Theoretical Numerical Analysis : A Functional Analysis Framework

by Atkinson, Kendall E.; Han, Weimin

• ISBN13: 9780387951423
• ISBN10: 0387951423
• Edition: 1st
• Format: Hardcover
• Copyright: 2001-03-01
• Publisher: Springer Verlag

Summary

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.

Table of Contents

Series Preface	p. vii
Preface	p. ix
Linear Spaces	p. 1
Linear spaces	p. 1
Normed spaces	p. 7
Convergence	p. 9
Banach spaces	p. 11
Completion of normed spaces	p. 12
Inner product spaces	p. 18
Hilbert spaces	p. 22
Orthogonality	p. 23
Spaces of continuously differentiable functions	p. 30
Holder spaces	p. 31
L[superscript p] spaces	p. 32
Compact sets	p. 35
Linear Operators on Normed Spaces	p. 38
Operators	p. 39
Continuous linear operators	p. 41
L(V,W) as a Banach space	p. 45
The geometric series theorem and its variants	p. 46
A generalization	p. 49
A perturbation result	p. 50
Some more results on linear operators	p. 55
An extension theorem	p. 55
Open mapping theorem	p. 57
Principle of uniform boundedness	p. 58
Convergence of numerical quadratures	p. 59
Linear functionals	p. 62
An extension theorem for linear functionals	p. 63
The Riesz representation theorem	p. 64
Adjoint operators	p. 67
Types of convergence	p. 72
Compact linear operators	p. 73
Compact integral operators on C(D)	p. 74
Properties of compact operators	p. 76
Integral operators on L[superscript 2](a,b)	p. 78
The Fredholm alternative theorem	p. 79
Additional results on Fredholm integral equations	p. 83
The resolvent operator	p. 87
R([lambda]) as a holomorphic function	p. 89
Approximation Theory	p. 92
Interpolation theory	p. 93
Lagrange polynomial interpolation	p. 94
Hermite polynomial interpolation	p. 98
Piecewise polynomial interpolation	p. 98
Trigonometric interpolation	p. 101
Best approximation	p. 105
Convexity, lower semicontinuity	p. 105
Some abstract existence results	p. 107
Existence of best approximation	p. 110
Uniqueness of best approximation	p. 111
Best approximations in inner product spaces	p. 113
Orthogonal polynomials	p. 117
Projection operators	p. 121
Uniform error bounds	p. 124
Uniform error bounds for L[superscript 2]-approximations	p. 126
Interpolatory projections and their convergence	p. 128
Nonlinear Equations and Their Solution by Iteration	p. 131
The Banach fixed-point theorem	p. 131
Applications to iterative methods	p. 135
Nonlinear equations	p. 135
Linear systems	p. 136
Linear and nonlinear integral equations	p. 139
Ordinary differential equations in Banach spaces	p. 143
Differential calculus for nonlinear operators	p. 146
Frechet and Gateaux derivatives	p. 146
Mean value theorems	p. 149
Partial derivatives	p. 151
The Gateaux derivative and convex minimization	p. 152
Newton's method	p. 154
Newton's method in a Banach space	p. 155
Applications	p. 157
Completely continuous vector fields	p. 159
The rotation of a completely continuous vector field	p. 161
Conjugate gradient iteration	p. 162
Finite Difference Method	p. 171
Finite difference approximations	p. 171
Lax equivalence theorem	p. 177
More on convergence	p. 186
Sobolev Spaces	p. 193
Weak derivatives	p. 193
Sobolev spaces	p. 198
Sobolev spaces of integer order	p. 199
Sobolev spaces of real order	p. 204
Sobolev spaces over boundaries	p. 206
Properties	p. 207
Approximation by smooth functions	p. 207
Extensions	p. 208
Sobolev embedding theorems	p. 208
Traces	p. 210
Equivalent norms	p. 211
A Sobolev quotient space	p. 215
Characterization of Sobolev spaces via the Fourier transform	p. 219
Periodic Sobolev spaces	p. 222
The dual space	p. 225
Embedding results	p. 226
Approximation results	p. 227
An illustrative example of an operator	p. 228
Spherical polynomials and spherical harmonics	p. 229
