Theoretical numerical analysis : a functional analysis framework / Kendall Atkinson, Weimin Han.

Format:

Book

Author/Creator:

Atkinson, Kendall E.

Contributor:

Han, Weimin.

Anne and Joseph Trachtman Memorial Book Fund.

Series:

Texts in applied mathematics ; 39.

Texts in applied mathematics ; 39

Language:

English

Subjects (All):

Functional analysis.

Physical Description:

xvi, 450 pages : illustrations ; 24 cm.

Place of Publication:

New York : Springer, [2001]

Summary:

An introduction to functional analysis intended for graduate students pursuing research involving numerical analysis at a more theoretical and abstract level. The book blends functional analysis, numerical analysis and computational techniques including extensive exercises.

Contents:

1 Linear Spaces 1

1.2 Normed spaces 7

1.2.1 Convergence 9

1.2.2 Banach spaces 11

1.2.3 Completion of normed spaces 12

1.3 Inner product spaces 18

1.3.1 Hilbert spaces 22

1.3.2 Orthogonality 23

1.4 Spaces of continuously differentiable functions 30

1.4.1 Holder spaces 31

1.5 L[superscript p] spaces 32

1.6 Compact sets 35

2 Linear Operators on Normed Spaces 38

2.1 Operators 39

2.2 Continuous linear operators 41

2.2.1 L(V,W) as a Banach space 45

2.3 The geometric series theorem and its variants 46

2.3.1 A generalization 49

2.3.2 A perturbation result 50

2.4 Some more results on linear operators 55

2.4.1 An extension theorem 55

2.4.2 Open mapping theorem 57

2.4.3 Principle of uniform boundedness 58

2.4.4 Convergence of numerical quadratures 59

2.5 Linear functionals 62

2.5.1 An extension theorem for linear functionals 63

2.5.2 The Riesz representation theorem 64

2.6 Adjoint operators 67

2.7 Types of convergence 72

2.8 Compact linear operators 73

2.8.1 Compact integral operators on C(D) 74

2.8.2 Properties of compact operators 76

2.8.3 Integral operators on L[superscript 2](a,b) 78

2.8.4 The Fredholm alternative theorem 79

2.8.5 Additional results on Fredholm integral equations 83

2.9 The resolvent operator 87

2.9.1 R([lambda]) as a holomorphic function 89

3 Approximation Theory 92

3.1 Interpolation theory 93

3.1.1 Lagrange polynomial interpolation 94

3.1.2 Hermite polynomial interpolation 98

3.1.3 Piecewise polynomial interpolation 98

3.1.4 Trigonometric interpolation 101

3.2 Best approximation 105

3.2.1 Convexity, lower semicontinuity 105

3.2.2 Some abstract existence results 107

3.2.3 Existence of best approximation 110

3.2.4 Uniqueness of best approximation 111

3.3 Best approximations in inner product spaces 113

3.4 Orthogonal polynomials 117

3.5 Projection operators 121

3.6 Uniform error bounds 124

3.6.1 Uniform error bounds for L[superscript 2]-approximations 126

3.6.2 Interpolatory projections and their convergence 128

4 Nonlinear Equations and Their Solution by Iteration 131

4.1 The Banach fixed-point theorem 131

4.2 Applications to iterative methods 135

4.2.1 Nonlinear equations 135

4.2.2 Linear systems 136

4.2.3 Linear and nonlinear integral equations 139

4.2.4 Ordinary differential equations in Banach spaces 143

4.3 Differential calculus for nonlinear operators 146

4.3.1 Frechet and Gateaux derivatives 146

4.3.2 Mean value theorems 149

4.3.3 Partial derivatives 151

4.3.4 The Gateaux derivative and convex minimization 152

4.4 Newton's method 154

4.4.1 Newton's method in a Banach space 155

4.4.2 Applications 157

4.5 Completely continuous vector fields 159

4.5.1 The rotation of a completely continuous vector field 161

4.6 Conjugate gradient iteration 162

5 Finite Difference Method 171

5.1 Finite difference approximations 171

5.2 Lax equivalence theorem 177

5.3 More on convergence 186

6 Sobolev Spaces 193

6.1 Weak derivatives 193

6.2.1 Sobolev spaces of integer order 199

6.2.2 Sobolev spaces of real order 204

6.2.3 Sobolev spaces over boundaries 206

6.3 Properties 207

6.3.1 Approximation by smooth functions 207

6.3.2 Extensions 208

6.3.3 Sobolev embedding theorems 208

6.3.4 Traces 210

6.3.5 Equivalent norms 211

6.3.6 A Sobolev quotient space 215

6.4 Characterization of Sobolev spaces via the Fourier transform 219

6.5 Periodic Sobolev spaces 222

