Theoretical Numerical Analysis: A Functional Analysis Framework

582 Pages·2005·5.861 MB·English

by  Kendall Atkinson, Weimin Han (auth.)

39
Texts in Applied Mathematics
Editors
J.E. Marsden
L. Sirovich
S.S. Antman
Advisors
G. Iooss
P. Holmes
D. Barkley
M. Dellnitz
P. Newton
Texts in Applied Mathematics
1. Sirovich:Introduction to Applied Mathematics.
2. Wiggins:Introduction to Applied Nonlinear Dynamical Systems and Chaos.
3. Hale/Koçak:Dynamics and Bifurcations.
4. Chorin/Marsden:A Mathematical Introduction to Fluid Mechanics, 3rd ed.
5. Hubbard/Weist:Differential Equations: A Dynamical Systems Approach:
Ordinary Differential Equations.
6. Sontag:Mathematical Control Theory: Deterministic Finite Dimensional
Systems, 2nd ed.
7. Perko:Differential Equations and Dynamical Systems, 3rd ed.
8. Seaborn:Hypergeometric Functions and Their Applications.
9. Pipkin:A Course on Integral Equations.
10. Hoppensteadt/Peskin:Modeling and Simulation in Medicine and
the Life Sciences, 2nd ed.
11. Braun:Differential Equations and Their Applications, 4th ed.
12. Stoer/Bulirsch:Introduction to Numerical Analysis, 3rd ed.
13. Renardy/Rogers:An Introduction to Partial Differential Equations.
14. Banks:Growth and Diffusion Phenomena: Mathematical Frameworks
and Applications.
15. Brenner/Scott:The Mathematical Theory of Finite Element Methods, 2nd ed.
16. Van de Velde:Concurrent Scientific Computing.
17. Marsden/Ratiu:Introduction to Mechanics and Symmetry, 2nd ed.
18. Hubbard/West:Differential Equations: A Dynamical Systems Approach:
Higher-Dimensional Systems.
19. Kaplan/Glass:Understanding Nonlinear Dynamics.
20. Holmes:Introduction to Perturbation Methods.
21. Curtain/Zwart:An Introduction to Infinite-Dimensional Linear Systems Theory.
22. Thomas:Numerical Partial Differential Equations: Finite Difference Methods.
23. Taylor:Partial Differential Equations: Basic Theory.
24. Merkin:Introduction to the Theory of Stability of Motion.
25. Naber:Topology, Geometry, and Gauge Fields: Foundations.
26. Polderman/Willems:Introduction to Mathematical Systems Theory:
A Behavioral Approach.
27. Reddy:Introductory Functional Analysis with Applications to Boundary-Value
Problems and Finite Elements.
28. Gustafson/Wilcox:Analytical and Computational Methods of Advanced
Engineering Mathematics.
29. Tveito/Winther:Introduction to Partial Differential Equations:
A Computational Approach.
30. Gasquet/Witomski:Fourier Analysis and Applications: Filtering,
Numerical Computation, Wavelets.
(continued after index)
Kendall Atkinson Weimin Han
Theoretical Numerical
Analysis
A Functional Analysis Framework
Second Edition
Kendall Atkinson Weimin Han
Department of Computer Science Department of Mathematics
Department of Mathematics University of Iowa
University of Iowa Iowa City, IA 52242
Iowa City, IA 52242 USA
USA [email protected]
[email protected]
Series Editors
J.E. Marsden L. Sirovich
Control and Dynamical Systems, 107–81 Division of Applied Mathematics
California Institute of Technology Brown University
Pasadena, CA 91125 Providence, RI 02912
USA USA
[email protected] [email protected]
S.S. Antman
Department of Mathematics
and
Institute for Physical Science
and Technology
University of Maryland
College Park, MD 20742-4015
USA
[email protected]
Mathematics Subject Classification (2000): 65-01
Library of Congress Control Number: 2005925512
ISBN-13: 978-0-387-25887-4 eISBN-13: 978-0-387-21526-6
Printed on acid-free paper.
