Table Of ContentNumerical Solution of Partial Differential Equations
NATO ADVANCED STUDY INSTITUTES SERIES
Proceedings of the Advanced Study Institute Programme, which aims
at the dissemination of advanced knowledge and
the formation of contacts among scientists from different countries
The series is published by an international board of publishers in conjunction
with NATO Scientific Affairs Division
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C Mathematical and D. Reidel Publishing Company
Physical Sciences Dordrecht and Boston
D Behavioral and Sijthoff International Publishing Company
Social Sciences Leiden
E Applied Sciences Noordhoff International Publishing
Leiden
Series C - Mathematical and Physical Sciences
Volume 4 - Reactions on Polymers
Numerical Solution of
Partial Differential Equations
Proceedings of the NATO Advanced Study Institute
held at Kjeller, Norway, August 20-24, 1973
edited by
J.G.GRAM
Institutt for Atomenergi, Kjeller, Norway
D. Reidel Publishing Company
Dordrecht-Holland / Boston-U.S.A.
Published in cooperation with NATO Scientific Affairs Division
First printing.' December 1973
Library of Congress Catalog Card Number 73-91204
ISBN -13 :978-94-0 10 -2674-1 e-ISBN -13 :978-94-0 10 -2672-7
DOl: 10.1007/978-94-010-2672-7
Published by D. Reidel Publishing Company
P.O. Box 17, Dordrecht, Holland
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All Rights Reserved
Copyright © 1973 by D. Reidel Publishing Company, Dordrecht
Softcover reprint of the hardcover 1s t edition 1973
No part of this book may be reproduced in any form, by print, photoprint, microfilm,
or any other means, without written permission from the publisher
CONTENTS
PREFACE v.n
L. Collatz:
METHODS FOR SOLUTION OF PARTIAL DIFFERENTIAL
EQUATIONS 1
A.R. Mitchell:
VARIATIONAL PRINCIPLES - A SURVEY 17
D.M. Young:
SOLUTION OF LINEAR SYSTEMS OF EQUATIONS 35
L.B. Rall:
SOLUTION OF NONLINEAR SYSTEMS OF EQUATIONS 55
A.R. Mitchell: 107
ELEMENT TYPES AND BASE FUNCTIONS
W.J. Kammerer, G.W. Reddien, R.S. Varga:
QUADRATIC INTERPOLATING SPLINES: THEORY AND
APPLICATIONS 151
O.C. Zienkiewicz:
SOME LINEAR AND NON-LINEAR PROBLEMS IN FLUID
MECHANICS. FEM FORMULATION 173
1. Ho1and:
APPLICATION OF FINITE ELEMENT METHODS TO STRESS
ANALYSIS 195
E.E. Madsen. G.E. F1admark:
SOME FINITE DIFFERENCE METHODS FOR SOLUTI ON OF HEAT
CONDUCTION PROBLEMS 223
E. Spreeuw:
FINITE ELEMENT COMPUTER PROGRAMS FOR HEAT
CONDUCTION PROBLEMS 241
LIST OF PARTICIPANTS 267
PREFACE
This book contains the transcripts of the invited lectures
presented at the NATO Advanced Study Institute on "Numerical
Solution of Partial Differential Equations". The Study Institute
was held at the Netherlands-Norwegian Reactor School, Institutt
for Atomenergi, Kjeller, Norway, 20th - 24th August 1973. The
members of the Scientific Advisory Committee were:
A.R. Mitchell, University of Dundee, Scotland
I. HoI and, University of Trondheim, Norway
T. Havie, UniverSity of Trondheim, Norway
The members of the Organizing Committee were:
E. Andersen, Institutt for Atomenergi, Kjeller, Norway
G.E. Fladmark, Institutt for Atomenergi, Kjeller, Norway
J.G. Gram, Institutt for Atomenergi, Kjeller, Norway
The aim of the Study Institute was to bring together mathe
maticians and engineers working with numerical methods. The
papers presented covered both theory and application of methods
for solution of partial differential equations. The topics were
finite element methods, finite difference methods, and methods
for solution of linear and nonlinear systems of equations with
application to continuum mechanics and heat transfer.
The total number of participants was 68. Their names are
given at the end of the book. The publication of these proceed
ings could be realized through the kind cooperation of the lec
turers. The Advanced Study Institute was financially sponsored
by NATO Scientific Affairs Division. The Organizing Committee
wishes to express its gratitude for this support. Valuable
assistance was given by Mrs. G. Jarrett who took care of many of
the practical arrangements during the meeting (like hiring air
planes for participants wanting to go Sightseeing in the Oslo-area).
