Theoretical Numerical Analysis: A Functional Analysis Framework, Količina 13

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Springer Science & Business Media, 1. jan. 2005 - 576 strani
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"This textbook prepares graduate students for research in numerical analysis/computational mathematics by giving to them a mathematical framework embedded in functional analysis and focused on numerical analysis. This helps the student to move rapidly into a research program. The presentation of each topic is meant to be an introduction with certain degree of depth. Comprehensive references on a particular topic are listed at the end of each chapter for further reading and study. In this new edition many sections from the first edition have been revised to varying degrees as well as over 140 new exercises added. A new chapter on Fourier Analysis and wavelets has been included."--BOOK JACKET.
  

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Vsebina

Linear Spaces
1
12 Normed spaces
7
121 Convergence
10
122 Banach spaces
13
123 Completion of normed spaces
14
13 Inner product spaces
21
131 Hilbert spaces
26
132 Orthogonality
28
74 Characterization of Sobolev spaces via the Fourier transform
303
75 Periodic Sobolev spaces
307
751 The dual space
309
752 Embedding results
310
753 Approximation results
311
754 An illustrative example of an operator
312
755 Spherical polynomials and spherical harmonics
313
76 Integration by parts formulas
319

14 Spaces of continuously differentiable functions
38
141 Hölder spaces
41
15 L𝛲 spaces
43
16 Compact sets
47
Linear Operators on Normed Spaces
50
21 Operators
52
22 Continuous linear operators
55
221 ℒVW as a Banach space
59
23 The geometric series theorem and its variants
60
231 A generalization
63
232 A perturbation result
66
24 Some more results on linear operators
71
241 An extension theorem
72
242 Open mapping theorem
73
243 Principle of uniform boundedness
75
244 Convergence of numerical quadratures
76
25 Linear functionals
79
251 An extension theorem for linear functionals
80
252 The Riesz representation theorem
82
26 Adjoint operators
85
27 Types of convergence
90
28 Compact linear operators
93
281 Compact integral operators on CD
94
282 Properties of compact operators
95
283 Integral operators on L²ab
97
284 The Fredholm alternative theorem
99
285 Additional results on Fredholm integral equations
103
29 The resolvent operator
107
291 Rλ as a holomorphic function
108
Approximation Theory
113
31 Approximation of continuous functions by polynomials
114
32 Interpolation theory
115
321 Lagrange polynomial interpolation
117
322 Hermite polynomial interpolation
121
324 Trigonometric interpolation
124
33 Best approximation
129
332 Some abstract existence results
131
333 Existence of best approximation
134
334 Uniqueness of best approximation
136
34 Best approximations in inner product spaces projection on closed convex sets
139
35 Orthogonal polynomials
146
36 Projection operators
150
37 Uniform error bounds
154
371 Uniform error bounds for L2approximations
156
372 Interpolatory projections and their convergence
158
Fourier Analysis and Wavelets
161
42 Fourier transform
175
43 Discrete Fourier transform
180
44 Haar wavelets
185
45 Multiresolution analysis
193
Nonlinear Equations and Their Solution by Iteration
200
51 The Banach fixedpoint theorem
202
52 Applications to iterative methods
206
521 Nonlinear equations
207
522 Linear systems
208
523 Linear and nonlinear integral equations
211
524 Ordinary differential equations in Banach spaces
215
53 Differential calculus for nonlinear operators
219
532 Mean value theorems
223
533 Partial derivatives
224
534 The Gateaux derivative and convex minimization
226
54 Newtons method
230
542 Applications
233
55 Completely continuous vector fields
236
551 The rotation of a completely continuous vector field
238
56 Conjugate gradient method for operator equations
239
Finite Difference Method
249
62 Lax equivalence theorem
256
63 More on convergence
265
Sobolev Spaces
272
72 Sobolev spaces
279
721 Sobolev spaces of integer order
280
722 Sobolev spaces of real order
286
723 Sobolev spaces over boundaries
