varga - functional analysis and approximations in numerical analysis

8/17/2019 Varga - Functional Analysis and Approximations in Numerical Analysis

1/87


2/87

CBMS NSF

REGIONAL CONFERENCE SERIES

IN APPLIED

MATHEMATICS

A series of lectures on topics of current

research

interest in

applied

mathematics under the direction of

the

Conference Board of the Mathematical Sciences, supported by the National Science Foundation and

published by SIAM.

G A R R E T T B IRKHOFF The Numerical Solution of Elliptic Equations

D. V. L I N D L E Y Bayesian Statistics A Review

R . S .

V A R G A

Functional Analysis an d Approximation Theory in Numerical Analysis

R R .

B A H A D U R Some Limit Th eorems

in

Statistics

P A T R I C K BILUNOSLEY Weak Convergence

of

Measures: Applications in Probability

]. L . L I O N S Some Aspects of th e Optimal Control of Distributed Parameter Systems

R O G E R PENROSE Techniques

of

Differential Topology in Relativity

H E R M A N

C H B R N O F F

Seque ntial Analysis

and

Optimal Design

3.

DURB IN Distribution

Theory fo r

Tests Based

on the

Sample Distribution Function

SO L I .

R U B I N O W Mathematical Problems

in the

Biological Sciences

P . D. LAX Hyperbolic Systems

of

Conservation Laws and the M athematical Theory of Shock

Waves

I. J. S C H O E N B E R G Card inal Spline Interpolation

I V A N SINGER The Theory

of Best Approximation

and

Functional Analysis

W E R N E R C. R H E I N B O L D T Methods

of

Solving Systems of Nonlinear Equations

H A N S

F .

W E IN B E R G E R Variational Methods for Eigenvalue Approximation

R .

T Y R R E L L R O C K A F E L L A R

Conjugate Duality

and

Optimization

SIR

J A M E S

LIGHTHILL Mathematical

Biqfluiddynamics

G E R A R D

SALTON

Theory of Indexing

C A T H L E E N

S.

M O R A W E T Z Notes on Time Decay and Scattering for Some Hyperbolic Problems

F .

H O P P E N S T E A D T

Mathematical Theories of Populations: Dem ographics Gene tics and Epidemics

R I C H A R D

ASKE Y

Orthogonal Polynomials and Special Functions

L . E . P A Y N E Improperly Posed Problems in Partial Differential Equations

S. R OSEN Lectures

on the

Measurement

an d

Evaluation

of the Performance of

Computing Systems

H E R B E R T

B .

K E L L E R Numerical Solution of Two Point Boundary Value Problems

J P .

LASALLE

T he

Stab ility of Dynamical Systems

- Z.

ARTSTEIN Appendix

A:

Limiting Equations

and Stability ofNonau tonomous Ordinary Differential Equations

D.

G O T T L I E B AND

S. A

ORSZAG Numerical Analysis of Spectral Methods: Theory and Applications

P E T E R J H U B E R Robust Statistical Procedures

H E R B E R T S O L O M O N Geometric Probability

F R E D S. R O B E R T S Graph Theory

and Its

Applications

to

Problems

of

Society

J U R I S H A R T M A N I S Feasible Computations and

rovable

Complexity Properties

Z O H A R M A N N A Lectures on the Logic of Computer Programming

E L L I S L . J O H N S O N

Integer Programming: Facets

Subadditivity and

Duality

for

Group

and

Semi-

Group

Problems

S H M U E L

W I N O G R A D Arithmetic Complexity of Computations

J. F. C. K I N G M A N Mathematics of Genetic Diversity

M O R T O N E .

GURTIN

Topics in Finite Elasticity

T H O M A S G . K U R T Z Approximation of Population Processes

continued

on inside back cover)


3/87

S.

Varga

Kent State

University

Kent Ohio

Functional

Analysis

and

Approximation Theory

in Numerical Analysis

SOCIETY FOR INDUSTRIAL AND APPLIED MATHEMATICS

PHILADELPHIA


4/87

1 0 9 8 7 6 5

All rights reserved. Printed in the U nited States of A m erica. No part of this book m ay be

reproduced stored or transm itted in any m anner withou t the written perm ission of the

publisher. For inform ation write to the Society for Industrial and A pplied Ma them atics

3600 University City Science Center Philadelphia

PA

19104-2688.

ISBN 0 89871 003 0

i m

is a registered

Copyright 97 by the society for industrial and applied

opyri ht


5/87

This

vo lume is

dedicated

to

G R R E T T I R K H O F F

on the occasion of his sixtieth birthday


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his page intentionally left blank


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FUNCTIONAL ANALYSIS AND APPROXIMATION THEORY

IN

NUME RIC L N LYSIS

Contents

Preface

ix

Chapter

L SPLINES 1

Chapter 2

G E N E R A L I Z A T I O N S

OF

L SPLINES

11

Chapter 3

I N T E R P O L A T I O N A N D A P P R O X I M A T I O N R E S U L T S F O R P IE C E

W I S E P O L Y N O M I A L S

IN H I G H E R D I M E N S I O N S 1 7

Chapter 4

T H E

R A Y L E I G H R I T Z G A L E R K I N M E T H O D F O R N O N L I N E A R

B O U N D A R Y V A L U E P R O B L E M S 25

Chapter 5

F O U R I E R A N A L Y S IS 35

Chapter

6

LEAST

S Q U A R E S M E T H O D S 43

Chapter 7

E I G E N V A L U E

P R O B L E M S 51

Chapter

8

P A R A B O L I C P R O B L E M S 59

Chapter

9

C HEB YSHEV

S E M I D I S C R E T E A P P R O X I M A T I O N S F O R L I N E A R

P A R A B O L I C P R O B L E M S

69

v ii


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Preface

The purp ose of these lectu re notes is to survey in par t the enormo us ly expa nding

l i te ra ture

on the nu m er ica l approximat ion o f solut ions o f e l l ip t ic bou nda ry v alue

problems by means o f var ia t ional and finite e leme nt me thods. Surv ey ing this area

will,

as we shall see, require almost constant application of results and techniques

from funct ional analysis

and

approximat ion theory

to the field of

n u m e r i c a l

analysis,

an d it is our

hope that

th e

m aterial presented here w ill serve

to

s t imula te

further

act ivi ty which will s t rengthen the t ies already connecting these fields.

Although our primary interest will concern the numerical approximat ion of

elliptic bou nda ry v alue problems, the methods to be described lend them selves as

well ra ther na tura l ly to discussions concerning eigenv alue problems and init ial

va lue

problems, such

as the

hea t equat ion .

On the

negative side,

it is

u n f o r t u n a te

t ha t

a lmos t noth ing will

be

said here about

scientific

computing i.e.,

the

real

problems of implementat ion of such mathematical theories to working programs

on h igh-speed computers , and the numerical experience which has al ready been

gained

on

such problems Fortun ately, scientific com puting

is one of the key

points

of the mo nog raph by Professor Ga rret t

B i rkhof f ,

1

and w e are g rateful to be able to

refer the

reader

to

th is w ork .

The in tent of these lecture notes is to make each port ion of the notes roughly

independent of the remaining material . This is why the references used in each

of

th e n ine chapters a re compiled separately at the end of each chapter .

It is a sincere pleasure to acknowledge th e suppor t of the National Science

Foundat ion under a g rant to the Conference Board of the Mathematical Sciences ,

fo r th e Regional Conference held a t Boston University July 20–24, 1970, and to

acknowledge Professor R obin Esch s superb ha ndl ing of even the most mi nu te

details

of this Conference in Boston. Without h is unt ir ing efforts the Conference

would

not have been a success.

It

is also a pleasure to acknowledge the fact that these notes benefi ted great ly

from suggest ions and comments by Garret t Birkhoff , James Dailey, George Fix,

John Pierce and B lair Sw artz. Finally, we tha nk M rs. Julia Froble for her careful

typing

of the

manuscr ip t .

