86) and (5.2.89) into (5.2.85), one can derive

(Ah-f1(u))(uh-u

+

J2^wi) »= 1

*=i

t=i

= £>?(**-A)«*+0(0 »=i

= O(h30+a),

infi^,

(5.2.90a)

and the boundary condition uh

-

u

s

+ ] C A i ?u;i = °'

o n dQh

'

(5.2.90b)

V. SEM for differential

194

(5.2.83) follows from the discrete maximum principle.

equations

□

If u G C7+
^2

h w

^ ^

= °( fc o + *)>

in

"»>

(5.2.91)

1
where /? = (ft, ••-,&) and wp G C 7 - 2 M+"(n) (1 < |/?| < 2) are indepen­ dent ofh. Proof. Since u G C 7+
Ahu-Au=

^2

h2pvp+0(hl+°),

(5.2.92)

1<|/?|<2

where vp G C5+<7~2'^(f2) are independent of h. Using Theorem 2.5 and (5.2.92), one can derive = / ( x , uh) - / ( x , u) - /i(t*)(tifc - ti) + Ati =

X)

^ 2 % + 0(^+
Ahu

inO fc>

(5.2.93)

1<WI<2

where wp G C5+<7~2I^I(Q) are independent of h. Construct an auxiliary equation ( A n - fi(u))wp = wp,

in Q,

<

(5.2.94) wp = 0,

on dft,

and the corresponding approximate equation r

( A * - / i («))«£ = £ , , <

innft, (5.2.95)

u>/j = 0,

on dfi/,.

Now we shall prove wp G C7+a-2M(Ti), 1 < |/?| < 2. First, consider the case \f}\ = 2. In order to prove u^ G C 3+ "(ft), by using Lemma 2.6, it is sufficient to prove wp(x) = 0, Vx G M, where M is the set of all k (< 5 - 2 ) dimensional boundaries. Comparing (5.2.93) with the identity f(x , « A ) - f(x, u) - h(u)(uh

- u) = 0, Vx € M,

(5.2.96)

2. SEM for finite-difference approximations

195

one obtains vp(z) = wp{x), V 1 6 M , 1 < \/3\ < 2.

(5.2.97)

But vp is a 6-th order partial derivative of u, hence, it follows from u\dn = 0 that vp(x) = 0, V* e M, and then wp € C 3+
= 0, V* €

= ^-(x)

5+
M, i = 1, • • .,*. By Lemma 2.6, wp G C (fi). Applying Theorem 2.5, the approximate solution w^ (\/3\ = 1) of wp has an asymptotic expansion h2aw0a

E

Wp - Wp =

+ 0(h30+°),

in Qh,

(5.2.98)

|o|=l

where wpa are independent of h. By (5.2.93) and (5.2.95), we have

E **S)

( A * - A («))(«*-«-

1
= ( A * - / ! ( « ) ) (tl* •-u-

E

i<\P\<2

^ % + £ V'frfi-WfiJ) i^l/^l^s

2

- « - E * % + £ E/i2(/J+0f)^

= (A* - / ! ( « ) ) ( « * ■

1
|/9|=l|o|=l

+<W)) = 0{hl+"),

(5.2.99)

infih

Grouping the terms with the same power of h in wp and u;^ 0 , from (5.2.99) one can derive ( A * - A ( u ) ) ( « 'k - « -

ft2%)=0(/«^).

E

infih,

1
(5.2.100a) and uh-

u-

£

h2/3wl) = 0, on<9Qh.

(5.2.100b)

1
(5.2.91) follows from the discrete maximum principle.

□

R e m a r k . Marchuk and Shaidurov a s s e r t e d ^ that the auxiliary prob­ lem may not satisfy the coherence condition, and then the expansion to

V. SEM for differential equations

196

higher order may not hold even for the case / G C* + a (Q), k > 4. However, the proof of Theorem 2.6 is under the assumption u E C 7 + a ( Q ) , hence, the expansion to higher order can also be proved provided that u is sufficiently smooth.

2.4. Numerical

examples

E x a m p l e 2 . 1 . Consider the two-dimensional Poisson equation - A n = /, u = 0,

in Q = (0, l ) 2 ,

(5.2.101)

on dfi,

with the exact solution u(x, y) = x(l—x)y(l—y) cos(7rx/2) cos(iry/2). Using the five-point finite difference scheme and solving the linear equation by the conjugate gradient method, the results of the Richardson extrapolation and Types 1, 2 splitting extrapolation methods are given in the following tables: Table 2.1. Step size hx = hy = J number of

Richardson extrapolation

extrapolations max error

CPU(sec)

Type 1 SEM

Type 2 SEM

max error CPU(sec) max error CPU(sec)

0

2.236E-3

0.01

2.236E-3

0.01

2.236E-3

0.01

1

1.230E-5

0.17

6.564E-5

0.12

6.564E-5

0.14

2

4.113E-8

1.59

6.159E-7

0.71

5.977E-7

0.52

3

7.008E-11

14.60

1.406E-8

3.80

1.754E-8

1.80

4

9.6E-14

125.13

3.215E-10

17.69

8.865E-9

4.35

8.739E-12

79.32

5.266E-9

9.84

5

Table 2.2. Step size hx = hy = ^ number of

Richardson extrapolation

extrapolations max error

CPU(sec)

Type 1 SEM

Type 2 SEM

max error CPU(sec) max error CPU(sec)

0

6.111E-4

0.14

6.111E-4

0.13

6.111E-4

0.13

1

8.825E-7

1.65

4.724E-6

1.43

4.724E-6

1.45

2

6.786E-10

15.10

1.168E-8

8.16

8.086E-9

5.74

3

1.729E-10

40.02

5.635E-9

17.60

4

3.803E-12

179.00

9.000E-9

42.83

3.0514E-9

93.20

5

The solution of this example is smooth, and so three kinds of extrapola­ tions are all effective. The Richardson extrapolation has the best stability but the amount of work and storage increase rapidly with the number of

2. SEM for finite-difference

approximations

197

extrapolations, while it increases much slower for the splitting extrapola­ tion, especially for the Type 2 methods. However, due to the poor stability, the accuracy will be limited when the number of splits is large. E x a m p l e 2.2. Consider the two-dimensional Poisson equation (5.2.101). Its exact solution u{xy y) = sin(16x -f 16y)x(l — x)y(l — y) is an oscillatory function. The numerical results are listed in Tables 2.3 and 2.4. Table 2.3. Step size hx = hy = J number of Richardson extrapolation Type 1 SEM Type 2 SEM extrapolations max error CPU(sec) max error CPU(sec) max error CPU(sec) 0 2.704E-1 0.02 2.704E-1 0.02 2.704E-1 0.02 1

6.110E-2

0.19

2.823E-1

0.14

2.823E-1

0.15

2

2.730E-2

1.95

9.970E-2

0.75

1.019E-1

0.56

3

1.733E-3

18.04

4.23

4

4.895E-4

152.11

1.591E-2 1.122E-3

19.92

1.927E-2 2.227E-3

4.69

4.064E-5

89.47

1.83

1.751E-4

10.69

6

9.860E-6

20.76

7

2.696E-6

37.83

5

Table 2.4. Step size hx = hy = ^ Type 1 SEM number of Richardson extrapolation Type 2 SEM extrapolations max error max error CPU(sec) max error CPU(sec) CPU(sec) 0.17 2.673E-2 0.16 0.16 2.673E-2 0 2.673E-2 1

1.485E-3

1.86

4.992E-3

1.79

4.992E-3

1.64

2

6.040E-3

17.28

3.538E-4

9.70

3.518E-4

6.53

3

1.551E-5

47.58

1.630E-5

20.25

4

3.538E-7

213.64

6.464E-7

48.49

5.578E-8

105.91

5

In this example, the accuracy of the first few approximations obtained by SEM is not improved significantly since the relatively large initial mesh widths hx and hy cannot reflect the oscillation of u(xy y). The accuracy in­ creases rapidly with the number of extrapolations. In addition, for the Type 2 methods, the CPU and storage only slightly increase with the number of extrapolations. By contrast, the accuracy of the approximations obtained by the Richardson method decreases with the number of extrapolations, because the global refinement yields a large linear system. E x a m p l e 2.3. Consider the 3-dimensional Poisson equation

f Au = f, u = 0,

inn = (0,l)3, on d£l,

V. SEM for differential equations

198 3

TTXi

t=i

2

with the exact solution u(xi, x2) x3) = ]\ [*«(! — x%) c o s (

)] • The com-

parison between the numerical results and storage of the Richardson ex­ trapolation and the splitting extrapolation are given respectively in Tables 2.5 and 2.6. And the comparison of CPU for solving subproblems with dif­ ferent degree of parallelism on a multiprocessor computer is given in Table 2.7. =fco= i Table 2.5. Step size h\ — = hh~ 2 = ^3 = J Type 2 SEM

Type 1 SEM

Richardson extrapolation

number of

extrapolations max error

max error CPU(sec) max error CPU(sec)

CPU(sec)

0

3.91E-4

0.04

3.91E-4

0.04

3.91E-4

0.04

1

2.44E-6

0.71

6.79E-6

0.52

6.79E-6

0.49

2

7.61E-9

13.6

1.90E-7

2.53

1.90E-7

2.09

3

3.23E-11

285.67

5.06E-9

12.98

5.32E-9

7.41

4

1.41E-10

21.94

5

7.89E-12

57.11

Table 2.6. Comparison of storages for three extrapolation methods

(hx = h2 = h3 = J.) number of extrapolations Richardson extrapolation Type 1 SEM Type 2 SEM 0

162

162

162

1

1769

339

339

2

16956

945

945

3

149063

2135

2135

4

1250370

4410

3430

5

2048543

9051

5411

Table 2.7. Comparison of degree of parallelism for three extrapolation methods number of

Richardson extrapolation Type 1 SEM Type 2 SEM

extrapolations N

max CPU

JV max CPU N max CPU

1

2

0.33

4

0.13

4

0.13

2

3

12.94

10

0.38

10

0.38

3

4

272.01

20

1.42

20

0.67

4

5

35

35

1.44

5

6

55

55

2.54

Here, N denotes the number of subproblems and max CPU denotes the CPU needed by the largest subproblem. Table 2.7 shows that the degree of parallelism of the SEM is high.

2. SEM for finite-difference

approximations

199

E x a m p l e 2.4. Consider the 4-dimensional Poisson equation AM

= /,

U = 0,

in Q = (0, l ) 4 ,

on 3ft, WXi

with the exact solution u(xi, x 2 , a:3, z 4 ) = fj[ [x,(l - a?,-) cos ( ) ] . For in*=i 2 formation on the accuracy, CPU time, storage and the degree of parallelism, see respectively the following tables: Table 2.8. Comparison of accuracy for three extrapolation methods (*! = h2 = h3 = fc4 = \) number of Richardson extrapolation Type 1 SEM Type 2 SEM extrapolations max error CPU(sec) max error CPU(sec) max error CPU(sec) 0 6.8E-5 0.18 6.8E-5 0.14 6.8E-5 0.14 4.76E-7 1 8.23 3.24E-6 1.85 3.24E-6 1.91 2.10E-9 2 407.30 8.21E-9 14.48 8.26E-8 12.73 2.00E-9

3

60.86

Table 2.9. Comparison of storage for three extrapolation methods (hX = h2 = fc3 = *4 = \) number of extrapolations Richardson extrapolation Type 1 SEM Type 2 SEM 0

486

486

486

1

12197

1350

1350

2

253368

1656

1656

3

4617929

7980

7980

Table 2.10. Comparison of degree of parallelism for three extrapolation methods (hi = h2 = h3 = /14 = - ) number of Richardson extrapolation Type 1 SEM N max CPU extrapolations N max CPU 5 0.46 8.04 2 1 1.47 15 399.05 3 2 3

31

Type 2 SEM N max CPU

5

0.46

15

1.32

31

3.52

This example shows that the higher the dimension, the more effective the SEM, and the higher the degree of parallelism. E x a m p l e 2.5. Consider the two-dimensional inhomogeneous boundary value problem of the divergence type d du d <9u -(e-5*—) + — (e-5*— ) = / , inQ = (0,l)2, dx on dQ, u = 9,

200

V. SEM for differential

equations

with the exact solution u = (1 — e 5 ^"" 1 )) sin(7ry). The numerical results are given in Table 2.11. T a b l e 2 . 1 1 . (hx = hy = number of

Richardson extrapolation

extrapolations max error

max CPU

\)

Type 1 SEM

Type 2 SEM

max error max CPU max error max C P U

0

2.22E-2

0.09

2.22E-2

0.02

2.22E-2

0.05

1

3.68E-4

0.23

1.48E-3

0.15

1.48E-3

0.20

2

1.73E-6

1.96

2.74E-5

1.22

2.72E-5

0.60

3

3.93E-9

105.13

3.96E-7

11.03

5.33E-7

2.08

4

2.68E-12

591.71

3.75E-9

47.42

8.57E-9

7.88

5

3.0E-15

1282.95

1.76E-11

274.8

1.14E-10

22.78

6

1.20E-12

52.85

7

3.00E-14

155.45

3. S E M for finite-element approximations The extrapolation of the finite element method is a new research field evolved in the last decade, it develops rapidly and has a wide scope of application. Recent works show that the extrapolation and asymptotic ex­ pansion of the finite element approximation have become the bases tone for studying the theory of finite element superconvergence, algorithms with a high order of accuracy and a posteriori error estimate. For details, see the monograph by Q. Lin and Q.D. Zhu^'^ and the summary by R. Rannacher^ 10 ^. The extrapolation of the finite element method has a great advantage over that of the finite difference method. First, the latter has its limita­ tion: for smooth domains, the boundary interpolation technique can help to derive asymptotic expansions, but at the same time, it also destroys the uniformity of the scheme and makes the coefficient matrix nonsymmetric. Second, the finite difference method has theoretical defects, e.g., in order to extrapolate once, it requires the exact solution u € C 5+
3. SEM for

finite-element

approximations

201

equal to the dimension of the problem, while in the finite element method, the independent mesh parameters can be set according to the shape and size of the problem and the characteristics of the computer. And a large prob­ lem can be converted into several smaller subproblems and then solved by SEM on a multiprocessor computer. It is concluded that the finite element splitting extrapolation is more suitable than the Richardson extrapolation in solving large scale problems on multiprocessor computers. 3.1. Finite-element equations

approximations

of second

order

elliptic

In the following we shall introduce briefly the three main parts of the finite element method, namely, the weak form of a problem, the subdivision of a domain and the Ritz-Galerkin method. Consider a second order elliptic equation: s

(

- ^2 DjiaijDiu)

+ du = / ,

in fi C JR5,

(5.3.1a)

on Ti C dQ,

(5.3.1b)

onr0 = an\ri,

(5.3.ic)

M=l I

\

5

^2 aijDiUUj = g, »,i=i

( ti = o,

where v = (i/i, • • •, i/s) is a unit outward normal derivative, (5.3.1b) is called the natural boundary condition, and (5.3.1c) the coercive boundary condi­ tion. In order to derive the weak form of (5.3.1), construct the following function space: V={vGH1(Q):v\rQ

(5.3.2)

= 0}.

Obviously, V is a Hilbert space, satisfying the inclusion relation Hi(Q) CVCH

= H*CV+

C

H'\il)y

where H = L2(Q,). Construct a bilinear form on V a(u,v) = I (Y\

aijDiuDjV + duv)dx.

(5.3.3)

a ( , •) is said to be V-elliptic, if there exists a constant y > 0, such that a f t i . t O ^ T l M f t , Vti€ V.

(5.3.4)

Obviously, if the matrix [a^-]. = 1 is symmetric and uniformly positive def­ inite on Q, and d G L°°(ft), d > 0, then (5.3.4) holds.

V. SEM for differential

202

Definition 3.1. ueV

equations

is called a weak solution of (5.3.1), if it satisfies a(u,v) = f(v),

VvG7,

(5.3.5)

where the right-hand side of (5.3.5) f(v) = / fvdx + /

Jo,

Jr1

gvdx

is a continuous linear functional on V. By using the Lax-Milgram theorem^231, the ^-ellipticity guarantees the existence and uniqueness of the weak solution. The Ritz-Galerkin method is to find a class of finite dimensional subspaces Vh! C Vh2 C V for hi > h2, and the corresponding approximate solution uh EVhy satisfying a(ti\i;) = /(«), V w € % .

(5.3.6)

Furthermore, by using the Lax-Milgram theorem, the approximate solution satisfies uh = Rhu,

(5.3.7)

where Rh : V —► Vh is a Ritz projection, i.e., the orthogonal projection takes a(-, •), defined in (5.3.3), as an inner product. In other words, uh G Vh is an optimal approximation to u in the sense of inner product a(-, •). Thus, the density of \J Vh in V implies the convergence of the Ritz-Galerkin method. /i>0

The finite element method is a special case of the Ritz-Galerkin method. The difference between them is that the finite element space Vh is a set of piecewise polynomials, and the classical Ritz-Galerkin method often regards Vh as the set of smooth functions. Hence, the partition Jh — {e} of a domain Q is the first step of the finite element method, where e stands for a partition element. However, the following restriction is imposed: suppose that t\ and e2 are two different elements, then either t\ D e2 is an empty set, or they have a common vertex, or a common side. The most important elements are the triangular and rectangular in two-dimensional problems and the cuboid in three-dimensional problems. If Jh is a triangulation of fi, one can then define linear finite element subspace Sh(Q) = {ve C(Tl) : v\e € Fi(e), Ve €

Jh}}

where P\(e) denotes the set of linear functions on e. If Jh is a rectangular partition of fi, then one can define a bilinear finite element subspace S h (fi) = {v e C(ty : v\e € Qi(e), Ve €

Jh),

3. SEM for

finite-element

203

approximations

where Q\{e) denotes the set of bilinear functions on e. For the threedimensional cuboid elements, the trilinear finite element subspace can be established similarly. Now take Vh = Sh(Q) C\ V> the corresponding finite element approximation uh G Vh satisfies a(uh,v) = f(v),

Vw€ % .

(5.3.8)

Applying the basis functions (pj, (5.3.8) can be expressed as an equivalent algebraic equation. There is an one-to-one correspondence between a basis function and a node. The j-ih basis function

= 6ij

=

0,

i±h

hj = l , - - - , m ,

where N{ are nodes, m is the number of nodes excluding those on To, and it is the dimension of Vh. Let m

"*(*) = 5>;WOO'

(5.3.9)

where rjj = uh(Nj), j = 1, • • •, m, are unknowns. Substituting (5.3.9) into (5.3.8), one obtains a linear system of 77 = (771, • • •, rjm)T Arj = 6,

(5.3.10)

where A = [a,j] is an m x m matrix with elements atJ- = a( i = 1, • • •, m. The monographs [23] and [132] discussed the error estimations of the fi­ nite element method. When considering the flexibility of the finite element method, one must note that the accuracy depends directly on the partition. For example, in order to guarantee the convergence, the partition usually has to be regular and quasi-uniform^. In order to guarantee that the er­ rors at the nodes have asymptotic expansions, the partition is required to be piecewise uniform^ 1 '. Recent research shows that an "optimal partition" should use as many rectangular elements or piecewise regular quadrilateral elements as possible (for details, see Q. Lin and Q.D. Zhu^'''). In the fol­ lowing, we discuss the SEM for finite element method by using the integral identity of rectangular elements. We assume that our readers are already familiar with the basic knowledge of the finite element method and only the relevant references are given.

204

V. SEM for differential

3.2. Basic

expansion

of bilinear

rectangular

equations

elements

For simplicity, consider the Dirichlet problem of Poisson equation - A n = /,

in ft, (5.3.11)

u = 0,

on dft,

where ft C M2. Suppose that Jh is a rectangular partition of ft, and So (ft) C ^ o ( ^ ) ls ^ n e corresponding space of bilinear elements, the finite element approximation uh G SQ satisfies a(uh,v)

:= / (DiuD\v + D2uD2v)dxdy

= / fvdxdy,

Vv G S#(ft),

(5.3.12)

d d with D\ = — and Di = — . Define the interpolation operator Ih : C(ft) —* 5g(ft), where the in­ terpolation basis points are the nodes of Jh. Let e G J ^ be an element defined by e = (xe - h e , X e + / l e ) X (T/C - * e , 2/e + * e ) ,

(5.3.13)

where (x e ,y e ) is the center of e, while /i e and ke are the halves of two edge-lengths of e respectively. Let hi = max{/i e },/i2 = max{ibe} and eeJk eeJh h0 = max{/ii, A 2 }. We shall prove the following integral identity: Lemma 3.1.1''' For any rectangular element e G Jh, the following integral identity holds: / Di(u — Ihu)Divdx.dy Vi; G S*(ft),

= I B{y)Dl{Dl{u^Ihu)Dlv)dxdy,

(5.3.14)

where (5.3.15)

B{y)^\[{y-yef^kl\. Proof. Since B"(y) = 1, IDi(u-

Ihu)Divdxdy

= I B"(y)Dl{u

-

Ihu)Divdxdy.

3. SEM for finite-element approximations

205

On the right-hand side, integrating by parts with respect to y twice, one obtains (5.3.14). a L e m m a 3.2.( 7 7 1 If u G % 3 ( f i ) (2 < p < oo), then there 15 a superconvergent estimate a(u - Ihu, v) < Ch20\\u\\3)P\\v\\iy,

Vt, G S£(fi),

(5.3.16)

where p' satisfies - + — = 1. p p P r o o f . Note t h a t F(y) = \[B{y)}2.

B{y) = F"(y) - V o

(5.3.17)

0

Substituting into the right-hand side of (5.3.14), integrating by p a r t s and applying the inverse estimate, one obtains

J B(y)D2(D1(u -

h^D^dxdy

= j Biy^DlDxuDiV

+ 2DiD2(u

= J B(y)D2D1uD1vdxdy =

+ 2DlD2v

I B{y)D\D1uDxvdxdy-2

< C*fc>| 3 ,p, e |t;|i, p ',e +

< Ck2e\u\3,pMhP>,',

-

Ihu)DlD2v]dxdy J B(y)D iZ?2(« — Ihv)dxdy

j F'\y)D1D2v

■

DiDJudxdy

Ck3e\u\3,Pte\vh,P',e

Vt;€50h(n),

(5.3.18)

where, and hereafter, C is a constant independent of the partition and the notation | • \m,p,e denotes a semi-norm of Wpm(e). Substitute (5.3.18) into (5.3.14), it becomes / Di(u - Ihu)Divdxdy

< Ch20\u\3)P>e\v\1)P>te.

(5.3.19a)

j D2(u - Ihu)D2vdxdy

< Chl\u\3,ptc\v\ilP;e.

(5.3.19b)

Similarly, there is

Finally, by the Holder's inequality, we obtain a(u-

IhU,v)

206

V. SEM tor differential

= £

r /

< Ch20 ]T)

equations

2

(.52Di(u-Iku)Div)dzdy \u\3,p,e\v\iy,e

e€Jh

< chK Y: \
e£jh

< Cfc2||n||slPln||t;||ilp'ln, Vu G 5j(0). T h e proof of (5.3.17) is thus complete.

O

It is known t h a t for a bilinear finite element space, t h e interpolation error satisfies \\u-Ihu\\1)P
(5.3.20)

For a linear finite element space in a uniform triangulation or in a piecewise uniform triangulation, a similar super convergent estimate as (5.3.16) is still valid, b u t t h e proof is n o t as simple as t h a t for t h e case of rectangular elements. In t h e following argument, t h e crux is t o apply an auxiliary function B(y) which was suggested by Q. Lin in [70]. By this technique, we have t h e following theorem: T h e o r e m 3.1.'''J Suppose that Jh is a regular rectangular partition, n SQ(Q) is a bilinear finite element subspace. If u G ( f l ^q(e)) #o(^)> eeJh (1 < q < oo), then forVv G SQ(£1), there is an asymptotic expansion a{u — u1 ,v) =

D

+

-2F'(y)D22D1uD1D2v]dxdy

{ / [B{y)D\DluDlv

I [A{X)D\D2UD2V

-

2E'{x)DlD2uD1D2v\dxdyy

(5.3.21a) ifu G ( n

< ( e ) ) Hi/oHQ), (1 < q < oo), then

e€Jh

a{u — u1 > v)

3. SEM for

finite-element

207

approximations

2

= 2

h { " J [%DlD2uD2vdxdy-

e£Jk

-1

e

j^DlDmD^dxdy e

+ / [ - D32D1uD2{F{y)Dlv)

+ 4F(y)D^D1uD1D2v]

+ j [- DfD2uD1(E(x)D2v)

+

dxdy

AE{x)D\D2uD1D2v]dxdyV (5.3.21b)

and ifu € ( n

W*(e)) n Wf(Q) n #<J(fi),
a(w — u1, v)

= £ { ? [(hl + k*)DlDl«-vdzdy + f [E(x)(D^DiuDiv +F(y)(D\DiuD2v

-

ADlDluDiv)

- AJ^2DluD2v)]dxdy^,

(5.3.21c)

where u1 = I^u, and E(x) = W ) ] 2 , o

A(x) = l-[(x - xef - h*\, 2

F(y) = W ) ] 2 ,

B(y) = \[(y - yef - k%

0

Ac = he/hi,

(5.3.22)

I

He =

ke/h2.

Proof. By using the integral identity (5.3.14), it follows that / D\(u — uI)D\vdxdy = j

B(y)Dl(D1(u-uI)Div)dxdy

= j B(y)DlD1uD1vdxdy+

I 2B(y)D1D2(u

-

uI)D1D2vdxdy

= Vi + J 2 , VvES£(fi).

(5.3.23)

By using (5.3.17) and integrating by parts, we have J2 = 2 jF'\y)DlD2{u

-

u^DxD^dxdy

208

V. SEM for differential

2 f —k 2 e / o

equations

DiD2(u-uI)D1D2vdxdy

Je

= - 2 /F'{y)DlD 1 uD l D 2 vdxdy.

(5.3.24)

Substitute into (5.3.23), it becomes / Di(u — uI)D\vdxdy =

IB{y)DlDiuDivdxdy - 2 f F\y)DlD1uD1D2vdxdy.

