Theoretical numerical analysis: a functional analysis framework
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, NY
Springer
2009
|
| Edition: | 3. ed. |
| Series: | Texts in applied mathematics
39 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 625 S. graph. Darst. |
| ISBN: | 9781441904577 9781441904584 |
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MARC
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| 100 | 1 | |a Atkinson, Kendall E. |d 1940- |e Verfasser |0 (DE-588)12286977X |4 aut | |
| 245 | 1 | 0 | |a Theoretical numerical analysis |b a functional analysis framework |c Kendall Atkinson ; Weimin Han |
| 250 | |a 3. ed. | ||
| 264 | 1 | |a New York, NY |b Springer |c 2009 | |
| 300 | |a XVI, 625 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Texts in applied mathematics |v 39 | |
| 650 | 4 | |a Functional analysis | |
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Record in the Search Index
| _version_ | 1804139243948212224 |
|---|---|
| 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
|
| any_adam_object | 1 |
| author | Atkinson, Kendall E. 1940- Han, Weimin 1963- |
| author_GND | (DE-588)12286977X (DE-588)121177971 |
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| ctrlnum | (OCoLC)602701096 (DE-599)BVBBV035584424 |
| dewey-full | 515 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515 |
| dewey-search | 515 |
| dewey-sort | 3515 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 3. ed. |
| format | Book |
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| id | DE-604.BV035584424 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T21:41:00Z |
| institution | BVB |
| isbn | 9781441904577 9781441904584 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017639765 |
| oclc_num | 602701096 |
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| owner_facet | DE-20 DE-91G DE-BY-TUM DE-11 DE-739 DE-898 DE-BY-UBR DE-83 DE-188 |
| physical | XVI, 625 S. graph. Darst. |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Springer |
| record_format | marc |
| series | Texts in applied mathematics |
| series2 | Texts in applied mathematics |
| spelling | Atkinson, Kendall E. 1940- Verfasser (DE-588)12286977X aut Theoretical numerical analysis a functional analysis framework Kendall Atkinson ; Weimin Han 3. ed. New York, NY Springer 2009 XVI, 625 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Texts in applied mathematics 39 Functional analysis Numerische Mathematik (DE-588)4042805-9 gnd rswk-swf Funktionalanalysis (DE-588)4018916-8 gnd rswk-swf Funktionalanalysis (DE-588)4018916-8 s Numerische Mathematik (DE-588)4042805-9 s DE-604 Han, Weimin 1963- Verfasser (DE-588)121177971 aut Texts in applied mathematics 39 (DE-604)BV002476038 39 Digitalisierung UB Passau application/pdf 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 Inhaltsverzeichnis |
| 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 |
| url | 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 |
| volume_link | (DE-604)BV002476038 |
| work_keys_str_mv | AT atkinsonkendalle theoreticalnumericalanalysisafunctionalanalysisframework AT hanweimin theoreticalnumericalanalysisafunctionalanalysisframework |