R I C H A R D S. V A R G A

1

Garret t B i rkhoff , Numerical Solution

of

Elliptic Partial

Differential

Equations SIAM Publ ica tions ,

1 9 7 1 78 pp.

ix


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11/87

CHAPTER 1

L-Splines

1 1 Basic theory Splines,

as is

well known, were effectively in t roduced

to the

mathematical world

by I. J. Scho enberg

[1.1]

in

1946,

an d

splines hav e since becom e

the focus of much

mathematical

ac t ivi ty . In

particular approximation theorists

and num erical analy sts hav e of late literally seized upon splines because of their

m any beau tiful properties and because of their w ide range of application to the

numerical approximation

of

solutions

of

differential equations.

It is

these beautiful

properties

and wide

range

of

applications

of

splines which

we

propose

to

cover

in

part in these lectures.

The m athem atical developm ent of the theo ry of splines since Schoenberg s

fundamental paper in 1946 has been both extremely diverse and extremely rapid.

Several recent books on splines (cf. Ah lberg, Nilson and W alsh [1.2], T. N. E.

Greville [1.3], I. J. Schoenberg [1.4]) indeed attest to this rapid development.

To

describe

the

development

of

spline theory,

w e

begin here with

a

s tudy

of

L-splines. This

is a

somew hat m iddle ground

in the

dev elopment ,

in

tha t

the

theory

of L-splines

is

cer ta inly

not

classical,

nor is it the

m ost general

to

da te. How ever ,

as

w e

shall see, most of what is obtained here for L-splines carries over to more

general splines recent ly investigated

b y

several authors.

To begin, for

—

cc < a < b a ,

b for w h ich

In par t icular , ^ a, b is the collection of all

uni form

par t i t ions of [a , b], and its

elements are denoted by A

u

.

For additonal nota t ion, if C

p

[a, b] is the set of all real-valued func t ions wh i ch

have c on t inuous deriva t ives of order at least p in [a , b], w e then recall that the

Sobolev space W

s

q

[a , b ], wh e r e 1 r g q oo and s is any nonnegat ive in teger , is

This research was supported in part by AEC Grant

(11-1)-2075.


12/87

C H A P T E R 1

def ined as the

completion

of the set of all

real-valued funct ions

eC°°[a, b]

w i th

respect to the norm:

Equivalent ly ,

W

s

q

[a ,

b] is the

collection

of all

real-valued funct ions

/

def ined

on

[a,b] w i t h fo r s > Q)D

s

~

l

f a b solu te ly co n t in u o u s on [a,b] an d D

s

feL

q

[a,b].

Clearly,

for s > 0,

W

s

q

[a , b] cC

s

~

l

[a, b ].

To describe L-splines, consider the linear

d i f ferent ia l

operator L of order m:

w h e r e c

}

e C

j

[a , b], 0

̂

j

^

m,

w i th c

m

x) ^ <

> 0 for all x e [a ,

b].

A n

imp o r ta n t

special case

is the

choice

L =

D

m

. Ne xt ,

let z be any

fixed) posit ive integer with

1 ̂

z

̂

m .

Then, Sp L,

A ,

z) ,

th e

L-spline space ,

is the

collection

of all

real-valued

funct ions

w def ined on [a,

6]

such that cf. Ah lberg, Nilson and Wa lsh [1.2, Chap. 6],

Greville [1.5], and Schultz and Varga [1.6])

where L* is the formal a d jo in t of

L, i.e.,

L*v

In other

w ords, each w e Sp L, A , z) is locally a solution

of

L*Lw = 0, pieced together at the

in ter ior

k n o t s

x, in

such

a

way, d e p e n d i n g

on z,

that

weC

2m

~

z

~

l

[a,b].

Thus,

Sp L,

A , z)

c: C

2m

~

z

~

l

[a, b] ,

but

because

of the

assumed smoothness

of the

coefficients

C j in

1.1.4),

w e can

sharpen this inclusion

to

Sp L,

A, z) cr W^~

z

[a,

b].

In

a d d i t io n , it can be

verified

that Sp L, A, z) is a linear space of d ime n sio n

2m + z N -

I).

In th e

important special case

L =

D

m

,

th e

elements

of

Sp D™,

A , z) are,

from

1.1.5), polynomials of degree 2m — 1 on each subinte rval of A, and as such are

called polynomial splines. More specially, when L = D

m

and z = m, elements of

the

associated

L-spline

space are called Hermite splines, and the collection of

such Hermite splines

is

denoted

by //

m )

A) .

From

1.1.5), tf

(m)

(A) c W^fab]

c

C

m

~

l

[a,

b] . Similar ly, whe n

L =

D

m

and z = 1, the

e lements

of the

associated

L-spline space are called simply

splines,

and the collection of such splines is den oted

by Sp

M )

A). From 1.1.5), Sp

B I)

A)

W

2

£~

l

[a,

b] c C

2m

2

[a, ] .

We now

discuss

th e

possibility

o f

interpolation

of

given func t ions

by

elements

in

Sp(L, A , z ) .

Giv e n

an y geC

m

~

l

[a,b], it can be

s h o w n

by

e lementary methods

cf. [1.6])

that there exists

a

unique s €

Sp L, A , z)

w hich in terpolates

g in the

sense

that


13/87


14/87

H PTER

It is

also in terestin g

to

remark that

th e

inequa l i ty

of

1.1.10)

can be

shown

to b e

quasi-optimal cf. BabuS ka, Prager , Vitasek [1.9, p. 232]) in the sense of

n-widths

of

Kolm ogorov cf. Lo rentz [1.10, Chap. 9]), and such related ideas for spl ines have

been studied by Aubin

[1.11]

and G olomb

[1.12].

Error bo und s analogous to Theorem 1.2 can be obtain ed for L-spline inter-

polation of smoother funct ions g. In part icular , if ge W l

m

[a,b] and s is its

Sp L,

A , z)- interpolant in the sense of

1 .1 .6 ) ,

an in tegrat ion by parts again shows

that the

second

integral relation cf. [1.2, p . 205])

is

va lid. This

is

sim ilarly used

in

p rov ing

cf.

[1.13]) the

er ror bound s

of the

following

theorem.

T O R M 1.3. Given ge

W\

m

[a b] and

given Ae2P

a

a,b),

let s be the

unique

element in Sp L, A, z) which interpolates g in the sense o/ 1.1.6).

Then,

for 2

̂

q

̂

oo,

For polynomial splines

can be replaced by in

1.1.12) .

For 0

^

j ̂ m — 1 , it is wor th no t ing that th e error bound s o f

1.1.12)

a re valid

for any A e £? a, b}. The exponent of

n

in 1.1.12) is again best possible for the space

W\

m

[a

b]

for general L-spl ine interpolat ion. H ow ever, in t e rms of error bound s for

g in

W

2

™[a, b ]

o r

W^[a b]

fo r p olynom ial spl ine interpolat ion, the exponent of

n

in

1.1.10) and 1.1.12) can

in

special cases

be

increased

by

when q

+0 0

cf .

Swartz

and Varga

[1.13]).

Next , we

also

mention that the results of Theorems

1.1-1.3 are k no w n to be valid for more general forms of bou nd ary interpolat ion

than considered in 1.1.6). In addition, it is also possible to vary the parameter z

from

knot to knot with n o change in the interpolat ion err or bounds. Such

refine-

ments

can be

found

for

example

in

[1.6]

and

[1.13].

From th e

in terpolat ion error bounds

of

Theorems

1.2 and

1.3,

one can

deduce,

via

the use of interpolation space theory to be described in § 1.2), analog ous inter-

polat ion error bounds

for

functions

g in

spaces in termediate

to

W™[a,b]

and

W|

m

[a

b].

But the desire is to obtain error b ound s for functions g even

less

smooth

than C

m

~

l

[a

b], and

this poses

a

problem. Clearly,

th e

interpolation

of g, as

defined in 1.1.6) needs the existence of derivat ives of g of o rder

m —

1 in

[a, b],

and

thus,

a

modificat ion

of the

definition

of

interpolation

in

1.1.6)

is

necessary.