(5.3.25a)

Similarly, / D2(u — uI)D2vdxdy -

j A(x)D\D2uD2vdxdy

- 2 I

E\x)D\D2uD1D2vdxdy.

Vt; E Sj(fi).

(5.3.25b)

Summing up the integrals for all e G Jh, the proof of (5.3.21a) is complete. For u G ( I ] W?(«)) n # £ ( 0 ) , substituting (5.3.17) into (5.3.23), and integrating F"(y) by parts twice, we obtain / Di(u — uI)D\vdxdy --k2e

= --JbJ I DlDmDivdxdy + j

D2D1(u-uI)D1D2vdxdy

F"(y)DlD1uD1vdxdy

+2 j F(y)Dl[D2D1(u

- tiJ)Z}iJD2t;]
= ~-k2efDlD1uD1vdxdy+ +ADlDiuDiD2v]dxdy = --k2e

I

JDlDxuDivdxdy

fFiy^DmDxv

3. SEM for finite-element approximations

209

+ [[-D$D1uD2(F(y)D1v)

+

AF{y)D\DluDlD2v]dxdy. (5.3.26a)

Similarly, / D2(u — uI)D2vdxdy = — h \ I D\D2uD2vdxdy o Je I[-D\D2uD1{E(x)D2v) +■ Je ->3 +4E(x)DfD 2uD1D2v]dxdy.

(5.3.26b)

Summing up the integrals for all e G Jh, the proof of (5.3.21b) is complete. For u G W*(Q) C\ ( U W*(e)) H H%(Q), noting that for the rectaneeJk gular partition, when integrating by parts in the y-direct ion, Ae remains unchanged, then ^2

I ^lD\D2uD2vdxdy

= - ^

I

\\D\Dluvdxdy.

Similarly, when integrating by parts in the ^-direction, fie remains un­ changed, and ^2

I nlDlDxuDxvdxdy

= - ^2

I&V\V\wdxdy.

Therefore, integrating (5.3.21b) by parts, (5.3.21c) is obtained. Corollary 1. If u G ( II

w

?(e))nHo(fy>

(? =

2

D

or oo), then there

are the following superconvergent estimates:

where ||u||' m>| = ( £

\\uh - u% < Chl\\u\\'3,

(5.3.27a)

\Wh ~ u'Hi.00 < Ch\\ lnfto|||«||^oo.

(5.3.27b)

IMIX,,,,.) 17 '

and

Nlm.oo = ™ « I H k o o . . .

210

V. SEM for differential

equations

Proof. Set v = uh — u1 in (5.3.21a). By using the inverse estimate, it follows that

ii« k -« 7 n?


< Ca(u-u\uh

-u1)


= chl\\u\\3\\uh-u%. Eliminating the common factor on both sides, the proof of (5.3.27a) is complete. In order to prove (5.3.27b), the estimate of the discrete Green function G\ in W\(Si) norm is needed. The discrete Green function Gj is defined to be the Ritz projection of the Green function GZi where z = (x,y) is a fixed point in Q, so Ghz satisfies a{Ghz,v) = v(z), Vv G 5j(Q).

(5.3.28)

Denote by dzGhz the partial differential of Ghz with respect to x or y. There is the following estimate*36)' I 132 !; Il«-G*|| ltl = 0(|lnAo|).

(5.3.29)

By using (5.3.28) and (5.3.27a), we have |0,(«*-«')(z)| =

\a{u-uI,diGhz)\

< CASHiillUP-GjlU,!
D

Corollary 2. Under the assumptions of Theorem 3.1, if u £ ( J~[ eeJh W*(e)) n i / £ ( Q ) , (g = 2,oo), then there exist wi,w2 € #o(ft), such that \\uh - u1 - h\Rhwl

- h\Rhw2\\i

< CAg||ii||i,

(5.3.30a)

and \\uh -u1

- h\Rhwl

- h22Rhw2\\liO0

< Chl\ lnfcolNIi, (5.3.30b)

3. SEM for finite-element approximations

211

where J?j, stands for the Ritz projection operator. Proof. Define linear functionals on Hl{Q) as follows: F v

^ ) = -» £ [%DlD2uD2vdxdy, 6

Je

eeJ»

and F

*(v)

= - : E

flilrftDiuDivdxdy.

1 Obviously, |F,(v)| < — ll^llsllvj|i» > = 1,2. By using the Lax-Milgram o theorem, there exist two functions

ti^w^njnflftn), • = 1,2, such that a(wiy v) = Fi(v),

Vi; G ffJ(fi), i = 1,2.

(5.3.31)

Substituting into (5.3.21b), we have a(u — u , v) = ft?a(u;i,i;) + /ila(w 2 ,t;) + 0(ftg)||ti||i||t;||i, Vw€5j(fi). Note that a(u;t-,t;) = a(i?/ju;t-,i;), Vv G 5 Q ( ^ ) . h\RhWi — h\RhW2, from (5.3.32), we get \\uh _

M/

(5.3.32) Setting i; = uh — u1 —

_ fc2/JfcU;i - &liifcu;2||i < C&g||ti||i.

By the definitions of w\ and W2, from (5.3.21b), it follows that a(y

_
w)

< c/igllixll^Ml!,!.

Let v = 9 2 G£. By the estimate (5.3.29), the proof of (5.3.30b) is complete. D

Theorem 3.2.1*'] Under the assumptions of Theorem 3.1, if Q, is a convex angular domain, and u G W*(Q) C\ ( J ] W*(e)) O H$(Q), then eeJh

there exist functions w\ and W2 defined by (5.3.31) such that \\uh _ ui _ h*wi _ h\w%

< Cht\\u\\'^

(5.3.33a)

212

V. SEM for differential

equations

and ||« h - u1 - h\w[ - fcX||o,oo < C^|ln/io||l«ll4,oo> (5.3.33b) where w[ and w2 stand for the interpolation functions of w\ and w2 respec­ tively. Proof. Using the Nitsche technique^], we shall prove I),!* _ , / _ hlRhW! - hlRhw2\\0

< CTiolMi;,

(5.3.34a)

and (In* - „ ' - hlRhw1 - hlRhw2\\0,oo < Cht\ lnfco||Mi;f00. (5.3.34b) Let

Vv e J5To(0).

Obviously, the following a priori estimate is valid 1Mb < C\\uh -u1-

h\Rhw1 - h\Rhw2\\«.

(5.3.35)

Hence, by using (5.3.30), it follows that \\uh^uI^h\Rhw1^h\Rhw2\\l = a{uh — u1 — h\RhWi — h2RhW2icp) = a(uh -u1

- hlRhivx - h\Rhw2)

+a(uh - u1 - hlRhWx - h\Rhw2, (p1)


- hlRhWx - h22Rhw2i(pI)i

where - + — = 1, q = 2,00. Applying (5.3.21b), c^u* - u J - h\Rhw1 - h\Rhw2y (p1) [DlD1uD2(F(y)D1
= Z) " + J2

I h

eeJ '

^F(y)D^D1uD1D2(pIdxdy

(5.3.36)

3. SEM for finite-element approximations

+ ^2 eej»

213

- IDlD2uD2(E(x)D2
+ ^2 I 4E(x)DlD2uD1D2tpIdxdy (5.3.37)

= Ji + J2 + Ja + JAEstimate J\ by integrating by parts, obtaining

Ji = - ^2 f DtDiuF{y)D1
Again, applying integration by parts to the first term of the right-hand side, it follows that

Ji = - Yl [DtuF(y)Dl(fI -
/

DtuF(y)D1(
-
Jde

eeJ"

(D$uF{y)D\
e€J»Je

*

^ |«k«,e(|v :, |2,«',e + / » o 1 | V / - V | l , 9 ' , e ) k ee^


< CAj|ti|i i f ||v|| 2 , f ..

(5.3.38)

Here, de2 stands for the two sides of the element e parallel to the t/-axis, and the trace theorem is applied in the derivation above. An estimate for J3 can be obtained similarly. The estimates of the right-hand side of (5.3.38) for J2 and J4 are obvious. Substitute the estimates of J i , J2J J3 and J4 into (5.3.37), it follows that h

a(u

- u1 - h\RhWl

- h22Rhw2iiPI)

< Cht\\u\&tq\\
Applying the a priori estimate (5.3.35), and using (5.3.39), the proof of (5.3.34a) is complete. In order to prove (5.3.34b), set q = 00 in (5.3.39), and take (p = g2 G 2 HQ(£1) H i / (ft), where gz is the regular Green function defined by FresheRannacher^"] it becomes |( M * _
-

h\Rhw2){z)\

214

V. SEM for differential

= a(uh -u1

- h\Rhwi

- h\Rhw2,

equations

G*)

< C&S||n|r4f00||^ - GJIU.1 + CfcJ||ti||l4f00||^||2|i
Hence, by the interpolation estimate of Sobolev space, there are \\RhWi - w{\\o < Chi

i = 1,2,

(5.3.40a)

and \\RhWi - "flkoo < Chm'™W"-£\

(5.3.40b)

where e > 0 is any positive number. Replacing Rh^i and RhW2 in (5.3.34) by w[ and w^ they become \\uh - u1 - h\w[ - hlwT2\\o < Ch40\\u\\'4i

(5.3.41a)

and \\uh - u i - h\w[

< Chmm^^^2-£h

_ h\w%iOQ

(5.3.41b)

Judging from the fact that a convex domain in which rectangular partitions can be implemented must be a rectangle, (5.3.41b) can be improved to (5.3.33b). D T h e o r e m 3.3.1*'] j

addition to the assumptions of Theorem 3.2, if uew?(n)n( n wf5(e)) ni^n), g = 2oroo, eeJh n

then \\uh -u1-

h\RhW! - hlRhw2\\i

||u* - u1 - h\RhWl

||„* _ ui - h\w[

< Chi\\u\\'5,

- h\Rhw2\\hoo

_ h l ^

(5.3.42a)

< Cfc$|lnfeo||Ml5,co> (5.3.42b)

< Chl\\u\\'s,

(5.3.43a)

3. SEM for finite-element approximations

215

and \\uh - u1 - h\w[ - ftXlli.00 < Ch%\ln/i0|||t/|r5. Proof. First, let v = uh - u1 — h\Rhwi

— h\Rhw2

(5.3.43b)

and use (5.3.21c),

IMI? = a(uh - u1 - h\Rhwi =

- h\Rhw2,

Z ) { /E(*)[DiIhuDiv ~ e€J"

v) ^DluDlV]dxdy

Je

+ I F{y)[D\DiuD2v

-

ADlD\uD2v]dxdy\

< C^llwlUlluH'j.

(5.3.44)

Eliminating the common factor on both sides, (5.3.42a) is obtained. Second, set v = dzGhz, we have dz (uh - u1 - hlRhwx = a(uh -u1

-

- hlRhwx

h22Rhw2){z) - hlRhw2i

G))

< C^Hiilli.ooll^Gjlli.i

(5.3.45)

and hence (5.3.42b) is proved. Finally, replace RhW\ and RhW2 in (5.3.42) by w[ and w\, and apply the finite element estimate, the proof of (5.3.43) is complete. □ R e m a r k 1. If the rectangular partition is uniform, then Ac and /i e are constants, these means w\ and w2 are functions independent of h. Hence (5.3.33a) shows that the mean square error of the splitting extrapolation value is 0 ( / I Q ) - (5.3.33b) shows that the maximum error of the splitting extrapolation at the nodes is O(fto|lnfto|). (5.3.43) reflects that the SEM is also effective when it is applied to derivatives of u. R e m a r k 2. It is possible to apply the splitting extrapolation twice, this depends on the smoothness of w\ and w2. If Q is a rectangle, one can judge the degree of smoothness of w\ and w2 by the construction of functionals Fi (i = 1,2) and Lemmas 2.5 and 2.6. R e m a r k 3. Suppose that Q is a smooth domain or a polygon, then there is no rectangular partition of Q. But if Do CC D C ^ , D is covered by a uniform rectangular partition, and D includes no corner point of Q, then by using the theory of interior estimate, one can prove ||,4* _ u1 - h\w[ - ^11^111,00,1)0 ^ C/^|ln/io|,

(5.3.46a)

216

V. SEM for differential

equations

and \\uh - ti J - h\w[ - h22wl\\ltDo < Ch3Q.

(5.3.46b)

Furthermore, if u G W*(D) H ( U W*(e)) n # £ ( 0 ) , then H ^ -
(5.3.47a)

||U* _ u1 - h{w[ - /iMlk°°,Do < Ch$\lnh0\.

(5.3.47b)

and

These facts imply that for some complicated domains where a uniform rect­ angular partition cannot be implemented, one can first apply the uniform partition to the interior of the domain, and then accelerate the finite ele­ ment approximation by SEM. The extrapolation method remains effective when it is applied on concave domains, but the convergence may be slow. R e m a r k 4. The bilinear form of a general second order elliptic equation is .

2

2

£ ( u , t ; ) = / (^^aijDiuDjV J*1 t=i j = i

2

+ yY^bjDjv

+

duv)dxdy)($.ZA$)

j=i

and the corresponding Dirichlet problem is: Find u G #o(fi),

sucn

B(u, v) = (/, v), Vr; G Hi(Q).

that (5.3.49)

However, finite element expansions under a uniform rectangular partition are still included in the discussion of the asymptotic expansions of B(u — ii 7 ,t;), Vv G S#(ft). By detailed derivation*771, one can prove that if u G W%{Q)C\Hl(ti)C\ ( [ ] Wj(e)), there exist fourth order differential operators e

L\ and Li independent offt,such that

B(u — u7, v) = hl

$ ^ ( i l W , v ) L 2 ( c ) + ft2 ^ ( L 2 W , v ) L 2 ( e )

+0(Aj)IH| / 5 i f ||t;||i f f ' l

V«6 5 j ( n ) , « = 2,oo.

(5.3.50)

3. SEM for finite-element approximations

3.3. Expansion

217

of three-dimensional

problems

In the following, we shall generalize the above results to three dimen­ sional cases. Consider the three-dimensional Poisson equation d2u dx2

d2u d2u , + + = / , in Q C M3, dy2 dz2 " u = 0, on afi.

(5.3.51a) (5.3.51b)

Suppose that there exists a cuboid partition Jh on Q. Let Sh = {v e C(Q) : t;|e € Oi(e), Ve € J h } , where SQ = Sh C\ satisfying

HQ(£1),

a(uh,v)

uh is a finite element approximation of (5.3.51)

= f VuhVv Jci

= f /w, Vi; G 5j(fi), Jet

(5.3.52)

where V = ( — , — , — ) stands for a gradient operator, and the integral dx dy dz element dxdydz is omitted. In order to derive an asymptotic expansion of the finite element ap­ proximation, it is necessary to prove the relevant integral identity similar to that of the two-dimensional cases. First, if e E Jh, let (xeiyeize) be the center of e, and 2hej2ke, 2re be the edge-lengths of e, hence, e = (xe-/ie,xe

+ he)x(ye-keiye

+ ke)x(ze-re,ze

+ r e ). (5.3.53)

Construct the following auxiliary functions:

E{x) =

\[{x-xey-h%

F(y) =

l[(y-ye?-kl],

and G{z)=l-[{z-zef-r%

(5.3.54)

which have the following properties: F'(y) = y-ye, (**)<»> = 6 ( » - y . ) , (F3)<4) = 4 5 ( y - i / e ) 2 - 9 f c e 2 .

(5.3.55)

V. SEM for differential

218

We denote by u1 the interpolation of u on Sh(ti).

equations

Let w = (u - u% =

1

by parts it is easy to prove the following —(u — u ). Using ntegration i dx integral identity:

=

fa fa­

F'(y)G'(z)wyM), (5.3.56a)

= J (F(y)wxx + G(z)w2Z ye)w = f (-(F\y))'wyy Je 6

-klG>(z)w9,), (5.3.56b)

3

ir-

■ ze)w = / (-(G2(z))'w„ Je 6

-r2eF'(y)wyz), (5.3.56c)

3

and

J (V-Ve)(z--

ze)w = f (-{F\y))'G\z)wyy Je 6

Lkl{G2)"wyz).

18

(5.3.56d) In the following, let = max/ie,

/i2 = maxfce)

= maxr c ,

h0 = max{/ii, /i 2 , ^3},

(5.3.57) and Ae,l = he/hi, Theorem 3.4.^

Ae,2 = fce/^2, Ae,3 =

If ueW^(Q)C\H^(Q)n(

rt/h^.

(5.3.58)

U Wf5(c)), 2 < q < oo,

Men Mere eris/ fourth order differential operators L\,Li

and L3, such that

a(u — t/,t;) = X > ? £ [Linv + e$J*Je » =1 =

0(ht)\\u\\'tJv\\lit,,

\/vesS(p). h

(5.3.59)

Especially, if J is a uniform partition, then the differential operators are independent ofh. Proof. First, we shall prove the following expansion:

f1

\u — u )—v

dx

3. SEM for finite-element approximations =

219

o 1 3 / (*eW**yy+ ^ * * « ) v + 0(&o)IIHk«||«||i l j', 3

fc 3 c€J^^ /rk ^e

VveSj(Q).

(5.3.60)

Note that v € 5j(f2). Applying Taylor's expansion on the element e, one obtains vx(x, y, z) = vx(x, y c , z c ) + (y - i/e)tf*y (*, y, * c ) +(z - *«)«**(*, yc *) + (y - Ve)(z - z c )v r y z , vx(x,ye>ze) = vx(x,y,z) - (y - y e )v x y (x,y,z) - ( z - ze)vxz(x, y, z) + (y - y c )(z - ze)vxyZi

(5.3.61b)

Vxy(x, y, Ze) = ^ ( x , y, z ) - ( z - Ze)Vxyz,

(5.3.61c)

vXz(x, ye,z) = vxz(x, y, z) - (y - ye)vxyz.

(5.3.61d)

(5.3.61a)

and

Applying (5.3.61a) and letting w = -Q—(u — w7), it follows that

WV-E

+ ( z - Z e ) t ; ^ ( x , yc, *) + ( y - Ve)(z - * e K y * ] •

Since i; x (x,y c ,z e ), v x y (x,y,z c ), vxz(x,ye,z) using (5.3.56) and (5.3.61), we derive

and v xyz are constants, by

/ wvx = / { [ ^ f o K v v + G(z)ti M 1 - F'(y)G'fc)u; y ,] **(*, y c , ze) 1 1 + [~(F 2 (y))Vyy 6 3 1 1 + [-(G\z))'uxzz 6 3 + [-(F2(y))'G'(z)uxyy 6

-hlG'{z)wyz\vxy{x,y,ze) -r2eF'(y)wyz]vX2(x,ye,z) - -kl{G2(z))"wyz}vxyz}. 18

(5.3.62)

V. SEM for differential

220

equations

Substituting

F(y) =

6

-(F\y))"--kl 3

and

G(z) = l-(G\z))" -

^

into (5.3.62) and integrating by parts, it becomes / wvx rf 1 = / { [~F2(y)u*yyyy Je K 6 1 —r\uxzz -

1 1 - - M * y y + -G2(z)uXZ222 3 6 F(y)G(z)uxyy2Z]vx(x,ye,ze)

o

1 -[-F2(y)UXyyy 6 1 -[-G2{z)uX222 6 1 + [-F2(y)G(z)uxyyy2 6 1 r

■ - / (k2uxxyy 3 */c

— { /

f r

3 1

^G(z)U X y «] VXy (x , V, Ze)

3

1

k2eG2(z}Uxy 18

zz ^ Vxy z

. | dxdydz }

)v(x,y€ize)dxdydz

}(fr2u*yy + r*ux^)t;(:r, t/c, ze)dydz

J S?

l

l

+ / {["^(yWyyyy + «/e ^ 6 1 1 + [-J,2(y)1l**yyy ~ 6 3 1 1 + [-G2(z)uXX22Z 6 3 1 2 + [--F (y)G(«)tiiCJflfJ

-\J

rlF(y)uXyy2]vX2(x,ye,z)

+ r^ti xxzz

- /

J Si

o

1

- G 2 ( z ) t i ^ ^ --F , (y)Cr(z)n a . yy „]v J .(x,y e ,^e) 6 'k2eG(z)uxxy22]vy(x,y,ze) r2eF(y)uXXyy2)v2{x,ye,z) 1

^ k2eG2(z)uxy2ZZ]vxyzjdxdydz

~ y }[-^2(y)M*yyy«y(*»y»«e)

3. SEM for finite-element

221

approximations

1 1 —k 2 e G{z)u xyzzz v y {x, y, ze) + -G2(z)uxzzz 3 6 12 —r e F{y)u xyyz v z {x y y ey z)]dydz.

vz(x, ye, z) (5.3 .63)

Here 5i and 52 stand for two boundary surfaces of e which are perpendicu­ lar to the x-axis. Summing up the integrals for all e G Jh, and noting that the surface integrals are cancelled in the summation procedure, we get / (u - u*)xvx Jet

= -3 X ) Je/ { [k*u**w + rlu****\ *ej»

\v -{y-

ye)vy - (z - ze)vz + (y - ye)(z -

ze)vyz]}dxdydz

+0(hi)\\u\&Jv\\1j = - X ) / {[fceU**yy + r2eUXX2Z]v 3

eeJh

Je

+ klF(y)UXXyyyVy 2

+r eG{z)uxxzzzvz

+ r\F'(V^ggy

g ZVy + k\G{z)UXXyy

-

k eF{y)G'{z)uxxyyyvyz

-r;F'(y)G(z)uxxzzzvyz

}dxdydz +

= ~ X ^ / [klu**yy +

ZVZ

2

0(ht)\\u\\f,jv\\liq>

r2euxxzz]vdxdydz

+0(^)ll«ll'5,,ll»l|l.,'.

(5.3 .64)

Thus the proof of (5.3.60) is complete. t

t

«// o il

•// o it

Similar expansions can be obtained for / wyvy and / ming all of them, we obtain (5.3.59).

wzvz. By sum-

□

h

Corollary 1. If J is a uniform partition, then there exist functions Wi,W2 and w% 6 HQ(Q,) independent ofh, such that 3

uh - u1 = J2^RhWi

+ O(fc*|lnfto| 2/3 ),

1= 1

where RhW% is the Ritz projection of W{. Proof. Since Jh is a uniform partition, Theorem 3.4 shows

a{uh - „ ' , « ) = W

/ ^«» + 0(*$)IM|'B|M|i,

(5.3 .65)

222

V. SEM for differential

Vv G S j ( 0 ) .

equations

(5.3.66)

Let Wi G #o(ft), (i = 1,2,3) satisfy a(wi,v) = [ Uuv,

Vv G # o ( ^ ) -

(5.3.67)

Substituting into (5.3.66), it follows that a(uh - u1 - £ t f i f c u ; , , „) = O ^ I M I s l M l i . 1= 1

Vv G Sj(t2).

(5.3.68)

3

Let v = uh — u1 — Y^ h?RhWi, then »'=i

||„» _ u / _ ^hfRnWiWi

= 0(ht).

(5.3.69)

Vt> € S%(Q),

(5-3-70)

Applying the inverse estimate, IMIo.oo < C\lnh0\V3\\v\\lt which leads to 3

\\uh -u1


-J2hlRhWi\\0)oo

(5.3.71)

t=i

Corollary 2. //u;,- G W*(fl) H H^O),

(i = 1,2,3), Men

3

tifc - ti 7 = ^2h?wi

+ O(&S|ln& 0 | 2/3 ).

(5.3.72)

»=i

Proof. If wi G W^(fi), by using the finite element estimate! 23 1, I K - RhWi\\0ioo < Chl\\wi\\2too. Replacing RhWi in (5.3.71) by wi9 (5.3.72) is obtained.

(5.3.73) D

Corollary 3. If w{ G W*+e(Q) 0 ffj(n), (i = 1,2,3), anrf 1 > £ > 0, Men 3

« * - « ' = 2 A , V + 0(&2 + *|lnfto| 2 / 3 ). 1= 1

(5.3.74)

3. SEM for

finite-element

223

approximations

P r o o f . By the inverse estimate, \\RhWi - titfHo.00 < C|Infc 0 | 2/3 ||iJfcti;,- 2 3

w{\\lt3

f


D

R e m a r k 1. Corollary 3 is only a global estimate. In fact, applying the theory of interior estimate, one can show t h a t at the interior points there exists an asymptotic expansion of the form (5.3.72). R e m a r k 2 . T h e results in this section can be generalized t o the mul­ tidimensional cases (cf. the work by A. Zhou et a/.^**^).

3.4. Piecewise parameter

uniform asymptotic

rectangular partition expansion

and

multi-

T h e uniform rectangular (cuboid) partition is the foundation of asymp­ totic expansions in the two previous sections, where the number of indepen­ dent mesh parameters is equal to the dimension of a domain. But for large scale problems, one can apply the piecewise uniform rectangular (cuboid) partition to increase the number of independent mesh parameters, which not only facilitates parallel evaluation, but also reduces the amount of work and storage. For example, the domain Q = ( 0 , 2 ) x ( 0 , l ) i s partitioned with three independent mesh widths /ii,/i2 and hs (Fig. 3.1). Fig. 3.1.

hs

hi

/i2 m

W i t h o u t loss of generality, suppose t h a t 0, = [J Q t , jf1

are uniform

»=i

rectangular partitions of Q,-, and the global partition Jh

m

= (J jj1

sat-

t=i

isfies the regular partition condition. Let 2/i t) i and 2/i t| 2 be the edgelengths of the rectangular element of j / 1 . Among the 2m mesh parameters

224

V. SEM for differential

equations

{hi)iihi2}^l=li there are only / independent mesh parameters, denoted by /ii,'.-,A/ ( 2 ~ < / < 2 m ) . Theorem 3.5. If Jh is a piecewise uniform rectangular partition of fi C M2, and suppose that Ai, ••-,/»/ are I independent mesh parame­ ters, and u G ( ]1 wq(e)) H #£(fi), (q = 2,oo), then there exist Wi G HQ(£1)

(i = 1, • • • ,p), such that hfRnwiW! < Ch30\\u\\'4i (5.3.75)

\\uh _ „/ _ h\Rhw1 and ||« h - u1 - hlRhWl

hfRhw,\\ii00

< C^|ln/io|||u||^

(5.3.76)

m

Proof. Since Jh = [J j / 1 , and each j / 1 is a uniform partition of fi,, *=i

then the identity (5.3.21b) can be rewritten as a(u — u1, v) = ^£{-9^,1 /

D\D2uD2vdxdy--hl2

+ J2 { f [~

/

DlD1uD2vdxdy\

DlD1uD2(F(y)D1v)^4F(y)DlD1uD1D2v]dxdy

+ j [ - D\D2uD1{E{x)D2v)

+

4E(x)DfD2uD1D2v]dxdyV

Vt;G5j(Q).

