Reliable methods for computer simulation: error control and a posteriori estimates
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Amsterdam [u.a.]
Elsevier
2004
|
| Ausgabe: | 1. ed. |
| Schriftenreihe: | Studies in mathematics and its applications
33 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturverz. S. 281 - 299 |
| Beschreibung: | X, 305 S. graph. Darst. |
| ISBN: | 0444513760 |
Internformat
MARC
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| 100 | 1 | |a Neittaanmäki, Pekka |d 1951- |e Verfasser |0 (DE-588)132647702 |4 aut | |
| 245 | 1 | 0 | |a Reliable methods for computer simulation |b error control and a posteriori estimates |
| 250 | |a 1. ed. | ||
| 264 | 1 | |a Amsterdam [u.a.] |b Elsevier |c 2004 | |
| 300 | |a X, 305 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Studies in mathematics and its applications |v 33 | |
| 500 | |a Literaturverz. S. 281 - 299 | ||
| 650 | 4 | |a Fehlerkorrekturcode - Computersimulation | |
| 650 | 4 | |a Approximation theory | |
| 650 | 4 | |a Error-correcting codes (Information theory) | |
| 650 | 4 | |a Numerical analysis | |
| 650 | 0 | 7 | |a A-posteriori-Abschätzung |0 (DE-588)4346907-3 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Computersimulation |0 (DE-588)4148259-1 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Computersimulation |0 (DE-588)4148259-1 |D s |
| 689 | 0 | |5 DE-604 | |
| 689 | 1 | 0 | |a A-posteriori-Abschätzung |0 (DE-588)4346907-3 |D s |
| 689 | 1 | |5 DE-604 | |
| 700 | 1 | |a Repin, Sergej I. |e Verfasser |4 aut | |
| 830 | 0 | |a Studies in mathematics and its applications |v 33 |w (DE-604)BV000000646 |9 33 | |
| 856 | 4 | 2 | |m Digitalisierung UB Bamberg |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015397556&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-015397556 | ||
Datensatz im Suchindex
| _version_ | 1804136158270062592 |
|---|---|
| adam_text | Contents
Introduction
1
1.1
Sources
of errors affecting the reliability of numerical
Solutions 1
1.2
The main approaches to error estimation
............ 3
1.2.1
A priori error estimates
................. 4
1.2.2
A posteriori error estimates
............... 5
Mathematical background
7
2.1
Basic notation
.......................... 7
2.1.1
Vectors and matrices
................... 7
2.1.2
Spaces of functions
.................... 8
2.2
Sobolev spaces
.......................... 11
2.2.1
Generalized derivatives and Sobolev spaces
...... 11
2.2.2
Sobolev spaces with negative indices
.......... 14
2.3
Generalized solutions and their approximation
......... 17
A posteriori estimates for iteration methods
23
3.1
The Banach fixed point theorem
................ 23
3.2
Iteration methods for bounded linear operators
........ 28
3.2.1
Two-sided estimates
................... 28
3.2.2
Iteration schemes
..................... 29
3.2.3
Example
.......................... 31
3.3
Iterational methods for integral equations
........... 32
3.4
Iteration methods for ordinary differential equations
..... 35
3.5
Notes for the Chapter
...................... 36
A posteriori estimates for finite element approximations
39
4.1
Methods based upon estimates of the negative norm of the
equation residual
......................... 40
4.1.1
Approximation errors and residuals
........... 40
4.1.2
Residual type estimates
................. 41
CONTENTS
4.2 Evaluation
of
negative
norms of residuals
...........45
4.2.1
Estimation of the residual in a one-dimensional problem
46
4.2.2
Estimation of the residual for linear elliptic problems
. 49
4.2.3
Estimation of the residual norm by implicit methods
. 51
4.3
Error estimation methods using adjoint problems
....... 54
4.4
Methods based upon post-processing of finite element ap¬
proximations
........................... 58
4.4.1
Post-processing by averaging
.............. 60
4.4.2
Superconvergence
..................... 62
4.4.3
Post-processing by equilibration
............ 63
4.5
Error estimation of goal-oriented quantities by gradient av¬
eraging techniques
........................ 64
4.5.1
General scheme
...................... 64
4.5.2
Example
.......................... 67
4.5.3
Error estimates in terms of seminorms
......... 71
4.6
Notes for the Chapter
...................... 74
Foundations of duality theory
79
5.1
Convex sets and functionals
