An Introduction to the Mathematical Theory of Finite Elements
By J. T. Oden and J. N. Reddy
()
About this ebook
J. T. Oden is Director of the Institute for Computational Engineering & Sciences (ICES) at the University of Texas at Austin, and J. N. Reddy is a Professor of Engineering at Texas A&M University. They developed this essentially self-contained text from their seminars and courses for students with diverse educational backgrounds. Their effective presentation begins with introductory accounts of the theory of distributions, Sobolev spaces, intermediate spaces and duality, the theory of elliptic equations, and variational boundary value problems. The second half of the text explores the theory of finite element interpolation, finite element methods for elliptic equations, and finite element methods for initial boundary value problems. Detailed proofs of the major theorems appear throughout the text, in addition to numerous examples.
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An Introduction to the Mathematical Theory of Finite Elements - J. T. Oden
An Introduction
to the Mathematical Theory
of Finite Elements
J. T. Oden
The University of Texas, Austin
J. N. Reddy
Texas A&M University
DOVER PUBLICATIONS, INC.
Mineola, New York
Copyright
Copyright © 1976, 2011 by J. T. Oden and J. N. Reddy
All rights reserved.
Bibliographical Note
This Dover edition, first published in 2011, is an unabridged and corrected republication of the work originally published in 1976 by John Wiley & Sons, In., New York. J. T. Oden has provided a new Preface for this edition.
Library of Congress Cataloging-in-Publication Data
Oden, John Tinsley
An introduction to the theory of finite elements / J. T. Oden and J. N. Reddy.
p. cm.
Originally published: New York : Wiley, 1976
Includes bibliographical references and index.
ISBN-13: 978-0-486-46299-8
ISBN-10: 0-486-462299-4
1. Boundary Value Problems-Numerical Solutions. 2. Approximation Theory. 3. Finite element Method. II. Reddy, J. N. III. Title.
QA379.032009
518.64—dc22
2009007827
Manufactured in the United States by Courier Corporation
446299401
www.doverpublications.com
To the memory of
JOHN JAMES ODEN
PREFACE TO THE DOVER EDITION
The advent of finite element methods has arguably been one of the most important advances in the history of applied sciences, making possible the analysis of thousands of physical phenomena and engineering systems modeled by partial differential equations. The original edition of this book was written as an attempt to put the subject on a solid mathematical foundation, and to demonstrate that the theory of finite elements rests comfortably within the modern theory of PDE's. Since it was published, the field has undergone enormous development and change, and many of the earlier theoretical results have been displaced by deeper and far more general developments. Yet, many of the basic ideas remain unchanged: the notion of weak or generalized solutions, properties of Sobolev spaces, the embedding theories, theorems on existence and uniqueness of solutions, a priori error estimates, affine element families, etc. For this reason, much of the older work is still hoped to have merit beyond its historical value.
In completing this Dover Edition, I have had the generous help of many friends who aided in correcting errors and typos in the original volume. I take full responsibility for deficiencies that may remain and give special thanks to my dear colleagues Ivo Babuska, Leszek Demkowicz, Serge Prudhomme, and Andrea Hawkins, who assisted in proofing the latest editions. I also thank Charlott Low for invaluable help in preparing the manuscript for publication.
JTO
Austin, 2010
PREFACE
During a remarkably short span of years the subject of finite elements has expanded from a collection of effective techniques for solving practical problems in engineering and science to a rich and exciting branch of applied mathematics. The aim of this book is to present the student of engineering science or applied mathematics an introductory account of this mathematical theory.
The book has developed as a result of seminars and courses on finite-element theory taught by the authors at five universities in recent years to students with diverse backgrounds and often modest mathematical preparation. For such an audience, we have found it effective to begin the study with basic mathematical concepts and to systematically build on these the elements of approximation theory, Hilbert spaces, and partial differential equations essential to an understanding of the most important aspects of linear finite-element theory. This book essentially follows this plan. To keep the size and scope of the work within reasonable limits, it has been necessary to omit several important topics in favor of more basic ones. For example, we have not included material on nonlinear problems, integral equations, or eigenvalue problems. However, some of these subjects should be easily mastered by the reader of this book; other subjects must await study in future works.