(5.3.77)

Let witj G # (t = 1, • • •, m, j = 1, 2), satisfy a(witl,v)

= -~

I D\D2uD2vdxdy,

Vt; G # £ ( 0 ) , (5.3.78a)

a(wit2iv)

= - - I DlDmDtvdxdy,

Vv G ftf(Q). (5.3.78b)

and

The Lax-Milgram theorem guarantees the existence and uniqueness of the solution Wij. Substituting (5.3.78) into (5.3.77), we have m

fl

i

,

(« -« -E(*'i^.i i=l

+ A 'jW-.j).«) < <7*oll«llilMli(5.3.79)

3. SEM for finite-element approximations

225

Noting that among the 2m mesh parameters {ft,,i, fc,-,2, * = l , - - - , m } , there are only / independent ones. Packing them, one obtains /

\\uh-uI-'£ih*Rhwi\\1
h

Similarly, let v = dzG

and substitute into (5.3.79), we have

z

/

11"* ~

uI

h RhWi

-Y, *

Wi>«> ^ CfegllnftollHli-

The proof of the theorem is complete. Corollary 1. If w^ G W^fa) expansion

D

H HQ(Q),

then there is an asymptotic

l

uh = u1 + Y,h?Wi + O(fcg|lnfc 0 |),

(5.3.80)

where w^ £ 11^(0,), (i = 1, • • • , / ) , are functions independent of the mesh parameters. Proof. By the inverse estimate and (5.3.75), / h u W" ~ * ~ J^tiRhWi\\0loo < C|ln/i 0 | 1 / 2 /i?.

(5.3.81)

•=i

In addition, if W{ £ W^(Q) C\ estimate:

HQ(Q),

noting the following finite element

\\RhWi - Wi\\ot00 < Ch0\lnhol and replacing Rh^i by it;,-, the proof of (5.3.80) is obtained.

(5.3.82) Q

Applying the Nitsche technique, similar to Theorem 3.2, it is possible to expand the remainder and improve the estimate to 0(/io|lnfto|), provided that u £ W*(n)C)H%(£l)n( [ ] W*(e)). All these results can be generalized to three-dimensional cases.

e£jh

Theorem 3.6. If Jh is a piecewise uniform cuboid partition ofCl C M3, and suppose that fei, ••*,/*/ are I independent mesh parameters, and u £ Wf(Q)r\Hl(n)C\( n W*(e)), then there exist functions w{ E H^Q), (i = e£jh

1, • • •, /), such that

l

\\uh - u1 - ^hlRHWiWi

= 0(AJ),

(5.3.83a)

226

V. SEM for differential

equations

and I

/3 \\uhh -ii -u1J--^fc?/Jfcti;,-||o X > ^ , | | o f ,oo~ = O(fcJ|lnfc O ( ^ | l n / i 0 0| |22/3 ). ||ti ).

(5.3.83b)

»=i m

Let Jh

Proof.

= |J j / 1 , and jj / /1 1 be uniform cuboid partitions of t=i

fit, i = l , - - , m . Assume that 2Af->i,2At-j2 and 2/it>3 are the three edgelengths of elements in J-1, then (5.3.60) can be expressed as f

d

dv dxdydz

T

{u u)

LTx - Yx =

5Z

Q

,-=i

d

(hl2u**yy + hl3uxxzz)vdxdydz

/

+

O^IMI'SJMIM'I

^

Vi;eS£(ft), VveSJfm,

(5.3.84)

and a(w — i / , t>) m

-I

1

»

r

= ££ *W/ {*iV>?(^ k i ^ M ++ DX»g)« Dl)u = l> ++^^, , 2a^(D? ^ ( ^ ++Dl)u ,=i •/«* ,=1

33

l

•/«* l

+hl3Dl{Dl +hl D»)«}t«tedy»)« V»€5j(Q). Vtr€S$(Q).

(5.3.85)

Let u;,-j 1.2,3),satisfy satisfy t^ij G € #o(fi), #o(^)> (*' ( l = 1> l i -*' *• i*mi!"*!i =i =1) 2,3), D

u vdxd dz

«(^,i<») ) u '• vdxdydz, «(£.•,;-») == ** // £>J(53 £>J(S^ *) v >

(fi), VVww Ge ^^(fi),

hence m m

33

h R a(u a(«h --u u1 - 53 53-^2^2hijRkWij,r) i,i h™i,j.") h

1

ft = V«€S$(n). = 0(ht)\\u\\' 0(Ao)ll«ll5,5fJv\\ ll»lli,.', 1>ql, V»GS 0 (Q).

(5.3.86)

But out of these 3m mesh parameters {/i»j, i = 1, • • •, m; j = 1, 2, 3}, only / parameters are independent. In (5.3.86), packing hij and setting g = 2, one obtains

||«ft - «*J' - Y,tfiRk*i\V o(h*). IK E*?^^lli = 0(h$). «t=i= 1

(5.3.87)

3. SEM for finite-element

227

approximations

Using the inverse estimate for
O

Corollary 1. If w( G W£(fi) n ff£(0), (i = 1, • , / ) , then i

uh = ui

+ J^h'?wi + O(h*\lnh0\2/3).

(5.3.88)

t=i

Corollary 2. If wt G W^1"1" n # £ ( 0 ) , (i = 1, • • •,/) and 1 > e > Q, then l h

1

u - u = ^h?w!

+ O(fc^|lnfc0|2/3).

(5.3.89)

»=i

These corollaries can be proved directly by the inverse estimate of the finite element theory. The results show that the accuracy of splitting ex­ trapolation depends on the smoothness of the solution Wij of the auxiliary problem, the smoothness of the solution u and the shape of the domain Q. However, there is always \\wj — RhWi\\ot00 = O(h€0)) hence, in many cases the SEM can be used to improve the accuracy of an approximation.

3.5. Piecewise

strongly-regular

partition

For general domains, instead of the rectangular (cuboid) partition, which cannot be globally implemented, one can apply the piecewise stronglyregular partition. A strongly-regular partition is obtained from joining the equally-spaced points on the opposite sides of a convex quadrilateral. If a domain is first decomposed into convex quadrilaterals, and each of them is subdivided into a strongly-regular partition, then the union is called a piecewise strongly-regular partition. For example, a plane polygon can be dem

composed into m convex quadrilaterals ft = (J ft,-, and the equally-spaced »=i

points on two opposite sides of each Qt- are joined to form quadrilateral ele­ ments. In Fig. 3.2, a pentagon is decomposed into 2 convex quadrilaterals, on which a piecewise strongly-regular partition can be constructed. Fig. 3.2.

228

V. SEM for differential

equations

Let $, : [0, l ] 2 —► £li be a bilinear mapping: f xxtf, f)) = a,-i(l- 0 ( 1 " *?) + « « * ( 1 - i?) + 0,-sf i| + a i 4 ( l - 0*?> ^ (5.3.90) ( *2«, 17) = M l - 0 ( 1 " l) + M ( l - 17) + M*7 + M l " 0*7, which maps four vertices of [0, l ] 2 onto four vertices of ft, with the coor­ dinates (a«j,k»j), (j = 1,2,3,4). Obviously, under the mapping $,, the square [0, l ] 2 is mapped onto the quadrilateral Qt. For a function v defined on fii, there is a corresponding function v, defined on [0, l] 2 , such that v = vo$i.

(5.3.91)

2

Similarly, for a function t; on [0, l] , one can also define the corresponding function v = v 0 S r 1 onfi,. The space of test function is SQ

(fi) = {v e

HQ(SI)

n C(fi) : v 0 $, are the piecewise

bilinear functions on [0, l] 2 , i = 1, • • •, m } .

(5.3.92)

Let IhU G S'o(fi) ^ e an interpolation function, IhU = IhU 0 $,- be the corresponding bilinear interpolation of 2 on [0, l ] 2 . Note that each £2,-, (1 = 1, • • •, m), has two mesh parameters /i t) i and hij, but only / (< 2m) are independent (e.g., only three independent mesh parameters in Fig. 3.2), which are denoted by / i i , - - , / i / . A piecewise strongly-regular quadrilateral partition works more flexibly than a piecewise rectangular partition, and there is an asymptotic expansion with respect to mesh-parameters h\, • • •, hi. Since f d dv / —(u — Ihu)—dxdy Jcti dx dx f

J[o,iP

ft

r L

ft

d£

dr,

ft

dr,

ft

8i

J

+P11—(« - Jh«)—v +P22—(w - ift«)—v }>d£dn, d£ d£ dr, dr, V» € Sj(n), where Pll = ( — X2(Z,V))2/J, Of] P22 = ( —

X2((iT]))2/Ji

or) ft ft

P12 = or,

-—X2{Lr,)—x2{i,ri)/j, d£

(5.3.93)

3. SEM for finite-element approximations

229

and \d d d d I J = — xxiCn)— x2(Cr))-— xi(£,i?) — x2(t,l) I *d£ orf or] o£ ' (5.3.93) converts the integral on £2, into that on [0, l ] 2 . In addition, hu is a bilinear interpolation on the uniform rectangular partition on [0, l ] 2 . Hence, by (5.3.50), there exist fourth order differential operators L{ti and Lt>2, such that /

V(u — Ihv)Vvdxdy

= T,hh / ^ j w ^ y + o(fcS)||nir5fnJHIi.ni. Vv G Sj(O).

(5.3.94)

Hence, a(u — u , v) m

= EE

2 f t

t=ii=i

.

'i/

i.-Jtinrfxdy+O^HiillilHl!,

7n

»

VvGS£(ft).

(5.3.95) m

Determine Wij G //o(ft), (* = 1> * > 5 3 a(wij,v) = /

=

Lijuvdxdy,

1> 2), such that Vv G #d(ft),

i = l , . , m , i = 1,2.

(5.3.96)

By using (5.3.95), it follows that m

2

II"* " « ' - E E ^ i ^ i l l = ° ( f t o ) .

(5.3.97)

Since only / mesh parameters are independent, packing them, one can ob­ tain /

ll«* - «' - Y,tiRkV>i^

= °(fto)-

(5-3.98)

1= 1

If the domain is convex, then Wij G H2(Q.) H HQ(£1). Using the inverse estimate, \\IhWij -RhWij\\0tOO

< C\lnh0\1,2\\IhWij
1/2

,

-RhWij\\i (5.3.99)

V. SEM for differential

230

equations

then, uh

=

/ i -^jThlhwi u

= 0(^|lnAo|1/2)

(5.3.100)

t=i

holds. For concave angular domains, the error has a different asymptotic expansion, but the accuracy can also be improved by the application of SEM. These results can be extended to three-dimensional cases. Suppose that m

ft C -R3, and there exists an initial partition : ft = (J ft,, where each ft, is »=i

a convex hexahedron, then there exists a trilinear mapping $,- : [0, l ] 3 —► ft,-. A uniform cuboid partition on [0, l ] 3 will be mapped by $, onto a stronglyregular partition on ft,-. Similar to the previous discussion, one can obtain an asymptotic expansion of uh — u1 with respect to the independent mesh parameters. For a domain with curved boundary, ft,- may be a quadrilateral with curved boundary. It is possible to implement similarly a piecewise stronglyregular partition, and there is also an asymptotic expansion of uh — u1 with respect to the independent mesh parameters. m

Cross points must be avoided in the initial partition ft = (J ft,, because *=i

at cross points, the error in finite element solutions cannot be accelerated by extrapolation methods.

3.6. Numerical

examples

E x a m p l e 3.1.t 131 l Solve the following equation by the SEM with multimesh-parameters, Atx = / ,

u = 0,

in ft = (0,2) x (0,1),

on 3ft.

(5.3.101)

The exact solution is u = 1 0 x j / ( l - x / 2 ) ( l - y ) e x + y . Decompose ft = ftiUft2, where ft2 = [0,1] x [0,1] and ft2 = [1,2] x [0,1] (Fig. 3.1). Partition ft with three independent mesh widths h1,h2yh3l and take the initial mesh widths as hi = /i 2 = h3 = —. Let t/0 denotes the finite element solution with the 16 partition (hi, /i 2 , /13), uuu2 and u 3 denote respectively the finite element solutions with the partition (/ii/2,h 2 } A3), (huh2/2,h3) and (hu ft2, ft3/2).

3. SEM for finite-element

231

approximations

The result of SEM is

uC

4

3

15

Ui

=~T,»=i

(5.3.102)

UQ.

3

6

Table 3.1. SEM with multi-mesh-parameters solution

max-error

root mean square error

1.380634E-2 1.246342E-2

6.585123E-3 6.426419E-3

«3

1.080608E-2 5.959988E-3

5.156504E-3 2.759510E-3

uc

3.051758E-5

9.569507E-6

"0 "1 «2

E x a m p l e 3 . 2 . ^ ^ Consider the following Poisson equation: f Ati = / , < I u = 0,

infi = (0,l) 2 , on dCl.

The exact solution is u = sin 7rx(cos 2iry — 1). The comparison between the results of SEM and Richardson extrapolation is shown in Table 3.2, where the initial partition is h\ = /12 = - . 4 Table 3.2. Comparison of SEM and Richardson extrapolation Point ( 1 , 1 ) Point ( 1 , 1) Point ( 1 , 1 )

Point (J, J)

Richardson extrapolation

4.9E-4

7.2E-4

6.9E-4

1.0E-3

Splitting extrapolation

2.5E-4

8.6E-4

3.5E-4

1.2E-3

These results show that the accuracy in both cases are of the same order, but the amount of work of the Richardson extrapolation is three times more than that of SEM. Without applying the extrapolation methods, the order of accuracy will be much lower, e.g., the maximum error oiuhlih2 is 1.1E-1, and the maximum error of uhl/2>h2/2 is 2.9E-2.

Chapter VI COMBINATION METHODS FOR ACCELERATING THE CONVERGENCE

Extrapolation and combination methods are two kinds of methods for accelerating the convergence. They both combine several independent ap­ proximate solutions to one with higher order of accuracy, but they are dif­ ferent in essence. The extrapolation method combines several approximate solutions obtained by applying the same algorithm on different grids, and hence it is not suitable to be evaluated in parallel. On the contrary, the com­ bination method combines approximate solutions obtained synchronously by applying different algorithms to grids with similar scales, and then ob­ tain an approximate solution with higher order of accuracy. In this way, the approximate equations of similar scales are suitable to be evaluated in parallel. The principle of combination methods was proposed by Q. Lin and T. Lilt 64 ). For the application in the solution of boundary integral equations, see T. Lii and C.Z. Mat 84 !. In this Chapter, we introduce first the combination principle, and then its applications to differential equations, integral equations and boundary integral equations. Zenger's combination techniques for the solution of sparse grid problems will be discussed in the next Chapter. 1. C o m b i n a t i o n m e t h o d s

1.1. Combination

principle

Consider a linear operator equation in =7,

(6.1.1)

where L maps a subspace of a Banach space U into the Banach space F. Assume that (6.1.1) has a unique solution u. It is usually impossible to solve (6.1.1) exactly. However, one can construct m approximate equations dependent on the mesh parameter h: L*u$ = f*,

f = l,...,m, 232

(6.1.2)

1. Combination methods

233

where V> : U-1 —► Fj1. U-1 and FJ1 are finite dimensional Banach spaces depending on mesh parameters, and u^ and //* are net functions defined on the grid ft^ C ft. In a special case, L* is also a linear operator mapping U into F, and so the right-hand side of (6.1.2) can be taken as //* = / . Assume that the approximate operators L* (i = l , - - , m ) satisfy the following conditions: A i ) Operators V> (i = 1, •, m) are invertible, and there exist constants d > 0 independent of ft, such that

\m)-l\\
(i=l,...,m).

(6.1.3)

A2) For the solution u of (6.1.1), there exist /t-(ti) G F such that = hP(h(u) + r?) -> 0,

L*u -Lu

as ft -» 0,

(6.1.4)

where p > 0, and r^ —► 0 (as /i —► 0). A3) The following problem Xt/;,- =/,-(ti)

(6.1.5)

has a unique solution iut-, and the corresponding approximate problem Llw* = U(u)

(6.1.6)

also has a unique solution w^, with ef = u£ - Wi — 0 (ft -> 0).

(6.1.7)

L e m m a 1 . 1 . ^ ' ( C o m b i n a t i o n principle) Under the assumptions A\), Ai) and A3), if there exist combination coefficients a,- (z = 1, • • -,771), such that

X > = i,

(6.1.8a)

»=i m

5>fc(«) = o,

(6.1.8b)

t=i

then

u-j2^i=hpJ2a'(ei+(Li)~lri)*=i

(6-L9)

»=i

Proof. By using A2), there is

L$(u-u$-hpWi) h.. - LuT„. PpLL-w h hh = L^u t = ft*(/i(ix)-L>t + rf), i = l , . . . , m .

(6.1.10)

234

VI. Combmation methods for accelerating the convergence

It follows from A3) that u — 11% — hpW{

= &"(e? + ( X ? ) " 1 ^ ) , i=l,.--,m.

(6.1.11)

Using (6.1.8), we have m

m

] T a,.«;,. = L " 1 ^ and (6.1.9) follows.

a,-Zf-(u) = 0,

(6.1.12)

D

For difference equation, the combination error estimate (6.1.9) is valid m

only at grid points x £ Q Q^. t=i

R e m a r k 1. There is a different property between the coefficients of the combination method and the extrapolation method. The combination coefficients usually satisfy at- > 0 (i = 1, • • •, m), while the signs of extrap­ olation coefficients are alternate. 1.2. Combination

methods

for integral

equations

Consider an integral equation f h{x,y)u{y)dy = f{x). (6.1.13) Jo Assume that the kernel k(x,y) and f(x) are sufficiently smooth, and 1 is not an eigenvalue of the integral operator u(x)-

I k(x,y)u(y)dy. (6.1.14) Jo Using the rectangular cubature rule, construct an approximate operator oiK: Ku=

n-1

K£u = h^*(*, i=o

1

j

(«' + «)fc)ti((t + -)&), z

(6.1.15)

z

meanwhile, using the trapezoidal cubature rule, construct another approx­ imate operator

K$u = - J2 [*(*> ift)w(ift) + *(«, (« + l)h)u((i + l)ft)], (6.1.16) 2 »=o

1. Combination methods

235

1 where h = —. n Again, construct two Nystrom's approximate equations ti* - Kful

= / , i = 1, 2.

(6.1.17)

Let L? = I - K?, (t = 1,2), and L = I - K. We shall now verify the conditions Ai), A2) and A3). By the theory of collective compactness: \\(K$ - K)K*\\ - 0, (h - 0), i = 1,2.

(6.1.18)

It follows that Ai) holds. A 2 ) and A 3 ) are derived from Taylor's expansions Kfu - Ku = h2li(u) + 0(h4),

i = 1,2,

(6.1.19)

with /

1 ,1 d2 i( ) = - — / —-Hx,y)u(y)dy, u

1 ,1 a 2 ^(«) = — / — 12 Jo dv

(6120)

k(x,y)u(y)dy.

Thus, there is 1 -h(u)

2 + -l2(u) = 0.

(6.1.21)

O

o

By Lemma 1.1, we obtain u - - ( i i j + 2w§) = 0 ( / i 4 ) . (6.1.22) o Note that u\ satisfies a linear system with n unknowns while u^ satisfies a linear system with n + 1 unknowns. It needs almost the same amount of work to solve these two systems. However, in order to reach the same order of accuracy by using extrapolation methods, solutions of linear systems with n + 1 and with 2n unknowns are needed. R e m a r k 2. For a weakly-singular kernel, (6.1.13) is equivalent to the following form (l-J

k{x,y)dy)u{x)~

J

k(x,y)(u(y)-u(x))dy

= f(x). (6.1.23)

This form has no singularity, and the Nystrom approximate solutions u\ and u\ can be obtained, hence, the accuracy of the combination approxi2 1 mation -u\-\—itf} can still be improved.

236

VI. Combination methods for accelerating the convergence

R e m a r k 3. The Simpson rule and the Gauss-Legendre two-point rule can also be combined to obtain approximations with higher order accuracy. It is explained as follows. By using the Simpson rule, one can construct a Nystrom approximate operator: 1 K%u = h"%2 [*(*> ih)u(ih) + 4fc(s, (i + -)h)u((i 2

t=o

1 + -)/i) 2

+ i ( x l ( t + l)ft)u((i + l)&)]/6.

(6.1.24)

And by using the Gauss-Legendre two-point rule, one can construct another Nystrom approximate operator:

KU =h-£ L [*(«, *±A*+±h)u(2l±lh + ±h) 2 ^

2

^

2

,/$

+*(*, ?i±±h - -U)„(?!±ifc _ _LA)1. z v3

*

V3

(6.1.25)

Since /^tx-^ii=

/i4

-/i4

K%u-Ku=

<94

+ 0(/i6),

fc(x,y) d4

135 dt/ 4

i(*,y) + 0 ( / i 6 ) ,

(6.1.26a) (6.1.26b)

which imply that if u?1 satisfy the following Nystrom approximate equations t i * - ^ t i ? = /,

z = 3,4,

(6.1.27)

then 4

3

7

7

and the unknowns of systems (6.1.27) are respectively 2n -f 1 and 2n. 1.3. Combination

methods

for difference

equations

Consider a three-dimensional Poisson equation 3

d2u

Lu = Y* = /, ttdxl u = (7,

in fi = ( - 1 , l ) 3 , K >' on dQ.

(6.1.28)

1. Combination

methods

237

For a given mesh width ft, one can construct two difference operators. The first one is a seven-point difference operator

= —2 { u(x1 + ft, x 2 , x 3 ) + ii(xi - ft, a?2, «3) ft ^ +ti(xi, x 2 + ft, x 3 ) + ti(xi, x 2 - ft, x 3 ) + ti(xi, x 2 , x 3 + ft) +ti(xi, x 2 , x 3 - ft) - 6ix(xi, x 2 , x 3 ) | ,

in 0 fc ,

(6.1.29)

and the second one is a nine-point difference operator LgU =

4ft 2

\ ti(xi + ft, x 2 -f ft, x 3 + ft) L

4-u(xi - ft, x 2 4- ft, x 3 4- ft) -f u(xi 4- ft, x 2 —ft,x 3 -f ft) +ti(xi - ft, x 2 - ft, x 3 4- ft) -f u(xi 4- ft, x 2 4- ft, x 3 - ft) 4-u(xi - ft, x 2 4- ft, x 3 - ft) 4- u(xi 4- ft, x 2 - ft, x 3 — ft) +u(xi - ft, x 2 - ft, x 3 - ft) - 8tc(xi, x 2 , x 3 ) | ,

in Qh. (6.1.30)

It is easy to derive that the truncation error is Llu-Lu = h2U{u) + 0(hA), 1 = 7,9, with *d4u

1

i7(U) =

-y/—, 4

« ^i 9 uU

1 ^

12 £J £?&*

^_^

34u ),

(6.1.31)

iZl^dxj

hence,

-/7(«) + -/„(«) = —^2« = L(—f).

3 3 12 Now construct two difference equations

12

inQA)

L*u* = f,

(6.1.33)

2

u? = g

(6.1.32)

h

/,

on dQh, * = 7,9.

12 It can be proved, by using the combination principle, that if u € C 6 (Q), then 1 h? u--(2u!} + 4) f = 0(hA). (6.1.34)

238

VI. Combination methods for accelerating the convergence

In fact, let W{ (i = 7,9) be the solution of an auxiliary problem Lwi = h(u),

in Q, (6.1.35)

Wi = — / , 12

on dQ, i = 7,9,

then L^u-u^ -h2Wi) = L* u — Lu — h2 LiW{ = h2U{u) - h2Lwi + h2(Lwi - L^Wi) + 0(hA) = 0(/i4),

inftfc, 2

u - u^ - /i wt- = 0,

(6.1.36a) on dftfc.

(6.1.36b)

It follows from the discrete maximum principle that u-u1>-h2wi

= 0(h4),

inftfc, 2 = 7,9,

hence, 1 u - -(2ti£ + ti§) o

h2 o

(2w7 + w9) = 0(h*)y

in f2h.

(6.1.37)

i Note that w = —(2wr + wa) satisfies 3 2

1 / 1 x Lw = -/7(u) + -/„(«) = L(—f),

3 1 w = —/, 12

3

in «,

12 on Oil, (6.1.38)

i.e., w = — / . The proof of (6.1.34) is complete. Remark 4. Bramble^ " constructed a 19-point difference operator Xf9 and a difference equation ii9«i9 = / + — A / , «19 = 9,

infi„,

(6.1.39a)

on 5Q,

(6.1.39b)

1. Combination methods

239

and obtained

u-u\z

= 0(hA),

inQ h .

As compared to the Bramble method, the combination method is sim­ pler as the evaluation of A / in Q^ is omitted. R e m a r k 5. The combination methods can be applied to eigenproblems, nonlinear elliptic equations and problems with curved boundaries. E x a m p l e 1.1. Consider a weakly-singular integral equation (cf. Baker and Miller^) u(x) - I \n\xJo

y\u{y)dy = g(x)}

(6.1.40)

with 0.5{x 2 In x + (1 - x 2 ) ln(l - x) - (x + 0.5)}.

g(x) = x-

In order to discretize (6.1.40) by using cubature rules, (6.1.40) can be expressed in an equivalent form u(x)(l -

- / l n | x - y\dy) In \x - y\ \u(y) - u(x)jdy

= g(x))

(6.1.41)

and the singularity is cancelled. By using the mid-rectangular cubature rule, the nodes are taken as 2 i +1 . r-m-i Xi = , i = 0, • • •, n — 1. The approximate solutions {^t/ l=0 satisfy 2n the equations f1 Hi (l - / In \xi -y\dy) Jo = 9(*i),

1

n

] T In |x,- - XJ \ (UJ - tl,-)dx

nJ=0

t = 0,.-.,n-l.