...................79
5.2
Dual functionals
.........................86
5.2.1
Nomenclature
.......................86
5.2.2
Properties of dual functionals
..............88
5.2.3
Examples
.........................90
5.3
Differentiation of convex functionals
..............96
5.3.1
Subdifferential
......................96
5.3.2
Properties of subdifferentials
..............99
5.4
Lower semicontinuous functionals and existence of minimizers
101
5.4.1
Lower semicontinuous functionals
............102
5.4.2
Existence of minimizers
.................106
5.4.3
Dual variational problems
................107
5.5
Uniformly convex functionals
..................113
5.6
Integral functionals
........................118
5.6.1
Lower semicontinuity
...................118
5.6.2
Convexity
.........................119
5.6.3
Conjugate functionals
..................120
5.6.4
Differentiation
......................122
Two-sided a posteriori estimates for linear elliptic problems
125
G.I Basic: relations
..........................125
6.2
Two-sided estimates of deviations
................131
6.2.1
Upper estimates
.....................131
CONTENTS
vii
6.2.2
Lower estimates
.....................132
6.2.3
Estimates of deviations in terms of the dual variable
. 134
6.3
Properties of two-sided estimates
................135
6.3.1
Exactness
.........................135
6.3.2
Computability
......................138
6.4
Linear elliptic equations of the second order
..........141
6.4.1
Dirichlet boundary conditions
..............142
6.4.2
Neumann boundary condition
..............144
6.4.3
Mixed boundary conditions
...............146
6.4.4
Lower estimates
.....................149
6.5
The linear elasticity problem
..................150
6.5.1
Upper estimates
.....................151
6.5.2
Lower estimates
.....................153
6.6
Linear elliptic equations of the fourth order
..........154
6.7
Relationship with other methods
................157
6.7.1
Residual based estimates
................158
6.7.2
Estimates based on the regularization of the dual
variable
..........................159
6.7.3
Estimates based on the equilibration of the dual
variable
..........................161
6.7.4
A priori projection type error estimates
........162
6.8
Error estimates taking account of the indeterminacy in the
data
................................163
6.8.1
General concept
.....................163
6.8.2
Upper bound of the error
................165
6.8.3
Lower bound of the error
................172
6.9
Error estimation in terms of
lineai
funcţionale.........
174
6.10
Practical implementation
....................176
6.10.1
General scheme
......................176
6.10.2
Minimization of the
majorant
..............178
6.10.3
Effectivity index and shape index
............180
6.10.4
Non-Galerkin approximations
..............180
6.10.5
Duality error
majorants
for finite element approxima¬
tions
............................193
6.10.6
Computational costs
...................202
6.10.7
Conclusion
........................204
6.11
Comments
.............................204
6.12
Notes for the Chapter
......................207
viii CONTENTS
7
A posteriori estimates for nonlinear variational problems
209
7.1
Nonlinear variational problems
.................209
7.2
General form of the error
majorant
for the functional J{v)
=
G(Av)
+
F{v)
...........................212
7.3
Variational problems with nonhomogeneous boundary condi¬
tions
................................216
7.4
Error
majorant
for the case F(v)
=
(l,v)
............221
7.5
Properties of compound
funcţionale
...............223
7.6
Two-sided error estimates
....................231
7.7
A class of nonlinear variational problems
............234
7.8
Comments
.............................236
7.8.1
Power growth functionals
................237
7.8.2
Nonlinear elasticity
....................238
7.8.3
Nonconvex variational problems
............240
8
A posteriori estimates for variational inequalities
245
8.1
Variational inequalities
......................245
8.2
Problems with obstacles
.....................246
8.2.1
Variational formulations
.................246
8.2.2
Perturbed problem
....................247
8.2.3
Dual perturbed problem
.................248
8.2.4
Estimates of the deviation
................249
8.3
The elasto-plastic torsion problem