© 2005, 2001 Springer Science+Business Media, LLC
All rights reserved. This work may not be translated or copied in whole or in part without the
written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring
Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or
scholarly analysis. Use in connection with any form of information storage and retrieval,
electronic adaptation, computer software, or by similar or dissimilar methodology now known
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The use in this publication of trade names, trademarks, service marks, and similar terms, even
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or not they are subject to proprietary rights.
9 8 7 6 5 4 3 2 (corrected as of the 2ndprinting, 2007)
springer.com
Dedicated to
Daisy and Clyde Atkinson
Hazel and Wray Fleming
and
Daqing Han, Suzhen Qin
Huidi Tang, Elizabeth and Michael
Series Preface
Mathematics is playing an ever more important role in the physical and
biological sciences, provoking a blurring of boundaries between scientific
disciplines and a resurgence of interest in the modern as well as the clas-
sical techniques of applied mathematics. This renewal of interest, both in
research and teaching, has led to the establishment of the series: Texts in
Applied Mathematics (TAM).
Thedevelopmentofnewcoursesisanaturalconsequenceofahighlevelof
excitement onthe researchfrontier as newer techniques, suchas numerical
and symbolic computer systems, dynamical systems, and chaos, mix with
and reinforce the traditional methods of applied mathematics. Thus, the
purpose of this textbook series is to meet the current and future needs of
these advances and to encourage the teaching of new courses.
TAM will publishtextbooks suitable for use in advancedundergraduate
and beginning graduate courses, and will complement the Applied Math-
ematical Sciences (AMS) series, which will focus on advanced textbooks
and research-levelmonographs.
Pasadena,California J.E. Marsden
Providence, Rhode Island L. Sirovich
College Park, Maryland S.S. Antman
Preface to the Second Edition
This textbook prepares graduate students theoretical background for re-
search in numerical analysis/computational mathematics and helps them
to move rapidly into a research program. It covers basic results of func-
tional analysis, approximation theory, Fourier analysis and wavelets, iter-
ation methods for nonlinear equations, finite difference methods, Sobolev
spaces and weak formulations of boundary value problems, finite element
methods, elliptic variational inequalities and their numerical solution, nu-
merical methods for solving integral equations of the second kind, and
boundary integral equations for planar regions with a smooth boundary
curve. The presentation of each topic is meant to be an introduction with
certaindegreeofdepth.Comprehensivereferencesonaparticulartopicare
listed at the end of each chapter for further reading and study.
The motivation and rationale for writing this textbook are explained in
Preface to the First Edition. For this second edition, we add a chapter on
Fourier analysis and wavelets, and three new sections in other chapters.
Many sections from the first edition have been revised to varying degrees.
Over 140 new exercises are included.
WethankB.Bialecki,R.Glowinski,andA.J.Meirfortheirhelpfulcom-
ments on the first edition of the book. It is a pleasure to acknowledge the
skillful editorial assistance from the Series Editor, Achi Dosanjh.
Preface to the First Edition
Thistextbookhasgrownoutofacoursewhichweteachperiodicallyatthe
University of Iowa. We have beginning graduate students in mathematics
whowishtoworkinnumericalanalysisfromatheoreticalperspective,and
theyneedabackgroundinthose“toolsofthetrade”whichwecoverinthis
text. Ordinarily, such students would begin with a one-year course in real
andcomplexanalysis,followedbyaoneortwosemestercourseinfunctional
analysis and possibly a graduate level course in ordinary differential equa-
tions, partial differential equations, or integral equations. We still expect
our students to take most of these standard courses, but we also want to
movethemmorerapidlyintoaresearchprogram.Thecoursebasedonthis
book allows these students to move more rapidly into a research program.
The textbook covers basic results of functional analysis and also some
additional topics which are needed in theoretical numerical analysis. Ap-
plications of this functional analysis are given by considering, at length,
numerical methods for solving partial differential equations and integral
equations.
Thematerialinthetextispresentedinamixedmanner.Sometopicsare
treatedwithcomplete rigour,whereasothersaresimply presentedwithout
proof and perhaps illustrated (e.g. the principle of uniform boundedness).