(6.1.42)

After obtaining {u{}, the Nystrom approximate solution is given as follows: 2f + 1 Ui, u(x) = {

for x = jn-i

, i = 0, • •, n — 1,

^(x)+~/Zmlx~xilwi'

*^{«o,---,*n-i}.

(6.1.43a)

(6.1.43b)

VI. Combination

240

methods for accelerating

the

convergence

By using the trapezoidal cubature rule, the nodes are taken as x,- — —, i = 0, • • •, n. The approximate solution {£, }" = 0 satisfies the equation ,1

=

!

» "

ui (1 - / In \xi - y\dy) - - ^ In |x,- - xs \ (UJ - u{)dx Jo ni=0 (6.1.44) 9(*i)i * = 0, ■

where n-l

£

U(i) =

«(0)/2 + «(n)/2 + 2 «(.'). »=1

i=0

After obtaining {tit}, the Nystrom approximate solution is: it,-,

for x = —, i = 0, n

u(x) = <

(6.1.45a)

>™,

i n "

9(*) + " ! " > I* - x i l " i - * ^ { - : i = 0, • • •, n}. (6.1.45b) And the combination approximate solution is 2_ 1_ uc{x) = -u(x) + -u(x).

(6.1.46)

Table 1.1 lists the errors of tZ(xt-), 2(x,-), and uc(x{). Since the nodal values obtained directly from (6.1.43a) and (6.1.45a) have a better accuracy than that of the nonnodal values obtained from (6.1.43b) and (6.1.45b), Table 1.1 lists only the nonnodal values. Table 1.1. h = 1/4 Xi

tx(xi) - u(*i) u(xi)-u(xi)

u(xi)-

uc(xi)

0.0625

1.03E-2

-1.31E-2

2.52E-3

0.1875

-2.04E-2

1.21E-2

-9.59E-3

0.3125

-1.32E-2

3.24E-3

-7.75E-3

0.4375

-4.22E-2

3.39E-2

-1.68E-2

0.5625

-3.40E-2

3.37E-2

-1.14E-2

0.6875

-6.16E-2

7.84E-2

-1.49E-2

0.8125

-5.02E-2

1.04E-1

1.07E-3

0.9375

-6.44E-2

2.07E-1

2.62E-2

E x a m p l e 1.2. Solve the equation (cf. Baker and Miller®, p. 358) 1 Z"1 1 «(*) " - / ; rr«(y)<*y = 1irJ-il + (x- y)2

(6-1-47)

2. Combination methods for collocation-method

solutions

241

The kernel of (6.1.47) is smooth. For the results with n = 8 and n = 16, see Table 1.2. The efficiency of the combination method in this case is obvious. Table 1.2. u(xi) Xi n = 8 ±1 -1.67E-3 ± 3 / 4 -2.54E-3 ± 1 / 2 -3.02E-3 ± 1 / 4 -3.17E-3 0 -3.21E-3

uj(xi)

ti(*i)-«J(*0

n = 16 -4.11E-4 -6.33E-4 -7.55E-4

n = 8 3.31E-3 5.00E-3 5.97E-3 -7.86E-4 6.30E-3 -8.00E-4 6.35E-3

n = 16 8.33E-4 1.25E-3 1.49E-3 1.58E-3 1.59E-3

e «w-« w n = 16 n = 8 -1.25E-5 -2.73E-5 -2.71E-5 -1.56E-5 -2.16E-5

3.72E-6 -6.09E-6 -6.17E-6 3.57E-6 -3.21E-6

E x a m p l e 1.3. Consider an eigenvalue problem (cf. Baker and Miller® p. 177) / e*yf(y)dy = \f(x). (6.1.48) Jo By using the trapezoidal cubature rule with h = 1/80, the maximal approx­ imate eigenvalue is 1.35306. Using a more accurate method, the maximal approximate eigenvalue is 1.35303. The results of the combination method, see Table 1.3. Table 1.3. n

A*

1.34756 1.36445 8 1.35164 1.35583 16 1.35268 1.35372 4

Ac = j-Ajvf + j AT 1.353187 1.35303 1.35303

These examples show that using the trapezoidal cubature rule and the mid-rectangular cubature rule, the errors of u\, u\ and Ajvf , XT have op­ posite signs. Using this property, we obtain the following a posteriori esti­ mations min{itf,w5} < u < max { u\, v% },

(6.1.49)

and min{Ajvf, XT} < A < max {AM, A T } .

(6.1.50)

2. Combination methods for collocation-method solutions of qua­ dratic and cubic splines Consider a nonlinear two-point boundary value problem ( u" = f(t,u), 0<*<1, { I u(0) = ti(l) = 0.

(6-2.1)

VI. Combination methods for accelerating the convergence

242

Assume that / is a smooth function on [0,1] x M, and the problem (6.2.1) has a unique solution. Let {t,}o be equally spaced nodes, ti = ih (i = 0 , - - - , n ) , h = —, n and rrii — ~(£,-i + *»), (i = 1, • • •, n), be the midpoints of It- = [£,_i, t,-]. We denote respectively by i?2(0> ^ 3 ( 0 the quadratic and cubic i? spline functions^ 4 1, and by 1*2 (0> u$(i) the quadratic and cubic spline collocationmethod solutions, i.e., "2(0 = ]C

C

«52(T--0'

n+1

u3(t)=

(6.2.2a)

t

^diB3(--i),

(6.2.2b)

t=-i

where the undetermined coefficients {c,-}^ and {d,}"!" 1 are respectively determined by the collocation equations u'2'{mi) = /(m t -,ii 2 (m t )), i = l, — ,n,

(6.2.3a)

ti2(0) = ti 2 (l) = 0, and

"aft) = /fa, «3(*.-)), i = l , - - - , n ,

(6.2.3b)

ti 3 (0) = n 3 (l) = 0. By (5.1.5) u2 and u 3 have the following error estimates respectively " < P r ( « - « 2 ) | | 0 0 = 0(A 2 ), i = 0 , l ,

(6.2.4a)

| £ ( U - U 3 ) L = 0 ( / > 2 ) , j = 0,l,2.

(6.2.4b)

T h e o r e m 2 . 1 . The combination approximation u(t) = 2u2(t) - u3(t) has the following error estimate at the nodes o m«|«(tO-«(i,-)|

= 0(A 4 ),

(6.2.5)

and the global estimate satisfies \\u-u\\oo=0(h3).

(6.2.6)

2. Combination methods for collocation-method

solutions

243

Proof. Let II2 be a piecewise constant interpolation operator with basis points {m t }J; and let II3 be a piecewise linear interpolation 0]aerator with basis points {£,}o • Then, the collocation equation (6.2.3) is equivalent to

< k

«2=na/(*,«2),

(6.2.7a)

«a(0) = «a(l) = 0,

•

< «S = ns/(*,t»3), k

(6.2.7b)

u3(0) = « 3 (i) = 0.

Let ek(t) = u(t) - uk(t) (k = 2,3). Subtracting (6.2.7) from (6.2.1), one obtains

< k

e* = /(*,«) - /(*, «*) + ( I - n*)/(*, «Jt),

(6.2.8)

e*(0) = e fc (l) = 0, i = 2,3.

Set wk = /(*, uk) - /(*, ti),
By using the estimate (6.2.4), it follows that

||^t* a || o o = O(fc 2 ),i = 0,l>

(6.2.9a)

| | ^ t « , L = o(* 2 ).i = o,i,2.

(6.2.9b)

We have /(t,ti) - /(t,ti*) = /i(t,ti)c fc + 0 ( / i 4 ) , Ar = 2,3, 9 where /^ = —f(t,u). Substituting into (6.2.8), we get du [ e'i - fi(t, u)ek = (/ - nfc)<7 + (/ " nk)wk + 0 ( / i 4 ) ,

[ efc (0) = c t ( l )

= 0 , 4 = 2,3.

(6.2.10)

VI. Combination methods for accelerating the convergence

244

Let 11(8,1) be the Green function of the boundary problem (6.2.10). It is well known that H(s, i) is continuous and piecewise smooth but its derivative is discontinuous at s = t, then «*(*) nk)g(t)dt ^ I = / H{8,t)(i-n .,*)(!--k)g(t)dt+ = ['His Jo

Jo 4 + 0(A 0(h4),), JtJt = 2,3.

= Ji(s) + JJ22(«) (s) = Ms) +

4 ff(, S)* t)(/-n + o(/i fcH(tyt(*)<*<+ °C» 4)) ) ( ' - - n*)™*

(6.2.11)

For the case k = 2, i.e.,

MU) = Y, / A(*0

H(ti,t)(I-n 2)g{t)dt 2)g{t)dt ,<)('-- n

~~ ^ 2 4 d<2

( f f ( I - -n 2 )ff)(m 'i) +

0(* 5 )]

= E [ ^2 ( ^ - n a ) i r ) ( m i ) + o(fc«)] " h n

^2

dH ,

1 ~bT

= A,yhr t

2

+ Hg"](m*) + 0(h*)

dH

9 'dt +

r dH ,

h2 f 1 —2 I Hg"dt]+0(h' ) 24 h. h r

+ Hg"dt + 0{h 1+ ), (/l) 9 dt + — / 2iX/^ = E^X-ar^ ° ~ 12 Jo -el 24 ./o 8H

/l 2

/•!

/l 2

/•! 0 #

4

y

ft2

rl

where H = H(ti,t). Integrating by +parts using the condition, = — / g'dt — and / Hg"dt + boundary 0(h4), one obtains 12 Jo dt 24 Jo where 7/ = H(ti,t). Integrating by parts and using the boundary condition, h2 /■! one obtains — / H(ti,t)g"dt + 0(h4). [6.2 .12) 2 MU) = h24 Jo/-i 4 MU) = / H(ti,t)g"dt + 0(h ). (6.2.12) Jo and s E /,-, since For the case s ^ti, (1 = 1, • •24 •, n),

lH{S

For the case s ^ ti,- (i = 1, • • •, n), and s G /,-, since Mi- n2)g(t)dt

y

ff(«,0(J-n3-H(s, M0
,0-

m , ) ) ( / - - H2)gdt + H(s,

mi) (i

L~

- n2)gdt

= (#(s, t) - H(s, rm))(I - H2)gdt + H(s, rm) J (I - U2)gdt = jf0(h*), we get we get

= oV). M») = J (s) = 1

h22 /•! h //•! H(s,t)g"(t)dt / H(s,t)g"(t)dt 24 Jo

+ 0(h3). + 0(h3).

(6.5>.13) (6.2.13)

2. Comb/nation methods for collocation-method

solutions

245

By using the estimate (6.2.9a), one obtains in a similar way h2 fldH h2 /•! J2(U)= — / w'2dt+— / HwUt + 12 Jo dt 24 Jo 2 h /-i 8H = — w'ndt 24 Jo dt = 0(h*), 3

J2(s) = 0(h ),

8±tit

0(h4)

(6.2.14) (6.2.15)

»=l,.-.,n.

Substituting (6.2.12)-(6.2.15) into (6.2.11), one obtains asymptotic expan­ sions h2 e2(ti) =

/-i / H(ti,t)g"(t)dt + 24 Jo t = l,---,n,

0(h*), (6.2.16a)

L2

e2(s) =

/ H(8,t)g"(t)dt + 24 Jo 8&{tu-',in}>

0(h3), (6.2.16b)

For the case k = 3, i.e., •M«) = Z ) /

(6.2.17)

H(s,t)(I-n3)g(t)dt.

i=iJl>

For ,;' / i, by using Simpson's rule and the estimate to the remainder, it follows that J

H(s,t)(I-n3)g(t)dt 4 g(tj) + 9(ti-i) = -MTfrmjlfrm,)-*" ]+0(/>5) 6 2 /i 3

= =

12

H(s,,mj)g"(mj)+0(h5)

/ H(s,t)g"(t)dt 12 Jij

+

(6.2.18)

0(h5).

For j = i, by using the following interpolation expansion (/ - n3)<7 = g"(U-i)(t - <,_!)(< - U)/2 + 0(h3),

t €U)

246

VI. Combination methods for accelerating the convergence

and the integral mean value theorem, one obtains f

H(8,t){I-V\3)g(t)dt

= ]-H(Sl09"(U-i) I (t - U-i)(t - U)dt + 0(/i4) Jii

2

2

h

r / H(s,t)g"(t)dt + 0{h*). 12 Jii Summing up with respect to j , there is

(6.2.19)

=

+ 0(h4).

Ji(s) = ~J*H(s,t)g"(t)dt

It follows immediately from the estimate (6.2.9b) that J2(s) =

0(h4),

and so one obtains the asymptotic expansion e3(s) =

/ H(s,t)g"(t)dt 12 Jo

+ 0(h*).

Combining (6.2.16) with (6.2.20), the proof of Theorem is complete.

(6.2.20) D

E x a m p l e 2 . 1 . Consider a two-point boundary-value problem u" — Au = 4cosh(l), i/(0) = ti(l) = 0, with an exact solution u(x) = cosh(2x —1)—cosh(l). The comparison among numerical results obtained from the quadratic and cubic spline collocationmethods and the combination method at the nodes with h = — are listed 10 in Table 2.1. Table 2 . 1 . Xi = ih ti(xi) xx *2 *3 *4 *5

u2(xi) u(xi) -

2.58E-4 4.36E-4 5.52E-4 6.17E-4 6.38E-4

u3(xi) u(xi) -

u(xi)

5.16E-4 8.72E-4

3.72E-7 6.68E-7

1.10E-3 1.23E-3

8.82E-7 1.01E-6 1.05E-6

1.27E-3

3. Combination methods for the Nystrom

247

solution

Table 2.1 shows that the combination methods are very effective, and that the quadratic spline collocation-method is better than the cubic spline collocation-method. 3. Combination methods for the Nystrom solution of boundary integral equations of the second kind 3.1. Boundary

integral

equations

of the second

kind

Partial differential equations can be converted usually to boundary in­ tegral equations by the potential theory. As an example, the Dirichlet problem of the plane Laplace equation Au = 0,

in ft, (6.3.1)

u = g,

on T = dft,

can be converted to the boundary integral equations of the second kind by using the double layer potential theory, i

a(z)$(z)

where x = (xux2),

-\

d f / $(x) ln]x - z\ds(x) = g(z), 2w Jr dnx z e T,

z = (zuz2),

(6.3.2)

\x - z\2 = (xx - zx)2 + (x2 - z2)2.

d

-— dnx denotes an outward normal derivative. Let a(z) = - , if z is a smooth point on T, otherwise, a(z) —

6(z)

, where 6(z) is an exterior angle. Once $(z)

2-K

is obtained, the following formula 1

f d / (x)-—ln\x-z\ds(x), z G ft (6.3.3) 27r Jr dnx gives the values of the solution u at interior points. The procedure mentioned above consists of two steps: to convert a differential equation to a boundary integral equation, then to find its ap­ proximate solution. This method is called the boundary element method (BEM), which is one of the active fields of numerical analysis at present. The BEM has many obvious advantages: it not only reduces the dimension of the problem to be evaluated, but also deals easily with some exterior problems which are very difficult to solve by the finite element method. However, the computation of BEM has some difficulties of its own, e.g., u(z) =

VI. Combination methods for accelerating the convergence

248

standard methods such as Galerkin's method and the collocation-method will consume a great deal of CPU in evaluating integrals; the discrete ma­ trix is full and nonsymmetric, and its order increases rapidly in successive refinements; for a nonsmooth boundary, the integral kernel is singular at the corners. In the following we shall discuss how to overcome these difficulties. One possible way is to replace these standard methods by the Nystrom method to save the evaluation of integrals (cf. R.A. Kress*50!). Another way is to replace the solution of a large system with mn unknowns by having m systems, each with n unknowns, then simply average their results, so as to reach the same order of accuracy as that obtained from the large system. Such a combination method for boundary integral equations is proposed by T. Lii and C.Z. Ma*84!. 3.2. Combination

methods

for the Nystrom

solutions

Assume that a boundary curve is expressed by a parameter s as

r = {*(*) = (*i(*),*2M): o < s < i}. If r is a closed curve, then x(s) is a periodic function with period 1. In particular, if T is a smooth curve, then x(s) is a smooth periodic function, and x(s) : [0,1] —► T is a one-to-one mapping. Hence, there exists a constant C > 0, such that (6.3.4)

|*'(*)| > c , v * € [o,i].

By means of the transformation, the equation (6.3.2) is expressed as the following integral equation on [0,1]
I k(t, s)
and k(t,s)

1 s'aMxifo) - »i(t)) - x'1(s)(x2(s) x

(Xl(s) -

2

Xl(t))

+ (x2(s) -

k(t, s) = <

(6.3.5)

has the following - ga(0)

x2(t))2 (6.3.6)

2

*

(*i(«)) +(*,(«))' = t.

3. Combination methods for the Nystrom solution

249

It can be seen that if x(s) G C / + 2 [0,1], then jfc(s,*) G C'([0, l]x[0,1]), where C'[0,1] denotes the set of continuously differentiable periodic functions of order /. For convenience, rewrite (6.3.5) in the following operator form (I-K)
(6.3.7)

= g.

Since the trapezoidal rule of periodic functions is highly accurate, it can be used to construct the Nystrom approximate operator *»('• ">(-)> n t=1 n n

(6.3.8)

and the approximate equation (6.3.9)

(I-Kn)
R.A. Kress^50! gave an error estimate of
J^'V = - ! > ( * >

M

n i=1 mn and m approximate equations

), 0 < j < m - l ,

(6.3.10)

mn

(/ - K&)
(6.3.11)

Solve in parallel
m —1

?mn = -Yl m

*&•

(6'312)

i=o The following theorem shows that (pmn and (pmn have almost the same order of error. T h e o r e m 3.1. Ifk(t,s) an error estimate

G C'([0,1] x [0,1]), y G C'[0,1], then there is

lk-£m„||oo = 0((mn)-' + n-2').

(6.3.13)

VI. Combination methods for accelerating the convergence

250

Proof. By (6.3.8) and (6.3.10), one obtains 1 m— 1

Kmn

= -Yj m

Jtf>.

(6.3.14)

i=o

Using the error expression of the trapezoidal rule of periodic function

r1

«

(KP - K)

d

'

)) — mn ds

(k(t,s)
where Pi(s) is a Bernoulli's periodic function with Fourier's series expansion ( Pi(s) = (-1)V-W

°° 2sin(27TJ5) £ 1 — 1 , for / odd, ~

fl(«) = (-l)P-2)/2 V i=i

2COS(2TTJS)

-, for / even.

(b.d.lbj

(2^i)'

Without loss of generality, assume that / is odd. Let c\ v)l\t)

f1 = (-l)('-3)/2 / Jo

dl 2sin

[27rjn(s - i/(mn))] — [*(*, s)
Substituting into (6.3.15), one obtains oo

( M ° - K)
(6.3.18)

Using the identity m-l

^ sin (27rjn(s - i/(mn))) •=o

= 0, for m Jtf,

it follows that m-l w 0

') = 2$ > jj ° (W = 0, 0, for mjlj. mj[j. »=o

(6.3.19)

Construct an auxiliary function wy satisfying the integral equation (I - K)wf

= vf\

0 < i < m - 1.

(6.3.20)

3. Combination methods for the Nystrom solution

Its approximate solution un

251

satisfies the equation = vf\ sf = ■*°. 0 < 1i < m --- 11..

(I - - K^)wf Jtf >)■ (I-

(6.3.21) 5.21) (6.:

l<

It follows from (6.3.7), (6.3.11) and (6.3.18) that

)

oo OO

,

(I-KP)bp(i - itf))(p _-^vm-n_ n-< ^ r ' 5si,f')))==0. o, D2J'i)"' 7=1

0 < i < m - 1.

(6.3.22) (6.: J.22)

Since Knl —► K, (n —► oo), and (6.3.7) has a unique solution, the inverse operators ( l — Kn ) , 0 < i < m — 1, exist and are uniformly bounded, hence, CO oo

- M -- n


<«f

i=i OO oo

= V -- V®

oo

+ ""

■ ' « , «

i=i

= 0,

- n

-

i=i

'(-f- -«S°)

0 < i < m - 1.

(6.3.23) (6.: J.23)

By using k(t,s) G C'([0,1] x [0,1]) and the error estimate of Nystrom's approximate solution (see Chapter 4), it is easy to prove that there exists a constant C independent of n,j and i such that

M° "

1 Ww^ - - wf\\ . iu^ll < CnCn~'.

5.24) (6.3.24) (6.:

Substituting into (6.3.23), we have oo oo

J . (0

- ' i i ; =f = =0 (0{n~ n "- 221 n ¥>" ')-).

(6.:(6.3.25) 5.25)

i=i

Noting the identity m- l m

£

w?\t)

= 0,

for mjlj, for mj[j,

(6.: 5.26) (6.3.26)

t=0

and summing up with respect to i in (6.3.25) and taking average, the proof of (6.3.13) is complete. □ In order to discuss the error estimate of the combination method for the solution of analytic periodic function, we denote by B„ the set of analytic functions with period 1 in a strip region = {zeC:\I : \Imz\ < 5S<7 , = {* G Cm:z\<
*}.

(6.3.27) 3.27) (6.:

252

VI. Combination methods for accelerating the convergence

where C is a complex plane. If f(s) is a restriction of f(z) E Ba on the real axis, then using the trapezoidal rule on the interval [0,1], the error has the following expansion^] ETn(f)

= £ e x P ( " 2™\k*\)ak(f), l*l>o

0 < |
(6.3.28)

where the sign of W £ (—
(6.3.29)

(6.3.30)

the solution

(6.3.31)

In the following we shall prove that the combination method can improve the order of error quadratically. T h e o r e m 3.2. Under the condition (6.3.30), we have \\
(6-3.32)

Proof. Using (6.3.28) and Fourier's expansion of (p, we have (K - KP)

(6.3.33)

where

obviously,

f1 vW(t) = / k(t, s)
1

{

))ds,

(6.3.34)

2n

= 0.

(6.3.35)

1

Let u/ ) and itA ) satisfy respectively the equations (7 - K)w& = «(«'), i = o, 1,

(6.3.36a)

(I - ffW)£(0 = t;W, i = 0,1.

(6.3.36b)

3. Combination methods for the Nystrom solution

253

By (6.3.31), ||™(° - £ W ||oo = 0(exp(-27m|<7|)), i = 0,1,

(6.3.37)

hence, = 0 ( e x p ( - 4 7 r n | ^ | ) ) , i = 0,1. (6.3.38)

(l-KP)(
Since (J — Kn ) _ 1 are uniformly bounded, it gives

(6.3.39)

By using (6.3.35), it follows that ^(i)

w(o) +

=

o.

Therefore,

= 0(exp(-47rn|F|)). □

(6.3.40)

In general, the combination of more than two approximate solutions cannot raise the order of accuracy. However, it can cancel more remainders in the error expansion and hence can still improve the accuracy. In addition, as the computation is done in parallel, the combination of more approximate solutions will not need much CPU.

3.3. The nonsmooth

cases

If T is a nonsmooth boundary, e.g., a polygon or a polygon with curved boundary, then the kernel of the boundary integral equation (6.3.2) is dis­ continuous or even singular. In this case, the combination methods men­ tioned above do not work. However, they can still be applied if the Mori double exponential formula is used. Consider an integral 1=

f f(s)ds. Jo

(6.3.41)

Using a transformation s = \j)(i) — — [tanh (-7rsinh(tf)) 4- l ] , we have oo

/

f(j>{t))xl>'{t)dt. -oo

If there exist positive integers ao, ai, a^ such that l / W W W I ~ flo exp ( - ai exp(a 2 |*|)), t — ±oo,

(6.3.42)

254

VI. Combination methods for accelerating the convergence

then taking an appropriate mesh width fc, the error of the truncated trape­ zoidal rule

/<"> = /, JT f{rl>{ih))j>'{ih)

(6.3.43)

i=-N

is I - Z ^ = O (exp ( - CN/ IniV) J , where C > 0 is a constant. Here the convergence rate is of exponential order (cf. [94]). In order to avoid the 4 overflow, the mesh width is usually chosen as h = —. N (6.3.43) can be regarded as the cubature rule on the interval (—hN, hN). Since the integrand and its derivatives of various order decrease to zero double exponentially as \t\ —» oo, it is possible to regard this integrand approximately as a smooth periodic function on (—hN, hN). Construct rn cubature rules

)¥'(h

4? = ^ E fW

)> 0 < t < m - 1,

m

J=-N

m (6.3.44)

and i

m —1

C°o — £«Pm

(6.3.45)

,=o If T is a nonsmooth boundary, e.g., a polygon with curved boundaries, r r , r = 1, •••,/, then the kernel of equation (6.3.2) is piecewise smooth, and the corner points may be the singular points of the kernel. However, the integral on T is still the summation of the integrals on T r . T r can be expressed as Tr = {xr(s) = (xri(s), xr2(s)) : 0 < s < l } . Set <pr(s) = 9(xr(s))

= $(z), z e T r ,

(6.3.46)

then 1 r d — / $(x) \n\x 2TT Jrj dnx =

/ Jo

z\ds(x)

kjr(t,s)(pr(s)ds,

zeTr)

j , r = l,...,/.

Define the integral operators Kjr
kjr(t,s)(p(s)ds,

j,r=l,...,/.

