Theoretical Numerical Analysis : A Functional Analysis Framework
by Atkinson, Kendall E.; Han, WeiminEdition: 1st
Format: Hardcover
Pub. Date: 2001-03-01
Publisher(s): Springer Verlag
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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 |
| Table of Contents provided by Rittenhouse. All Rights Reserved. |
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