To do

this,

we

make

use of the

fam il iar not ion

ofLagrange

polynomial interpolation,

as

described

in

Davis [1.14,

Chap. 2 ].

If A 6̂ 0,

b)

has at least 2m knots, let

J ^ m - i . og

denote th e Lagrange poly-

nomial interpolation of degree 2m — 1) of g in the knots x

0

x

:

• • • , x

2m

-1 i.e.,


15/87

L-SPLINES

Then, a l though

g

need

not

possess derivatives th ro ug h order

m — 1 at x — a ,

< ^ 2 m - i,o£ does, and we can define interpolat ion by an s e Sp L, A , z) at x = a n ow

by means o f

Similar definitions of

interpolation

can be

used

at

o ther knots

of A. W e now

state

a

result

of

Swartz

and

V arga [1.13] see also Sc hu ltz [1.15]

for a

related

use of

Lagrange polynomial

interpolat ion) .

T H E O R E M

1.4. Given g e

C

k

[a ,

b ] with 0 rg k < 2m and

given

A e

a

(a , b)

with

at

least 2m knots, le t s be the unique element in Sp L, A , z) which interpolates g in the

sense that

where

^

2m

-\ iS

*

s

tne

Lagrange

polynom ial interpolation ofg

in the 2m consecutive

knots X j . , x

j i l

,

••• ,

x

ji

2m

-i where

x,-e

[ X j . ,

x

ji 2m

- i ] - Then,

for 2

q oo,

For

polynom ial splines

(L = D

m

), the term

involving

H g H ^ ^ b ] in

1.1 .16)

va n be

deleted.

In

1.1.16),

w e h a v e used the

notation

to

denote

the

usua l modulus of continuity

of any

bounded function

/ d e f i n e d

on

[a,b].

For the

extension

of the

result

of

Theorem

1.4 to

Sobolev spaces,

w e

have

th e

following coro llary cf. [1.13]).

C O R O L L A R Y

1.5. With

th e

hypotheses

of

Theorem 1.4,

if

geW

i

[a, b] with

1

r oo and 0

k < 2m, then for

m a x r ,

2) f q oo,

For polynomial splines, \ \ g \ \ w

k

r l[ a , b ] can oe replaced

by \\D

k+ 1

g\\

Lr[atb]

in

1.1.18).

W e note tha t w hen k = m —

1

and r = 2, the first inequ ali ty o f 1.1.18) reduces

to the inequ ality of 1.1.10). Similarly, when k = 2m — 1 and r = 2, the first

inequal i ty of 1.1.18)

reduces

to the

inequali ty

o f

1.1.12). Th us Theorem

1.4 and

Corollary 1 .5

generalize

th e

results

of

Theorems

1.2 and

1.3, even though

th e

process

of

interpolat ion

is d i f f e r e n t in

both cases.


16/87

6

C H A P T E R 1

To

summarize, this section introduces

L-splines and

gives representative error

bounds

fo r

L-spline

interpolation. For the extension of these

error

bounds, we

shall

find it

useful

to

describe

in the

next section

th e

idea

of

interpolating spaces.

1 2 Interpolation spaces

and

applications

The

results

of

Theorem

1.4 and

Corollary

1.5 can be

extended

to

more general spaces, using

th e

idea ofinterpolation

spaces (cf. Bu tzer an d Berens [1.16,

Chap.

3]), which we

briefly

describe.

Let X

0

and

X^

be two Banach spaces with norms || • ||

0

and || • ||

1}

respectively,

which are

contained

in a

linear Hausdorff space # , such that

the

identi ty mapping

of

X

i

in

3 C

is

continuous

for

i =

0 and

i

= 1. If

X

0

+ X

1

= { e

# :/ =

0

+

j\

where /) e

X

t

, i =

0,1}, then X

0

r\ X

1

and X

0

+

X

l

are iBanach spaces under the

norms:

It follows that

where

inclusion

is understood in

this section

to

mean that

th e

identity

mapping is

continuous.

Any Banach space X

c3 C

is said to be an intermediate

space

of X

0

and

Xi

if it

satisfies

th e

inclusion

We now

give

Peetre's

real-variable method (cf. [1.16]

and

Peetre [1.17])

for

constructing

intermediate spaces of

X

0

and

Xi.

For each positive

t

and each

f E X

0

+ Xi), define

Then, for any 9 with 0


17/87

L-SPLINES

i . e . , T is a bound ed linear transform ation from X

t

to Y

t

with norm at most

M,,

i = 0,1. A gain, the following is know n (cf. [1.16, p.

180]

and

[1.17]).

T H E O R E M 1.7.

Let T be any

linear transformation from (X

0

+ XJ to (Y

0

+

Y

which

satisfies

(1.2.6).

Then,

for any

Q


18/87

C H A P T E R 1

W e now explicitly show how the theory of interpolation spaces can be used to

extend

the

L-spline error bounds

of

Theorems

1.2 and

1.3.

For j a

nonnegative

integer

with

0

̂

j

^

m — 1 and 2

̂

q

̂

oo,

define

the

linear transformation

T

W?[a,

b] -»

W{\a

t

b] by

means

of

where

s is the

unique

Sp L, A ,

z)-interpolant

of g in the

sense

of 1.1.6).

With this

definition

of

T

and the

definition

of the

Sobolev norm

in 1.1.3), th e

error bounds

1.1.10)

and

1.1.12) respectively

can be

expressed

as

Thus, if Y

0

= Y

l

= W

J

q

[a,b], and X

0

=

W^[a,b]

and

X

l

= W

2

2

m

[a,b], then

Tis

from 1.2.15) a

bounded linear transformation from

X

(

to

f

with norm

at

most M

t

,

i = 0,1, where

Hence, as X̂ X̂ =

(W?[a,b], Wl

m

[a,

b])

9tr

= B °

2

-

r

[a, b], a = 1 +

0)m,

from

1 .2.13), and as

(Y

0

,

Y^)

9tq

=

W

J

q

[a, b], then from Theorem 1.7, Tis a bounded linear

t ransformation from

B ?

r

[a,

b ] to W

j

[a, b] with norm at most

i.e.,

fo r

any m < a < 2m and any 1

̂

r

̂

oo.

The error bounds o f 1.2.17) for L-spline interpolation, while extending th e

results

of

Theorems

1.2 and

1.3, were obtained

by

interpolating

th e

right-hand

sides

of

1.1.10)

and

1.1.12).

On the

other hand,

the

error bounds

of

1.1.10)

and

1.1.12)

also hold

for different

values

of), and

this permits analogous interpolation

of the

left-hand sides

of

1.1.10)

and 1.1.12). The combination of these

results

can

be

formulated cf. Hedstrom

and

Varga [1.20])

as the

following theorem.

THEOREM 1.9. Let

feB2

2

r

[a,b],

where m


19/87

L- S P LI N ES

fo r

0

j

m - 1

and any 2 - ^ p - ^

< x >

In

particular,

if f

e W ^ IX ^]

WI

̂

=

ff

2m, then for 0 7

m,

The extension in Theorem 1.9 of

Theorems

1.2 and 1.3 can be further generalized

i f

we apply the theory of interpolation

spaces

to Corollary 1.5, w here Lag rang e

interpolation polynomials

are

used

to

define interpolation cf.

1.1.15)). From

1.1.18)

w e obtain the following result .

T H E O R E M

1.11. Given any feB°

q

[a, b], 1 <

o


20/87

10 CHAPTER 1

[1.16] P. L. B U T Z E R AN D H. B E R E N S Semi-Groups o f Operators an d Approximations, Springer-Verlag,

N ew

York, 1967.

[1.17] J. P E E T R E

Introduction

to Interpolation,

Lecture notes, D epartment

of

Mathematics, Lund, 1966.

In

Swedish.)

[1.18] P.

G R I S V A R D

Commutativite

de

deux foncteurs d'interpolation et applications, J. Math. Pures

Appl.,

45

1966),

pp . 143 290.