(6.3.47)

3. Combination methods for the Nystrom solution

255

The boundary integral equation (6.3.2) can be expressed by the integral system

"j(*)
j = l, ••,/,

(6.3.48)

r=l

where 1 _ I g'

for t G (0,1),

aj(t) =

0(*)/(2TT),

for t = 0 or t = 1,

and 6(0) and 0(1) are the exterior angles at two end points of Tj respectively. Let (p = ( A = diag(Qfi,...,af/), and T = [Kjr] • r = 1 , then (6.3.48) is equivalent to the operator equation A
(6.3.49)

Using the cubature rule (6.3.44), construct Nystrom's approximation T)[,of T and m equations

(A - T ^ V i ? = 9,0
(6.3.50)

After obtaining m approximate solutions independently, take an average i

f

m— 1

= -I>1?,

(6-3.51)

which is just the combination approximation. Similarly, when JV, h are appropriately chosen, the contributions of the integral (6.3.42) on (—oo, —Nh) and (Nh, oo) can be omitted. The error of the cubature rule (6.3.43) is 0 ( e ~ c / / l ) , and the error of the cubature rule (6.3.45) is 0 ( e ~ c m / l ) , where c is a positive constant independent of h. The value Ip evaluated from (6.3.50) and (6.3.51) is more accurate than each tp^h) (i = 0, • • •, m — 1), moreover, its error will decrease quadratically. For details, see the following numerical examples.

3.4. Neumann

boundary value

problems

Consider the following Neumann boundary value problem of Laplace equation Au = 0,

in fi,

8u — = 9, on

on r ,

r f g(x)ds(x) = 0.

( 6 - 3 - 52 )

VI. Combination methods for accelerating the convergence

256

By using the single layer potential theory, (6.3.52) can be converted to the boundary integral equation of the second kind i ±a(y)a(y)

f

/ a(x)

2TT JT

d

In \x - y\ds(x) = g(y),

(6.3.53)

dny

where the sign of the first term on the left-hand side depends on that (6.3.52) is an interior problem or an exterior problem. Certainly, for the exterior problem, in order to guarantee the uniqueness of the solution, the values of the solution at infinity have to be restricted. For the interior problem, the trivial solution is a constant, hence the solutions of (6.3.52) are not unique. In order to guarantee the uniqueness, usually

L

a(x)ds(x) = 0

(6.3.54)

is taken as a constraint condition. Once
u(y) =

Vy G

ft.

(6.3.55)

Similar to the case of Dirichlet problem (6.3.2), if T is a smooth bound­ ary, then the kernel of the integral operator is also smooth; if T is a polygon, then the kernel is piecewise smooth. For smooth kernels, one can obtain approximation by using Nystrom's method, and improve the accuracy by the combination methods; for piecewise smooth kernels, before these pro­ cedures, one has to apply the double exponential formulae. All these algo­ rithms have the same efficiency as those for DirichletJs problems. In the following we shall consider the solution of the discrete equation of the Neumann's problem. Assume that, by using the quadrature rule, (6.3.53) and (6.3.54) are converted to the following discrete equation Ax = 6,

(6.3.56a)

n

lTx = Y,Uxi = 0,

(6.3.56b)

»=i

which is an inconsistent system. By the least square method, we can derive a normal system BTBx T

T

T

= BTb, T

(6.3.57)

where B = [A : /], b = [b : 0] . However, to solve (6.3.57) directly is expensive: considerable amount of work is needed for evaluating BTB, moreover, the condition number of BTB is the square of that of B.

3. Combination

methods for the Nystrom

solution

257

In order to avoid solving equation (6.3.57), it is possible to regard (6.3.56) as finding the Galerkin approximation in the subspace H = {x £ Mn : lTx = 0}, i.e., find x G H satisfying yTAx = yTbi

Vy G H,

(6.3.58)

which is equivalent to the following bordered system \ (6.3.59) Obviously, the solutions of two systems (6.3.57) and (6.3.59) are almost identical, while solving (6.3.59) is easier than solving (6.3.57).

3.5. Numerical

examples

E x a m p l e 3 . 1 . Consider the Dirichlet problem of Laplace's equation An = 0, u = g, x

2

in ft,

(6.3.60)

on T,

y 2

where H is an ellipse: (—) + (—) < 1. The exact solution u = In [(x — 5 4 10)2 -f y 2 ], and so both T and g are smooth. Table 3.1 lists the errors of the relevant Nystrom approximation and the combination approximation at interior points, where m denotes the combination number. Table 3 . 1 . n = 4

n = 8

m = 1

m = 2

m = 3

m = 1 m = 2 m = 3

(0,0)

1.4E-1

9.1E-4

3.8E-4

1.6E-3

(i.o)

7.1E-2 -2.5E-3

(0,1)

2.7E-1

(*i>yO

(2,0)

l.OE-2

0

9.3E-5 -1.3E-3 4.7E-7 l.OE-3

-9.8E-2 -7.0E-4 -3.1E-4

0 0

1.1E-2 2.3E-5 9.5E-7 5.6E-4

0

0

The results show that the combination method is very efficient. For example, the result of n = 4, m = 2 is almost as good as that of n = 8, m = 1, which coincides with Theorem 3.1. E x a m p l e 3.2. Consider again the Dirichlet problem (6.3.60), here the boundary T is expressed in parametric form: T = { (xi(t) = —— sin - , #2(0 v3 2 = — sint) : 0 < t < 2TT}. It is easy to verify that (0,0) is a corner point

258

VI. Combination methods for accelerating the convergence

2

3 3 2

of T with an interior angle -w. The exact solution is u = r / cos (-0). Construct the Nystrom approximate equation by using double exponential formula ip(t) = - [tanh(sinh(*) + 1)] and the cubature rule (6.3.44). For the errors of Nystrom's approximate solution and combination solutions, see Table 3.2. Table 3.2. m = 1 4.9E-3 (0.2,0) 3.6E-2 (0.6,0) (0.2,0.2) -1.8E-2 (0.5,-0.1) 1.4E-2 (*i>yi)

n = 20 m = 2 2.0E-5 4.2E-4 -2.6E-2 6.3E-5

m = 3 1.6E-6 1.8E-6 -4.2E-3 -1.9E-6

n = m = 1 -7.0E-6 8.5E-4

40 m = 2 2.2E-8 2.9E-7

-3.0E-2 -5.2E-4 5.2E-5 6.0E-8

The solution u in this example is a nonsmooth function, but it can be seen from Table 3.2 that the combination method based on the double exponential formula is very efficient. E x a m p l e 3.3. Assume that the domain ft = (0,1) x (0,1) of the problem (6.3.52) is a square. The exact solution is u = y. For four sides of this square, construct a cubature rule by using the double exponential formula. For the errors of Nystrom's approximate solution and combination solutions, see Table 3.3. Table 3.3. (xi,Vi)

n = 4X 6 n = 4 x 12 m = 1 m = 2 m = 3 m = 1 m = 2

(0.25,0.25) 4.8E-3 -3.4E-5 -2.6E-5 1.9E-5 -2.2E-7 (0.75,0.75) 4.9E-3 -1.1E-5 -1.0E-5 -2.0E-5 -4.0E-7 (0.5,0.5) 9.7E-3 -1.9E-5 -3.7E-5 9.5E-7 -3.0E-8

E x a m p l e 3.4. Consider the Neumann problem (6.3.52) with a domain ft which is the ellipse in Example 3.1. The exact solution is u = In [(x — 10)2 + y2]. One can treat this problem in two different ways: solving the bordered equation (6.3.59), or solving the normal equation (6.3.57). Both methods have almost identical results, but (6.3.59) needs much less amount of work. Table 3.4 shows the errors of Nystrom's approximation and the combination approximation.

3. Combination methods for the Nystrom solution

259

Table 3.4. n = 8

n = 16

m = l

m = 2

m = 3

m = 1

m = 2

(2,0)

-1.2E-4

8.5E-6

1.5E-6

8.4E-6

0

(2,2)

-1.6E-2

3.2E-3

l.OE-4 3.2E-3 -6.2E-6

(4,1)

2.5E-1

4.1E-2

4.0E-2 4.1E-2 -3.5E-5

(0,0)

1.3E-5 -1.2E-6 -4.2E-7 -9.6E-7 -3.7E-9

(*i,yi)

These four examples show t h a t the combination m e t h o d is very pow­ erful. Since the discrete m a t r i x of an integral equation is full, the amount of work and the storage for solving a full m a t r i x with mn unknowns are 0(m3n3 ) , on the contrary, those for solving m systems with n unknowns are 0(mn3). Furthermore, these m systems can be solved in parallel without communication among processors. R e m a r k 1. In principle, the combination m e t h o d can be applied to three-dimensional problems. However, the integral operators of threedimensional problems are weakly-singular, which have to be treated with special m e t h o d s . R e m a r k 2 . So far, the error estimation of boundary integral equations obtained from the finite element m e t h o d or the collocation-method is often conservative. In fact, the b o u n d a r y element m e t h o d has a very high order of accuracy. R e m a r k 3 . T h e accuracy of the collocation m e t h o d can also be im­ proved by using the combination m e t h o d . T h e collocation m e t h o d with piece wise constants is equivalent to the Nystrom approximation of the midrectangular rule, and the collocation m e t h o d with linear elements is equiv­ alent to the Nystrom approximation of the trapezoidal rule. Therefore, a simple average of the results by using the collocation m e t h o d with piecewise constants and the collocation method with piecewise linear functions can improve the accuracy. R e m a r k 4 . T h e Richardson m e t h o d m a y be invalid in some cases, since it is based on the polynomial expansion of errors, which is not available for analytic s m o o t h data. In these four examples, the errors decrease rapidly with n. It is known t h a t the computational complexity increases rapidly with the order of the full m a t r i x and hence the combination m e t h o d s are recommended to avoid this puzzle. R e m a r k 5 . For general integral equations, the double exponential formulae and the combination methods can also be applied.

Chapter VII SPARSE GRID METHODS AND COMBINATION TECHNIQUES

In the numerical solution of an s-dimensional partial differential equa­ tion, in order to obtain an error of order 0(h2) by using either finite ele­ ment methods or finite difference methods, the number of grids is 0(h~s). If s is large, the cost will be very high. For example, assuming that the band-width of a discrete matrix is 0 ( / i ~ s + 1 ) , a banded solver will have the computational complexity 0(h~3s+2) and storage complexity 0(h~2s+1). Even if better solvers are applied, the amount of work and storage needed by a problem with dimension greater than three are hard to bear. However, the recent development of multilevel algorithms makes the solution of mul­ tidimensional problems accessible. In 1990, C. Zenger proposed a sparse grid method (cf. [130]), and proved that only 0(h~1\\nh\s~1) nodes are needed to reach an accuracy of order 0 ( / i 2 | l n / i | 5 _ 1 ) . Comparing with full grid methods, the order of accuracy is slightly deteriorated by a logarithmic factor. The number of unknowns for a sparse grid method is almost inde­ pendent of the dimension. Unfortunately, sparse grid methods also have difficulties of their own. First, the coefficient matrix is no more band type and the condition number increases rapidly with the dimension. Second, for the solution of full grid matrices there are many available solvers, but little can be used for sparse grid matrices. M. Griebel, M. Schneider and C. Zenger^ y l proposed combination techniques based on the sparse grid prin­ ciple. U. Riidel 107 ! discussed the relationship between the combination techniques and the splitting extrapolation. The combination techniques mean that if the error has a splitting expansion, then by solving indepen­ dently some coarse grid equations and combining their solutions, one can obtain a solution with the same order of accuracy as that obtained from a globally refined grid. The combination principle is based on the approx­ imation theory of sparse grids, the union of these coarse grids is just the Zenger sparse grid. The difference of the combination techniques and the SEM shows that both are based on an asymptotic expansion of the error, 260

1. Sparse grids

261

b u t they differ in t h a t the combination techniques require the error expan­ sion in the splitting form while the SEM requires the expansion in a more strict form to enable the extrapolation values to reach a higher order of accuracy. T h e purpose of the combination techniques is by combining the coarse grid solutions to reach the same order of accuracy as t h a t obtained from the globally refined grid. In general, for smooth problems, the SEM is more efficient t h a n the combination techniques. But for singular problems, where the SEM m a y be ineffective, the combination techniques remain effi­ cient. M. Griebel and V. Thurnert 4 2 ] have obtained numerical results using the combination techniques for hydrodynamic problems and problems with complicated domains. T h e algorithms of the combination techniques are naturally parallel. All approximate equations on coarse grids are mutually independent and the order of their matrices are almost the same, hence, very little communi­ cation among processors is required when implementing on multiprocessor computers. In this Chapter, we discuss sparse grid m e t h o d s in §1 and the combina­ tion techniques with numerical examples in §2. We also introduce briefly the methods of implicit extrapolation which have been developing fast in recent years. 1. S p a r s e g r i d s

1.1. Multilevel

splitting

of finite

element

spaces

In the s t a n d a r d finite element method, each interpolation basis point Z{ corresponds to a basis function (fi satisfying
l
(7.1.1)

where n is the dimension of Sh, the space of basis functions. T h e G r a m m a t r i x [a((pi, ipjj\ ? of the basis function {<£>,} in the sense of energy inner product is called the stiffness matrix. It is a symmetric sparse m a t r i x , because the support of the basis function has a diameter of 0{h). The basic procedure of the s t a n d a r d finite element m e t h o d is to solve the linear equations with stiffness matrix. But the condition number of the stiffness m a t r i x is 0 ( / i ~ 2 ) which may cause a great difficulty in evaluation. O.C. Zienkiewicz and H. Yserentant proposed the concept of hierarchical basis based on the multilevel splitting techniques. Y s e r e n t a n t ^ y J proved t h a t the condition number of the stiffness m a t r i x constructed by hierarchical basis is only 0 ( | l o g / i | 2 ) , and by means of the conjugate gradient m e t h o d (CG), only 0{n log n log - ) operations will be required in order to make the e error of an approximate solution in energy norm less t h a n e. T h e multilevel

262

VII. Sparse grid methods and combination

techniques

theory is now an important branch of numerical mathematics, and a basis for studying the multi-grid methods as well as the domain decomposition methods. For details, see the monograph by J. Bramble^ 12 !. The concept of the hierarchical basis is introduced in the following. It is known that the automatic mesh refinement of the finite element method means that the initial partition Jo is subdivided automatically to generate the refined grid required. If Jo is a triangulation, then such a refinement, obtained by joining the three midpoints of each triangle to form four subtriangles, thus generates Ji, and so on. If Jo is a convex quadrilateral partition, then joining the midpoints of two opposite sides to form four sub quadrilaterals, generates J i , and so on. In general, for an s-dimensional domain A, if Jo is a partition consists of cuboid elements, then such a refine­ ment is to subdivide each cuboid into 2s sub-cuboids by the s hyperplanes through the center of element and perpendicular to the corresponding co­ ordinate axis. Denote by Afk the set of all vertices, and by Sk the subspace of functions which are continuous in Cl and piecewise s-linear on elements of Jk- Obviously, according to the definition, we have MkCXk+u SfcCW

(7.1.2)

For any u G C(Q), denote by Iku the piecewise s-linear interpolation function of u in Sk, i.e., Iku G Sk) and Jfcti(x) = u(x), VxGA/i.

(7.1.3)

In particular, if u G Skl then u = IjU, j > k. Hence, for a fixed n, under the assumption i_iti = 0, there is a decomposition n

u = £ ( J * t * - I t - i t i ) , VueSn,

(7.1.4)

Jb=0

The decomposition expression (7.1.4) is very important—it is easy to see that hu-h-xu G Sk\ and (Iku - I jb _ 1 ii)(x) = 0, Vx G -A4-1- Hence, if we define a sequence of subspace as follows: Tk = {veSk:

w(*) = 0, VxGA/]b-i}, k = lr..,n,

(7.1.5)

then Sn has a direct sum decomposition 5 n = 5 0 + T 1 + ...-hf n .

(7.1.6)

By (7.1.5), the hierarchical basis can be defined inductively as follows: First, take the ordinary nodal basis to be the hierarchical basis of So, and sec­ ond, suppose that the hierarchical basis on Sk has been defined, then the

I. Sparse grids

263

hierarchical basis on 5jt+i consists of the hierarchical basis on Sk and the nodal basis on Tfc+i. For the one-dimensional case, this concept can be illustrated as follows: Let Q = [0,1]. We denote by Sn a (2 n + 1)-dimensional piecewise linear function subspace with a partition of mesh width hn = 2~ n . Tn is the 2 n _ 1 dimensional subspace, and the functions in Tn vanish at the grid nodes corresponding to S n - i . It is easy to see that the support of the basis functions in TJ (i = l , - - - , n ) must belong to one of the intervals [(j — l ) / 2 , ' - 1 ) i / 2 * " 1 ] , 1 < j < 2*- 1 . The basis functions of TUT2 and T3 are shown in Fig. 1.1.

Fig. 1.1.

Basis functions of Ti, T 2 ,T 3 .

Take any u G C 2 [0,1]. u1 denotes the interpolation of u on Sn. By the standard error analysis, <e=

ti — t / '

n

-4" -

1

,d 2 u, dx2

(7.1.7)

Since u1 G 5 n , by (7.1.6), there is a direct sum decomposition n

u1 = uo 4- ^ u , - , w0 € S0l Ui eT{, i = 1, • -,n.

(7.1.8)

*=i

L e m m a l . l . t 1 3 0 ! / / t i G C 2 [0,1], then 1

.||d 2w

Nl-fij^-lsr

_

e4n-.+l

(7.1.9)

Proof. Obviously, U{ can be expressed by the basis function of 7}. Taking any basis function of Ti, one may assume that the support of this

VII. Sparse grid methods and combination techniques

264

basis function is (a - ft,-, a + ft,-), where ft,- = 2~'. Define the function ' -(hi - 0 / 2 ,

MO = < L

for£>0,

-(hi + 0 / 2 ,

(7.1.10)

for£<0.

j-0.5 . , Hence, for a = ———, i = 1, • • •, 21 \ we have 2«-i

/*' *,-(0«"(« + 0<*e = / ' -*i(0*'(« + *K i r°

l /^

= u(a) - [w(a + ft,) + u(a - hi)] / 2 = I,-ti(a) -

Ii-iu(a)

= iit-(a).

On the other hand, it is obvious that 1 / ' ki(t)u"(a + t)dt\ < |||ti"||oo / ' i , - « K = y | | « " | | o o . (7.1.11) Noting that the maximum values of U{(x) occur at the nodes of Tj, by (7.1.10) and (7.1.11), the proof of (7.1.9) is complete. D (7.1.9) shows that ti,- —► 0 as i —► oo, this is the main advantage of using the hierarchical basis approximation: when i becomes larger, the contribution of the subspace T} to u will be less. For a given £, there exists n > 0 such that for i > n, it is not necessary to evaluate 7}. In fact the error satisfies ll« J - «Hoo <

oo £ IMIoo »'=n+l

1

~

\\d2un

<- 2 Y 4 - ll

" Ai

^ 2 »~

i4

1 ,,d u.

23

1

= --4-"-

2

loo

4 which is deteriorated by a factor of — than the estimate in (7 .1.7). 3

265

1. Sparse grids

1.2. Two-dimensional

sparse

grids

Let Q = [0,1] x [0,1]. Divide Q into rectangular grids by the mesh widths h\ = 2~ n , /12 = 2 ~ m . Let 5 n j m denote the subspace of piecewise bilinear functions on the equidistant rectangular grid. Let 5° m = 5 n > m H HQ(Q), then the dimension of 5 ° m is (2 n - l)(2 m - 1). Let T n>m be the subspace of S„ m with functions vanishing on all nodes corresponding to the grids of S„ m_1 and SjJ_ ljm . T n>m is a 2 m + n _ 2 -dimensional subspace, and the support of its basis function is one of the intervals [ ( j - l ) / 2 n - 1 , j / 2 n - 1 ] x [(i - l)/2m-1,k/2m-1], 1 < j < 2 n " \ 1 < ib < 2 m " 1 . Fig. 1.2 shows some examples of the supports of the basis functions in T^*. Fig. 1.2.

T h e s u p p o r t of t h e basis functions in Tj f *.

For the bilinear finite element space S° n , there is a direct sum decom­ position

3U = Ei>.*> *=1

(7112)

Jk=l

i.e., each u £ S„ n can be uniquely decomposed into n

n

t = l

Jb=l

u = ^2J2Uik'

with u r t € 3i,k> l
(7.1.13)

In order to study the properties of u,i, define a function space X, which d4u is a set of functions on Q such that exists and is continuous. Define 8x2dy2

266

VII. Sparse grid methods and combination

techniques

a seminorm on X as follows: li/l — 1 u

dAu 2

,

(7.1.14)

2

\ \ - 1 0x dy 1

oo

For u € X, we denote by u1 the interpolation function in S„ n. By using (7.1.13), the unique decomposition of u1 is 7

n

n

u = 5 3 ]£!*.-*, UtteTU, l < « , i < n .

(7.1.15)

Lemma 1.2. If u E X, then there is MOP^-'-''-1!*!.

(7.1.16)

Proof. Let &,(£) be a function defined by (7.1.10), (x 0 ,i/o) be of TJ- j . Integrating by parts, one obtains

a n

°de

rhi rhj d4u(x0 + x,2/o + 2/) / / &i(x)fej(x) dxoh/

y. fci y_ fci

v

'

JV

5x25y2

= w(«o,yo)- r[ M ( x o,yo + ftj) + ti(so,!/o- A i) +ti(x 0 + ft,-, 2/o) + w(x0 - ft,-, yo)] 1 + - [u(x 0 4-ft,-,t/o + ftj) + u(xQ - fti, 2/0 + hj) 4 +u(x 0 + ft,-, yo - ftj) + u{xQ - ft,-, yo - hj)] = c.

(7.1.17)

Note that c = u,-;(x0,2/o), and |i/, ; (x, T/)| can attain its maxima only at the nodes of Ty. On the other hand, using (7.1.17), we have 1 fhi 1

/

fhi

/

*'(*)*i(*>

d4u(x0 + x, 2/0 4- y)

-***v < M—•

r^i

4

./-**J-fci dx2dy2 The proof of (7.1.16) is thus complete. Corollary. Ifu£X,

D

the following will be the estimate of Sn,n \\u-uJ\

1

W-

|oo<—1

18

(7.1.18)

1. Sparse grids

267

Proof. By using the estimate (7.1.16), one can derive oo

oo

n

Ui

n

ll« - "'lleo < I 5 3 X ) > - 1 3 1 3 UiJ II t=ij=i

»=ij=i oo

oo

<EEiKiiico+ E EIK»=lj=l

oo

oo

i=n+lj=l

oo

<2|«|£ ^ 4-<->"1 »=1j=n+l oo

oo

t=l

j=n+l

2 <

-4"n-1|ti|

9 = -hl\u\. 18

D

Using Lemma 1.2, the component of u1 in 7^*, denoted by i/,*, will decrease exponentially as i + k increases. Hence, the strategy of Zenger is to ignore the terms of £ ° n in TJ^ (i + k > n), and to replace S ° n by the following sparse grid space

t = EE^= »=1 fc = l

E

T

^-

(7119)

*+fc
The dimension of 5° n is (n — l)2 n -f 1, i.e., it is of order (^(/i" 1 log ft"1), however the dimension of 5° n is 0 ( / i ~ 2 ) . Furthermore the accuracy of the approximation of u at S„ n is only slightly deteriorated than that at S° n . In fact, we have the following T h e o r e m l . l . I 1 3 0 ! 7/ u G l , and u on S„ n , Men

denotes the interpolation of u

\W - S ' l U < - A » (log, A" 1 + - ) |«|. 48 6

(7.1.20)

Proof. Let n n —1+1 J

« = X) E «=i

t=i

"•>■ " o - e i i j .

(7.1.21)

VII. Sparse grid methods and combination

268

techniques

Hence, oo

oo

n

t=ij=i n

oo

^||E E i=lj=n-i+2 00 oo

oo

Uii+ 4

n—t-fl

»=i i = i oo

^ ^HL 1

i=n+lj=l oo

oo

* ( £ E "'-''- + EH*-'-*-->i t=lj=n-i+2

i=n+lj

1 4 = (n-4-n-2+-4-"-2)|u| o

=l

y

D = - ^ ( i o g 2 * » 1 + -)MWe denote by48Q n>n the full grid3 of S^n, and by Qsn>n the grid of S nj „, called the sparse grid. Fig. 1.3 shows the picture of a full grid and its corresponding sparse grid. (7.1.20) is only an estimate in L^ norm, and in Li norm the similar argument holds. Moreover, in the energy norm (H1 norm), the sparse grid method can reach the same order of accuracy, i.e., 0(h) as that by the full grid.

Fig. 1.3.

3 3 x 3 3 full grid and its c o r r e s p o n d i n g sparse grid

1.3. Higher dimensional

sparse

grids

The two-dimensional sparse grid can be generalized to higher dimensional cases with better efficiency. For an s-dimensional problem, the dimension of a full grid finite element space is 0(ft~ 5 ) while for the cor­ responding sparse grid, the dimension is reduced to 0(ft x (log ft"- i r n Moreover, the order of the interpolation errors is only reduced slightly from

1. Sparse

269

grids

0(h?) to 0 ( / i 2 l o g ( / i " 1 ) ) . Obviously, the higher the dimension of the prob­ lem, the more efficient the sparse grid method. Let Q = [0,1]* be a cube of Ms. We denote by Qn ... n the full grid with 1 ' ' mesh width — = / i n , and by 5 n , . ,n the corresponding piecewise s-linear finite element space. Let If' be a piecewise linear interpolation operator with respect to the variable x*. T h e interpolation basis points are taken at % — , i = 0, • • •, 2 J . Let X be a function space on ft, in which functions have 02su

continuous partial derivatives

\ii\

—

\u\ —

1

. Define a seminorm on X by

d2 u

'

1

U«? - - - a*?loo 1

(7.1.22)

Let u1 be the interpolation of u on 5 n > ... ) n . Obviously, u1 = l£)'-I{n)u. Define the subspace of Sn,-,n J-a

=

(7.1.23)

as follows:

-*Ofi, •••,<*•

^((^-eo-^-ei)). 0 < c*t- < n , i = 1 , . . . , 5 ,

(7.1.24) (k)

where 3R(A) denotes the range of the operator A. Again, define I)_{ = 0 (k = 1, • • •, s). It is easy to derive the following direct sum decomposition T Sn,..;n=c r E"-5-) «i.-.«.0 a,=0

(7-L25)

1=

In fact, suppose u E 5'nJ--,nj then « = «' = # > # > . . . # > «

= n[E(^ ) -e 1 )]« Jk = l

afc=0

= E--E(^ ) -e 1 )-(^ ) -ei)« a!=0 n

a,=0 n

or1

or,=0

i,•••)««> =

0

(7.1.26)

270

VII. Sparse grid methods and combination

techniques

and the decomposition of u is unique. By the estimate (7.1.9), it is easy to obtain an estimate of « a i ) ... ) t t |

IK,.,.. Iloo = | (/£> - i

OO

< -~4- , a , |w|, V u E X , V

(7.1.27)

it means that the components of u in Ta (|a| > n) can be omitted. Hence one needs only to consider the following sparse grid space 5 n ,.., n = ^

T«.