We owe a great deal to those who developed the mathematical theory of finite elements in recent years. We have been particularly influenced by the work of Ivo Babuška and J. P. Aubin, and we have profited not only in writing this book but also in our own research, from the writings of Philippe Ciarlet, P. A. Raviart, J. L. Lions, and others, and from numerous discussions with our colleague, Ralph Showalter. The first author registers a special note of thanks to the Finite-Element Circus and to certain members of the Circus who have patiently discussed the subject with him; particularly Ivo Babuška, Jim Douglas, Ridgway Scott, Gilbert Strang, Al Schatz, Bruce Kellogg, Mary Wheeler, and James Bramble. We are also thankful for the advice we have received from several colleagues who read an early draft of the manuscript. In particular, we have benefited from the suggestions of John Cannon and Linda Hayes, who read the entire manuscript, and from the comments of Philippe Ciarlet. We also express thanks to M. G. Sheu, N. Kikuchi, and C. T. Reddy who helped with the proofreading. Much of our work on finite-element methods has been supported through the Air Force Office of Scientific Research and the U. S. National Science Foundation. We express our sincere gratitude for this support.
J. T. ODEN
J. N. REDDY
Austin, Texas
Norman, Oklahoma
January 1976
CONTENTS
Chapter 1Introduction
1.1 The Finite-Element Method
1.2 The Mathematics of Finite Elements
1.3 The Present Study
1.4 Notations and Preliminaries
References
PART IMATHEMATICAL FOUNDATIONS
Chapter 2Distributions, Mollifiers, and Mean Functions
2.1 Introduction
2.2 Functionals and Test Functions on One-Dimensional Domains
2.3 Distributions
2.4 Locally Integrable Generators, Regular and Singular Distributions
2.5 Some Properties of Distributions
2.6 Distributional Differential Equations
2.7 Distributions and Generalized Functions in Rn
2.8 Fourier Transforms, Rapidly Decaying Functions, and Tempered Distributions
2.9 Weak and Strong Derivatives in Lp(Ω)
2.10Mollifiers and Mean Functions
References
Chapter 3Theory of Sobolev Spaces
3.1 Introduction
3.3 Partitions of Unity, Boundaries, and Cone Conditions
3.5 The Sobolev Integral Identity
3.6 The Sobolev Embedding Theorems
References
Chapter 4Hilbert Space Theory of Traces and Intermediate Spaces
4.1 Introduction
4.2 Hilbert Spaces Hm(Ω) of Integer Order
4.3 Hilbert Spaces Hs(Rn) for Real s 0
4.4 Duals of Hilbert Spaces
4.5 Duals of Spaces Hs(Rn) and Hm(Ω)
4.7 Intermediate and Interpolation Spaces
4.8 Interpolation Theory in Hilbert Spaces
4.9 Hilbert Spaces Hs(∂Ω)
4.10 The Trace Theorem for Hs(Ω)
References
Chapter 5Some Elements of Elliptic Theory
5.1 Introduction
5.2 Linear Elliptic Operators
5.3 Boundary Conditions
5.4 Green's Formulas
5.5 Regularity Theory in Hs(Ω), s 2m
5.6 Compatibility Conditions—Existence and Uniqueness in Hs (Ω), s 2m
5.7 Existence and Regularity Theory in Hs(Ω), s < 2m
References
PART IITHE THEORY OF FINITE ELEMENTS
Chapter 6Finite-Element Interpolation
6.1 Introduction
6.3 Local and Global Representations of Functions
6.4 Restrictions, Prolongations, and Projections
6.5 Conjugate Basis Functions
6.6 Finite-Element Families
6.7 Accuracy of Finite-Element Interpolations
References
Chapter 7Variational Boundary-Value Problems
7.1 Introduction
7.2 Formulation of Variational Boundary-Value Problems
7.3 Coercive Bilinear Forms
7.4 Weak Coerciveness
7.5 Existence and Uniqueness of Solutions
References
Chapter 8Finite-Element Approximations of Elliptic Boundary-Value Problems
8.1 Introduction
8.2 Galerkin Approximations
8.3 Existence and Uniqueness of Galerkin Approximations
8.4 Finite-Element Approximations
8.5 Properties of Finite-Element Subspaces
8.6 Error Estimates
8.7 Pointwise and L∞(Ω) Error Estimates
8.8 Quadrature, Boundary, and Data Errors
8.9 H−1 Finite-Element Methods
8.10Hybrid and Mixed Finite-Element Methods
References
Chapter 9Time-Dependent Problems
9.1 Introduction
9.2 Finite-Element Models of the Diffusion Equation
9.3 Semidiscrete L2 Galerkin Approximations
9.4 Elements of Semigroup Theory
9.5 Semigroup Methods for Galerkin Approximations
9.6 Hyperbolic Equations of Second Order
9.7 First-Order Hyperbolic Equations
References
Author Index
Subject Index
An Introduction
to the Mathematical Theory
of Finite Elements
1
INTRODUCTION
1.1THE FINITE-ELEMENT METHOD
The finite-element method emerged from the engineering literature of the 1950's as one of the most powerful methods ever devised for the approximate solution of boundary-value problems. A relatively complete historical account is given in [1.1]. It is a variational method of approximation, making use of global or variational statements of physical problems and employing the Rayleigh–Ritz–Galerkin philosophy of constructing coordinate functions whose linear combinations represent the unknown solutions. The key to the success of the method is the unique way in which these coordinate functions are constructed: A given domain is represented as a collection of a number of geometrically simple subdomains (finite elements) connected together at certain nodal points. The variational problem is formulated approximately, for arbitrary boundary conditions, over each subdomain, necessitating only a choice of simple local coordinate functions for each element. These are generally polynomials. The global model of the problem is then obtained by simply fitting the elements together to depict a given domain, and summing, so to speak, the local contributions furnished by each element. We describe these ideas more precisely in Chapter 8.
