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[资源] Theoretical Numerical Analysis A Functional Analysis Framework（Third Edition）

Series Preface vii
Preface ix
1 Linear Spaces 1
1.1 Linear spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Normed spaces . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.1 Convergence . . . . . . . . . . . . . . . . . . . . . . 10
1.2.2 Banach spaces . . . . . . . . . . . . . . . . . . . . . 13
1.2.3 Completion of normed spaces . . . . . . . . . . . . . 15
1.3 Inner product spaces . . . . . . . . . . . . . . . . . . . . . . 22
1.3.1 Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . 27
1.3.2 Orthogonality . . . . . . . . . . . . . . . . . . . . . . 28
1.4 Spaces of continuously differentiable functions . . . . . . . 39
1.4.1 H¨older spaces . . . . . . . . . . . . . . . . . . . . . . 41
1.5 Lp spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
1.6 Compact sets . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2 Linear Operators on Normed Spaces 51
2.1 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.2 Continuous linear operators . . . . . . . . . . . . . . . . . . 55
2.2.1 L(V,W) as a Banach space . . . . . . . . . . . . . . 59
2.3 The geometric series theorem and its variants . . . . . . . . 60
2.3.1 A generalization . . . . . . . . . . . . . . . . . . . . 64
xii Contents
2.3.2 A perturbation result . . . . . . . . . . . . . . . . . 66
2.4 Some more results on linear operators . . . . . . . . . . . . 72
2.4.1 An extension theorem . . . . . . . . . . . . . . . . . 72
2.4.2 Open mapping theorem . . . . . . . . . . . . . . . . 74
2.4.3 Principle of uniform boundedness . . . . . . . . . . . 75
2.4.4 Convergence of numerical quadratures . . . . . . . . 76
2.5 Linear functionals . . . . . . . . . . . . . . . . . . . . . . . 79
2.5.1 An extension theorem for linear functionals . . . . . 80
2.5.2 The Riesz representation theorem . . . . . . . . . . 82
2.6 Adjoint operators . . . . . . . . . . . . . . . . . . . . . . . . 85
2.7 Weak convergence and weak compactness . . . . . . . . . . 90
2.8 Compact linear operators . . . . . . . . . . . . . . . . . . . 95
2.8.1 Compact integral operators on C(D) . . . . . . . . . 96
2.8.2 Properties of compact operators . . . . . . . . . . . 97
2.8.3 Integral operators on L2(a, b) . . . . . . . . . . . . . 99
2.8.4 The Fredholm alternative theorem . . . . . . . . . . 101
2.8.5 Additional results on Fredholm integral equations . 105
2.9 The resolvent operator . . . . . . . . . . . . . . . . . . . . 109
2.9.1 R(λ) as a holomorphic function . . . . . . . . . . . . 110
3 Approximation Theory 115
3.1 Approximation of continuous functions by polynomials . . . 116
3.2 Interpolation theory . . . . . . . . . . . . . . . . . . . . . . 118
3.2.1 Lagrange polynomial interpolation . . . . . . . . . . 120
3.2.2 Hermite polynomial interpolation . . . . . . . . . . . 122
3.2.3 Piecewise polynomial interpolation . . . . . . . . . . 124
3.2.4 Trigonometric interpolation . . . . . . . . . . . . . . 126
3.3 Best approximation . . . . . . . . . . . . . . . . . . . . . . . 131
3.3.1 Convexity, lower semicontinuity . . . . . . . . . . . . 132
3.3.2 Some abstract existence results . . . . . . . . . . . . 134
3.3.3 Existence of best approximation . . . . . . . . . . . 137
3.3.4 Uniqueness of best approximation . . . . . . . . . . 138
3.4 Best approximations in inner product spaces, projection on
closed convex sets . . . . . . . . . . . . . . . . . . . . . . . . 142
3.5 Orthogonal polynomials . . . . . . . . . . . . . . . . . . . . 149
3.6 Projection operators . . . . . . . . . . . . . . . . . . . . . . 154
3.7 Uniform error bounds . . . . . . . . . . . . . . . . . . . . . 157
3.7.1 Uniform error bounds for L2-approximations . . . . 160
3.7.2 L2-approximations using polynomials . . . . . . . . 162
3.7.3 Interpolatory projections and their convergence . . . 164
4 Fourier Analysis and Wavelets 167
4.1 Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . 167
4.2 Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . 181
4.3 Discrete Fourier transform . . . . . . . . . . . . . . . . . . . 187
Contents xiii
4.4 Haar wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . 191
4.5 Multiresolution analysis . . . . . . . . . . . . . . . . . . . . 199
5 Nonlinear Equations and Their Solution by Iteration 207
5.1 The Banach fixed-point theorem . . . . . . . . . . . . . . . 208
5.2 Applications to iterative methods . . . . . . . . . . . . . . . 212