(7.1.28)

\a\
Let u1 denote the interpolation of u G X in S^.-.-.n- The following theorem gives an error estimate of the approximation in the sparse grid space. Theorem 1.2. If u G X, Men Mere eztsfe a constant Cs dependent on s, such that \\u - S'Hoo < CshKlogh-1)-1^.

(7.1.29)

Proof. By (7.1.27),

|TX — G^Hoo = || 5 3

Ua

W°°

\a\>n

< £ Halloo |a|>n

oo

4-fc

s ( E - E I)M fc=n + l *

|or|=fc

4

£ E T ( t i l ')M <

4-nn-1C,|u|

^C/i^log/i-1)5"1^!.

D

(7.1.30)

We now estimate the dimension of S n , .,n- Obviously, Qp C ft<>, A < a,- (i = 1, • • • ,5), hence,

dim5n,...,n < 5 3 diimla \a\=n

1. Sparse

271

grids

= 2 " £ l = 2»(" + *T 1 ) = 0(2nn'-1)

= Oih-^logh-1)'-1),

(7.1.31)

which shows that the dimension of the sparse grid space is much less than the dimension of the full grid space 0 ( / i ~ 5 ) , while the order of accuracy in Loo norm is only slightly deteriorated by a logarithmic factor. Moreover, the order of accuracy in H1 norm is the same as that of the full grid. For details, see [18]. 1.4. Finite

element

equations

on a sparse

grid

Let the initial mesh widths be hx and /i y , and let Clij denote the rect­ angular grid with mesh widths hx2~% and /i y 2~ J . Obviously,

«* -

k

fc+l-t

U

U n<j

t=l

(7-1.32)

j=l

is a sparse grid with k refinements. k

k + l-i

k

k+l-mNij

^=EE ^ = £ £ E^>. i=i

j=i

»=i

j=i

( 7 - 133 )

i=i

is a sparse grid space. Vij)Zl =sp&n{(pj } denotes the one-dimensional space spanned by the nodal basis functions corresponding to the node z\. Nij is the number of nodes of fiij, but {^/ : i + j
for i = j = 1,

{ ^ € ^ 1 , ^ ^ 0 , - 1 , 1 } ,

for i > 1, j = 1,

{^GBij^^fiij-i},

for i = 1, j > 1,

{^J)Gflij,^jfflMjU".j-i}.

for i > 1, j > 1,

Bij = {

(7.1.34) where Bij are the sets of the standard nodal basis functions of Qij. Take the hierarchical basis of Sk as k

k+l-i

„

m ~ u u B^. t=l

j=l

(7.1.35)

VII. Sparse grid methods

272

and combination

The finite element equation on the sparse grid becomes:

techniques

Find u e

Sk

satisfying

, v«effj,

(7.1.36)

a(u ,«) = / ( « ) .

or equivalently, the stiffness equation for the sparse; grid L»Su

fHS

=

(7.1.37)

with L

K

*)» V y m > V n eJTjE, J m , n= a(
(f?s)m

= (f,
Fig. 1.4. Sparsity pattern for sparse grid stiffness matrix

Table 1.1. Eigenvalues and condition numbers of sparse grid matrices *

0

1

dimension

1

5

2

4

Ml/2,l/2v

8.000

12.162

Ml/2,l/2x

8.000

5.837

2.208

1.000

2.083

8.272

17

5

49

6

129

m a t r i x w i t h o u t scaling A A

.

/2 1/2

*K * )

18.272

32.715

64.290

1.785 18.322

128.134

1.469 1.199 43.739

106.857

m a t r i x after diagonal scaling U ^ x C P - 1 / ^ ^ ' 1 / ^ - 1 ^ ) 1.000

1.335

1.769 2.115 2.553

1.000

0.664

0.248

1.000

2.009

7.108 11.352

UMID-WAI'WD-W) [

KiD-^Al'^^D-1'2)

0.186

2.903

0.135 0.098 18.890

|

321 |

29.482

273

2. Combination techniques

But the stiffness matrix of (7.1.37) has complicated structure. Fig. 1.4 plots the sparsity pattern of a 49x49 stiffness matrix for a sparse grid' 1 ^. In addition, unlike the hierarchical basis stiffness matrix for a full grid, the condition number of the stiffness matrix for a sparse grid increases rapidly with k. Table 1.1^^ '^ shows that the condition number of the sparse grid increases as fast as 0(2 f c / 2 ), it is faster than 0(k)y the condition number of the full stiffness matrix for the hierarchical basisl 129 !. Hence, direct appli­ cation of the iterative methods (e.g., the Gauss-Seidel method, CG method) cannot attain the optimal complexity. By means of the Bramble-PasciakXu (BPX) multilevel method, M. Griebel, C. Zenger and S. Zimmert41^ suggested the multilevel Gauss-Seidel algorithms, where the number of iter­ ations is independent of k. Detailed computational methods and numerical results can be found, e.g., in [41]. 2. C o m b i n a t i o n techniques Although the dimension of the sparse grid space is reduced from 0(h~s) to 0(/i~ 1 (log/i~ 1 ) 5 - 1 ), the interpolation error only slightly deteriorates from 0(h„) to © ( / ^ ( l o g / i " 1 ) 5 - 1 ) . However, the sparsity of the stiffness matrix generated by the sparse grid is destroyed, and the condition num­ ber increases exponentially with k. Furthermore, there are many efficient software packages available for the standard finite element methods, but so far, good software packages for solving equations with sparse grid stiffness matrix derived from sparse grid methods are still lacking. For this reason, M. Griebel, M. Schneider and C. Zenger^^ proposed the combination tech­ niques for solving the sparse grid equations. The combination techniques are based on the following relation between the sparse grid and the full grid interpolations uTn n and u\ -:

«».» = E

i+j=n + l


( 721 )

This relation shows that the interpolation u„ n in the sparse grid space 5° n can be regarded as the linear combination of interpolations ujj in the full grid spaces Sij (i+j — n, n + 1). In two-dimensional case, this combination is shown in Fig. 2.1. We denote by Uij the finite element approximation on the full grid space

S?j. Let

"n,» = £ t+j=n+l

"U - £ "V i+j=n

(7'2-2)

274

VII. Sparse grid methods and combination

techniques

be a combined approximation. We shall now prove that if the solution is sufficiently smooth, and the error of the discrete solution Uij has the following expansion « u - uij uu = = d(hi)h? d(hi)hl

+ C22{hj)h] (hj)h] + + Dihuhfihlh], D(hi: hj)h?h],

(7.2.3)

then the error u — G£ n is of the same order as that of u — u„ n . Here, the coefficients C\{hi),C2{hj) and D{hi,hj) are bounded functions of x,y and the parameters hi (or/and /ij), i.e., there exists a constant K such that \Cx(hi)\ < K, f \Ci(hi)\ { I \D(hi,hj)\
\C2(hj)\ < K, \d{hj)\

(7.2.4)

Fig. 2 . 1 . I n d e x diagram of linear c o m b i n a t i o n for t h e two-dimensional case

2.1. Two-dimensional

combination

techniques

39

T h e o r e m 2.1.[ 1 If the error of the discrete solution Uij has an expansion of the form (7.2.3), and the coefficients satisfy (7.2.4), then the error of the combined approximation u„ n has the following estimate lltl--K,n\\0o
~K,n ~ ucn,n—=U ti---

u■

2sJ2 i+j=n+l »+j=n+l

UiW

1>

» ++ Yl -M

Ui

>i ,3

i+j=n

ui ) fa -~ < Jj) =E = S fa(M -- uuij) u) -- J2 £ fa = E (CiMhj+ 02^)1$+ Dih hi)tihj) h )h?h]) =«'+i=n+l E M

*+j=n + lL

»+j=n i+j=n

it j

i+i=n+lL

(7.2.5)

2. Combination

275 275

techniques techniques

£ (Ci^w{Cxih^hl YC + C^h^ 2{hj)h) - E

++ Dihuh^hfh)) D{hi,hj)hfh'])

i+j=n i+j=n

= Eci(W»=1 »=i

n-l

n

h{hi)h? + Ec 2 (Ai)ftJ

1=1 1= 1

3j = = l1

n-1

-Ec2(ft,-)fe? + E 2 i =l

+j=n+l »'+.;'=n+l

ft D(hi>ithj)h]h] D(h hj)h?h]-'£ - E ^(DfahjWh] .- ,A;)W «+j=n

1 hj)]hl (huhj)- £E ^C D^.fc,-)]^■- »-+ Ej = nE+ l D^c^o*+j=n

= = [Ci(/i„) ++ CC2(M + 2 (A„)-f

44

»+j=n+l

t+j=n

Applying (7.2.4), it leads to an estimate 2

W-K,n\<\Cl(hn)\h2n W-K,n\<\Cl(hn)\h I

1

+-

n

+ \C,(h \C,(hn)\hl )\hl

^

^

.

D h h h DWi,N)h h E E ( i> i)\ l +|- *+j=n E + l ( i, >)- E i+j=n ^ . ^ l ^

*+j=n + l \c (h )\hl < |C!(A„)|AS + \C22(hnn)\hl

< \Cl(hn)\h2n +

»+j=n

1

+- E 4

x+j=n+l

\D(hi,hj)\h2 + E p(^,fti)^+

-|-j=n t*+j=n

w^hMhi *i)l*n

1 2 1 < Kh Khl + -ifnft^ Khl + tfft* + A"(n.-1)A» - l)h2n + 4 -Knhl+K(r 5 = Ki a/ £*((l l ++ - n ) 4 5 = **»(! ia*(i + o g2(fc„ ). 2 ( 0J ))+ -- ilog 4 4 Taking the maximum norm and the proof of (7.2.5) is complete.

(7.2.6) (7-52.6) D

Theorem 2.1 shows that the terms depending onft,-or hj with i,j / n are cancelled, and just the ftn-terms remain. However, the error of the combined approximation is only slightly deteriorated by a logarithmic factor than that of the full grid approximation. The discrete error U{j has an expansion of the form (7.2.3) if and only if u is sufficiently smooth. In general, instead of (7.2.3), we assume that the discrete approximation satisfies the following expansion uitj = = dihiW Ci(hi)h? + + C*{hj)h) C2{hj)h) + + D(hi, D(hi, hjWq, u - UiJ hj)h\h),

(7.2.7)

where v is a positive number which is independent offt,-,hj, x and y, and the coefficients satisfy the condition (7.2.4). Hence, by repeating the proof of

276

VII. Sparse grid methods and combination

techniques

Theorem 2.1, the combination solution defined by (7.2.2) has the following error estimate 2" + l \\u

U

n,n\\oo

v. iVAl n ^l -f

!Og2^n

;;

= 0(Klog(h-1)).

(7.2.8)

By comparing this estimate with \\u — i/n,n||oo = 0(h„), the difference is only a logarithmic factor. Note that the combination techniques have an advantage that the com­ bination pattern is independent of the exponent v in the asymptotic expan­ sion. Therefore, in the expansion (7.2.7), the exponent v is not necessary to be exactly known and hence comparing with the extrapolation methods, the combination techniques are more suitable for non-smooth problems, since the extrapolation methods are applicable only when the value of v is known exactly. 2.2. Three-dimensional

combination

techniques

Two-dimensional combination techniques can be generalized to threedimensional case. Let Q be a unit cube, /i, = 2~\hj = 2~J', hk = 2~k be mesh widths in the x-, y- and ^-directions respectively. We denote by Uitjtk the finite element approximate solution at the grid Qij}k. Suppose that the solution is sufficiently smooth, and the error has an expansion of the following form u - ui)j}k = Ci(A,-)A,? 4- C2(hj)h] +C3(hk)h2k + D^hi.h^hfh] +D3(hj1hk)h]hl

+ D2(hi, hk)h?h2k (7.2.9)

+ E(hiyhjihk)h?h]hl

where the coefficients are bounded, i.e., there exists a positive constant K independent of ft,, hj,hk and x, y, z, such that |A(/i,>i)|<^,

\G(hi)\
/ = 1,2,3,

and (7.2.10)

\E{hi,hhhk)\
T h e o r e m 2.2.^ y l Assume that the error of a full grid solution itt,j,fc has an expansion as (7.2.9), and the coefficients satisfy (7.2.10), then the combined approximation

"ntn,n =

Ui

>3>k

J2 i+j+k=n+2

-2

£ »+j+fc=n+l

«U,*+

£ ,+j+fc=n

W

M>*

(7-211)

2. Combination techniques

277 211

has the following error estimate

\\ul l « -
=r

) i n ^ e g ra -l solutions of the Diophantine

2

n 1 i == ( n +2 )4\ J- --(2 ( 2

4

2

(7.2.12) (7.2 .12)

n

3

2 n 2) +J (+ l2 2)" >) ■

then W U

-

U

n,n,n

U ui k 53 "•".)>,*) 53 ((u - ~ J> i+j+k=n+2 i+j+k=n+2

--2 2

£YL

t+j+Jb=n+l

(u-uij,*)+ + 5533 ((*w - wW.\i.*). »'J.*)0*-^,*) i+j+k=n t+j+fc=n

Substituting (7 (7.2.9) (7.2.13), Substituting .2.9) into (7.2 .13), we get

uU -—S nu° ,n,n n

n

n

J^(n -- * i ++ l)d(MA? + E ( n - i' ++ ^i)cC(fci)*? W l)Ci(M*. +X> == 53( t=i ii==ii ?

2

n-l

+ 5 > -- ** + [ j > --i)Ci(hi)hf i)Ci(ft,-)/i? + l)C l)C33(^)/il (^)/il -- 22[53(n fc=i n-l

»= 1

n-l

- JMhfih) + 53(n - *)C3(fcfc)*2] fc=l

n-2

+5>1=1

n-2

- » + l)Ci(A.-)A? + E ( " " J J=l

t= l n-2

l c

) *ihi)h}

h h -k-l)C3(h I>i(fc.-,fchj) k)hl+ + 53 i)ft?ft? + 53(n-*-l)C E Di(hi> l] s (M*k « +i+j
k=l Jfe = l

+ J2 x+Jb
+ D (hi,h k)hjhl+ MW

2 I»2(/»i

2 53 (hj,hk)h]h 2 DD33(hj,h k)h]hik j+k
E + E a(fci,fc *)W] + E i,M^fc*]+ + 53 E ^i>s(ft -2[ 2

ti+j
j+k
++^53 D )h}hi D1(hi,hft>)ft?A? Dz{h h**)W I»2(^,i 2(h{u,h j)h^hj+ k)h}hl I>i(A,-, *iW i+k
■1 i+j
,/.,)/»?/»? Mhi,hj)h]h]

7 2 13 ((7.2.13) -- )

278

VII. Sparse VJI. Sparse grid grid methods and combination

techniques

Yl DD(h(hi,h )h?h ++ E 53 D {h,ft*)ft| h )h]h\ hi ,MW E tt+fc
+

22 i:

2

k

k

3

22

k

2

k

2

jt k

* T J T * — T*T*

+ £ + £

h k

■1

■l

*-rj -rn,—T*TX

E(hi9Eihi^M^hlhl hjthk)h^h]hl

(7.2.14)

i+j+k=n

In the above derivation, the equalities n 53 d(Mtf = £ > -- t + l)Ci(A,-)ft?, «'+J+*=n+2 «=1 t+i+*=n+2

and and

t=l

h)h]h) A,-)fc?A? £ Di(hi, D (h ,h )h]h == = E J2 i?i(A*, DifcWih] E i+j+Jb=n+2 i+i+fc=n+2 » + j < n + lL »+i
i

j

2

j

are applied. Combining all the Ci-terms in (7.2.14), we have n

5 ^ ( n -- tt + i)Ci(fcO&? l)Ci(fci)A? 5>n-l

n-2

-2 53(n - i)Ci(ft,-)ft? + 53(n - i - l)Ci(fc)*? ««=1 = 1

»»== 11

n-2

? = *' +J!) 1)] d(hi)h} = £E i(Knn "~*+ ) ~" 22(« ( n "~0o++ (n ( »-- *t ---l)]Ci(A,-)A, tt = = ll

l))]C 1 (/i„_ 1 )^_ 1 -mc^h^hi^

+ ( n_- (( „n_- il )) + ( n --- ((„ » -+ [[(„ + il )) _- 22 („ +(n n+ + +(n -- n C!(hn)hl =■■ C!(hn)hl.

l)Ci(hn)hl l)Ci(hn)hl

(7.2 1.15) (7.2.15)

Therefore, all terms with hi, h{, hj, h /i*k (i, ( t ,jj, ,fc k ^^ n) n) are are cancelled cancelled in in this this comcombining procedure. Similarly, combining all the £)i-terms, Di-terms, we obtain Di(hi, D {h ,hhj)hlh] )hih E
53

l

i

j

2

j

E

-2 53 Dx{hi Diihi^hfh] ,hj)h?h] + + 53 -2E

i + j < n _ Ii *'+i
i+j
.*i)W

D^hi DtfahjWh]

( A,A,hj)h?h] 5 )A?A?++ Ea5 3 {l-2)D {h .A,)*?*? h )h h - 2)D (ft i - - 2 + l ) D(A - *+>
=

11

it

i+i=n+l

i il i

11

u j
2

i

2

j

2. Combination techniques

=

£

279

D^hi^hfh) - £

»+j=n+l

= [-

Dxihi^h^h]

i+j=n

£

£

D1(hi,hj)-

*+j=n + l

i?i(A.-,A,-)]^

»+j=n

1 < [-ntf + (n - l)tf]A*. (7.2.16) 4 Similarly, C2-, C3-, A r and Z)3-terms can be combined and we have the same results as those for Ci-terms and Di-terms. Finally, for 2?i-terms, we have the following estimate:

I

£

Eihi.hjMWtfhW

i+j+k=n+2

£

<

|^(Ai,fci,M|An+2

i+j+k=n+2

n(n + l) < K

(7.2.17)

hl+2.

Substituting the results of (7.2.15)—(7.2.17) into (7.2.14), it follows 1

l« - «n,n,nl < ?>Khl + *Khl ("« + « ~ l) 4

ln(n+l)

+Kh* [

l(n-l)n

+

(n - 2)(n - 1)

+

16 2 2 2 65 25 = Kh2n(l+— n+—n2) 32 32 65 25 = Khl(l+-log2(h-l)+-(\og2(h-1))2). The proof of Theorem 2.2 is complete.

] 2

(7.2.18)

□

Corollary. Instead of expansion (7.2.9), if the error ofuij^

satisfies

u - uiJ}k = Ci(/i,-)fc? + C2(hj)h) + Ca(hk)hl

+D1(hi,hj)h»h"j + D2(hit hk)h?hl + +E(hi,hj,hk)hWhl,

D^hjMWjK (7.2.19)

where v is a positive real number, the coefficients Ci,Di and E satisfy the condition (7.2.10), then, the combined approximation defined by (7.2.11) has an error estimate ll"-
(7.2.20)

VII. Sparse grid methods and combination

280

techniques

Since the coefficients of the combination will not change when the ex­ ponent v varies, the combination method is more effective than the extrap­ olation method for nonsmooth problems. R e m a r k 1. By means of the following identity^

gc-tfCj1)^!1)-1it is easy to derive the s-dimensional combined approximation

<••■» ='£(-#(•I1) £ «J=0

V

' |a|=n+*-j-l

R e m a r k 2. The combination techniques are also successful in evalua­ tion of multiple integrals. For details, see Basenzenski and Delves [6]. R e m a r k 3. The number of independent mesh parameters for combi­ nation techniques can be larger than the dimension of the domain. This is similar to the multi-parametric expansion considered in §3.4, Chapter V. R e m a r k 4. The exponent v in the expansion (7.2.19) may depend on the nodes (x,y, z) (e.g., in the case that the solution has singular points), but the combined approximation can still be almost as accurate as the solution obtained from a full grid method.

2.3. Numerical

examples

For two-dimensional problems, the idea of the combination technique is to solve independently n problems with 2 n unknowns and (n — 1) problems with 2 n _ 1 unknowns. And for three-dimensional problems, it is to solve in­ dependently n(n + l ) / 2 problems with 2 n unknowns, (n — l ) n / 2 problems with 2 n _ 1 unknowns, and (n - l)(n - 2)/2 problems with 2 n ~ 2 unknowns. All these problems can be solved in parallel. Moreover, the size of each problem is dependent only on the parameter n, and independent of the di­ mension of the domain Q. Hence the combination technique is suitable for working on multiprocessor computers. The results from different proces­ sors will be combined without any communication among these processors. The detailed discussion can be found in the work of M. Griebel et a / . ^ . The comparison between the errors of the full grid approximation and the combined approximation is given in the following examples. E x a m p l e 2 . 1 . ( S m o o t h s o l u t i o n ) ^ Consider the Laplace equation Au = 0,

infi = (0,1) x (0,1),

2. Combination

techniques

281

with Dirichlet's boundary condition. The exact solution equals u = sin(7rz/) •sinh(7r(l — x))/sinh (7r). Table 2.1 shows the errors of the full grid so­ lution and the combined solution at points Pi = (0.5,0.5) and Pi — (0.25,0.25). Table 2.2 shows the errors in Li norm and Loo norm. The ratio | e n _ i ) n _ i | / | e n ) n | shows that the error is of 0(h?) for the full grid solution and is of O (h2\ogh~1) for the combined solution. Table 2.1. n 1 2 3 4 5 6 7 8 9 10 11 12

e

P o i n t w i s e e r r o r , full g r i d o n t h e l e f t , c o m b i n e d s o l u t i o n o n t h e r i g h t .

i , n ( j » j ) ~ l«n,nl

-

7.4268E-2 1.5498E-2 3.7333E-3 9.2504E-4 2.3075E-4 5.7655E-5 1.4412E-5 3.6028E-6 9.0070E-7 2.2517E-7

4.792 4.151 4.036 4.009 4.002 4.001 4.000 4.000 4.000

-l,n-ll -cn,n\%(I J)W' nl*n,nl e

}

-

1.3252E-2 3.1526E-3 7.7889E-4 1.9415E-4 4.8503E-5 1.2123E-5 3.0307E-6 7.5768E-7 1.8942E-7

4.204 4.048 4.012 4.003 4.001 4.000 4.000 4.000

[l3

<,.(*.*)- n - l , n - l I-1.J 7.4268E-2 9.9913E-3 1.2929E-3 5.5787E-5 -5.1448E-5 -2.9150E-5 -1.1357E-5 -3.8568E-6 -1.2186E-6 -3.6824E-7 -1.0796E-7 -3.0964E-8 -8.7346E-9

-

7.433 7.727 23.176 1.084 1.764 2.566 2.944 3.165 3.309 3.410 3.486 3.544

-9.3480E-3 6.1846E-3 1.2064E-3 2.2655E-4 3.8414E-5 5.0772E-6 1.3935E-7 -2.4756E-7 -1.3249E-7 -5.0770E-8 -1.7105E-8 -5.3792E-9

ZIZ^J] K.J 1.511 5.126 5.325 5.897 7.566 3.643 0.562 1.868 2.609 2.968 3.179

T a b l e 2 . 2 . E r r o r s i n L2- a n d L > full g r i d o n t h e l e f t , c o m b i n e d s o l u t i o n o n t h e r i g h t . n 1 2 3 4 5 6 7 8 9 10

1

,

l e n - l , n - l 2 1| l*n - l , n - l l o o |Cn,n|oo |cn,nloo |«n,»| 2

I6*'*'2 7.4268E-2 1.2047E-2 2.5072E-3 5.7950E-4 1.3985E-4 3.4385E-5 8.5272E-6 2.1234E-6 5.2980E-7 1.3232E-7

-

6.165 4.805 4.326 4.144 4.067 4.032 4.016 4.008 4.004

7.4268E-2 1.8742E-2 4.4585E-3 1.1169E-3 2.7847E-4 6.9659E-5 1.7412E-5 4.3528E-6 1.0882E-6 2.7205E-7

-

3.963 4.204 3.992 4.011 3.998 4.001 4.001 4.000 4.000

- l , n - l * ' \rc 1 lCn,nl«>

7.4268E-2 1.4123E-2 3.2023E-3 7.9926E-4 2.1322E-4 5.9058E-5 1.6610E-5 4.6816E-6 1.3141E-6 3.6642E-7

-

7.4268E-2 3.2387E-2 1.2887E-2 4.5351E-3 1.4838E-3 5.0056E-4 1.5518E-4 4.6647E-5 1.3667E-5 3.9669E-6

5.258 4.410 4.007 3.748 3.610 3.556 3.548 3.563 3.586

n

-l,n-l|o° | an.nl°°

-

2.293 2.513 2.841 3.056 2.964 3.226 3.327 3.413 3.445

E x a m p l e 2.2. (Singular s o l u t i o n ) ^ ' Consider the Dirichlet prob­ lem of the Laplace equation Ati = 0,

i n O = ( 0 , l ) x (0.5,0.5),

with an exact solution u = (x2 -f y2)

cos ( a r c t a n ( - ) / 2 ) , which has a y

singular point (0,0). Table 2.3 shows the comparison between the errors of the full grid solution and the combined solution at points Pi = (0.5,0.0) and P 2 = (0.25,-0.25). Table 2.4 compares the L2 norm and Loo norm

282

VII. Sparse

grid methods

and combination

techniques

of these errors. Table 2.3 shows |e n _i > n _i

l/|c»,n| — 2 3 / 2 «2.828, hence, the full grid error is of 0(h / ) at points far away from the singular point. Table 2.4 shows len-i^-iloo/le^nloo —► 2 1 / 2 «1.414, hence, the error is only of Ofi1/2) in the neighborhood of the singular point. The theory of the combination techniques for nonsmooth problems is not yet perfect, but Tables 2.3 and 2.4 illustrate that, similar to the case of smooth solutions, the error of the combined solution is only slightly deteriorated than that of the full grid solution by a logarithmic factor \ogh~1. 3 2

Table 2.3.