The importance of this process of building a global approximation from a number of local ones is that it embodies a systematic procedure for constructing coordinate functions for arbitrary domains—thereby finally freeing the analyst of traditional difficulties due to irregular geometries and boundary conditions. Moreover, the coordinate functions generated in this way are usually piecewise polynomials with local compact support; i.e., they assume nonzero values only in a relatively small neighborhood of certain nodal points. This property normally leads to stable, well-conditioned equations for the properly posed boundary-value problem. Indeed, for linear elliptic problems, the method leads to a banded system of linear algebraic equations which can be solved using any of a variety of well-known techniques. All these features combine to make a quite general and effective method which has found fruitful application in practically every area of mathematical physics.
1.2THE MATHEMATICS OF FINITE ELEMENTS
To the structural engineer, the idea of building up a structure by fitting a number of structural elements together is quite a natural one. That is precisely how he visualizes a structure as being built, and engineers have developed and used stress analysis techniques based on this simple fact for many decades. The translation of these ideas into concrete mathematical statements, however, is not a trivial task and has come about only in relatively recent times.
It is not uncommon in the history of applied mathematics for a mathematical concept or method to be used successfully for years before its mathematical basis is completely understood. Expanding technology demands solutions to physical problems, and the competent technician must develop methods to solve them. When his methods fail or are too fragile for wide application, they are abandoned and forgotten. When they succeed and are sufficiently broad, they eventually attract the attention of mathematicians who are equipped to dissect them and unravel their intrinsic properties. Such was the case with complex variable theory, operational calculus and the theory of distributions, much of probability theory, and numerous other branches of mathematics.
The mathematical theory of finite elements was no exception. By 1965, hundreds of papers on the method had appeared in the engineering literature; it was widely used in industrial applications, was the subject of courses at most technical universities, and was firmly established as a general and powerful method of analysis. However, only a handful of mathematically oriented papers on the method appeared in the engineering literature of the 1960's, and only one or two purely mathematical papers on the subject appeared in 1968 and 1969. A complete mathematical theory began to be pieced together in the 1970's. It was about this time that the strength and elegance of the method and its relation to contemporary research in interpolation theory, splines, and differential equations began to be appreciated by the mathematical world. In a remarkably short time, a virtual flood of mathematical literature appeared on the subject, and a variety of important aspects of the method was quickly put on a sound mathematical footing.
Today the theory has reached a fairly high degree of development, at least as it applies to linear elliptic boundary-value problems, and its foundations are now recognized to be a natural union of spline theory and the modern theory of partial differential equations. In addition, finite-element methods occupy an increasingly important place in modern numerical analysis, and their implementation continues to prompt developments in computational methods and computer software.
1.3THE PRESENT STUDY
As the mathematical theory of finite elements continues to be developed in the framework described above, it becomes more inaccessible to the very practitioners who developed it, not to mention the beginning student who wishes to learn the underlying mathematical features. This book was written with these readers in mind. It represents a systematic introduction to the mathematical theory of finite elements, starting with relatively elementary mathematical concepts and proceeding to an exposé of basic mathematical properties of finite-element approximations of linear boundary-value problems.