5.2.1 Nonlinear algebraic equations . . . . . . . . . . . . . 213
5.2.2 Linear algebraic systems . . . . . . . . . . . . . . . . 214
5.2.3 Linear and nonlinear integral equations . . . . . . . 216
5.2.4 Ordinary differential equations in Banach spaces . . 221
5.3 Differential calculus for nonlinear operators . . . . . . . . . 225
5.3.1 Fr′echet and Gˆateaux derivatives . . . . . . . . . . . 225
5.3.2 Mean value theorems . . . . . . . . . . . . . . . . . . 229
5.3.3 Partial derivatives . . . . . . . . . . . . . . . . . . . 230
5.3.4 The Gˆateaux derivative and convex minimization . . 231
5.4 Newton’s method . . . . . . . . . . . . . . . . . . . . . . . . 236
5.4.1 Newton’s method in Banach spaces . . . . . . . . . . 236
5.4.2 Applications . . . . . . . . . . . . . . . . . . . . . . 239
5.5 Completely continuous vector fields . . . . . . . . . . . . . . 241
5.5.1 The rotation of a completely continuous vector field 243
5.6 Conjugate gradient method for operator equations . . . . . 245
6 Finite Difference Method 253
6.1 Finite difference approximations . . . . . . . . . . . . . . . 253
6.2 Lax equivalence theorem . . . . . . . . . . . . . . . . . . . . 260
6.3 More on convergence . . . . . . . . . . . . . . . . . . . . . . 269
7 Sobolev Spaces 277
7.1 Weak derivatives . . . . . . . . . . . . . . . . . . . . . . . . 277
7.2 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . 283
7.2.1 Sobolev spaces of integer order . . . . . . . . . . . . 284
7.2.2 Sobolev spaces of real order . . . . . . . . . . . . . . 290
7.2.3 Sobolev spaces over boundaries . . . . . . . . . . . . 292
7.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
7.3.1 Approximation by smooth functions . . . . . . . . . 293
7.3.2 Extensions . . . . . . . . . . . . . . . . . . . . . . . 294
7.3.3 Sobolev embedding theorems . . . . . . . . . . . . . 295
7.3.4 Traces . . . . . . . . . . . . . . . . . . . . . . . . . . 297
7.3.5 Equivalent norms . . . . . . . . . . . . . . . . . . . . 298
7.3.6 A Sobolev quotient space . . . . . . . . . . . . . . . 302
7.4 Characterization of Sobolev spaces via the Fourier transform 308
7.5 Periodic Sobolev spaces . . . . . . . . . . . . . . . . . . . . 311
7.5.1 The dual space . . . . . . . . . . . . . . . . . . . . . 314
7.5.2 Embedding results . . . . . . . . . . . . . . . . . . . 315
7.5.3 Approximation results . . . . . . . . . . . . . . . . . 316xiv Contents
7.5.4 An illustrative example of an operator . . . . . . . . 317
7.5.5 Spherical polynomials and spherical harmonics . . . 318
7.6 Integration by parts formulas . . . . . . . . . . . . . . . . . 323
8 Weak Formulations of Elliptic Boundary Value Problems 327
8.1 A model boundary value problem . . . . . . . . . . . . . . . 328
8.2 Some general results on existence and uniqueness . . . . . . 330
8.3 The Lax-Milgram Lemma . . . . . . . . . . . . . . . . . . . 334
8.4 Weak formulations of linear elliptic boundary value problems 338
8.4.1 Problems with homogeneous Dirichlet boundary conditions
. . . . . . . . . . . . . . . . . . . . . . . . . . 338
8.4.2 Problems with non-homogeneous Dirichlet boundary
conditions . . . . . . . . . . . . . . . . . . . . . . . . 339
8.4.3 Problems with Neumann boundary conditions . . . . 341
8.4.4 Problems with mixed boundary conditions . . . . . . 343
8.4.5 A general linear second-order elliptic boundary value
problem . . . . . . . . . . . . . . . . . . . . . . . . . 344
8.5 A boundary value problem of linearized elasticity . . . . . . 348
8.6 Mixed and dual formulations . . . . . . . . . . . . . . . . . 354
8.7 Generalized Lax-Milgram Lemma . . . . . . . . . . . . . . . 359
8.8 A nonlinear problem . . . . . . . . . . . . . . . . . . . . . . 361
9 The Galerkin Method and Its Variants 367
9.1 The Galerkin method . . . . . . . . . . . . . . . . . . . . . 367
9.2 The Petrov-Galerkin method . . . . . . . . . . . . . . . . . 374
9.3 Generalized Galerkin method . . . . . . . . . . . . . . . . . 376
9.4 Conjugate gradient method: variational formulation . . . . 378
10 Finite Element Analysis 383
10.1 One-dimensional examples . . . . . . . . . . . . . . . . . . . 384
10.1.1 Linear elements for a second-order problem . . . . . 384
10.1.2 High order elements and the condensation technique 389
10.1.3 Reference element technique . . . . . . . . . . . . . . 390
10.2 Basics of the finite element method . . . . . . . . . . . . . . 393
10.2.1 Continuous linear elements . . . . . . . . . . . . . . 394
11 Elliptic Variational Inequalities and Their Numerical Ap-
proximations 423