[1.19] J.

P E E T R E Espaces a interpolation, generalisations, applications, Rend. Sem . Mat. Fiz. Milano ,

34 1964), pp.133 164.

[ 1.20] G E R A L D W . H E D S T R O M AN D R I C H A R D S. V A R G A

Application

ofBesov

spaces

to spline

approxima-

tion, J . Ap prox. Theory, to

appear.


21/87

CHAPTER

Generalizations

of L-Splines

2 £ -splines.

There

are a

variety

of

general izat ions

of

L-splines

and it is of

interest to see how

they extend

the

L-spl ine theory.

W e

begin this sect ion with

results

of

Jerome

and Schumaker

[2.1].

Let A =

{ A , - } *

= 1

be any set of l inearly independent , bounded l inear funct ionals

on W ™ [a , b] , and let r = r

l 5

r

2

, • • • , r

k

)

T

denote

an y

vector

o f

real Eucl idean

fc-space,

R

k

.

If L is the l inear

differential

opera tor of

1.1.4),

then cf.

[2.1])

se

W™[a,b] is an

Lg-spline

interpolating r with respect to A, i .e. ,

A , - s )

=

r , ,

1

i

:g

k , provided

it

solves

th e

fol lowing

m inimiza t ion

p r o b l e m :

The

relat ionship with L-splines

in

1.1.9)

of

Theorem

1.1 is

clear; while

th e

linear

differential opera tor L remains unchanged, the manner of interpolat ion, now by

means of A, is generalized. For notation, the space of all Lg-splines such tha t

s

satisfies 2.1.1)

fo r

some

r e

R

k

is

denoted

by Sp L, A ) .

Based on resul ts of Golomb [2.2], Jerome and Schumaker [2.1] have proved,

in

the spirit of Anselone and Laurent [2.3] , the fo l lowing theorem.

T H E O R E M 2.1.

Given any

reR

k

there exists an

seW^[a b]

satisfying

2.1.1).

A function s e t/

A

r) satisfies 2 .1.1) if and

only

if

Moreover,

2.1.1)

possesses

a

unique solution

if and

only

if 91 n

t/

A

0)

= {0},

where

91 is the

null space

of L.

Finally,

Sp L, A) is a

l inear space

of

dimension

k + dim{9t n

C 7

A

0 ) } in

W ^[ a , b ] .

W e n o w assume that each A

;

e A is of the form

A

;

/)

=/^ /(x,), w h e r e

0 ji m

—

1 and x,-e [a ,b] .

Such

a A = { A , - } *

= 1

generates

a Hermite-Birkhoff

problem. For such Hermite-Birkhoff problems, th e so lu t ion s of the min imiza t ion

problem

2.1.1)

satisfies, as in

1.1.5),

Next , one can assign a nonnegat ive in teger t A) , which counts the n u m b e r of

consecutive deriv ative point fu nc tion als in A for details, see Jerome and V arga

[2.4]).

W ith this, the follo w ing can readily be shown cf. [2.4]) .


22/87

C H A P T E R 2

T H E O R E M

2.2. // A =

generates

a Hermite-Birkhoff

problem with

partition

e^

assume that m , assume that

, and

assume that

the

second integral relation holds

fo r A , i.e-

cf. 1.1.11)) ,

is valid

for any g G

Wl

m

[a,

b] and s is the

unique Lg-spline which interpolates

g in the

sense that

Then the error bounds of 1.1.10) of Theorem 1.2, as well as those of 1.1.12) of

Theorem

1 .3 are

valid.

M ore broa dly in terpre ted, the result of T heorem 2.2 can be c learly extended to

Besov spaces

exactly

as in Theorem 1.9 and Corollary 1.10, with identical error

bounds, thereby generalizing the results of [2.4]. T h us , Lg-splines offer generaliza-

tions in the area of interp ola tion cf. 2.1.5)), but do not generalize th e type of

differential

operator L considered.

2.2

y splines.

T he ne xt generaliza tion considered here is due to S chultz [2.5]

and Lucas [2.6]. If

where

p,-e W{[a,

b]

n L

x

[a, b ],

0 ̂j ̂m and

p

m

(x)

^

6

> 0 in

[a,b], assume

tha t E is W™[a, b]-elliptic, i.e., there exists a

constant

y > 0 such that

where

W™[a,

b]

denotes

the

subspace

of funct ions

u(x)

of

W™[a,

b]

of

§ 1 .1

sa t isfying

the homogeneous bou nda ry conditions

D

k

u(a)

= D

k

u(b) = 0 , 0 ^ / c ^ m — 1 .

A s

in § 1.1, let A

e (a,

b ), and let z again be a positive integer s atisfy ing 1

^

z

g

m .

T h e n , S(E,

A , z) , th e

y-spline space,

is the

collection

of

real-valued functions

w

defined on [a, b]

such that, relative

to A

cf .

1.1.5)),

Ew(x)

= 0 almost

everywhere

in

each

sub in te rva l

x

i 5

x

i

j ) ,

G iven any g eC

m

^[a^b], it is easily seen cf. [2.5]) that there exists a unique

s E

S(E.

A, z) which interpolates g in the sense of 1 .1.6), i.e.,


23/87


O F

L - S P L I N E S

13

Because the in terpolat ion of 2.2.4) a t the bou ndar ies en sures the second integral

relation cf. 1 .1 .11)) ,

th e

following contains

th e

upper bounds

of Theorems

2.4-

2 . 7 o f S c h u l t z [ 2 . 5 ] .

THEOREM

2.3.

Given any g e

C

m

~

*[a,

b

and any A €

3?(a,

b),

let s be the

unique

element in S(E, A , z) which interpolates g in the sense of 2 .2 .4) . Then, the error

bounds

of

1 .1 .10)

of

Theorem

1 .2 are

valid. Similarly,

if g€\Vl

m

[a,b] and

A 6 ^

a

(a, b), then th e bounds of 1 .1 .12 ) of Theorem 1 .3 are valid.

As in §2 .1 , we can more broadly interpret th e result of Theorem

2.3

since it s

extension to Besov spaces, exactly as in Theorem 1 .9 and Corollary 1.10, is now

immediate, thereby generalizing the results of [2.5] and [2.6].

Noting that the generalization of

Lg-splines

works through more general

collections o f boun ded l inear functionals A = { A J f

= 1

, while th e generalization

of

y-splines

w orks throug h more genera l

differential

operators, one can combine

these two ideas s imul taneou sly, and obtain the error b oun ds of 1.1.10) of T heorem

1 .2 and

1 .1 .12)

of Theorem 1.3. This has in fact been considered by Lucas [2.7].

The extensio n of these results to Besov spaces is also immediate.

2 3 Singular splines

One of the

more interesting developments with respect

to

one-dimensional spline theory

is

due ,

in its

generalized form,

to

Jerome

and

Pierce

[2.8]. We first

give

a brief

discussion

of the

background

fo r

this problem. Jamet

[2.9], using

finite-differences,

considered

th e

numerical approximat ion

of the

solution

of the

singular boundary value

problem:

where 0

^

a


24/87

C H A P T E R

2

where

i t was

assumed th a t

and approximate solutions of 2.3.3) w ere made up of solutions of

on subin tervals defined by a part i t ion A of

[0,1].

While it is true that the above

equation can be expressed as

L* Lw x ) = 0, w h ere

note th at since p 0)

can be

zero

in

2.3.4), these splines

are not in

general

L -splines

or y-splines. Generalizations to higher order singular Hermite splines were also

considered

by

D ailey [2.11].

To

describe

th e

singular

A -splines of

Jerome

and

Pierce [2.8] w hich gene ralizes

[2.10]),

let A be the

form ally self-adjoint operator defined

by

where

it is assumed that cf.

