Numerical Treatment of Integral Equations / Numerische Behandlung von Integralgleichungen: Workshop on Numerical Treatment of Integral Equations Oberwolfach, November 18–24, 1979 / Tagung über Numerische Behandlung von Integralgleichungen Oberwolfach, 18.

283 Pages·1980·5.777 MB·German

by  K. E. Atkinson (auth.), J. Albrecht, L. Collatz (eds.)

Numerical Treatment of Integral Equations
Numerische Behandlung von Integralgleichungen
ISNM 53
International Series of Numerical Mathematics
Internationale Schriftenreihe zur Numerischen Mathematik
Serie internationale d' Analyse numerique
Vol. 53
Editors
Ch. Blanc, Lausanne; A. Ghizzetti, Roma; R. Glowinski, Paris;
G. Golub, Stanford; P. Henrici, ZUrich; H. O. Kreiss, Pasadena;
A. Ostrowski, Montagnola, and J. Todd, Pasadena
Numerical Treatment
of Integral Equations
Workshop on Numerical Treatment of Integral Equations
Oberwolfach, November 18-24, 1979
Edited by
J. Albrecht, Clausthal-Zellerfeld and
L. Collatz, Hamburg
Numerische Behandlung
von Integralgleichungen
Tagung tiber Numerische Behandlung von Integralgleichungen
Oberwolfach, 18.-24. November 1979
Herausgegeben von
J. Albrecht, Clausthal-Zellerfeld und
L. Collatz, Hamburg
1980
Springer Basel AG
ISBN 978-3-7643-1105-6 ISBN 978-3-0348-6314-8 (eBook)
DOI 10.1007/978-3-0348-6314-8
CIP-Kurztitelaufnahme Library of Congress
der Deutschen Bibliothek CataIoging In Publicatiou Data
Numerical treatment of integral equaöons = Main entry under tide:
Numerische Behandlung von Integralgleichungen / Numerical treatment of integral equations.
Workshop on Numer. Treatment of Integral (International series of numerical
Equations, Oberwolfach, November 18-24, 1979. mathematics;
Ed. by J. Albrecht and L. Collatz. - v. 53)
Basel, Boston, Stuttgart : Birkhäuser, 1980. English or German.
(International series of numerical 1. Integral equations -Numerical solutions
mathematics; Vol. 53) Congresses I. Albrecht, Julius
11. Collatz, Lothar, 1910- 111. Series
NE: Albrecht, Julius [Rrsg.]; Workshop on QA431.N86 515.4'5 80-17801
Numerical Treatment of Integral Equations
(1979, Oberwolfach); PT
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@ Springer Basel AG 1980
Originally published by Birkhäuser Verlag
Basel in 1980.
Preface
This volume contains the manuscripts of lectures which were delivered at a
symposium on >T he numerical treatment of integral equations{ held in
November 1979 at the Mathematical Research Institute, Oberwolfach. In
terest in the symposium was very large, with about half of the participants
coming from abroad.
The following topics were covered: Volterra integral equations, Fredholm
integral equations of the second kind, Fredholm integral equations of the
first kind in connection with ill-posed problems, integro-differential equations
as well as branching problems with non-linear integral equations. The follo
wing methods of numerical treatment were referred to: Runge-Kutta methods,
kernel approximation methods, fixed point theorems, iteration methods,
bounds for eigenvalues, finite elements, quadrature methods and interval
analysis.
We are very grateful to Prof. M. Barner and the members of the Mathema
tical Research Institute for the use of their facilities and for helping to make
the symposium possible. Special thanks are due to Birkhauser Verlag for
once again providing us with the means of publishing the material.
Julius Albrecht Lothar Collatz
Clausthal-Zellerfeld Hamburg
Vorwort
Der Band enthiilt Manuskripte zu Vortriigen, die auf einer Tagung am Mathe
matischen Forschungsinstitut Oberwolfach tiber »Numerische Behandlung von
Integralgleichungen« im November 1979 gehalten wurden. Das Interesse an
der Tagung war sehr groG, etwa die Hiilfte der Teilnehmer kam aus dem
Ausland.
Folgende Gebiete kamen in den Vortriigen zur Sprache: Volterrasche Inte
gralgleichungen, Fredholmsche Integralgleichungen zweiter Art, Fredholm
sche Integralgleichungen erster Art in Verbindung mit inkorrekt gestellten
Problemen, Integrodifferentialgleichungen sowie Verzweigungsprobleme bei
nichtlinearen Integralgleichungen. Zur numerischen Behandlung wurden als
Methoden herangezogen: Runge-Kutta-Verfahren, Ersatzkernverfahren, Fix
punktsiitze, Iterationsverfahren, EinschlieGungssiitze, Finite Elemente, Qua
draturverfahren, Intervallanalysis.
Herrn Prof. Dr. M. Barner und den Mitarbeitern des Mathematischen For
schungsinstituts sei herzlich flir die Ermoglichung und die Durchftihrung der
Tagung gedankt.
Unserer besonderer Dank gilt dem Birkhiiuser Verlag flir die auch bei diesem
Band wieder sehr gute Ausstattung.
Julius Albrecht Lothar Collatz
Clausthal-Zellerfeld Hamburg
Index
Atkinson, K. E.: The Numerical Solution of Laplace's Equation in
Three Dimensions-II. . . . . . . . . . . 1
Baker, C. T. H.; Riddell, I. J.; Keech, M. S., and Amini, S.: Runge-
Kutta Methods with Error Estimates for Volterra Integral
Equations of the Second Kind . . . . . . .. 24
Bestehorn, M.: Die Berechnung pharmakokinetischer Parameter. Ein
Bericht aus der Praxis. . . . . . . . . . . .. 43
Brunner, H.: Superconvergence in Collocation and Implicit Runge-
Kutta Methods for Volterra-Type Integral Equations of the
Second Kind . . . . . . . . . . .. 54
Christiansen, S.: Numerical Treatment of an Integral Equation Origi-
nating from a Two-Dimensional Dirichlet Boundary Value
Problem. . . . . . . . . . . . . . . . .. 73
Eckhardt, U. and Mika, K.: Numerical Treatment of Incorrectly Posed