Integration by parts formulas	p. 234
Variational Formulations of Elliptic Boundary Value Problems	p. 238
A model boundary value problem	p. 239
Some general results on existence and uniqueness	p. 241
The Lax-Milgram lemma	p. 244
Weak formulations of linear elliptic boundary value problems	p. 248
Problems with homogeneous Dirichlet boundary conditions	p. 249
Problems with non-homogeneous Dirichlet boundary conditions	p. 249
Problems with Neumann boundary conditions	p. 251
Problems with mixed boundary conditions	p. 253
A general linear second-order elliptic boundary value problem	p. 254
A boundary value problem of linearized elasticity	p. 257
Mixed and dual formulations	p. 260
Generalized Lax-Milgram lemma	p. 264
A nonlinear problem	p. 265
The Galerkin Method and Its Variants	p. 270
The Galerkin method	p. 270
The Petrov-Galerkin method	p. 276
Generalized Galerkin method	p. 278
Finite Element Analysis	p. 281
One-dimensional examples	p. 283
Linear elements for a second-order problem	p. 283
High-order elements and the condensation technique	p. 286
Reference element technique, non-conforming method	p. 288
Basics of the finite element method	p. 291
Triangulation	p. 291
Polynomial spaces on the reference elements	p. 293
Affine-equivalent finite elements	p. 295
Finite element spaces	p. 296
Interpolation	p. 298
Error estimates of finite element interpolations	p. 300
Interpolation error estimates on the reference element	p. 300
Local interpolation error estimates	p. 301
Global interpolation error estimates	p. 304
Convergence and error estimates	p. 308
Elliptic Variational Inequalities and Their Numerical Approximations	p. 313
Introductory examples	p. 313
Elliptic variational inequalities of the first kind	p. 319
Approximation of EVIs of the first kind	p. 323
Elliptic variational inequalities of the second kind	p. 326
Approximation of EVIs of the second kind	p. 331
Regularization technique	p. 333
Method of Lagrangian multipliers	p. 335
Method of numerical integration	p. 337
Numerical Solution of Fredholm Integral Equations of the Second Kind	p. 342
Projection methods: General theory	p. 343
Collocation methods	p. 343
Galerkin methods	p. 345
A general theoretical framework	p. 346
Examples	p. 351
Piecewise linear collocation	p. 351
Trigonometric polynomial collocation	p. 354
A piecewise linear Galerkin method	p. 356
A Galerkin method with trigonometric polynomials	p. 358
Iterated projection methods	p. 362
The iterated Galerkin method	p. 364
The iterated collocation solution	p. 366
The Nystrom method	p. 372
The Nystrom method for continuous kernel functions	p. 373
Properties and error analysis of the Nystrom method	p. 376
Collectively compact operator approximations	p. 383
Product integration	p. 385
Error analysis	p. 388
Generalizations to other kernel functions	p. 390
Improved error results for special kernels	p. 392
Product integration with graded meshes	p. 392
The relationship of product integration and collocation methods	p. 396
Projection methods for nonlinear equations	p. 398
Linearization	p. 398
A homotopy argument	p. 401
The approximating finite-dimensional problem	p. 402
Boundary Integral Equations	p. 405
Boundary integral equations	p. 406
Green's identities and representation formula	p. 407
The Kelvin transformation and exterior problems	p. 409
Boundary integral equations of direct type	p. 413
Boundary integral equations of the second kind	p. 419
Evaluation of the double layer potential	p. 421
The exterior Neumann problem	p. 425
A boundary integral equation of the first kind	p. 431
A numerical method	p. 433
References	p. 436
Index	p. 445
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