2.3.4))

Next , le t H denote th e weighted Sobolov space of all real-valued functions f

defined on [a, b such that D

m

~ */ is absolutely con tinuous and

with n o r m

and consider the bil inear form B u, v) associated w ith A :

on H x H . W e

assume that

th e

operator

A is

H -elliptic, i.e.,

a

positive constant

p

exists such that


25/87


O F

L - S P L I N E S

where H denotes the closed subspace of H of all

funct ions

/ w i t h /^/(a)

=

D

j

f b )

= 0 for 0 ̂ j m - 1. From 2 .3 .8) and 2 .3 .9) , it read i ly fo llows tha t H

is

a

Hilbert space under

the

inner product

Next ,

let M

= be any set of

bounded l inear func t ion a ls wh ich

are

l inearly independent

on

H . Then cf . [2.8]) , seH

is a

\-spline interpolat ing

r

— (

r

i

r

2

• »

r

J

^

fc

i f

s

solves th e m inimization problem:

Because

o f the

vanish ing boundary da ta

fo r

/eH

i t

follows

tha t ,

fo r any

/£//

there exists a unique A-spline s wh ich interpolates / in the sense that

As before, Sp A, M ) ,

th e

class

of all s

w h ich satisfies 2.3.11)

for

some

r e

R

k

is a

linear space.

In order to obtain error es t imates for the interpolat ion of 2 .3 .12 ) , w e ne xt

assume,

as in §2.1 ,

that M =

( A / } j = i

generates

a

Hermite-Birkhoff problem, i .e . ,

each

A - 6 M is of the

form

A, / ) = D

j

/(*i)

wi th

0

̂ ;, ̂

m - 1 and x, e [a

b ].

In

th is case,

any

A-spline

s is a

solution

of As

=

0 on

subintervals def ined

f rom

M,

jus t as in 2.2.3) . Because we are considering H a second-integral relat ion holds,

and the fol low ing error b ou nd s can be proved cf. [2 .8]) .

THEOREM

2.4. // M generates a H ermite-Birkhof f prob lem with partition

A e 0 *

a

a, b ) , assume that t M )

̂

n, and

for

a ny

/e

H let s e H b e i ts unique A-spline

interpolation cf .

2.3.12)) .

Then, for any 2 ̂ q ̂ oo,

where d . 2.3.\0)) , and where

In addition, if

then for 2 5 S q ̂ oo,

forO

£j

g m - 1.

It

is

wor th remark ing tha t

th e

Sobolev space

W{[a b]

for; ^

m , are al l

sub-

spaces of H . However , to extend the resul ts of Theorem 2.4 via the theory of

interme diate spaces, as in the p revi ous theorems of this section, w e w ou ld necessarily


26/87

16

CHAPTER 2

work with spaces intermediate to H and, say, W\

m

[a

b],

and these are not, as in

previous cases, Besov spaces.

It is also worth no t ing that the results o f Jerome and Pierce [2.8] go beyond th e

assum ption of 2.3.9), i.e., w eak er assum ptions are m ade in [2.8] corresponding to

2.3.9), and existence and uniqueness of interpolation plus error bounds for the

more general extended Hermite-Birkhoff problem are treated there. Since the

case

a

m

x)

6

> 0 on [a,b] is not ruled out in 2.3.6), th e results of [2.8] thus

simul taneously generalize

Lg-splines

and y-splines and give, as a special case, th e

k n o w n results for er ror bounds for spline interpolation of Theorems 2.2 and 2.3.

R E F E R E N C E S

[ 2 . 1 ]

J . W.

J E R O M E

A ND

L. L .

S C H U M A K E R

O n

L g-splines,

J . A pprox . T heo ry , 2 1969) , pp.

29̂ 9.

[2.2]

M.

G O L O M B

Splines,

n-widths, and

optimal approximation, MRC Tech.

Summ. Rep. 784,

Mathematics Research Center, United States Army, Univers i ty

of

W isconsin, Mad ison,

1 9 6 7 .

[2.3] P. M. ANSELONE

A ND

P. J. L A U R E N T

A general method for the construction of interpolating or

smoothing spline-functions,

N um er . Math.,

12

1968),

pp.

66-82.

[2.4] J. W. J E R O M E A ND R. S. V A R G A

Generalizations

of spline

functions

a nd

applications

to

nonlinear

boundary

value a nd eigenvalue problems, Theory and

Applications

of

Spline

Functions,

T. N. E.

Greville, ed., Academic

Press, New

York, 1969,

pp. 103-155.

[2.5] M. H. S C H U L T Z

Elliptic

spline

functions

and the Rayleigh-Ritz-Galerkin

method,

Math. Comp.,

24

1970), pp.

65-80.

[2.6]

T. R.

L U C A S

A

generalization of L-splines, N um er . Math.,

15

1970),

pp.

359-370.

[2.7]

, A theory

of

generalized splines with applications to nonlinear boundary value problems,

Thesis, Georgia

Inst i tute

of Technology, 1970.

[2.8]

J.

J E R O M E

AND J.

P I E R C E

On spline functions determined by singular self-adjo int differential

operators, J. Approx. Theory, to

appear.

[2.9]

P.

J A M E T On the

convergence

of finite-difference

approximations

to

one-dimensional singular

boundary-value

problems

N u m e r . M a t h . , 1 4 1 970 ) ,

pp. 355-378.

[2.10] P. G.

CIARLET,

F. N A T T E R E R A ND R. S. V A R G A

Numerical methods

of

high-order accuracy

fo r

singular

nonlinear boundary value problems,

Ibid., 15

1970),

pp.

87-99.

[2.11] J . W. D A I L E Y Approximation

by

spline-type functions

a nd

related problems Thesis Case

W estern

Reserve University, 1969.


27/87

CHAPTER

Interpolation and Approximation Results for

Piecewise-Polynomials in

H igher Dimensions

3.1. Tensor products of one dimensional polynomial splines. For many applica-

tions, it is desirable to generalize the re sults of Ch apters 1 and 2 for o ne-d ime nsion al

piecewise-polynomial func tions or splines to n-dimensional analogues. The easiest

of such extensions is

obtained

by simply considering the tensor product of one-

dimensional spline spaces. This was considered in BirkhofT, Schultz and Varga

[3.1], specifically for the tensor prod uct of H erm ite polynom ial splines in two-space

variables.

To

describe briefly

the

results

of

[3.1],

let A =

^

x A

2

,

given

by

denote

a

parti t ion

of the

rectangle

Q = [a , b ] x [c,

d ]

in R

2

. If H

m )

A ; Q ) is the set

of

all

real-valued piecewise-polynomial

funct ions

v v x , y ) defined

on Q

such that

D

(ltj)

w

= D

l

x

D

j

y

w is

continuous

in Q for all 0 ̂ i, j ^

m

— 1, and

such that w x,

y

is a polynomial of degree 2m — 1 in each of the variables x and y in each sub-

rectangle [ - X ; , x , - + i ]

x

[ y j , y

j l

] defined

on fi b y A , we can

define

a

un ique in ter -

polation 5 in

H

m )

A ;

Q) of a real-valued

function

such that D

{p

q}

f is cont inuous

in

Q for all 0 ̂ p, q ̂ m — 1, by

means

of

I f

and

have th e

analogous me aning cf.

§ 1.1) for the

part i t ion

of

we

assume that A is an element of ̂ Q), with finite a

^

1,

i . e . , c f . § l . l ) ,

Then, using th e idea of the

Peano

k ernel theorem cf . Sard [3.2]) in two-space

variables,

the

following

w as

sh o w n

in

[3.1].

We use the

nota t ion S^ Q)

to

denote

th e

set of all real-valued funct ions / defined on Q such that

D

p

~

M)

/e L

2

Q )

for

all 0 <

i

<

p.

and such that Z)

M )

/ is cont inuous in

Q

for all 0 <

i +

i

< p.

T H E O R E M 3.1.

Given

where

= • a, b ] x c, d), and

given

17


28/87

1 8

C H A P T E R 3

le t

be

th e unique interpolation of f in the sense

o f

(3.1.2). Then

for allQ^h l̂ m

with

0

̂

h + ̂ m .

