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