11.1 From variational equations to variational inequalities . . . . 423
11.2 Existence and uniqueness based on convex minimization . . 428
11.3 Existence and uniqueness results for a family of EVIs . . . . 430
11.4 Numerical approximations . . . . . . . . . . . . . . . . . . . 442
11.5 Some contact problems in elasticity . . . . . . . . . . . . . . 458
11.5.1 A frictional contact problem . . . . . . . . . . . . . . 460
11.5.2 A Signorini frictionless contact problem . . . . . . . 465
12 Numerical Solution of Fredholm Integral Equations of the
Second Kind 473
12.1 Projection methods: General theory . . . . . . . . . . . . . 474
12.1.1 Collocation methods . . . . . . . . . . . . . . . . . . 474
12.1.2 Galerkin methods . . . . . . . . . . . . . . . . . . . 476
12.1.3 A general theoretical framework . . . . . . . . . . . 477
12.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483
12.2.1 Piecewise linear collocation . . . . . . . . . . . . . . 483
12.2.2 Trigonometric polynomial collocation . . . . . . . . 486
12.2.3 A piecewise linear Galerkin method . . . . . . . . . 488
12.2.4 A Galerkin method with trigonometric polynomials . 490
12.3 Iterated projection methods . . . . . . . . . . . . . . . . . . 494
12.3.1 The iterated Galerkin method . . . . . . . . . . . . . 497
12.3.2 The iterated collocation solution . . . . . . . . . . . 498
12.4 The Nystr¨om method . . . . . . . . . . . . . . . . . . . . . 504
12.4.1 The Nystr¨om method for continuous kernel functions 505
12.4.2 Properties and error analysis of the Nystr¨om method 507
12.4.3 Collectively compact operator approximations . . . . 516
12.5 Product integration . . . . . . . . . . . . . . . . . . . . . . . 518
12.5.1 Error analysis . . . . . . . . . . . . . . . . . . . . . . 520
12.5.2 Generalizations to other kernel functions . . . . . . . 523
12.5.3 Improved error results for special kernels . . . . . . . 525
12.5.4 Product integration with graded meshes . . . . . . . 525
12.5.5 The relationship of product integration and collocation
methods . . . . . . . . . . . . . . . . . . . . . . 529
12.6 Iteration methods . . . . . . . . . . . . . . . . . . . . . . . . 531
12.6.1 A two-grid iteration method for the Nystr¨om method 532
12.6.2 Convergence analysis . . . . . . . . . . . . . . . . . . 535
12.6.3 The iteration method for the linear system . . . . . 538
12.6.4 An operations count . . . . . . . . . . . . . . . . . . 540
12.7 Projection methods for nonlinear equations . . . . . . . . . 542
12.7.1 Linearization . . . . . . . . . . . . . . . . . . . . . . 542
12.7.2 A homotopy argument . . . . . . . . . . . . . . . . . 545
12.7.3 The approximating finite-dimensional problem . . . 547
10.2.2 Affine-equivalent finite elements . . . . . . . . . . . . 400
10.2.3 Finite element spaces . . . . . . . . . . . . . . . . . 404
10.3 Error estimates of finite element interpolations . . . . . . . 406
10.3.1 Local interpolations . . . . . . . . . . . . . . . . . . 407
10.3.2 Interpolation error estimates on the reference element 408
10.3.3 Local interpolation error estimates . . . . . . . . . . 409
10.3.4 Global interpolation error estimates . . . . . . . . . 412
10.4 Convergence and error estimates . . . . . . . . . . . . . . . 415
13 Boundary Integral Equations 551
13.1 Boundary integral equations . . . . . . . . . . . . . . . . . 552
13.1.1 Green’s identities and representation formula . . . . 553
13.1.2 The Kelvin transformation and exterior problems . 555
13.1.3 Boundary integral equations of direct type . . . . . 559
13.2 Boundary integral equations of the second kind . . . . . . . 565
13.2.1 Evaluation of the double layer potential . . . . . . . 568
13.2.2 The exterior Neumann problem . . . . . . . . . . . 571
13.3 A boundary integral equation of the first kind . . . . . . . 577
13.3.1 A numerical method . . . . . . . . . . . . . . . . . . 579
14 Multivariable Polynomial Approximations 583
14.1 Notation and best approximation results . . . . . . . . . . . 583
14.2 Orthogonal polynomials . . . . . . . . . . . . . . . . . . . . 585
14.2.1 Triple recursion relation . . . . . . . . . . . . . . . . 588
14.2.2 The orthogonal projection operator and its error . . 590
14.3 Hyperinterpolation . . . . . . . . . . . . . . . . . . . . . . . 592
14.3.1 The norm of the hyperinterpolation operator . . . . 593
14.4 A Galerkin method for elliptic equations . . . . . . . . . . . 593
14.4.1 The Galerkin method and its convergence . . . . . . 595
References 601
Index 617

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