The study is divided into two parts. Part I involves mathematical foundations and contains Chapters 2 through 5. Here all the groundwork for a theory of finite elements is laid. It begins with an elementary introduction to distribution theory, generalized derivatives, and mollifiers, and proceeds to an introductory account of the theory of Sobolev spaces. This is followed by a chapter on interpolation spaces and trace theorems, and a chapter on the theory of elliptic equations. Part II of the study is devoted to the theory of finite elements. It begins with Chapter 6, which is a chapter on finite-element interpolation in which many of the basic properties of finite-element models are described. Here we develop the special local and global properties of finite-element representations that deal with the connectivity of collections of elements and the decomposition of a given domain into a collection of elements. We also cite examples of several important families of finite elements, and we present a collection of theorems that establish the error of finite-element interpolations of smooth functions and of functions in certain Sobolev spaces. These theorems, which involve a priori error estimates for finite-element interpolations, are fundamental to the approximation theory taken up in Chapters 7 through 9. Chapter 7 contains an introduction to the theory of elliptic variational boundary-value problems, with some emphasis on properties of coercive bilinear forms and the equivalence of certain variational problems to classical ones. We also establish an extended version of the Lax–Milgram theorem, following Babuška, which establishes conditions for the existence of unique solutions to an abstract class of elliptic variational problems. We then show that practically all linear elliptic problems of interest fall within those covered by the theorem. In Chapter 8, all the theory developed in previous chapters is brought to bear on the central problem of the book: the approximation of linear elliptic boundary-value problems using finite elements. The basic elements of an approximation theory for finite elements is established, including criteria for stability, consistency, and convergence, as well as a priori error estimates. In Chapter 8 we also consider several special topics in finite-element theory such as boundary errors, L∞-and L2-error estimates, and quadrature errors, and we present a brief account of the theory of mixed and hybrid finite-element models. Finally, in Chapter 9 we present a summary account of finite-element methods applied to time-dependent problems wherein we confine ourselves to the simplest linear parabolic and hyperbolic problems.
1.4NOTATIONS AND PRELIMINARIES
As a prerequisite to reading this book, the reader should be equipped with an introductory course in functional analysis of a level roughly comparable to that covered in the texts of Kolmogorov and Fomin [1.2, 1.3]. However, much of what we cover in the earlier chapters assumes no more background than calculus and real analysis.
is read for every
or every,
a set A of elements a having property P is described as A = (a: a has property Pn = Πn (n times) denotes nto denote the conclusion of a logical unit, such as the completion of an example or the proof of a theorem.
As usual, Lpp ∞, denotes the space of all equivalence classes of real-valued (or complex-valued) Lebesgue-measurable functions u, whose pth powers |u|p are Lebesgue-integrable over Ω. The norm on Lp(Ω) is given by
(1.1)
dx = mes(dx1 dx2 · · · dxn). The dual of Lp(Ω) is Lq(Ω), where 1/p + 1/q , they satisfy Hölder's inequality
(1.2)
There are several basic theorems concerning properties of normed linear spaces, Hilbert spaces, and linear operators that play an important role in some of the developments to be presented later. For example, we mention
1. The Hahn–Banach theorem
2. The Riesz representation theorem
3. The projection theorem for Hilbert spaces
4. The closed graph theorem and the Banach theorem for linear operators on Banach spaces
5. The spectral theorem for linear operators.
Statements and proofs of these theorems can be found in standard texts on functional analysis (see, e.g., [1.2–1.7]). However, we record statements of some of these theorems in appropriate places in the chapters to follow. In the sequel, other notations and conventions are defined where they first appear.
REFERENCES
1.1. Oden, J. T., Finite Elements of Nonlinear Continua, McGraw-Hill, New York, 1972.
1.2. Kolmogorov, A. N. and Fomin, S. V., Elements of the Theory of Functions and Functional Analysis, Vol. I: Metric and Normed Spaces, Translated from the 1954 Russian edition by L. F. Boron, Graylock Press, Rochester, N.Y., 1957.
1.3. Kolmogorov, A. N. and Fomin, S. V., Elements of the Theory of Functions and Functional Analysis, Vol. II: Measure. The Lebesque Integral. Hilbert Space, Translated from the 1960 Russian edition by H. Kamel and H. Kromm, Graylock Press, Albany, N.Y., 1960.