288
73 Properties
289
731 Approximation by smooth functions
290
732 Extensions
291
734 Traces
293
735 Equivalent norms
294
736 A Sobolev quotient space
298
Variational Formulations of Elliptic Boundary Value Problems
323
81 A model boundary value problem
324
82 Some general results on existence and uniqueness
326
83 The LaxMilgram Lemma
330
84 Weak formulations of linear elliptic boundary value problems
334
842 Problems with nonhomogeneous Dirichlet boundary conditions
335
843 Problems with Neumann boundary conditions
337
844 Problems with mixed boundary conditions
339
845 A general linear secondorder elliptic boundary value problem
340
85 A boundary value problem of linearized elasticity
343
86 Mixed and dual formulations
348
87 Generalized LaxMilgram Lemma
353
88 A nonlinear problem
355
The Galerkin Method and Its Variants
361
92 The PetrovGalerkin method
367
93 Generalized Galerkin method
370
variational formulation
372
Finite Element Analysis
377
101 Onedimensional examples
379
1012 High order elements and the condensation technique
382
1013 Reference element technique nonconforming method
384
102 Basics of the finite element method
387
1022 Polynomial spaces on the reference elements
389
1023 Affineequivalent finite elements
391
1024 Finite element spaces
392
1025 Interpolation
395
103 Error estimates of finite element interpolations
396
1031 Interpolation error estimates on the reference element
397
1032 Local interpolation error estimates
398
1033 Global interpolation error estimates
401
104 Convergence and error estimates
404
Elliptic Variational Inequalities and Their Numerical Approximations
412
112 Elliptic variational inequalities of the first kind
420
113 Approximation of EVIs of the first kind
425
114 Elliptic variational inequalities of the second kind
428
115 Approximation of EVIs of the second kind
434
1151 Regularization technique
436
1152 Method of Lagrangian multipliers
438
1153 Method of numerical integration
440
Numerical Solution of Fredholm Integral Equations of the Second Kind
447
General theory
448
1212 Galerkin methods
450
1213 A general theoretical framework
451
122 Examples
456
1221 Piecewise linear collocation
457
1222 Trigonometric polynomial collocation
459
1223 A piecewise linear Galerkin method
461
1224 A Galerkin method with trigonometric polynomials
463
123 Iterated projection methods
468
1231 The iterated Galerkin method
470
1232 The iterated collocation solution
472
124 The Nyström method
478
1242 Properties and error analysis of the Nyström method
481
1243 Collectively compact operator approximations
489
125 Product integration
492
1251 Error analysis
494
1252 Generalizations to other kernel functions
496
1253 Improved error results for special kernels
498
1255 The relationship of product integration and collocation methods
503
126 Iteration methods
504
1261 A twogrid iteration method for the Nyström method
505
1262 Convergence analysis
508
1263 The iteration method for the linear system
511
1264 An operations count
513
127 Projection methods for nonlinear equations
515
1272 A homotopy argument
518
1273 The approximating finitedimensional problem
520
Boundary Integral Equations
523
131 Boundary integral equations
524
1311 Greens identities and representation formula
525
1312 The Kelvin transformation and exterior problems
527
1313 Boundary integral equations of direct type
531
132 Boundary integral equations of the second kind
537
1321 Evaluation of the double layer potential
540
1322 The exterior Neumann problem
543
133 A boundary integral equation of the first kind
549
1331 A numerical method
551
References
554
Index
569
Avtorske pravice

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Priljubljeni odlomki

Stran 559 - W. Freeden, T. Gervens, and M. Schreiner, Constructive Approximation on the Sphere, with Applications to Geomathematics, Oxford
Stran 559 - G. Fichera, Problemi elastostatici con vincoli unilateral!: il problema di Signorini con ambiguë condizioni al contorno, Mem. Accad. Naz. Lincei
Stran 561 - M. Jaswon and G. Symm, Integral Equation Methods in Potential Theory and Elastostatics, Academic Press, 1977.

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