Because the

Hermi te interpolation

of

(3.1.2)

is

local,

the

result

of

Theorem

3 .1

is

actually valid for any rectangular polygon, i.e., any polygon whose sides are

parallel to the co ordinate axes in the plane, such as an L--shaped region.

For our future needs, we now in troduc e the fol lowing no ta t ion . With n any

positive integer, let Q be a bounded region in Euclidean rc-space, R . We assume

that th e bounded region Q in R satisfies a restricted cone condition (cf. Agmon

[3.3,

p.

11]),

i.e., there exist

a finite

open cover

{ 0 , -} ? = i

of the

bo u n d a r y

dQ ,

of

where each 9

{

is an open subset of R , and associated open truncated cones

{C,}™

=

with vertices at the origin such that for any i and any xe 0 ,- n Q, then

x + C

{

= {w w = x + y, where ye CJ lies in Q. Next, if a =

( a

t

,

a

2

• • • , a

n

) is

any

n-tuple of nonnegative integers, then

denotes the

differential

operator

of order

The

space

of all

real-

valued funct ions which have continuous der ivatives

of all

orders

a

with |a|

m

in

Q

is

denoted

by C

m

(Q). T he

space C^(Q)

is the

collection

of all

infinitely differ-

entiable functions u in Q which vanish identically outside some compact set

contained

in

Q . Similarly, CQ(R )

is the

collection

of all

infinitely differentiable

funct ions in R which vanish identically outside some compact set in R .

The Sobolev spaces W ™ £ 1 ) and W ^( Q . ) , m a nonn egative integer, are the n defined

as the respective completions of C

CO

(Q) in the n o r m :

Similarly,

W%(R )

and W ^(R ) are the completions of

C$(R )

in the above norms,

and

W^(fJ)

is the completion of Co (Q) in the first norm of (3.1.5).

T he

result

of

Theorem 3.1

can be

interpreted

in

terms

of

Sobolev norms

as

follows.

C O R O L L A R Y 3.2. With the hypotheses of Theorem 3.1,

for

any k wi th 0

k

m

There are two inherent shortcomings of Theorem 3.1 and its Corollary 3.2.

O n e i s that the results as stated pertain only to rectangular polygons in

K

2

,

and

not to

higher dimensions.

T he

o ther

is

that

the

function space Sf

m

(Q) used

in


29/87

P I E C E W I S E P O L Y N O M I A L S

I N

H I G H E R

D IM E N S I O N S

19

these results is not w h a t one would expect , namely the usual Sobolev space

M ^ 2

m

(n) .

These sho rtcomings of [3.1] have been more than adequately covered by

Bramble and Hilbert ' s general izat ion in [3.4], [3.5] and

[3.6],

w h i c h we now

describe.

Consider

any closed

hypercube

Q in

/? , wi th

it s

2 vertices denoted

by

x ,,

1

i 5 s

2 . For u

€ C

2m

~

(Q), the m th Hermite interpolation u

m

of

u in Q is define d

as a polynomia l of degree 2m - 1 in each of its n va riables w hich sat isf ies at each

vertex

x,

for any y = ( y , , • • • , yj wi th 0

y

}

m - 1 for al l 1 j n. Because (3.1 .7)

is a

nonsingular l inear system

of

(2m)

n

equat ions

in

(2m)

n

u n k n o w n s ,

it is

readi ly

seen that the funct ion

u

m

,

so d efined, is uniqu e. This

approach,

in fact, generalizes

th e Hermite - in te rpola t ion of (3.1.2) in two dimensions .

Next, let

R

be

decomposed

in to hypercubes with sides

of

length h, i.e.,

R

=

l^JA, wh ere Q ,

n Q, is, for any

i

j,

e i ther em pty

or a

pa r t

of the

bounda ry

of Q,.

Because th e interpolation of

(3.1.7)

is local, then given any ue C

2 m

~

( /? ) ,

a

u n iq u e in te rpolan t u

m

of u can be found w hich satisfies

(3.1.7)

at every vertex of every

Q - . A s is read ily seen from

(3 .1 .7) ,

D *u

m

is c o n t i n u o u s in

R

for any a = ( a , , • • • ,

a j

with

0

a,

m - 1 , j = 1 , 2 , • • • , n.

Un l ik e

the

one-dimensional case,

an

a rb i t r a ry

function

u

e

W \

m

(R ) need n ot

have well-defined

der iva t ives

at the

vertices

of the Q , for the

in te rpola t ion procedure

of

(3.1 .7) . Thu s, it is necessary to smooth or

mollify

u to obtain a u

h

eC^(R ) for

which

th e

interpolat ion

of

(3.1 .7)

is

m eaningful . (This

is the

ana logue

of the use of

Lagrange p olynom ial interpolat ion in § 1 .1 . ) I t has been sh ow n by B ram ble and

Hilbert that

a

mollified

u

h

e C$(R ) , for

u

e

W l

m

(R ) ,

can be

found such tha t

fo r all

h

>

0,

there exists

a

cons tan t

C,

independent

of

h

and u, wi th

Next , wi th

u

h

e

Co(R ), le t

u

hm

be the H erm ite in te rpola t ion

of u

h

over hypercubes

of

side h,

in the

sense

of

(3 .1 .7 ) .

Then,

based

on a

general izat ion

of the Peano

kernel theorem, Bramble

and

Hi lber t have shown tha t

fo r any a = (a,, a

2

, • • • , aj wi th |a| 2m and 0 a, m - 1 for all 1 g

/

n.

Thus, combining

(3.1.8)

and

(3.1.9)

gives

for

any a

wi th

|a| 2m and 0

a,

m — 1 for all 1

/

n.

To extend this result of (3.1 .10) to a bounded region Q

c = /? ,

we use the

Calderon

extension theorem (cf. [3.3, p. 171]) . Specifically, if Q satisfies a restricted cone


30/87

20

CHAPTER

3

property, there exists a bounded linear transformation

with $v = v on

Q, i.e.,

for some positive constant C,

Thus, given

any ue W

2

2

m

(ty,

then

^w is an

element

of

W

2

2

m

(R ),

and

(3.1.10) then

can be

applied

to

S'u, i.e., w ith

(3.1.10) and

(3.1.11),

However , since by

definition I M I j r ,

2

< R n

^ I M I z .

2

< n )

f°

r an

y

VE

L

2

(R )

an d since

u = u on

Q,

it necessarily follow s that th e above in equ ality gives us that , for any

with

|a|

^

2m

and 0

^

a,

^

m

—

1 for all 1

̂

i

̂

n ,

where if

H

(

^\R )

is the subspace of

C

m

l

(R ) of all functions which are poly-

nomials

of

degree

2m — 1 in

each variable

on

each hypercube

Q, of /? =

of side h , then

H

(

^\Q.) is

simply

the

restriction of

H

(

^\R ) to Q. It

turns

out

that

not

all

terms

in

of

l l u j l ^ ^ n )

are

needed,

as

conjectured

by G.

Birkhoff.

The

improved

form of

(3.1.12) of B ramble and Hilbert [3.4] is given in the follow ing theorem.

T H E O R E M 3.3. For any

for all

a

with |a| ^

2m

and

0

̂ a, ^

m

—

1

for all ^ ̂ «,

where

K is the set

of all indices T — T J ,

T

2

,

• • • , t

n

)

with

\T\ = 2m such that the polynomial X

T

is not identically its own mth Hermite interpolation. The result is

also true for Q = R .

Note that the set K of Theorem 3.3 always contains the indices (2m, 0, • • • , 0),

(0 ,2m, 0, • • • ,

0),

• • • , (0,0, • • • , 2m), but for n > 2, K

contains other indices

as

well.

The results given thu s far can b e viewe d as results concerning the interpo lation

and approximation by the tensor product of one-dimensional Herm ite splines.

