Atkinson / Han Theoretical Numerical Analysis

A Functional Analysis Framework
Third Auflage 2009
ISBN: 978-1-4419-0458-4
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 625 Seiten

Reihe: Mathematics and Statistics (R0)

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 text covers basic results of functional analysis, approximation theory, Fourier analysis and wavelets, iteration 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, numerical methods for solving integral equations of the second kind, and boundary integral equations for planar regions. 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.  Because of the relevance in solving real world problems, multivariable polynomials are playing an ever more important role in research and applications. In this third editon, a new chapter on this topic has been included and some major changes are made on two chapters from the previous edition. In addition, there are numerous minor changes throughout the entire text and new exercises are added. Review of earlier edition: '...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, 2003

Autoren/Hrsg.

Atkinson, Kendall

Han, Weimin

Inhaltsverzeichnis

1;Series Preface;6
2;Preface;7
3;Contents;9
4;1 Linear Spaces;15
4.1;1.1 Linear spaces;15
4.2;1.2 Normed spaces;21
4.3;1.3 Inner product spaces;36
4.4;1.4 Spaces of continuously di erentiable functions;53
4.5;1.6 Compact sets;63
5;2 Linear Operators on Normed Spaces;65
5.1;2.1 Operators;66
5.2;2.2 Continuous linear operators;69
5.3;2.3 The geometric series theorem and its variants;74
5.4;2.4 Some more results on linear operators;86
5.5;2.5 Linear functionals;93
5.6;2.6 Adjoint operators;99
5.7;2.7 Weak convergence and weak compactness;104
5.8;2.8 Compact linear operators;109
5.9;2.9 The resolvent operator;123
6;3 Approximation Theory;128
6.1;3.1 Approximation of continuous functions by polynomials;129
6.2;3.2 Interpolation theory;131
6.3;3.3 Best approximation;144
6.4;3.4 Best approximations in inner product spaces, projection on closed convex sets;155
6.5;3.5 Orthogonal polynomials;162
6.6;3.6 Projection operators;167
6.7;3.7 Uniform error bounds;170
7;4 Fourier Analysis and Wavelets;180
7.1;4.1 Fourier series;180
7.2;4.2 Fourier transform;194
7.3;4.3 Discrete Fourier transform;200
7.4;4.4 Haar wavelets;204
7.5;4.5 Multiresolution analysis;212
8;5 Nonlinear Equations and Their Solution by Iteration;220
8.1;5.1 The Banach fixed-point theorem;221
8.2;5.2 Applications to iterative methods;225
8.3;5.3 Diferential calculus for nonlinear operators;238
8.4;5.4 Newton’s method;249
8.5;5.5 Completely continuous vector fields;254
8.6;5.6 Conjugate gradient method for operator equations;258
9;6 Finite Difference Method;266
9.1;6.1 Finite di erence approximations;266
9.2;6.2 Lax equivalence theorem;273
9.3;6.3 More on convergence;282
10;7 Sobolev Spaces;289
10.1;7.1 Weak derivatives;289
10.2;7.2 Sobolev spaces;295
10.3;7.3 Properties;305
10.4;7.4 Characterization of Sobolev spaces via the Fourier transform;320
10.5;7.5 Periodic Sobolev spaces;323
11;8 Weak Formulations of Elliptic Boundary Value Problems;339
11.1;8.1 A model boundary value problem;340
11.2;8.2 Some general results on existence and uniqueness;342
11.3;8.3 The Lax-Milgram Lemma;346
11.4;8.4 Weak formulations of linear elliptic boundary value problems;350
11.5;8.5 A boundary value problem of linearized elasticity;360
11.6;8.6 Mixed and dual formulations;366
11.7;8.7 Generalized Lax-Milgram Lemma;371
11.8;8.8 A nonlinear problem;373
12;9 The Galerkin Method and Its Variants;378
12.1;9.1 The Galerkin method;378
12.2;9.2 The Petrov-Galerkin method;385
12.3;9.3 Generalized Galerkin method;387
12.4;9.4 Conjugate gradient method: variational formulation;389
13;10 Finite Element Analysis;394
13.1;10.1 One-dimensional examples;395
13.2;10.2 Basics of the finite element method;404
13.3;10.3 Error estimates of finite element interpolations;417
13.4;10.4 Convergence and error estimates;426
14;11 Elliptic Variational Inequalities and Their Numerical Approximation;434
14.1;11.1 From variational equations to variational inequalities;434
14.2;11.2 Existence and uniqueness based on convex minimization;439
14.3;11.3 Existence and uniqueness results for a family of EVIs;441
14.4;11.4 Numerical approximations;453
14.5;11.5 Some contact problems in elasticity;469
15;12 Numerical Solution of Fredholm Integral Equations of the Second Kind;483
15.1;12.1 Projection methods: General theory;484
15.2;12.2 Examples;493
15.3;12.3 Iterated projection methods;504
15.4;12.4 The Nyström method;514
15.5;12.5 Product integration;528
15.6;12.6 Iteration methods;541
15.7;12.7 Projection methods for nonlinear equations;552
16;13 Boundary Integral Equations;560
16.1;13.1 Boundary integral equations;561
16.2;13.2 Boundary integral equations of the second kind;574
16.3;13.3 A boundary integral equation of the first kind;586
17;14 Multivariable Polynomial Approximations;591
17.1;14.1 Notation and best approximation results;591
17.2;14.2 Orthogonal polynomials;593
17.3;14.3 Hyperinterpolation;600
18;References;608
19;Index;623