Problems. A Case Study. . . . . . . . . . . .. 92
Hock, W.: Ein Extrapolationsverfahren flir Volterra-Integralgleichun-
gen 2. Art . . . . . . . . . . . . . . . . . 102
Hoffmann, K.-H. und Kornstaedt, H.-J.: Zum inversen Stefan-Pro-
blem. . . . . . . . . . . . . 115
Kardestuncer, H.: The Simultaneous Use of Differential and Integral
Equations in One Physical Problem. . . . . . 144
Kershaw, D.: An Error Analysis for a Numerical Solution of the Ei-
genvalue Problem for Compact Positive Operators . . . 151
Knauff, W. and Kress, R.: A Modified Integral Equation Method for
the Electric Boundary-Value Problem for the Vector Helm-
holtz Equation . . . . . . . . . . . . . . . 157
Marti, J. T.: On the Numerical Stability in Solving Ill-Posed Problems 171
McCormick, S.: Mesh Refinement Methods for Integral Equations . 183
Opfer, G.: Evaluation of Weakly Singular Integrals . . . . .. 191
Spence,A. and Moore, G.: A Convergence Analysis for Turning Points
of Nonlinear Compact Operator Equations . . . 203
Tesei, A.: Approximation Results for Volterra Integro-Partial Diffe-
rential Equations. . . . . . . . . . . . . . . 213
Topfer, H.-J. und Volk, W.: Die numerische Behandlung von Integral-
gleichungen zweiter Art mittels Splinefunktionen . . . . 228
Wendland, W. L.: On Galerkin Collocation Methods for Integral
Equations of Elliptic Boundary Value Problems. . . 244
Atkinson 1
THE NUMERICAL SOLUTION OF LAPLACE'S EQUATION IN THREE DIMENSIONS--II
by
K. E. Atkinson
1. Introduction. A numerical procedure will be described and analyzed
for the solution of various boundary value problems for Laplace's equa
tion
~u - 0
in three dimensional regions. This paper will discuss both the Dirichlet
and Neumann problems, on regions both interior and exterior to a smooth
simple closed surface S. Each such problem is reduced to an equivalent
integral equation over S by representing the solution u as a single
or double layer potential; and the resulting integral equation is then
reformulated as an integral equation on the unit sphere U in m3,
using a simple change of variables. This final equation over U is
solved numerically using Galerkin's method, with spherical harmonics as
the ~asis functions. The resulting numerical method converges rapidly,
although great care must be taken to evaluate the Galerkin coefficients
as efficiently as possible.
The use of integral equations is a well-known approach to the devel
opment of the existence theory for Laplace's equation; for example, see
[llJ, [12], and [16, Chap. 12J. Integral equations have also been used
for quite some time as a basis for developing numerical methods for solv
ing Laplace's equation. For examples of more recent work, see [6J, [7J,
[8J, [12J, [13J, [21J, and [22J. Much of the recent work uses a finite
element framework in solving the integral equation, often using a low
order approximation (for a major exception see [22]). The resulting nu
merical methods are quite flexible for a large variety of surfaces; but
they are often slowly convergent. They also lead to relatively large
linear systems which must be solved by iteration.
The approach of the present work is to use high order global approx
imations which converge rapidly if the boundary S and the boundary data
are sufficiently smooth. The resulting method converges very quickly;
and the associated linear systems are much smaller and can be solved
without iteration. This global approximation procedure will be less
flexible or general than the finite element method, because of the
Atkinson 2
restriction to regions with a smooth boundary; but it will be better
suited to regions with a smooth boundary.
The next section contains preliminary definitions and results,
needed for the later work. Section 3 discusses both the interior and ex
terior Dirichlet problems, with computational considerations and numeri
cal examples given in section 4. The exterior Neumann problem is dis
cussed in section 5; and the interior Neumann problem and associated ca
pacitance problem are given in section 6.
2. Preliminary Results. Let Di denote an open, bounded, simply-con
nected region in m3 with a smooth boundary S; and let D denote the
e
region exterior to D. U S. We assume S is a Lyapunov surface. This
~
means there is a tangent plane at every point P of the surface; and
furthermore, at each P on the surface, there is a local representation
of the surface,
(2.1)
with c having partial derivatives which are Holder continuous with ex
ponent 0< AsL We say S E Ll,A' In addition, if the function c has
k~ order partial derivatives, all of which are Holder continuous with
exponent A, we say S E ~,A' For more precise definitions, see Atkin
son [4] or Gunter [11, pp. 1, 99].
For functions f(x,y,z) defined on S, consider evaluating them in
a neighborhood of a point P using the representation for points on S
near P given by (2.1). This results in an equivalent function F(S ,Tl)
=
f(x,y,z). If the function F is n times continuously differentiable
with all the nth derivatives satisfying a Holder condition with expo
nent A, then we say f E ~ , A(S), For more details, see Gunter [11,
p. 98].
Since the numerical method will be defined for integral equations
over U, the unit sphere in m3, we assume there is a smooth mapping
(2.1) "':1oUn-~ 10t S.
For f defined on S, let
Q E u.
m
We will assume that the mapping is so defined that