We have chosen to avoid introducing a formalized framework for Lebesgue
measure and integration and also for distribution theory. Instead we use
standard results on the completion of normed spaces and the unique ex-
tension of densely defined bounded linear operators. This permits us to
introducethe Lebesguespacesformallyandwithouttheir concrete realiza-
tionusingmeasuretheory.Theweakderivativecanbeintroducedsimilarly
xii Preface to theFirst Edition
usingtheuniqueextensionofdenselydefinedlinearoperators,avoidingthe
need for a formal development of distribution theory. We describe some of
thestandardmaterialonmeasuretheoryanddistributiontheoryinanintu-
itive manner,believing this is sufficientformuchofsubsequentmathemat-
ical development. In addition, we give a number of deeper results without
proof,citingthe existingliterature.Examplesofthis aretheopen mapping
theorem, Hahn-Banach theorem, the principle of uniform boundedness,and
a number of the results on Sobolev spaces.
Thechoiceoftopicshasbeenshapedbyourresearchprogramandinter-
ests at the University of Iowa. These topics are important elsewhere, and
we believe this text will be useful to students at other universities as well.
Thebookisdividedintochapters,sections,andsubsectionswhereappro-
priate. Mathematical relations (equalities and inequalities) are numbered
by chapter, section and their order of occurrence. For example, (1.2.3) is
thethirdnumberedmathematicalrelationinSection1.2ofChapter1.Defi-
nitions,examples,theorems,lemmas,propositions,corollariesandremarks
arenumberedconsecutivelywithineachsection,bychapterandsection.For
example, in Section 1.1, Definition 1.1.1 is followed by an example labeled
as Example 1.1.2.
The first three chapters cover basic results of functional analysis and
approximation theory that are needed in theoretical numerical analysis.
Earlyon,inChapter4,weintroducemethodsofnonlinearanalysis,asstu-
dentsshouldbeginearlytothinkaboutbothlinearandnonlinearproblems.
Chapter 5 is a short introduction to finite difference methods for solving
time-dependent problems. Chapter 6 is an introduction to Sobolev spaces,
giving different perspectives of them. Chapters 7 through 10 cover mater-
ialrelatedto elliptic boundaryvalueproblemsandvariationalinequalities.
Chapter 11 is a general introduction to numerical methods for solving in-
tegral equations of the second kind, and Chapter 12 gives an introduction
to boundaryintegralequationsfor planarregionswith a smoothboundary
curve.
Wegiveexercisesattheendofmostsections.Theexercisesarenumbered
consecutivelybychapterandsection.Attheendofeachchapter,weprovide
some short discussions of the literature, including recommendations for
additional reading.
During the preparation of the book, we received helpful suggestions
from numerous colleagues and friends. We particularly thank P.G. Ciar-
let, William A. Kirk, Wenbin Liu, and David Stewart. We also thank the
anonymousrefereeswhosesuggestionsleadtoanimprovementofthebook.
Contents
Series Preface vii
Preface to the Second Edition ix
Preface to the First Edition xi
1 Linear Spaces 1
1.1 Linear spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Normed spaces . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.1 Convergence . . . . . . . . . . . . . . . . . . . . . . 10
1.2.2 Banach spaces . . . . . . . . . . . . . . . . . . . . . 13
1.2.3 Completion of normed spaces . . . . . . . . . . . . . 14
1.3 Inner product spaces . . . . . . . . . . . . . . . . . . . . . . 21
1.3.1 Hilbert spaces. . . . . . . . . . . . . . . . . . . . . . 26
1.3.2 Orthogonality . . . . . . . . . . . . . . . . . . . . . . 28
1.4 Spaces of continuously differentiable functions . . . . . . . 38
1.4.1 H¨older spaces . . . . . . . . . . . . . . . . . . . . . . 41
1.5 Lp spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
1.6 Compact sets . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2 Linear Operators on Normed Spaces 51
2.1 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.2 Continuous linear operators . . . . . . . . . . . . . . . . . . 55
2.2.1 L(V,W) as a Banach space . . . . . . . . . . . . . . 59