General results concerning approximation by tensor products of one-dimensional

splines in higher dimensions have also been established by several authors. In

analogy

with

the notation of § 1.1, le t S p

(m )

(A,, [a,, & J) be the collection of splines

defined

on [a,-,

bj,

of local degree 2m — 1 on subdivisions defined by A , - . For

notation,

le t

Q = n?=i(

a

i ^i)

be a

rectangular

parallelepiped in /? , and let

where A,-6^(0, , b

t

). Defining n = max

l

^

i

^

n

n

i

, n =

min

we say that A

e

̂ (Q) if n/n ̂a. Then Schultz [3.7] has proved the

following

theorem.


31/87

P IE C E W I S E P O L Y N O M I A L S

I N

H I G H E R D I M E N S I O N S

21

T H E O R E M 3.4.

Given where

an d given

then there exists a such that

where 0 ̂p ̂ min r, 2m —

1 ).

Using an

approach

of Harrick [3.8], suitable extensions can be made cf. [3.7])

for n-dimen sional

regions

Q

f|

=1

[«;,/?,] by considering approximations in

where 9 x) is positive in Q and vanishes suitably on dQ

f o r details,

see

[3.7]).

It

is

interesting that Bramble

and Hilbert

[3.5], [3.6] also have approximation

results, analogous

to

Theorem 3.3,

for

splines, which

are

effectively treated

as

tensor

products

of

one-dimensional splines

on a

uniform mesh

of

side

h. If S^,k

^

2,

is

the

collection

of all

funct ions u

which have continuous partial derivatives

of

order k — 2 in R , and, on any hypercube of side h, u is a polynomial of degree

k — 1 in each variable, then it can be shown actually v ia interpolation) that given

anyiie W£ /n,

From this, using

the Calderon

extension theorem

as in the

proof

of

Theorem 3.3,

one obtains cf. [3.5],

[3.6])

the following.

T H E O R E M

3.5. For any u e W*^),

Based on notions of quasi-interpolation, de Boor and Fix [3.9] have obtained

deep results which are like those of 3.1.16), but in the norms t t ^ H ) cf. (5.1.2 0)).

3.2. Zlamal type extensions. Taking the tensor product to attack higher-

dimensional approximation problems

for

piece wise-polynomial

func t ions

is

just

one way of extending one-dimensional results. Another approach, more closely

allied

to finite-element

methods

is due to M. Zlamal

[3.10]-[3.12],

and can be

described as

follows.

Assume that Q is a bounded region in

R

2

which can be triangulated, i.e., Q

can be exactly decomposed into a finite number of triangular subregions 7],

1 /̂ ̂N. Fixing

i and

calling

T

= 7], consider

any

polynomial

of

degree

two

in each variable:

If P i are the

vertices

of T

see Fig.

1 ) and

Q

t

are the

midpoints

of the

sides

o f T ,

1

̂ i ̂ 3, then for any

constants

/ ^ , , C T , , 1 ̂ i ̂ 3, it is easy to see that there

exists

a unique p ( x j , x

2

) of the form

(3.2.1)

such that


32/87

C H A P T E R

3

Thus ,

if/is

a cont inuous

function

on T, there exists a

un i que

polynomial

p(x

l

,

x

2

)

interpolating

on

T in the sense tha t

Zlamal has sh ow n cf. [3.10]) th e following.

T H E O R E M

3.6. Given

any

Q

cR

2

which

can be triangulated, i.e.,

let h

denote the largest length of any side of any T

{

, and let 0 denote the smallest

interior angle of any T

{

. Iffe

C

3

(Q),

and s is the unique piecewise-polynomial which

interpolates f in the sense o f

3.2.2),

then

for

any triangulation with 9

^

6

0

> 0, where K is independent of f and the geom etry.

Similar results have been obtained by Zlamal for piecewise-polynomial of

degree

3)

interpolations, defined

by

interpolating

/ and its two first

partial

derivatives at each vertex and interpolating / at the center of gravi ty of each

t

Since an interval and a triangle are

n -simplices

for n = 1 and n = 2, respectively,

it

would seem natural to generalize

Zlamal s

results to an n-dimensional setting

via n-simplices an n-simplex is the convex hu ll of n + 1 noncoplanar

points

in R ),

as

done

recently by Ciarlet and Wagschal

[3.13].

The inherent shortcomings of

these results of [3.10]-[3.13] can, however, be seen by the typical result of

Theorem

3.6 above. As in

Theorem 3.1

and its Co rollary 3.2, the function

space

setting for the

error bou nd of 3.2.3) is again not w hat one would natu ral ly

expect;

one would

expect

h

2

accuracy

for eW^ft) in

3.2.3). Fortu nately, Zlamal

and

Bramble

[3.14] have established an improved and generalized version of Th eorem 3.6,

which we now

describe.

Given

a

bounded region Q

in

R , whose boundary

5Q is a

simplicial complex

a generalization to R of a polygon in R

2

), assume a generalized triangulation

T

over

ft,

i.e., Q .

is the set-theoretic

union

of a finite

number

of

n-simplices S,,

1

^

i ̂N,

whose interiors

are

pair-wise disjoint

and

such that, given

any

n-simplex S,

of the

triangulation, each

one of its n — 1 )-faces is

either

a

portion

of the

boundary < Q

or

else

is also an n —

l)-face

of

another n-simplex

of the


33/87

P I E C E W I S E - P O L Y N O M I A L S

IN

H I G H E R D I M E N S I O N S

tr iangulation.

For ease of

description

here,

assume

that all

simplices

S,

are

equilateral

and

that

the

region Q

can be

triangulated

by

such reg ular simplices

S? for a

sequence of

h -* 0 ,

where

h is the

length

of a common edge of an

equilateral

simplex.

Next,

if

T

h

(R )

is the

subspace

of

C° (R )

of all fun ctio ns which are poly-

nomials

o f

degree

tw o in

each variable

on each

regular) simplex

5{ of

R

n

of

edge

h ,

then

T

h

G l ) is

defined

as the

restriction

of T

h

(R ) to

Q.

Given

there

is, in the manner of

3.2.2),

a

unique w

h

e T

h

(Q ) which interpolates

u at

each

vertex

and

midpoint

of

each

edge of the

triangulation

of Q.

Then,

in the

manner

of 3.1.8H3.1.12),

one

obtains

th e

following theorem cf.

[3.14]).

T H E O R E M

3.7. Let Q be a bounded region in R which can be triangulated by

regular

simplices Sf for a sequence of h

->

0.

Then

for

where C is independent ofh and u.

It is important to note that the results of Theorems 3.3 and 3.7, while proved

either fo r hypercubes or regular

simplices

in R , do extend to

more general regions.

In the case o f Theorem 3.3,

rectangular

parallelepipeds may be used,

provided

that

the

ratio

of the lengths of any

edges

remains bounded

above

and below for any h .

The same is true of Theorem

3.7.

For

computer implementat ion

of

these so-called Zlamal-type finite element

methods,

see

George

[3.15].

REFERENCES

[ 3 . 1 ] G. B I R K H O F F , M. H. S C H U L T Z AND R. S. V A R G A , Piecewise Herm ite interpolation in one and two

variables

with applications

to

partial

differential

equations, Num er . Math . ,

1 1

1968),

pp.

232-

256.

[3.2] A. S A R D , Linear

Approximation,

Math. Survey 9 , Am erican Mathe m atical Society, Providence,

Rhode Island, 1963.

[3.3]

S.

A G M O N , Lectures

o n

Elliptic Bound ary Value Problems,

V an

Nostrand, Princeton,

New

Jersey,

1 9 6 5 .

[ 3 . 4 ] J. H. B R A M B L E AND S. R. H I L B E R T , Bounds for a class o f linear functional with applications to

Hermite interpolation, Numer . Math . , 16

1971) ,

pp. 362-369.

[3.5] , Estimation

of linear

functional on

Sobolev

spaces with applications to Fourier transforms,

SIAM J. Nu m er. An al . , 7 1970), pp. 112-124.

[3.6] S. R. H I L B E R T , Numerical methods for elliptic boundary value problem s, Thesis, University of

Mar y l and , 1969.