P o i n t w i s e e r r o r , full g r i d o n t h e l e f t , c o m b i n e d s o l u t i o n o n t h e r i g h t .

n

en,n(h°)-

1 2 3 4 5 6 7 8 9 10 11 12 13 14

3.9695E-3 6.3484E-3 2.7436E-3 1.0995E-3 4.1708E-4 1.5408E-4 5.6081E-5 2.0225E-5 7.2498E-6 2.5879E-6

c

en,n§-\)]-

n-l.n-l

l«n,nl

-

l«n,nl

-

0.625 2.314 2.495 2.674 2.707 2.747 2.773 2.789 2.801

n-l,n-ll

-

1.2359E-2 3.9539E-3 1.3877E-3 5.0905E-4 1.8525E-4 6.6799E-5 2.3929E-5 8.5351E-6 3.0357E-6

3.126 2.849 2.726 2.748 2.773 2.792 2.804 2.812

115

1 2 3 4 5 6 7 8 9

110

i-l,n-l

i « - . . | a ^ I e n,nl2 3.9695E-3 1.1000E-2 4.6957E-3 1.9065E-3 7.5691E-4 2.9572E-4 1.1405E-4 4.3523E-5 1.6463E-5 6.1820E-6

1 n - l , n - l 's n,nl>T'~T' I'n.nl

n,nVjiUJ

3.9695E-3 -7.5628E-3 0.5248 -1.4178E-3 -3.4260E-3 2.2075 1.1336E-2 1.3736E-3 5.7404E-4 5.9681 7.4114E-4 1.2047E-3 0.4765 7.6178E-4 1.5814 -2.2430E-6 4.1336E-4 1.8429 4.5173E-6 1.8820E-4 2.1964 -5.4202E-6 7.7368E-5 2.4325 -3.1142E-6 3.0123E-5 2.5684 -1.4430E-6 1.1331E-5 2.6585 -5.9843E-7 4.1710E-6 2.7165 -2.3432E-7 1.5146E-6 2.7538 -8.8687E-8 5.4518E-7 2.7781 -3.2851E-8 1.9512E-7 2.7940 -1.1997E-8

n-l.n-l1 1

l-l.nl

-

0.125 8 1 330.429 0 0. 1. 2 2.411 2.554 2.642 2.700 2.738

E r r o r s i n L>2- a n d L o o - n o r m , full g r i d o n t h e l e f t , combined solution on the right.

Table 2.4.

n

\*1:

i e

2

0.361 2.343 2.463 2.519 2.559 2.593 2.621 2.644 2.663

lc

1

-l,n-lloo 1cn,nloo

|Cn

\^n,n\oo

3.9695E-3 2.7093E-2 2.1732E-2 1.5744E-2 1.1195E-2 7.9270E-3 5.6071E-3 3.9651E-3 2.8038E-3 1.9826E-3

-

0.147 1.247 1.380 1.406 1.413 1.414 1.414 1.414 1.414

C

\r lL l c n,nl2

I« C - l , n - l " K.nb

3.9695E-3 1.9356E-2 1.2190E-2 7.5137E-3 4.6105E-3 2.8254E-3 1.7246E-3 1.0491E-3 6.3508E-4 3.8318E-4

-

0.205 1.588 1.622 1.629 1.632 1.638 1.644 1.652 1.657

2

\rc 1 I'-n.nloo

l«°n - l , n - l » ~ 1 c n.nloo

3.9695E-3 3.6331E-2 5.3937E-2 5.7539E-2 5.3173E-2 4.5382E-2 3.7325E-2 3.1997E-2 2.6971E-2 2.2621E-2

-

0.109 0.674 0.937 1.082 1.172 1.216 1.167 1.186 1.192

E x a m p l e 2.3. (Nonlinear h e a t t r a n s f e r ) ^ ! Consider the Dirichlet boundary problem of the nonlinear heat conduction equation du

d dx

w

ft .

+

■ fe> 5<'-

<9ix

oy

= /,

i n n = (0,l)x(0,l),

2. Combination

techniques

283

where t h e material property function is

r I-^-I^)1^ eQ = <

1,

u>

1/2

ti = l / 2

I l + (l/2- u )

1/a

,

u < 1/2.

sinh(7r(l — . — has a singular point at ( 0 , 0 ) . ; sinh(7r) In order to find a combined solution, it is necessary to solve the nonlinear discrete equation T h e exact solution u = sin(7ry)

= fij , l -f J = n , n - h 1.

LijUij

Table 2.5 lists the error in Li n o r m and L{ x) n o r m for the full grid solution and the combined solution. Table 2.5. E r r o r s in L^- and Loo-norm, full grid on t h e left, combined solution on t h e r i g h t . n

• |Cn

| l«n-l,n-lb| | 'n|2 l«n,nl2 \Cn>n*°°

a = 0.125 3 2.3171E-3 4 5.4442E-4 5 6 7 8 a = 0.25 3 1.2328E-3 4 3.2187E-4 5 6 7 8 a = 0.5 3 1.7520E-3 4 3.9924E-4 5 6 7 8 a = 0.75 3 2.2734E-3 4 6.3935E-4 5 6 7 8 a - 1.0 3 1.5987E-3 4 3.7171E-4 5 6 7 8

l«n-l,n-lloo |Cn,nloo

, c , ' n>nl2

|c

n-l,n-l'2 \<,J2 ^n,n\oo

I ^ l V n ^ 0 ^ ] ^ J ^

4.696 4.256

4.1210E-3 1.0569E-3

3.978 3.899

3.4773E-3 8.5692E-4 2.2023E-4 6.0845E-5 1.7488E-5 5.0685E-6

4.062 4.058 3.891 3.619 3.479 3.450

1.1681E-2 4.0742E-3 1.4670E-3 4.5330E-4 1.3810E-4 4.1357E-5

2.496 2.876 2.777 3.236 3.282 3.339

3.408 3.830

2.3066E-3 6.5162E-4

3.351 3.540

5.1882E-3 1.4933E-3 5.1843E-4 1.7537E-4 5.6677E-5 1.7737E-5

3.594 3.474 2.880 2.956 3.094 3.195

1.2641E-2 5.9064E-3 2.3389E-3 7.6980E-4 2.7278E-4 8.9370E-5

2.267 2.140 2.525 3.038 2.822 3.052

4.886 4.388

5.9662E-3 1.3116E-3

3.077 4.549

8.5599E-3 2.7436E-3 1.0472E-3 3.3565E-4 1.3102E-4 4.4540E-5

2.735 3.120 2.620 3.120 2.562 2.942

2.8296E-2 1.1184E-2 4.3777E-3 1.6041E-3 6.0695E-4 2.2305E-4

1.411 2.530 2.555 2.729 2.643 2.721

4.301 3.556

7.1804E-3 1.8969E-3

2.923 3.785

1.0241E-2 2.6358E-3 1.0987E-3 3.6211E-4 1.9252E-4 1.7071E-4

2.078 3.885 2.399 3.034 1.881 1.128

3.0782E-2 1.1040E-2 4.4042E-3 1.8599E-3 7.4955E-4 9.6088E-4

1.365 2.788 2.507 2.368 2.481 0.780

4.360 4.301

5.0857E-3 1.2175E-3

3.151 4.177

7.9072E-3 2.8952E-3 1.0115E-3 3.3684E-4 1.0732E-4 3.2960E-5

2.567 2.731 2.862 3.003 3.138 3.256

2.2445E-2 9.1664E-3 3.9667E-3 1.7222E-3 6.3854E-4 2.2181E-4

1.809 2.449 2.311 2.303 2.697 2.879

VII. Sparse grid methods and combination

284

techniques

Applications of the combination techniques to b o u n d a r y layer problems and problems with non-rectangular domains can be found in [39]. There are also applications to hydrodynamics in the work by M. Griebel and V. T h u r n e r ^ 4 ^ . T h e numerical experiments of the combination techniques under various conditions can also be found in these references.

2.4. Comparison sparse grid

of combination methods

techniques,

SEM

and

U. Rude expounded in [107] and [106] the relation between the combi­ nation techniques and the SEM, and compared the numerical results. First, it is obvious t h a t both are based on the asymptotic expansion of the error, with the aim of eliminating as m a n y terms as possible. Second, there is also a big difference between them: the former tries to obtain an approximate solution of the fine grid, usually the order of accuracy is 0(h2), therefore this is a 0(h2) m e t h o d in essence, while the latter can have an arbitrary order of accuracy, provided there is a suitable asymptotic expansion. Of course, this higher order of accuracy is limited on the coarsest grids, b u t it can be extended to other points by interpolation (an algorithm for ob­ taining a higher order accuracy on the finer grids is given in Appendix III). Hence, as far as the smooth solution is concerned, the SEM is better t h a n the combination techniques. But for singular problems, the combination techniques may be more popular because they require weaker conditions for error expansion. Furthermore, the coefficients of the combination do not change when the exponents of the expansion vary. Hence, for singu­ lar problems, the combination techniques may be more effective t h a n the SEM. Finally, b o t h methods split the problem into mutually independent subproblems, and are suitable for parallel computation. T h e higher the dimension, the better the parallelism. T h e following examples are quoted from U. Riide's works [106] and [109]. E x a m p l e 2 . 4 . Consider the Dirichlet problem of Poisson's equation f -Au |^

= f(x,y), u = 0,

infi = (0,l)2, on 3fi,

(7.2.21)

with an exact solution 7rx wy u = x(l — x ) c o s ( — ) y ( l — y)cos ( — ) . z z T h e numerical results by the Richardson extrapolation, the SEM, the combination techniques and the sparse grid m e t h o d are listed in the follow­ ing four tables.

2. Combination

techniques

285

Table 2.6. Richardson extrapolation for a smooth model problem h Error u2 "0 "1 "3 "4 1 Point 5.916E-3 -1.163E-4 3.093E-7 -9.569E-11 -5.121E-11 2.020E-1 4.918E-3 3.994E-5 4.656E-8 1.363E-7 1/4 L2 Energy 2.121E-1 5.894E-3 5.768E-5 6.438E-8 1.850E-7 Point 1.391E-3 -6.979E-6 4.739E-9 -5.139E-11 4.739E-2 2.949E-4 9.185E-7 1/8 L2 1.737E-7 \ Energy 5.076E-2 3.995E-4 2.136E-6 5.116E-7 Point 3.427E-4 -4.317E-7 1.Q97E-10 1/16 L2 1.164E-2 1.807E-5 2.423E-8 1 Energy 1.254E-2 2.642E-5 1.118E-7 Point 8.535E-5 -2.696E-8 1/32 L2 2.899E-3 1.140E-6 1 Energy 3.127E-3 1.911E-6 Point 2.131E-5 7.241E-4 1/64 L2 1 Energy 7.813E-4

Table 2.7. Splitting extrapolation for a smooth model problem Error Point 1/4 L2 Energy Point 1/8 L2 Energy Point 1/16 L2 \ Energy Point 1/32 L2 1 Energy Point 1/64 L2 1 Energy h

"0

5.916E-3 2.020E-1 2.121E-1 1.391E-3 4.739E-2 5.076E-2 3.427E-4 1.164E-2 1.254E-2 8.535E-5 2.899E-3 3.127E-3 2.131E-5 7.241E-4 7.813E-4

"1

-3.270E-4 1.397E-2 1.683E-2 -1.927E-5 8.097E-4 1.095E-3 -1.187E-6 4.918E-5 7.125E-5 -7.393E-8 3.047E-6 4.633E-6

u2 u3 1 5.118E-6 -6.066E-8 7.938E-5 9.439E-5 1.157E-3 1.396E-4 | 7.527E-8-1.354E-10 1.993E-5 3.046E-6 5.224E-5 8.579E-6 1.162E-9 5.683E-7 2.966E-6

_ ..

i

286

VII. Sparse grid methods and combination

techniques

Table 2.8. Combination technique for a smooth model problem hx = hy

Error Point L2 Energy Point L2 Energy Point L2 Energy Point L2 Energy Point L2 Energy Point L2 Energy

1/2

1/4 1/8

1/16 1 1/32 1 1/64 1

tin 3.008E-2 5.299E-1 8.189E-1 5.916E-3 1.096E-1 3.618E-1 1.391E-3 2.635E-2 1.711E-1 3.427E-4 6.943E-3 8.236E-2 8.535E-5 2.096E-3 3.671E-2 2.131E-5 7.241E-4 7.813E-4

tli "2 "3 "4 u$ 2.717E-3 -1.840E-4 -2.439E-4 -1.092E-4 -3.935E-5 1.404E-1 3.491E-2 9.575E-3 3.051E-3 1.124E-3 4.235E-1 2.086E-1 1.013E-1 4.703E-2 1.470E-2 1.233E-3 2.571E-4 5.207E-5 1.000E-5 2.527E-2 6.408E-3 1.958E-3 7.130E-4 1.724E-1 8.326E-2 3.724E-2 3.280E-3 3.335E-4 8.021E-5 1.929E-5 6.864E-3 2.063E-3 7.170E-4 8.239E-2 3.673E-2 1.085E-3 8.479E-5 2.100E-5 2.092E-3 7.228E-4 3.671E-2 8.015E-4 2.128E-5 7.239E-4 7.823E-4

Table 2.9. Method of sparse grid for a smooth model problem hx = hy

1/2

1/4

1/8 \

1/16 1

1/32 \ 1 1/64 |

Error size Point L2 Energy size Point L2 Energy size Point L2 Energy size Point L2 Energy size Point L2 Energy size Point L2 Energy

qh,X,hy

"Vo

1 6.407E-3 4.488E-1 6.739E-1 9 1.663E-3 1.138E-1 3.379E-1 49 4.247E-4 2.818E-2 1.679E-1 225 1.171E-4 6.836E-3 8.193E-2 961 4.048E-5 1.622E-3 3.664E-2 3969 2.131E-5 7.241E-4 7.813E-4

q^x,hy

*Vl

5 1.045E-3 1.482E-1 3.732E-1 33 3.682E-4 2.849E-2 1.686E-1 161 1.138E-4 6.839E-3 8.194E-2 705 1.078E-4 2.626E-3 4.658E-2 2945 2.130E-5 7.241E-4 7.816E-4

Q^Xj^y

^0.2

Q^Xt^y

^0.3

Q^Xj^y

^0.4

17 49 129 7.211E-6 -6.145E-5 -2.215E-5 4.157E-2 1.081E-2 2.693E-3 1.917E-1 9.493E-2 4.390E-2 97 2 641 8.590E-5 2.829E-5 1.717E-5 6.956E-3 1.658E-3 7.288E-4 8.241E-2 3.693E-2 2.439E-3 449 1153 3.862E-5 2.058E-5 1.623E-3 7.243E-4 3.665E-2 9.232E-4 1921 2.121E-5 7.241E-4 7.889E-4

Q^Xt^y

^0.5

1

321 1.259E-6 8.816E-4 1.213E-2|

|

2. Combination

techniques

287

Table 2.6 lists the errors of the Richardson extrapolation in the L2 norm, the energy norm and the error at point P = ( - , - ) . Here, uk (k = 0,1, • • •, 4) denote the results after k extrapolations. Table 2.7 lists the errors of Type 1 SEM, where uk (k = 1, 2, 3) stands for the result after k split. Table 2.8 lists the errors of the combination techniques, where UQ stands for the result evaluated on the full grids, and the results of the combination k

techniques are uk = ^T uhx2-i}hy2i-k 1=0

k-1

- ] T uhx2-i)hy2i-k+i

(k = 1, • • •, 5).

»=o

Table 2.9 lists the errors of the method of sparse grid, where S0xk'

y

=

k

U 5°. x_i ofc_ijB_i denotes the sparse grid, and the "size" denotes the order y

i=o

'

of the stiffness matrix. These tables show that the Richardson extrapolation converges fastest but has the highest cost; the method of sparse grid are slightly better than the combination techniques and is unsuitable for performing in parallel; the SEM has lower cost and higher degree of parallelism, furthermore, it has the same order of error as the Richardson extrapolation (only with larger coefficients). Therefore it seems that, among these methods, SEM is superlative. E x a m p l e 2.5. Consider again the problem (7.2.21) with another solu­ tion u = sin(16x 4- 16y)x(l — x)y(l — y) which is an oscillatory function. The numerical results from the four dif­ ferent methods are listed in Tables 2 . 1 0 - 2 . 1 3 . It can be seen that when the mesh width is larger than the oscillatory period, the result obtained by a few steps of Richardson's extrapolation is not as good as the combina­ tion techniques. The reason is that when the mesh width is large, the high frequency oscillations cannot be correctly reflected. But when the grids are refined step by step, the accuracy of the Richardson extrapolation and SEM will be rapidly improved. The results of Type 1 SEM methods are listed in Table 2.11. Comparing with Type 1 methods, the amount of work and storage of Type 2 methods are more economical and the accuracy is only slightly worse, which are verified by the examples shown in Chapter V.

288

VIL Sparse grid methods and combination

techniques

Table 2.10. The Richardson extrapolation for an oscillatory model problem Error u0 ui u2 ti 3 -0.1910 2.163E-2 2.349E-3 -9.073E-5 Point 10.7775 1.028E+0 1.788E-1 6.366E-3 1/4 L2 Energy 1.061E+1 9.919E-1 1.771E-1 6.264E-3 -0.0315 3.554E-3 -5.260E-5 -6.584E-S Point 1.8661 2.182E-1 3.085E-3 1.483E-5 1/8 L2 1.505E-5 \ Energy 1.872E+0 2.196E-1 3.092E-3 Point -5.215E-3 1.728E-4 -8.870E-7 1/16 L2 3.066E-1 1.103E-2 6.186E-5 1 Energy 3.201E-1 1.184E-2 6.854E-5 Point -1.174E-3 9.970E-6 1/32 L2 6.858E-2 6.346E-4 1 Energy 7.222E-2 6.953E-4 Point -2.860E-4 1/64 L2 1.667E-2 1 Energy 1.759E-2 h

Table 2.11. Splitting extrapolation for an oscillatory h Error UQ ui U2 Point -0.1910 1.483E-1 -4.577E-2 1/4 L2 10.7775 8.696E+0 3.358E+0 Energy 1.061E+1 8.277E+0 3.063E+0 Point -3.150E-2 1.077E-2 -9.635E-4 1/8 L2 1.866E+0 6.669E-1 5.847E-2 Energy 1.872E+0 6.691E-1 5.857E-2 Point -5.215E-3 4.672E-4 -1.348E-5 3.066E-1 3.004E-2 9.573E-4 1/16 L2 1.061E-3 Energy 3.201E-1 3.234E-2 Point -1.174E-3 2.607E-5 1/32 L2 6.858E-2 1.665E-3 Energy 7.222E-2 1.831E-3 Point -2.860E-4 1/64 L2 1.667E-2 Energy 1.759E-2

u4 2.897E-7 1.868E-5 1.833E-5

model problem U3 ] 1.590E-2 1.229E+0 1.146E+0 2.085E-5 1.297E-3 1.303E-3

2. Combination techniques Table 2.12. hx = hy Error Point 1/2 L2 Energy Point 1/4 L2 Energy Point 1/8 L2 1 Energy Point 1/16 L2 \ Energy Point 1/32 L2 1 Energy Point 1/64 L2 1 Energy

289

Combination technique for an oscillatory model problem UQ

-8.727E-1 1.264E+1 2.876E+0 -1.910E-1 6.258E+0 3.372E+0 -3.152E-2 6.960E-1 1.131E+0 -5.215E-3 1.377E-1 3.675E-1 -1.174E-3 4.223E-2 1.466E-1 -2.860E-4 1.667E-2 1.759E-2

ui t*2 u3 u4 2.298E-1 3.345E-1 1.050E-1 2.254E-2 1.719E+1 1.396E+1 5.778E+0 9.796E-1 5.544E+0 4.988E+0 3.306E+0 1.640E+0 6.349E-2 1.439E-2 9.538E-3 3.441E-3 5.722E+0 9.649E-1 2.274E-1 6.492E-2 3.075E+0 1.523E+0 6.578E-1 2.984E-1 1.980E-4 1.230E-3 5.601E-4 1.851E-1 4.947E-2 1.613E-2 4.550E-1 1.904E-1 6.072E-2 -9.534E-4 -1.707E-4 4.069E-2 1.558E-2 1.470E-1 1.959E-2 -2.740E-^ 1.654E-2 1.757E-2

us 1 1.146E-2 2.337E-1 7.213E-l|

|

Table 2.13. Sparse grid method for an oscillatory model problem hx = hy

1/2 1

1/4 1

1/8 1 1 1/16 1 1 1/32 j

1/64 \

Error width Point L2 Energy width Point L2 Energy width Point L2 Energy width Point L2 Energy width Point L2 Energy width Point L2 Energy

ofti>hy

•Vo

1 1.746E-2 1.000E+0 9.999E-1 9 9.550E-3 1.001E+0 9.947E-1 49 -5.882E-3 4.442E-1 6.183E-1 225 -1.704E-3 1.059E-1 3.035E-1 961 -5.647E-4 2.518E-2 1.374E-1 3969 -2.860E-4 1.667E-2 1.759E-2

Q f c i» f t y

^0.1

ch*>hy

^0.2

cfcx,fcy

^0.3

Qhx>hy

"^0.4

Q^x^y

1

"^0.5 5 17 49 129 321 1.617E-2 9.183E-3 2.881E-4 -4.409E-3 -4.374E-4 1.000E+0 1.002E+0 8.671E-1 4.101E-1 1.308E-1 9.998E-1 9.934E-1 8.895E-1 5.817E-1 3.178E-l| 33 97 257 641 2.674E-4 -4.416E-3 -4.395E-4 1.661E-4 8.669E-1 4.101E-1 1.308E-1 3.359E-2 8.898E-1 5.817E-1 3.179E-1 1.501E-1 | 161 449 1153 -9.617E-4 2.927E-5 -3.083E-5 1.373E-1 3.469E-2 1.712E-2 3.336E-1 1.587E-1 4.422E-2 705 1921 -4.877E-4 -2.425E-4 2.538E-2 1.669E-2 1.380E-1 1.907E-2 2945 -2.813E-4 1.667E-2 1.765E-2

290

VII. Sparse grid methods and combination

techniques

3. Outline of t h e implicit extrapolation T h e Richardson extrapolation, the SEM, the sparse grid m e t h o d s and the combination techniques have a common characteristic, i.e., combine ap­ proximate solutions obtained on different grids to obtain a solution of higher order of accuracy. T h e theoretical foundation of this idea comes from the existence of an asymptotic expansion of the error, which are called the ex­ plicit extrapolation methods by [18], [107]. Recently, a kind of implicit extrapolation methods was studied. For implicit extrapolation, the idea of extrapolation is applied to get approximate equations with higher order ac­ curacy. Hence, neither a global expansion of the discrete error nor a uniform mesh is required. Problems with nonuniform meshes, locally refined grids, or the related h- and p-adaptive methods, can all be treated by the frame­ work of the implicit extrapolation m e t h o d s . At present, the research on m e t h o d s such as truncation error extrapolation, energy extrapolation and T-extrapolation is very active. In the following we shall introduce briefly the development of the implicit extrapolation m e t h o d s . • r - e x t r a p o l a t i o n . Many optimal algorithms for solving partial dif­ ferential equations, such as the multigrid method, can be considered as applications of the multilevel principle. These m e t h o d s require evaluations on various grids with different widths, and extrapolation will be imple­ mented economically. A. B r a n d t ^ 3 ) called this kind of extrapolations on multigrid method as r-extrapolation. For details, see W . Hackbuscht 4 3 ! and U. R i i d e l 1 0 3 ] . However, all these extrapolations basically belong to 113 the Richardson type. A. Schiiller and Q. iw i applied SEM to multigrid m e t h o d and constructed a new form of r-extrapolation. • T r u n c a t i o n error e x t r a p o l a t i o n . T h e truncation error extrapola­ tion is based on the asymptotic expansion of the discrete truncation error. By means of the idea of extrapolation, several discrete equations can be combined into an equation with higher order of error after eliminating the truncation error terms with lower order. Of course, in order to make such a combination possible, it is necessary to transfer the truncation errors on different grids to a common space, at the same time, it is also necessary to transfer the extrapolation solution to the original grid space. T h e imple­ mentation and numerical experiments of the truncation error extrapolation can be found in papers by U. Ridel 107 ! , and the application of truncation error extrapolation in multigrid m e t h o d can be found in the monograph by W . Hackbuscht 4 3 ). • E n e r g y e x t r a p o l a t i o n . T h e finite element solution can be ex­ plained as the approximation of a minimization problem. For example, the homogeneous Dirichlet problem of the Poisson equation is equivalent to the

3. Outline of the implicit

extrapolation

291

minimization problem of the quadratic functional u e

^n)(^(")"(«./)).

(7-3.1)

where En(u) = - I{Vufdxdy. (7.3.2) 2 Jn Denote by SQ the space of test functions. In order to evaluate approxi­ mately (7.3.2), replace u by the interpolation function u1, and correspond­ ingly, define the approximate energy by E&u) = E^u1)

= \vFAhu,

(7.3.3)

where u is the vector composed by the nodal values of u and Ah is the stiff­ ness matrix. U. Riidet 109 ! proved that under an appropriate assumption, the following asymptotic expansion exists, E&(u) - En(u) = /i 2 ei + h4e2 + .. •.

(7.3.4)

Hence, carrying out one extrapolation, one gets the approximate functional

££(«) = ^n / 2 («)-^fc(«).