We also wish to thank our colleagues at Institutt for Atomenergi
for their help in arranging the Study Institute.
Kjeller, September 1973. The Organizing Committee
METHODS FOR SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS
L. Collatz
Universitat Hamburg,
Hamburg, Germany
SUMMARY
In this survey we do not intend to mention all types of
numerical procedures but to look only on some special methods,
which were used frequently in recent years or which deserved
perhaps to be used more than hitherto.
I. DISCRETIZATION METHODS
1. General Formulation
Let us formulate the problems not in greatest possible
generality, but so general, that many applications are contained
in this formulation.
Let B be a domain in the n-dimensional point space Rn with
coordinates x , ..• ,x and OB the boundary of B. u(x.) may be an
unknown functton or R vector with m components u(l)(X~), .•• ,
u(m)(xj ); the linear or nonlinear differential equati~n
Mu = 0 in B (1.1)
and the boundary conditions
Su = 0 on OB (1. 2 )
may be prescribed, M and S may be vectors. In problems with free
boundaries B is not given.
2 L.COLLATZ
There are many different kinds of discretization. We mention
four methods
Finite difference methods
Finite differences of higher approximation
Hermitean methods (tlMehrstellenverfabrentl)
Finite element methods.
In all cases we have the question of
consistency
convergence
stability
of the discretization methods.
2. Finite Difference Methods
It is not necessary to explain this very often used and
very well known method (compare Isaacson-Keller (66) Mitchell
(69) a.o.). Only for comparing with the other methods we con
sider the classical case of Laplaces equation for a function
u(x,y)
o (2.1)
in the square grid with meshsize h
(2.2)
and we have in the well known form of patterns the formula
this method was used for treating complicated problems; with the
aid of solving large systems of linear or nonlinear equations,
for boundary and initial value problems; (Varga (62) Ortega
Rheinboldt (70) a.o.). The many improved methods for solving
big systems of equations are well known. A further method was
discussed by Young (73).
In the following simple example we will compare dis
cretization methods with parametric methods.
METHODS FOR SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS 3
y
1
¢=o
Fig. 1.
We consider the ideal flow of a liquid over a sill, fig. 1.
The streamlines are given by ~ = const, where ~(x,y) satisfies
the conditions
1
M = 0 in B (- 00 < x < 00, 'f(x) < y < 00, 'f(x) (2.4)
~ = 0 for y = 'f(x)
lim r~(xJy)-yl = 0 for every fixed x. (2.5)
y4X>
The calculation of by discretization methods causes in my
~
opinion more work than by parametric methods in No. 10.
3. Finite Differences Of Higher Approximation
Here one is using other patterns for which the remainder term
is of higher order. For the case (2.1) (2.2) we have for instance
the formula
- 1 0
0
16 0
1
16 -60 16 u - 0 12 0
h2
16 0
- 1 0
= o (h4)
4 L. COLLATZ
The formulas of this type have often the disadvantage of
difficulties in the neighbourhood of the boundaries (Collatz
(60)) .
4. Hermi tean Methods ("Mehrstellenverfahren")
Instead of (2.1) we consider a differential equation of the
form
Lu = f(x.,u) in B ( 4.1)
J
where L is a linear differential operator and the given linear or
nonlinear function does not depend on partial derivatives of u.
One chooses in B and OS a finite set of points P1, ••. Pq,
often as gridpoints of a regular grid and writes down for each of
these pOints an equation of the form
s
[a u(p ) + b Lu(P ) + r: c L Lu(Pp)] (4.2)
p p p P 0=1 pa a
Usually in the equation for the point P the sum contains
k
non zero terms only for the values of p for wruch Pp is in the
0:
neighbourhood Pk. TheoLa are fixed chosen differential
operators, for l.nstance dx:l' Do,... In the Simplest case one
puts c = 0 (compare Albrecht (57)(62), Collatz (60)(72».
pa
The values aQ,bp,c a are to be determined in such a. way that
one gets a remainder te~m of an order in h as high as possible if
one develops by Taylor expansion with respect to u and the
~
partial derivatives of u at Then we substitute Lu by f,
P~.
u(Pp) by approximate values U(Pp)' put ~=O and have one of the
equations for the U(Pa).
For illustration: We have for the case (2.1)(2.2) the formula
-2 -8 -2 1
1 40 -8 u + 8 III = 0 (h4) (4.3)
h2
-8 -2 1
For parabolic equations Wirz (72) has got good numerical
results. Hermitean methods for initial value problems see Collatz
(72 ) •