...............254
8.4
A model problem with a friction type boundary condition
. . 255
8.5
Variational problems in the theory of viscous fluids
......259
8.5.1
Preliminaries
.......................259
8.5.2
Variational problems
...................262
8.5.3
Estimates for solenoidal fields
..............265
8.5.4
Special cases
.......................268
8.5.5
Estimates taking into account violations of the incom-
pressibility condition
...................269
8.6
Examples
.............................272
8.7
Notes for the Chapter
......................279
Bibliography
281
Notation
301
Index
303
|
| adam_txt |
Contents
Introduction
1
1.1
Sources
of errors affecting the reliability of numerical
Solutions 1
1.2
The main approaches to error estimation
. 3
1.2.1
A priori error estimates
. 4
1.2.2
A posteriori error estimates
. 5
Mathematical background
7
2.1
Basic notation
. 7
2.1.1
Vectors and matrices
. 7
2.1.2
Spaces of functions
. 8
2.2
Sobolev spaces
. 11
2.2.1
Generalized derivatives and Sobolev spaces
. 11
2.2.2
Sobolev spaces with negative indices
. 14
2.3
Generalized solutions and their approximation
. 17
A posteriori estimates for iteration methods
23
3.1
The Banach fixed point theorem
. 23
3.2
Iteration methods for bounded linear operators
. 28
3.2.1
Two-sided estimates
. 28
3.2.2
Iteration schemes
. 29
3.2.3
Example
. 31
3.3
Iterational methods for integral equations
. 32
3.4
Iteration methods for ordinary differential equations
. 35
3.5
Notes for the Chapter
. 36
A posteriori estimates for finite element approximations
39
4.1
Methods based upon estimates of the negative norm of the
equation residual
. 40
4.1.1
Approximation errors and residuals
. 40
4.1.2
Residual type estimates
. 41
CONTENTS
4.2 Evaluation
of
negative
norms of residuals
.45
4.2.1
Estimation of the residual in a one-dimensional problem
46
4.2.2
Estimation of the residual for linear elliptic problems
. 49
4.2.3
Estimation of the residual norm by implicit methods
. 51
4.3
Error estimation methods using adjoint problems
. 54
4.4
Methods based upon post-processing of finite element ap¬
proximations
. 58
4.4.1
Post-processing by averaging
. 60
4.4.2
Superconvergence
. 62
4.4.3
Post-processing by equilibration
. 63
4.5
Error estimation of goal-oriented quantities by gradient av¬
eraging techniques
. 64
4.5.1
General scheme
. 64
4.5.2
Example
. 67
4.5.3
Error estimates in terms of seminorms
. 71
4.6
Notes for the Chapter
. 74
Foundations of duality theory
79
5.1
Convex sets and functionals
.79
5.2
Dual functionals
.86
5.2.1
Nomenclature
.86
5.2.2
Properties of dual functionals
.88
5.2.3
Examples
.90
5.3
Differentiation of convex functionals
.96
5.3.1
Subdifferential
.96
5.3.2
Properties of subdifferentials
.99
5.4
Lower semicontinuous functionals and existence of minimizers
101
5.4.1
Lower semicontinuous functionals
.102
5.4.2
Existence of minimizers
.106
5.4.3
Dual variational problems
.107
5.5
Uniformly convex functionals
.113
5.6
Integral functionals
.118
5.6.1
Lower semicontinuity
.118
5.6.2
Convexity
.119
5.6.3
Conjugate functionals
.120
5.6.4
Differentiation
.122
Two-sided a posteriori estimates for linear elliptic problems
125
G.I Basic: relations
.125
6.2
Two-sided estimates of deviations
.131
6.2.1
Upper estimates
.131
CONTENTS
vii
6.2.2
Lower estimates
.132
6.2.3
Estimates of deviations in terms of' the dual variable
. 134
6.3
Properties of' two-sided estimates
.135
6.3.1
Exactness
.135
6.3.2
Computability
.138
6.4
Linear elliptic equations of the second order
.141
6.4.1
Dirichlet boundary conditions
.142
6.4.2
Neumann boundary condition
.144
6.4.3
Mixed boundary conditions
.146
6.4.4
Lower estimates
.149
6.5
The linear elasticity problem
.150
6.5.1
Upper estimates
.151
6.5.2
Lower estimates
.153
6.6
Linear elliptic equations of' the fourth order
.154
6.7
Relationship with other methods
.157
6.7.1
Residual based estimates
.158
6.7.2
Estimates based on the "regularization" of the dual
variable
.159
6.7.3
Estimates based on the "equilibration" of the dual
variable
.161
6.7.4
A priori projection type error estimates
.162
6.8
Error estimates taking account of the indeterminacy in the
data
.163
6.8.1
General concept
.163
6.8.2
Upper bound of the error
.165
6.8.3
Lower bound of the error
.172
6.9
Error estimation in terms of
lineai
funcţionale.