(7-3.5)

which approximates En(u) to order 0 ( / i 4 ) . The stiffness matrix given by

£»(«) is ^

= ^"

/ 2

-^«/2)

T

^^/2.

(7-3-6)

where l\,2 is the restricted operator. Thus the numerical examplefluy^ has an order of error 0 ( / i 4 ) , and the method is simpler than the finite element method with piecewise polynomials of higher degree. It is also possible to derive methods of higher order approximation in the sense of energy extrapolation by applying the multivariate expansion of mesh parameters and using the coefficients of SEM. The numerical examples in [109] show that this is very effective. Apart from the implicit extrapolation methods discussed above, the ex­ trapolation method for hierarchical basis is very attractive. It is the special case of the energy extrapolation under the framework of finite element with hierarchical basis, and its idea is based on a purely local element-by-element analysis. Hence, it can be applied to irregular and unstructural grids. More­ over, it has special effect for the composite grids generated by the adaptive

292

VII. Sparse grid methods and combination

techniques

local refinement. For details, see U. Rude' ' and S. McCormick and U. Riidef91!. However, the research on the theoretical aspect of the implicit extrapo­ lation methods is far from perfect. Up to now, there are only results from the numerical experiments on the model problems. Further research on the link in the extrapolation techniques, multilevel methods and the fast adaptive composite grid methods (FAC) is recommended.

Appendix I TABLES OF SPLITTING EXTRAPOLATION COEFFICIENTS s m

P

Type 1

1

Type 2

1

2 2 (0,0)

37/45

31/36

(1.0)

-20/9

-128/45

(1.1)

16/9

16/9

(2,0)

64/45

81/40

2 3 (0,0)

-97/567

-41/180

(1.0)

148/135

41/18

(1.1)

-80/27

-208/45

(2,0)

-64/27

-297/56

(2,1)

256/135

27/10

(3,0)

4096/2835

1024/315

1

(o.o)

11749/722925

107/2880

1

(1,0)

-388/1701

-946/945

(1,1)

592/405

3328/675

(2,0)

2368/2025

729/128

(2,1)

-256/81

-1458/175

(2,2)

4096/2025

6561/1600

(3,0)

-4096/1701

-4096/405

;(3,1)

16384/8505

4096/945

(4,0)

1048576/722925

390625/72576

|2 4

2 5 (0,0)

-110689/147910455

-3751/907200

(1,0)

46996/2168775

9181/32400

(1,1)

-1552/5103

-8576/2835

(2,0)

-6208/25515

-11223/3200

(2,1)

9472/6075

1773/160

(2,2)

-4096/1215

-6561/448

(3,0)

151552/127575

82304/6075

(3,1) (3,2)I

-94208/6075

-16384/5103

1

262144/127575

293

1152/175

1

294

s\m ,2| 5

2 6

Appendix

P

Typel

Type 2

1

(4,0)

-1048576/433755

-6640625/342144

1

(4,1)

4194304/2168775

390625/54432

(5,0)

1073741824/739552275

17496/1925

(0,0)

50844133/3028466566125

2431/7257600

(1.0)

-442756/443731365

-2402/42525

(1,1) (2,0)

187984/6506325

832/675

751936/32531625

64131/44800

(2,1)

-24832/76545

-1527/175

(2,2)

151552/91125

14585103/627200

(3,0)

-397312/1607445

-557056/51975

(3,1)

606208/382725

1048576/42525

(3,2)

-262144/76545

-32768/1225

(3,3)

16777216/8037225

1048576/99225

(4,0)

38797312/32531625

49609375/1596672

(4,1)

-4194304/1301265

-78125/2673

(4,2)

67108864/32531625

78125/7168

(5,0)

-1073741824/443731365

-186624/5005

(5,1)

4294967296/2218656825

23328/1925

(6,0)

4398046511104/3028466566125

13841287201/889574400

3 2 (0,0,0)

17/5

83/24

(1,0,0)

-4/1

-208/45

(1,1,0)

16/9

16/9

(2,0,0)

64/45

81/40

-593/315

-2209/1080

3 3 (0,0,0) (1,0,0)

68/15

887/135

(1,1,0)

-16/3

-944/135

(1,1,1)

64/27

64/27

(2,0,0)

-64/15

-2241/280

(2,1,0)

256/135

27/10

(3,0,0)

4096/2835

1024/315

3 4 (0,0,0)

5129/8925

20/27

(1,0,0)

-2372/945

-976/189

(1,1,0)

272/45

294/25

(1,1,1)

-64/9

-1408/135

[(2,0,0)

1088/225

60597/4480

(2,1,0)

-256/45

-2088/175

[(2,1,1)

1024/405

18/5

|

i

Appendix

295

s \m\

P

Type 1

Type 2

3 4

(2,2,0)

4096/2025

6561/1600

(3,0,0)

-4096/945

-8192/567

(3,1,0)

16384/8505

4096/945

(4,0,0) 1

1048576/722925

390625/72576

(0,0,0)

-934921/9130275

-2297/12600

(1,0,0)

20516/26775

196753/75600

(1,1,0)

-9488/2835

-52918/4725

(1,1,1)

1088/135

4544/225

(2,0,0)

-37952/14175

-295227/22400

(2,1,0)

4352/675

129933/5600

(2,1,1)

-1024/135

-3078/175

3 5

(2,2,0)

-4096/675

-225261/11200

(2,2,1)

16384/6075

2187/400

(3,0,0)

69632/14175

1201408/42525

(3,1,0)

-16384/2835

-905216/42525

(3,2,0)

262144/127575

1152/175

(4,0,0)

-1048576/240975

-63671875/2395008

(4,1,0)

4194304/2168775

390625/54432

(5,0,0)

1073741824/739552275

17496/1925

349/45

47/6

(1,0,0,0)

-52/9

-32/5

(1,1,0,0)

16/9

16/9

(2,0,0,0)

64/45

81/40

-21277/2835

-2113/270

4 2 (0,0,0,0)

4 3 (0,0,0,0) (1,0,0,0)

1396/135

397/30

(1,1,0,0)

-208/27

-1264/135

(1,1,1,0)

64/27

64/27

(2,0,0,0)

-832/135

-2997/280

(2,1,0,0)

256/135

27/10

(3,0,0,0)

4096/2835

1024/315

3153181/722925

63517/12960

(1,0,0,0)

-85108/8505

-44558/2835

(1,1,0,0)

5584/405

44044/2025

(1,1,1,0)

-832/81

-5504/405

(1,1,1,1)

256/81

256/81

(2,0,0,0)

22336/2025

111807/4480

(2,1,0,0)

-3328/405

-2718/175

(2,1,1,0)

1024/405

18/5

4 4 (0,0,0,0)

1

|

|

1

Appendix

296 s m

P

Type 1

Type 2

4 4

(2,2,0,0)

4096/2025

6561/1600

(3,0,0,0)

-53248/8505

-53248/2835

(3,1,0,0)

16384/8505

4096/945

(4,0,0,0)

1048576/722925

390625/72576

(0,0,0,0,0)

125/9

1007/72 -368/45

5 2

5 3

6 2

(1,0,0,0,0)

-68/9

(1,1,0,0,0)

16/9

16/9

(2,0,0,0,0)

64/45

81/40

(0,0,0,0,0)

-11005/567

-4307/216

(1,0,0,0,0)

500/27

334/15

(1,1,0,0,0)

-272/27

-176/15

(1,1,1,0,0)

64/27

64/27

(2,0,0,0,0)

-1088/135

-3753/280

(2,1,0,0,0)

256/135

27/10

(3,0,0,0,0)

4096/2835

1024/315

(0,0,0,0,0)

109/5

263/12

(1,0,0,0,0)

-28/3

-448/45

(1,1,0,0,0)

16/9

16/9

(2,0,0,0,0)

64/45

81/40

-12589/315

-22009/540

6 3 (0,0,0,0,0,0) (1,0,0,0,0,0)

436/15

9091/270

(1,1,0,0,0,0)

-112/9

-1904/135

(1,1,1,0,0,0)

64/27

64/27

(2,0,0,0,0,0)

-448/45

-4509/280

(2,1,0,0,0,0)

256/135

27/10

(3,0,0,0,0,0)

4096/2835

!

1

|

1

1024/315

where s, m and /? denote the dimension, splitting number and splitting index, respectively. Remark 1. For m = 1,/? = (1,0, • • • ,0),ap = 4/3;/? = (0,0, • • - , 0 ) , ^ = -{As - 3)/3. Remark 2. ap = a 7 if /? and 7 have the same component but different permutations.

Appendix

297

Appendix II TABLES OF SUCCESSIVE SPLITTING EXTRAPOLATION COEFFICIENTS Type 1 s \m 2 1

2 2

Coefficients

P

(1 + 2P)/(1 - 2P)

(0,0) | (0,1)

2 P / ( - l + 2p)

(o,o)

l / ( l - 22P) 2

(1.1)

2 3

2P/(_I +

2 2P)

(0,2)

0

(o,o)

{(1 + 2 * ) 3 ( - l + 2P - 2 2 P)(1 + 2 2 P)} / { - 1 - 2(2P) - 2(2 2 P) 3

4

7

9

10

1

n

-3(2 P) - 3(2 P) -|- 3(2 P) + 4(2 P) + 2 ( 2 P ) + 2 ( 2 P ) } (0,3)

{ 2 8 P } / { l + 2(2P) + 2(2 2 P) + 3(2 3 P) + 3(2 4 P) -3(2 7 P) - 4(2 9 P) - 2(2 10 P) - 2 ( 2 n P ) }

2 4

2 5P(X

_ 2 P + 2 2 P)(1 + 2P + 22P)

(2,1)

- 1 _ 2 P - 2(2 3 P) - 24P + 3(2 5 P) - 2(2 6 P) + 2(2 7 P) + 2(2 9 P)

(o,o)

l / ( l - 28P)

(2,2)

2»P/(_1 +

0

(1,3)

(0,0,1) 3 2 (0,0,0) (0,0,2) ' (0,1,1)

28P)

0

(0,4)

3 1 (0,0,0)

|

(1 + 2 P

+1

)/(1-2P)

2 P / ( - l + 2P) ( - 1 - 3(2P) + 2 J P)(1 + 2P + 22P) - 1 - 4(2P) + 4(2 3 P) + 2 4 p 24P 1 + 4(2P) - 4(2 3 P) - 2 4 p 2 2 p ( l + 2P) - 1 - 3(2P) + 3(2 2 P) + 23P

'

Appendix

298

s m

P

Coefficients

4 1

(0,0,0,0)

(1 + 3 ( 2 P ) ) / ( 1 - 2P) 2 P / ( - l + 2P)

(0,0,0,1) p

14 2

(0,0,0,0) (0,0,0,2) (0,0,1,1)

5 1 (0,0,0,0,0)

(0,0,0,0,1) 5 2 (0,0,0,0,0)

(0,0,0,0,2) (0,0,0,1,1)

- 1 - 6(2 ) - 6(2 2 p ) - 6(2 3 P) + 3(2 4 P) - 1 -- 6(2P) + 6(2 3 P) + 24P 2(2 4 P) 1 + 6(2P) - 6(2 3 P) - 24P 2 2 P(1 + 2P) - 1 - 5(2P) + 5(2 2 P) + 23P (1 + 4 ( 2 P ) ) / ( 1 - 2 P ) 2 P / ( - l -I- 2P) - 1 - 8(2P) - 10(2 J P) - 12(2 3 P) + 6(2 4 P) - 1 - 8(2 p ) + 8(2 3 P) + 24P 3(2*P) 1 + 8(2P) - 8(2 3 P) - 24P 2 2 P(1 + 2P) - 1 - 7(2P) + 7(2'2P) -|- 2 3 p

Type 2 s TO

P

2 2 (0,0)

(1.1) (0,2) 12 3 (0,0)

Coefficients l / ( l - 22P) 22P/(-1 +

2 2 P)

0 | ( 2 P + 3P)(1 + 2 p )(-2P3P + 24P + 2 2 P3 2 P - 2 5 P3P)} / { ( - 1 + 2P)(22P3P + 2P32P - 2(2 3 P3 3 P) - 25P -f 24P3P + 2 3 p 3 2 P _ 2 2p 3 3p _ 2 (2 7 P) + 26P3P + 2 5 p 3 2 P - 2(2 4 p 3 p ) + 2(2 7 p 3 3 P)}

(0,3)

{ 2 6 P ( - 2 P + 3P) 2 (2P + 3P)} / {(1 - 2 2 p )(2 2 P3 p -|- 2 p 3 2 p - 2(2 3 P3 3 P) - 25P + 2 4 p 3P + 2 3 P3 2 P _ 2 2p 3 3p _ 2(2 7 P) + 2 6 P3 p + 2 5 p 3 2 P - 2(2 4 P3 p ) -f 2(2 7 P3 3 P)}

(2,1)

{2 3 P3 3 P(-1 + 2 2 P)(1 + 2 2 P } / / 2 2 p 3 p _j_ 2P32P - 2(2 3 p 3 3 p ) - 2 5 P + 2 4 P3 P + 2 3 p 3 2 p __22p33p _ 2(2 7 P) + 26P3P + 2 5 p 3 2 P - 2(2 4 p 3 p ) + 2(2 7 P3 3 P)}

2 4 (0,0)

l / ( l - 3 4 P)

(0,4)

0

(1-3)

0

(2,2)

34P/(-14-34P)

Appendix

5

771

13 2

299

£

Coefficients ( - 1 + 2*3*)(l + 2(2?) - 4(3^) + 2P3P) ( - 1 + 2 P ) ( - 1 + 3 P ) ( - 1 - 3(2^) + 3(3^) + 2P3P) 32P(-1 + 2P) (1 - 3 P ) ( - 1 - 3(2^) + 3(3^) + 2P3P) 2 2 P ( - 1 + 3 P) ( - 1 + 2P)(-1 - 3(2P) + 3(3^) + 2P3P)

(0,0,0) (0,0,2) (0,1,1)

3 3

l / ( l - 2 P)

(14,1)

3

3 P/(-1 +

(0,0,3)

23P)

0

(0,1,2) 4 2

\

3

(0,0,0)

0 2

{ - 1 - 4(2P) - 2 P + 6(3P) + 6(2P3P) + 3(3 2 P) - 12(2 2 P3 2 P)

(0,0,0,0)

+3(2 2 P3 2 P)} / { ( - l + 2P)(-1 + 3 P ) ( - 1 - 5(2^) + 5(3^) + 2 ^ ) } 2 ( 3 2 p ) ( " l + 2P) (1 - 3 P ) ( - 1 - 5(2P) + 5(3P) + 2P3P) 2 2 P ( - 1 + 3 P) ( - 1 + 2P)(-1 - 5(2P) + 5(3P) + 2P3P)

(0,0,0,2) (0,0,1,1) 5 2

|

{ - 1 - 6(2P) - 3(2 2 P) + 8(3P) + 12(22P3P) + 8(3^) - 24(2^32P)

(0,0,0,0,0)

+6(2 2 P3 2 P)} / { ( - l + 2P)(-1 + 3P)(-1 - 7(2P) + 7(3P) + 2P3P)} 3 ( 3 2 P ) ( - l - f 2P) (1 - 3 P ) ( - 1 - 7(2P) + 7(3P) + 2P3P)

(0,0,0,0,2)

2

(0,0,0,0,1)

2P(_I +

3 P)

( - 1 + 2P)(-1 - 7(2P) + 7(3P) + 2P3P) { - 1 - 8(2P) - 6(2 2 P) + 10(3P) + 20(22P3P) + 15(3P) - 40(2P32P)

6 2 (0,0,0,0,0,0)

+10(2 2 P3 2 P)} / { ( - l + 2P)(-1 + 3 P ) ( - 1 - 9(2P) + 9(3P) + 2P3P)} (0,0,0,0,0,2) (0,0,0,0,1,1)

4(3 2 P)(-1 + 2P) (1 - 3 ) ( - l - 9(2P) + 9(3 p ) + 2P3P) 2 2 P ( - 1 + 3P) ( - 1 + 2P)(-1 - 9(2P) + 9(3P) + 2P3P) p

1

Remark 3. There are some terms with coefficient 0, because when both of s, m are even or odd, the homogeneous terms can be eliminated only by global refinement. Remark 4. The successive splitting extrapolation is much less than the Romberg extrapolation but is larger than SEM in the amount of work. The successive splitting extrapolation can be applied to the expansion with fractional power, and has a large scope of applications.

Appendix

300

Appendix III EXTRAPOLATION AT FINE GRID POINTS

It is well known that the extrapolation methods only offer the values of high order accuracy at coarse grid points and, in general, by using interpo­ lation formulae of higher degree, one can obtain higher order accuracy at finer grid points. C M . Chen and Q. proposed a fine grid extrapola­ tion formula, which can give directly the higher order accuracy at finer grid points, and can also be applied to SEM. The algorithm will be introduced briefly as follows: Let Jh be a triangulation of Q C M2, K be a triangular element, and z be a vertex of K. Suppose that for the error of finite element approximate solution there is an asymptotic expansion (ti* - u){z) = h2$(z) + 0(/i 4 |log/i|).

(1)

Let Z12 be the midpoint of the line joined two vertices, z\ and 2 2 , of K as well as a vertex of the refined partition. The extrapolation formula, with z\2 as the finer grid point, is «*(*«) = « h / 2 (zi 2 ) +-(uhl\Zl) - uh(Zl)) + -{uhl\z2) - uh(z2)). 6 6 It can be proved that for any e > 0, there exists an error estimate ""0*12)- u{z12) = 0 ( / i 4 - ) .

(2)

(3)

In fact, by (1), 9(zt) = 3 ^ ( « " - «*/»)(*) + 0(h*\\ogh\),

i = 1,2.

(4)

Applying the error estimate of the linear interpolation of $,

*(*„) = !*(*!) + |*(z 2 ) + 0(fc2-*),

(5)

Appendix

301

hence, we have * ( * » ) = ^ j ( u f c - n fc/2 )(2n) + ^ - Z( n * - ufc/a)(Z2) + 0(h*-). on

(6)

3AI

Since uh'\zl2)

= ti(z ia ) + £ * ( * « ) + 0(/» 4 | log&l),

(7)

by using (6), the formula (2) and estimate (3) are obtained. For a rectangular partition, the bilinear finite element approximate so­ lution also has an asymptotic expansion of the form (1). In Fig. 1, Pi (i — 1,2,3,4) are coarse grid points, M,- (i = 0, • • •, 4) are refined grid points, and the extrapolation formula of refined grid points is f « h (Mi) = u h / 2 (Mi) + \\(u*l\Px) «"(M 0 ) = uhl\M0)

- u\Px))

+ ^ £

+ ( U "/2(p 2 ) _

(uh'2(pi)

uh(P2))},

(8)

~ «*(*))•

1= 1

The proof follows immediately by replacing $ in expansion (1) with the bilinear interpolation $ 7 of $.

Fig. 1

Fig. 2

We shall derive the results at refined grid points of SEM, by taking twodimensional case as an example. Suppose that with two independent mesh widths hi and h2, the asymptotic expansion of the bilinear finite element approximation is uhlM{z)-u{z)

= $! {z)h\ + $2(2)ft2 + o (AJ) , z e ahl M,

(9)

Appendix

302

where 2z is a grid point. Refining the mesh width h\, hi, « * ! / * . * » ( * ) - tUi ( zZ)) «*!/*.*»(*)

A? = # ! ( * ) — + *2(z)hl 4

+ 0(hi),

2: 6G f i&h W1/2,h 2 ,2,

(10)

hence, ** li(W «)

= = Similarly, Similarly,

A4 ( ^( «ik^l , h(az( )z ) >--«»»/».»»(z))+0(^), uh*'2>h>(z))+o(ht), ^ ze fel hni .fcl|fca fcj. .

( i i(ii) )

3/if *2(*) *a(*) = — j ( « h l , f c a W - uhlM,2(zj)+0(ht), 3/»l 3/15

z G n fcl|fca .

(12)

Let 3>f (z) (i = 1, 2) be bilinear interpolation functions of $,(2). $i(z). In the SEM, only Mi (i = 1, • • •, 4) are refined grid points obtained after refinement (see Fig. 2). Hence, 4 i ( f 2t )))) *{(Jfl) *{(M 1 ) ==: --( (*$i i(Pi) ( P 1 ) + *i(P 2V 2 1 2 u W 2 p w h l , hh l 2

== _lJ_V( - ^ E ( u"i/2,i( ^(Pi) -- « '(* ^" ( *)))) ,, ha

<> " 1

*£(Mi) =

t=i

™A£(« hi,fca/a (*)-« hi,ka (^))3 Ai2

t = 1

Thus, h2 a U(h(M!) ^ M i ) == ufc./a.i t i f c l ' 2 » f c(Ma) (Mi)

1 2 hl 1/2,) k a + " £ (" f c l / 2 , *»(p*)(^) - - utihl'Mfca(pi)) (P<))

+-E(«" .=i 6

«=i

2 2 kl,ka/2 a +- E D (« (uhlM,2 (Pi)(#) - « tihlfcl'* ' fca (P0). + (pi))=1 3 » i=i

(13)

3

Note that for M 2 G tthl)h2/2, there is a similar extrapolation formula. In other words, for any point in the refined grid ^ ^ ^ U Q ^ ^ , the higher order accuracy can be obtained directly.

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Index

collectively compact convergent 102, 143 collocation method 105, 241 combination technique 260 combination principle 233, 260 compact operator 99, 125, 145 comparison theorem 176 complexity 44, 103, 259, 273 composite grid 291 conjugate Banach space 149 consistency condition 170 continuous kernel 99 coarse grid solution 260 coherence condition 190

A algebraic precision 24, 31, 69, 124 Ascoli-Arzela Theorem 100 asymptotic expansion 2

B Banach-Steinhaus Theorem 104, 145 basis function 3, 64, 145, 163, 203, 261 Bernoulli number 21 Bernoulli polynomial 20, 74 bilinear finite element space 202 bordered equation 258 boundary element method 33 boundary integral equation 118 BPX multilevel method 273 Bramble-Hilbert Lemma 157 Bulirsch-Stoer extrapolation algorithm 12

D degree of parallelism 44, 118, 198, 287 direct sum decomposition 262 Dirichlet problem 170, 204, 247 256, 281 discontinuous kernel 99, 253 discrete maximum principle 172 discrete Green's function 210 double exponential formula 80, 253

c central difference operator 179

313

Index

314

double layer potential theory 247 Duffy transform 83

E energy extrapolation 290 energy norm 261, 268, 287 Euler-Maclaurin asymptotic ex­ pansion 2, 136 explicit extrapolation 290 extrapolation coefficients 9, 51, 234

F FAC 292 Fibonacci number 14, 50, 75 finite element extrapolation 43 Fredholm integral equation of the second kind 99 fractional power expansion 61 full grid 260, 268 full grid finite element space 268

H hierarchical basis 261 homogeneous decomposition 37 homogeneous function 33, 83 homogeneous splitting extrapo­ lation Q6 Holder continuous 191

I implicit extrapolation 261, 290 improper integral 61, 79 interpolation polynomial 2, 47, 63, 179 interpolation projection 144 inverse estimate 147, 205 irregular grid 173, 179 isometric partition 169 iterated kernel 101 iterative solution 146

K

G Galerkin method 105, 144 Gauss-Seidel algorithm 273 GC condition 64, Q6 generalized Euler-Maclaurin ex­ pansion 25 global expansion 290 gradual partition 169 Green's function 162, 210, 244

kernel of the integral operator 99, 256

L Lagrange interpolation formula 3, 47, 173 locally refined grid 290 logarithmic singular 31, 92, 143 Lax-Milgram theorem 202

Index

315

M

0

mean value theorem 5, 29, 163, 246 mesh parameter 6, 44, 117 169, 200, 223, 280, 291 modified quadrature method 141 Monte-Carlo method 69 multidimensional improper inte­ grals 61, 79 multidimensional weaklysingular integral equation 99 multigrid method 290 multilevel algorithm 260 multilevel splitting 261 multi-parameter asymptotic ex­ pansion 44 multiprocessor computer 44,135, 160, 198, 280 multivariate asymptotic expan­ sion 44, 62, 71, 109, 140, 159 multivariate Newton's interpola­ tion formula 48, 66

order of error 14,112,134,249, 287 orthogonal projection 144, 202

P piecewise continuously differen­ t i a t e 166 piecewise linear continuous func­ tion 144 piecewise linear interpolation 161, 243, 269 piecewise strongly-regular parti­ tion 227 piecewise uniform cuboid parti­ tion 225 piecewise uniform rectangular 223 polynomial extrapolation 6 point wise convergence 102 product integration method 143 projection method 105

N Neville's interpolation formula 4 Nitsche technique 212, 225 nodal basis 262, 271 nonsmooth kernel 132 nonstationary problem 160 nonuniform mesh 290 number-theoretical method 69 Nystrom's approximation 235

Q quasilinear ordinary differential 160

R rectangular partition 203 rectangular rule 26, 70, 109, 259

Index

316

recurrence algorithm 4, 44, 63, 92 refinement 13 reflexive Banach space 148 Richardson's extrapolation 2 Riemann Zeta function 30, 30 Ritz-Galerkin method 201 Ritz projection 202 Romberg's extrapolation 5

s seminorm 205, 266, 269 simplex 90 Simpson's rule 28, 124, 162, 236, 245 singular boundary value problem 169 single layer potential theory 256 sparse grid 260 sparse grid space 267 sparse grid stiffness matrix 272 stability factor 9, 52 stationary problem 160 stiffness matrix 261 Sturm-Liouville type eigenvalue problem 163 subdivision 201 successive elimination 61 super convergence 43, 200 support 261 symmetric property 58 symmetric quadrature 31

T Toeplitz theorem 13 trapezoidal rule 20 triangular polynomial 24 truncation error extrapolation 290 r-extrapolation 290

u uniformly convergent 101 uniformly bounded 104, 251 uniformly positive definite 201

v Vandemonde determinant 3 F-ellipticity 201

w weak form 201 weak solution 201 weakly-singular kernel 61, 101, 143, 153, 235 weakly-singular integral equation 143, 153, 239