6.5.1 The dual space 225

6.5.2 Embedding results 226

6.5.3 Approximation results 227

6.5.4 An illustrative example of an operator 228

6.5.5 Spherical polynomials and spherical harmonics 229

6.6 Integration by parts formulas 234

7 Variational Formulations of Elliptic Boundary Value Problems 238

7.1 A model boundary value problem 239

7.2 Some general results on existence and uniqueness 241

7.3 The Lax-Milgram lemma 244

7.4 Weak formulations of linear elliptic boundary value problems 248

7.4.1 Problems with homogeneous Dirichlet boundary conditions 249

7.4.2 Problems with non-homogeneous Dirichlet boundary conditions 249

7.4.3 Problems with Neumann boundary conditions 251

7.4.4 Problems with mixed boundary conditions 253

7.4.5 A general linear second-order elliptic boundary value problem 254

7.5 A boundary value problem of linearized elasticity 257

7.6 Mixed and dual formulations 260

7.7 Generalized Lax-Milgram lemma 264

7.8 A nonlinear problem 265

8 The Galerkin Method and Its Variants 270

8.1 The Galerkin method 270

8.2 The Petrov-Galerkin method 276

8.3 Generalized Galerkin method 278

9 Finite Element Analysis 281

9.1 One-dimensional examples 283

9.1.1 Linear elements for a second-order problem 283

9.1.2 High-order elements and the condensation technique 286

9.1.3 Reference element technique, non-conforming method 288

9.2 Basics of the finite element method 291

9.2.1 Triangulation 291

9.2.2 Polynomial spaces on the reference elements 293

9.2.3 Affine-equivalent finite elements 295

9.2.4 Finite element spaces 296

9.2.5 Interpolation 298

9.3 Error estimates of finite element interpolations 300

9.3.1 Interpolation error estimates on the reference element 300

9.3.2 Local interpolation error estimates 301

9.3.3 Global interpolation error estimates 304

9.4 Convergence and error estimates 308

10 Elliptic Variational Inequalities and Their Numerical Approximations 313

10.2 Elliptic variational inequalities of the first kind 319

10.3 Approximation of EVIs of the first kind 323

10.4 Elliptic variational inequalities of the second kind 326

10.5 Approximation of EVIs of the second kind 331

10.5.1 Regularization technique 333

10.5.2 Method of Lagrangian multipliers 335

10.5.3 Method of numerical integration 337

11 Numerical Solution of Fredholm Integral Equations of the Second Kind 342

11.1 Projection methods: General theory 343

11.1.1 Collocation methods 343

11.1.2 Galerkin methods 345

11.1.3 A general theoretical framework 346

11.2.1 Piecewise linear collocation 351

11.2.2 Trigonometric polynomial collocation 354

11.2.3 A piecewise linear Galerkin method 356

11.2.4 A Galerkin method with trigonometric polynomials 358

11.3 Iterated projection methods 362

11.3.1 The iterated Galerkin method 364

11.3.2 The iterated collocation solution 366

11.4 The Nystrom method 372

11.4.1 The Nystrom method for continuous kernel functions 373

11.4.2 Properties and error analysis of the Nystrom method 376

11.4.3 Collectively compact operator approximations 383

11.5 Product integration 385

11.5.1 Error analysis 388

11.5.2 Generalizations to other kernel functions 390

11.5.3 Improved error results for special kernels 392

11.5.4 Product integration with graded meshes 392

11.5.5 The relationship of product integration and collocation methods 396

11.6 Projection methods for nonlinear equations 398

11.6.1 Linearization 398

11.6.2 A homotopy argument 401

11.6.3 The approximating finite-dimensional problem 402

12 Boundary Integral Equations 405

12.1 Boundary integral equations 406

12.1.1 Green's identities and representation formula 407

12.1.2 The Kelvin transformation and exterior problems 409

12.1.3 Boundary integral equations of direct type 413

12.2 Boundary integral equations of the second kind 419

12.2.1 Evaluation of the double layer potential 421

12.2.2 The exterior Neumann problem 425

12.3 A boundary integral equation of the first kind 431

12.3.1 A numerical method 433.

Notes:

Includes bibliographical references (pages [436]-444) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Anne and Joseph Trachtman Memorial Book Fund.

ISBN:

0387951423

OCLC:

44750759