Theoretical Numerical Analysis: A Functional Analysis Framework (Texts in Applied Mathematics)

The book presents an abstract point of view of Numerical Analysis (as one can immediatly see by the title!). It is written by a master in the topic, author of more than 70 publications at the higher levels, well known for his contributions in Integral and Partial Differential Equations. If one is interested on the basic aspects of numerical analysis, I also suggest to consider his well known manual "Elementary Numerical Analysis".The present book presents several aspects that are not covered by most of the manuals in Numerical Analysis and highly contributes to have a wider idea of convergence and stability of some well known methods. Series Preface......Page 7 Preface to the Second Edition......Page 9 Preface to the First Edition......Page 11 Contents......Page 13 1.1 Linear spaces......Page 19 1.2 Normed spaces......Page 25 1.3 Inner product spaces......Page 39 1.4 Spaces of continuously di.erentiable functions......Page 56 1.5 L^p spaces......Page 61 1.6 Compact sets......Page 65 2 Linear Operators on Normed Spaces......Page 69 2.1 Operators......Page 70 2.2 Continuous linear operators......Page 73 2.3 The geometric series theorem and its variants......Page 78 2.4 Some more results on linear operators......Page 89 2.5 Linear functionals......Page 97 2.6 Adjoint operators......Page 103 2.7 Types of convergence......Page 108 2.8 Compact linear operators......Page 111 2.9 The resolvent operator......Page 125 3 Approximation Theory......Page 131 3.1 Approximation of continuous functions by polynomials......Page 132 3.2 Interpolation theory......Page 133 3.3 Best approximation......Page 147 3.4 Best approximations in inner product spaces, projection on closed convex sets......Page 157 3.5 Orthogonal polynomials......Page 164 3.6 Projection operators......Page 168 3.7 Uniform error bounds......Page 172 4.1 Fourier series......Page 179 4.2 Fourier transform......Page 193 4.3 Discrete Fourier transform......Page 198 4.4 Haar wavelets......Page 203 4.5 Multiresolution analysis......Page 211 5 Nonlinear Equations and Their Solution by Iteration......Page 219 5.1 The Banach .xed-point theorem......Page 220 5.2 Applications to iterative methods......Page 224 5.3 Di.erential calculus for nonlinear operators......Page 237 5.4 Newton’s method......Page 248 5.5 Completely continuous vector .elds......Page 254 5.6 Conjugate gradient method for operator equations......Page 257 6.1 Finite di.erence approximations......Page 267 6.2 Lax equivalence theorem......Page 274 6.3 More on convergence......Page 283 7.1 Weak derivatives......Page 291 7.2 Sobolev spaces......Page 297 7.3 Properties......Page 307 7.4 Characterization of Sobolev spaces via the Fourier transform......Page 321 7.5 Periodic Sobolev spaces......Page 325 7.6 Integration by parts formulas......Page 337 8 Variational Formulations of Elliptic Boundary Value Problems......Page 341 8.1 A model boundary value problem......Page 342 8.2 Some general results on existence and uniqueness......Page 344 8.3 The Lax-Milgram Lemma......Page 348 8.4 Weak formulations of linear elliptic boundary value problems......Page 352 8.5 A boundary value problem of linearized elasticity......Page 361 8.6 Mixed and dual formulations......Page 366 8.7 Generalized Lax-Milgram Lemma......Page 371 8.8 A nonlinear problem......Page 373 9.1 The Galerkin method......Page 379 9.2 The Petrov-Galerkin method......Page 385 9.3 Generalized Galerkin method......Page 388 9.4 Conjugate gradient method: variational formulation......Page 390 10 Finite Element Analysis......Page 395 10.1 One-dimensional examples......Page 397 10.2 Basics of the .nite element method......Page 405 10.3 Error estimates of .nite element interpolations......Page 414 10.4 Convergence and error estimates......Page 422 11.1 Introductory examples......Page 431 11.2 Elliptic variational inequalities of the .rst kind......Page 438 11.3 Approximation of EVIs of the .rst kind......Page 443 11.4 Elliptic variational inequalities of the second kind......Page 446 11.5 Approximation of EVIs of the second kind......Page 452 12 Numerical Solution of Fredholm Integral Equations of the Second Kind......Page 465 12.1 Projection methods: General theory......Page 466 12.2 Examples......Page 474 12.3 Iterated projection methods......Page 486 12.4 The Nystr ̈om method......Page 496 12.5 Product integration......Page 510 12.6 Iteration methods......Page 522 12.7 Projection methods for nonlinear equations......Page 533 13 Boundary Integral Equations......Page 541 13.1 Boundary integral equations......Page 542 13.2 Boundary integral equations of the second kind......Page 555 13.3 A boundary integral equation of the .rst kind......Page 567 References......Page 573 Index......Page 587 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). The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this text book series is to meet the current and future needs of these advances and encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Math­ ematical Sciences (AMS) series, which will focus on advanced textbooks and research level monographs. "This textbook prepares graduate students for research in numerical analysis/computational mathematics by giving to them a mathematical framework embedded in functional analysis and focused on numerical analysis. This helps the student to move rapidly into a research program. The presentation of each topic is meant to be an introduction with certain degree of depth. Comprehensive references on a particular topic are listed at the end of each chapter for further reading and study. In this new edition many sections from the first edition have been revised to varying degrees as well as over 140 new exercises added. A new chapter on Fourier Analysis and wavelets has been included."--Jacket This book gives an introduction to functional analysis in a way that is tailored to fit the needs of the researcher or student. The book explains the basic results of functional analysis as well as relevant topics in numerical analysis. Applications of functional analysis are given by considering numerical methods for solving partial differential equations and integral equations. The material is especially useful for researchers and students who wish to work in theoretical numerical analysis and seek a background in the "tools of the trade" covered in this book. This textbook covers basic results of functional analysis and also some additional topics which are needed in theoretical numerical analysis. For this second edition, a new chapter on Fourier analysis and wavelets and over 140 new exercises have been added, almost doubling the exercise amount from the last edition. Many sections from the first edition have been revised. Some of the other topics covered in this book are functional analysis and approximation theory, nonlinear analysis, Sobolev spaces, elliptic boundary value problems and variational inequalities. Overall, the book is clearly written, quite pleasant to read, and contains a lot of important material; and the authors have done an excellent job at balancing theoretical developments, interesting examples and exercises, numerical experiments, and bibliographical references. - R. Glowinski, SIAM Review