1.4. Naylor, A. W. and Sell, G. R., Linear Operator Theory in Engineering and Science, Holt, N.Y., 1971.
1.5. Taylor, A. E., Introduction to Functional Analysis, John Wiley, N.Y., 1958.
1.6. Dunford, N. and Schwartz, J. T., Linear Operators, Part I: General Theory, Interscience, N.Y., 1958.
1.7. Yosida, K., Functional Analysis, 3rd ed., Springer-Verlag, New York, 1971.
PART I
MATHEMATICAL
FOUNDATIONS
2
DISTRIBUTIONS,
MOLLIFIERS, AND
MEAN FUNCTIONS
2.1INTRODUCTION
In many problems in mathematical physics we encounter relations that can be defined precisely only in terms of an integral—or, more generally, in terms of the action of some operator on functions of a given class. For example, the Dirac delta function δ (x) has the property
(2.1)
where f . Attempts at actually defining δ(x) as a function of x are at best only symbolic; δ(x) is simply not a function in the usual sense. Its nonzero values do not exist for any x, and we can interpret it only in terms of the numbers f(a) it produces when it operates, via (2.1), on functions f of a given class.
The need for handling such mathematical objects as the Dirac delta almost always arises in problems with discontinuous data and especially point sources, e.g., points of concentrated nonhomogeneous data, discontinuous coefficients or, more importantly, in attempts to differentiate discontinuous functions. Concentrated forces on elastic bodies, impulsive forces on bodies in dynamics, and point electric charges in electrostatics are examples of point sources encountered in physical situations that have mathematical descriptions akin to (2.1), and all of which lead to mathematical difficulties if they are interpreted merely as functions in the classical sense. All these difficulties in handling point sources are resolved by defining a given source by its action on a special class of test functions. The source
δ(x − a) in (2.1), for example, is defined by f(a); the symbolism δ(x − a) really signifies that we have some sort of operation whose effect on a member f is the real number f(a). Effectively, δ(x − aa real number f(a). We refer to functionals of this type as distributions, a term we define more precisely below. For convenience in discussing properties of distributions, we often assign to each distribution, such as the Dirac delta δ, a special symbolism that makes it appear to be a function, e.g., δ(x); the symbolic δ(x) is called a generalized function. We now set out to generalize and make more precise all these concepts.
2.2FUNCTIONALS AND TEST FUNCTIONS ON ONE-DIMENSIONAL DOMAINS
= (− ∞, ∞). Let Cas a mapping fdefined by some sort of rule, y = f(x). Here y [or f(x)] is the value of f at x or the image of x under the mapping f, and the rule or law of correspondence of y and x is established by a specific formula for f(x). Geometrically, we can plot a curve showing the variation of y with x by preparing a table of values such as
Then, with each point xi , a definite value f(xi) of y exists, and f is viewed as a collection of pairs of points in the xy plane.
Now the function f(x(x, the exact character of which will be described later, and consider the scalar product s: fdefined by the Lebesgue integral,
(2.2)
) is well defined, we can construct a new representation of f ) via a table of the type
In other words, f is now described by a collection of weighted averages (x).
Now we recall that a mapping q of a linear vector space V a real number q) is called a functional on V) can be easily endowed with the structure of a linear space, it is now clear that the second tabular representation of f given above describes f (x). This latter visualization of f (x) have been established.
Obviously, if the functional representation of f (x(x) must be chosen so that two different continuous functions f1(x) and f2(x(x) to belong to a class of functions called test functions. To describe this class more precisely, we first introduce the following definition.
Support of a Function
The support of a function f on which f(x= supp f(x).
We refer to the subset of C) consisting of functions with compact support . That is, the set
(2.3)
.
Test Functions
= (− ∞, ∞) called the class of test functions ), which has the following properties:
.
converges if
n(x.
(ii.2) The derivative of any given order r n(x) converge uniformly, as n→∞, to the corresponding derivative of order r (x).
n n(Ω) to describe these classes.
(x(x.
) is a fairly delicate property designed (as we soon see) to provide a broad generalization of the concept of derivatives and differentiation. To better appreciate property (ii) consider what is meant by a null sequence n(xn(xn(xn(x) and all its derivatives with respect to x must approach zero uniformly as n→∞:
(2.4)
is a linear vector space. Indeed, if α and β 1(x2(x) are two test functions, α 1(x) + β 2(x.
n(K. The functions
(2.5)
(K(K)'s as K )], however, is not metrizable.
Fortunately, we