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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)
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S.
Varga
Kent State
University
Kent Ohio
Functional
Analysis
and
Approximation Theory
in Numerical Analysis
SOCIETY FOR INDUSTRIAL AND APPLIED MATHEMATICS
PHILADELPHIA
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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
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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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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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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.
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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
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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.,
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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.
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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
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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
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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
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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
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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.
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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]) .
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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.,
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G E N E R A L I Z A T I O N S
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
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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
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G E N E R A L I Z A T I O N S
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
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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.
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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
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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
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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
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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.
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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
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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
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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.
8/17/2019 Varga - Functional Analysis and Approximations in Numerical Analysis
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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.
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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
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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
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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):
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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],