[3.7] M. H. S C H U L T Z ,

Multivariate

spline functions and elliptic problems, Approximat ions wi th

Special Emphasis on Spline Functions, I. J. Schoenberg, ed., Academic Press, New York,

1 9 6 9 , pp.279-347.

[3.8]

I. I.

H A R R I C K ,

Approximation

of

functions

which

vanish

on the

boundary

of

a

region, together with

their partial derivatives,

by

functions of

special

type, A kad . Nau k . SSSR I zv . Sibirsk . Otd. ,

4

1963) , pp. 408^25.

[3.9]

C. DE

B O O R AND

G.

Fix, Spline approximation

by

quasi-interpolants,

J.

Approx. Theory,

to

appear .

[3.10] M. Z L A M A L , On the

finite element method, N u m e r .

M a th ., 12

1968),

pp. 394-409.

[3.11]

,

O n

some

finite

element procedures

for

solving second order boundary value problems,

Ibid., 14 1969), pp. 42-48.


34/87

24 C H A P T E R 3

[3.12]

A finite

element procedure

of the

second order accuracy.

Ibid., 14

1970),

pp .

394-402.

[3.13] P. G. CIARLET AND C. WAGSCHAL, Multipoint Taylor formulas an d applications to the finite

element method Ibid.,

17

1971),

pp.

84-100.

[3.14] J. B R A M B L E

A ND

M.

Z L A M A L

Triangular elements in the finite element method M a t h .

Comp.

24

1970) ,

pp.

809-820.

[3.15] J. A.

GEORGE, Computer implementation

of the finite

element

method.

Thesis Rep. CS208,

Computer Science Department, Stanford University, California, 1971.


35/87

CHAPTER

The Rayleigh-Ritz-Galerkin

Method

for

Nonlinear Boundary Value Problems

4.1. One dimensional problem. To show how theorems about interpolation and

approximation by piecewise-polynomial functions can be used to deduce results

about approximate solutions

of

nonlinear boundary value problems,

we first

discuss

two-point boundary value problems, as thoroughly considered in Keller

[4.1].

Specifically, we

shall consider problems

of the

form

with

homogeneous Dirichlet boundary conditions

where

the

differential

operator

in self-adjoint

form

is given by

For

one-dimensional problems, nonhomogeneous bound ary conditions D

k

u a ) —

a

k

D

k

u b)

= / ?

f c

0

k

m — 1, can alway s be reduced to the form

4.1.2)

by means

of

a

suitable change

of

dependent variable. Other types

of

boundary conditions,

such as nonlinear, Neumann, and mixed boundary conditions in one dimension

can

also be trea ted cf. [4.2], [4.3] and [4.4]).

For

specific assumptions about

< £

we

assume that

all P are

bounded

on

[a ,

b ]

,

0

j m and that the operator of 4.1.3) is W^a, b ]-elliptic, i.e., there exists a

positive constant K such that

where we recall that W™[a ,b] is the collection of all real-valued functions w x),

where D

m

~

1

w x) is absolutely continuous on [a,

b ] ,

D

m

w e L

2

[ a , b ] , and where

iyw a) = iyw b) = 0 for 0 j m — 1. In a ddition, we assume that

f x , u )

is a

real-valued

measurable

function

on

[a,b]

x R

such that

f x ,

v(x))eL

2

[a,

b ] fo r

all

v

e W™ [a ,

b ], and such that / satisfies


36/87

26

C H A P T E R 4

fo r a lmost all x e [a, b] and all — oo <

«,

v 0,

there exists

a

posi t ive constant

M(c)

such that

Defining

th e

qua si-bilinear form

fo r all u, V E W ™ [a,b], we say that the bounda ry va lue p rob l em o f (4.1.1 )- 4.1.2)

admits

a

generalized so lu tion

u in

W™[a ,

b]

(c f. Bro wder

4.5]) if

A

general result , based

on the

theory

o f

m o no t o ne o p e r a to r s

to be

discussed

in §

4.2,

is the

fo l l owing theo rem.

T O R M 4.1. W ith the assu mptions o/ 4.1.4)- 4.1.6), then the nonlinear boundary

value problem (4.1.1)-(4.1.2) admits a u nique generalized solution in W [a, b].

Next , if

S M

is any finite-dimensio nal subspace o f

W ™ [ a .

b], then, in analog y wi th

(4.1.8),

we would cal l W

M

in S

M

the Galerkin approximation of the generalized

so lu t ion u of (4.1.1 H4.1.2) if

The next resul t shows that

W

M

,

so def ined, is uniquely determined, and gives

e r r o r b o u nds fo r u — W

M

.

T O R M

4.2. W ith

the

assumptions

of

4.1.4)- 4.1.6), there

is a

unique

W

M

in

S

M

which

satisfies

(4.1.9). M oreover, there exist constants K and K', independent of

the choice of S

M

such that

for all 0 ^7

m — 1, where u is the unique generalized solution of (4.1.1)-(4.1.2)

inW^[a b].

W e

shall show

in

§4.2 that

th e

second inequal i ty

o f

(4.1.10)

is a

co nsequence

o f

Theorem

4.6 on

mono tone ope ra t o r s .

The first

inequality

o f

(4.1.10)

is

elem entary

to

establish,

and we

give

a

di rect proof .

For any v e W™[a , b] and any

nonnegat ive


37/87

NONLINE R

BOUND RY

V LUE P R OBLEMS

27

integer; with 0 g ̂ m — 1, the

fact

that iyv a) =

D*v b)

= 0 allows us to write

that

Hence,

where sgn y = 1 for any y ̂0 and sgn y = — 1 if y

j,] are

not, basically because they were derived

as

consequences

of

error bounds

in || •

H ^ i a b ]-

To be

more specific, consider

the

following special

caseof(4.1.1H4.1.2):


38/87

28 CHAPTER 4

where f x, w)eC°([0,1]

x R), and where / satisfies the hypotheses of (4.1.5)-

(4.1.6),

with

A = n

2

.

Using

the

Hermite subspace Hj^AJ c

tf^[0,1], it follows

from

Theorem 4.3 that if the unique generalized solution

u

is an element of

C

2

[0,1],

then

from

(4.1.12)

(with

a = 2, m =

1),

Ciarlet [4.7] improved the above error bound to

and this has subsequently been generalized in Perrin, Price and Varga [4.8].

We sketch these developments below.

Given the nonlinear boundary value problem of (4.1.1)-(4.1.2), choose the

specific y-spline

subspace

S

0

(Jzf,

A,

z),

where

is

given

by

(4.1.3).

In

this develop-

ment,

it is important that the differential operator Z£ of (4.1.3) be chosen equal to

the operator E of (2.2.1) defining the y-spline space. Then, if u is the generalized

solution of (4.1.1 H4.1-2) in ̂[a, b], and if V V

is its approximation in S

0

(J ?, A, z)

(cf. Theorem 4.3), let w be the interpolation of u in S

0

(J5f, A, z) , in the sense of (2.2.4).

Following the

construction

of

[4.8],

it can be

shown that

If

ue

W

a

2

[a,b], where

m

̂ a ^

2m, it

follows

from

Theorem

2.3 and

Corollary

1.10

that

so that the inequality of (4.1.14), using (4.1.14') with; = 0, reduces to

Hence, from the triangle inequality and the inequalities of (4.1.14') and (4.1.14 ),

for any 0 ĵ ^ m. Similar results in || •

||

Loo

[«,6]

c n

also be established, and are

stated in the

following

theorem.

THEOREM 4.4.

With the

assumptions

of (4.1.4)-(4.1.6), let u be the

unique

generalized

solution

of 4A.l)- 4.l.2) in

̂ ^[a^b],

and assume further

thatueW

2

{a,b],

where m ̂ a ^

2m. //

S

M

= S

0

^f,

A, z)

and W

M

is the unique element in S

M

which

satisfies

(4.1.9),

then

Furthermore, if u

e

C

ff

[a, b],

varga - functional analysis and approximations in numerical analysis

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