Theoretical numerical analysis : a functional analysis framework

Başlık:

Yazar:

Atkinson, Kendall

ISBN:

9780387951423

Ek Yazar:

Atkinson, Kendall

Yayın Bilgisi:

New York : Springer Verlag , 2001.

Fiziksel Tanım:

xvi, 450 s. : ill. ; 25 sm.

Seriler:

Texts in applied mathematics ; 39

Seri Başlığı:

Texts in applied mathematics ; 39

Genel Not:

Bibliyografik kaynakça ([436]-444 s.) ve indeks var.

Özet:

Contents Series Preface vii Preface ix 1 Linear Spaces 1 1.1 Linear spaces ........................ 1 1.2 Normed spaces ........................ 7 1.2.1 Convergence ..................... 9 1.2.2 Danach spaces .................... 11 1.2.3 Completion of normed spaces ........... 12 1.3 Inner product spaces .................... 18 1.3.1 Hubert spaces .................... 22 1.3.2 Orthogonality .................... 23 1.4 Spaces of continuously differentiable functions ...... 30 1.4.1 Holder spaces .................... 31 1.5 LP 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 £(V,W) as a Danach 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 I/2 (a, 6) ........... 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(A) 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 ......... Ill 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 L2-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 Finite Difference Method 171 5.1 Finite difference approximations .............. 171 5.2 Lax equivalence theorem .................. 177 5.3 More on convergence .................... 186 Sobolev Spaces 193 6.1 Weak derivatives ...................... 193 6.2 Sobolev spaces ........................ 198 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 10.1 Introductory examples ................... 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 Examples........................... 351 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 References 436 Index 445

Konu Terimi:

Fonksiyonel analiz

Functional analysis

Mevcut:*

Library	Materyal Türü	Barkod	Yer Numarası	Durumu/İade Tarihi
Arıyor... Ferit Melen Library	Book	049567	515 ATKt 2001 k.1	Arıyor... Unknown

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