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Theoretical numerical analysis: a functional analysis framework

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Bibliographische Detailangaben
Beteiligte Personen:	Atkinson, Kendall E. 1940- (VerfasserIn), Han, Weimin 1963- (VerfasserIn)
Format:	Buch
Sprache:	Englisch
Veröffentlicht:	New York, NY Springer 2009
Ausgabe:	3. ed.
Schriftenreihe:	Texts in applied mathematics 39
Schlagwörter:	Functional analysis Numerische Mathematik Funktionalanalysis
Links:	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017639765&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
Umfang:	XVI, 625 S. graph. Darst.
ISBN:	9781441904577 9781441904584

Internformat

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Datensatz im Suchindex

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adam_text	Contents 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 Hubert spaces ...................... 27 1.3.2 Orthogonality ...................... 28 1.4 Spaces of continuously differentiable functions ....... 39 1.4.1 Holder spaces ...................... 41 1.5 U 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 C(V,W) as a Banach space .............. 59 2.3 The geometric series theorem and its variants ........ 60 2.3.1 A generalization .................... 64 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 botmdedness ........... 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 Ĺ2 (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(A) as a holomorphic function ............ 110 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 /^-approximations using polynomials ........ 162 3.7.3 Interpolatory projections and their convergence . . . 164 Fourier Analysis and Wavelets 167 4.1 Fourier series .......................... 167 4.2 Fourier transform ........................ 181 4.3 Discrete Fourier transform ................... 187 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 Fréchet and Gâteaux derivatives ........... 225 5.3.2 Mean value theorems .................. 229 5.3.3 Partial derivatives ................... 230 5.3.4 The Gâteaux 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 БоЬоіел- 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 ................. 316 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 con¬ ditions .......................... 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 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 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 Nyström method ..................... 504 12.4.1 The Nyström method for continuous kernel functions 505 12.4.2 Properties and error analysis of the Nyström 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 colloca¬ tion methods ...................... 529 12.6 Iteration methods ........................ 531 12.6.1 A two-grid iteration method for the Nyström method 532 12.6.2 Convergence analysis .................. 535 12.6.3 The iteration method for the linear sj stem ..... 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 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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spellingShingle	Atkinson, Kendall E. 1940- Han, Weimin 1963- Theoretical numerical analysis a functional analysis framework Texts in applied mathematics Functional analysis Numerische Mathematik (DE-588)4042805-9 gnd Funktionalanalysis (DE-588)4018916-8 gnd
subject_GND	(DE-588)4042805-9 (DE-588)4018916-8
title	Theoretical numerical analysis a functional analysis framework
title_auth	Theoretical numerical analysis a functional analysis framework
title_exact_search	Theoretical numerical analysis a functional analysis framework
title_full	Theoretical numerical analysis a functional analysis framework Kendall Atkinson ; Weimin Han
title_fullStr	Theoretical numerical analysis a functional analysis framework Kendall Atkinson ; Weimin Han
title_full_unstemmed	Theoretical numerical analysis a functional analysis framework Kendall Atkinson ; Weimin Han
title_short	Theoretical numerical analysis
title_sort	theoretical numerical analysis a functional analysis framework
title_sub	a functional analysis framework
topic	Functional analysis Numerische Mathematik (DE-588)4042805-9 gnd Funktionalanalysis (DE-588)4018916-8 gnd
topic_facet	Functional analysis Numerische Mathematik Funktionalanalysis
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