174
6.10
Practical implementation
.176
6.10.1
General scheme
.176
6.10.2
Minimization of the
majorant
.178
6.10.3
Effectivity index and shape index
.180
6.10.4
Non-Galerkin approximations
.180
6.10.5
Duality error
majorants
for finite element approxima¬
tions
.193
6.10.6
Computational costs
.202
6.10.7
Conclusion
.204
6.11
Comments
.204
6.12
Notes for the Chapter
.207
viii CONTENTS
7
A posteriori estimates for nonlinear variational problems
209
7.1
Nonlinear variational problems
.209
7.2
General form of the error
majorant
for the functional J{v)
=
G(Av)
+
F{v)
.212
7.3
Variational problems with nonhomogeneous boundary condi¬
tions
.216
7.4
Error
majorant
for the case F(v)
=
(l,v)
.221
7.5
Properties of compound
funcţionale
.223
7.6
Two-sided error estimates
.231
7.7
A class of nonlinear variational problems
.234
7.8
Comments
.236
7.8.1
Power growth functionals
.237
7.8.2
Nonlinear elasticity
.238
7.8.3
Nonconvex variational problems
.240
8
A posteriori estimates for variational inequalities
245
8.1
Variational inequalities
.245
8.2
Problems with obstacles
.246
8.2.1
Variational formulations
.246
8.2.2
Perturbed problem
.247
8.2.3
Dual perturbed problem
.248
8.2.4
Estimates of the deviation
.249
8.3
The elasto-plastic torsion problem
.254
8.4
A model problem with a friction type boundary condition
. . 255
8.5
Variational problems in the theory of viscous fluids
.259
8.5.1
Preliminaries
.259
8.5.2
Variational problems
.262
8.5.3
Estimates for solenoidal fields
.265
8.5.4
Special cases
.268
8.5.5
Estimates taking into account violations of the incom-
pressibility condition
.269
8.6
Examples
.272
8.7
Notes for the Chapter
.279
Bibliography
281
Notation
301
Index
303 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Neittaanmäki, Pekka 1951- Repin, Sergej I. |
| author_GND | (DE-588)132647702 |
| author_facet | Neittaanmäki, Pekka 1951- Repin, Sergej I. |
| author_role | aut aut |
| author_sort | Neittaanmäki, Pekka 1951- |
| author_variant | p n pn s i r si sir |
| building | Verbundindex |
| bvnumber | BV022182795 |
| callnumber-first | M - Music |
| callnumber-label | MLCM 2006/41023 (Q) |
| callnumber-raw | MLCM 2006/41023 (Q) |
| callnumber-search | MLCM 2006/41023 (Q) |
| callnumber-sort | MLCM 42006 541023 Q |
| classification_rvk | SK 910 ST 340 |
| ctrlnum | (OCoLC)249888589 (DE-599)BVBBV022182795 |
| dewey-full | 003.3 |
| dewey-hundreds | 000 - Computer science, information, general works |
| dewey-ones | 003 - Systems |
| dewey-raw | 003.3 |
| dewey-search | 003.3 |
| dewey-sort | 13.3 |
| dewey-tens | 000 - Computer science, information, general works |
| discipline | Informatik Mathematik |
| discipline_str_mv | Informatik Mathematik |
| edition | 1. ed. |
| format | Book |
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| id | DE-604.BV022182795 |
| illustrated | Illustrated |
| index_date | 2024-07-02T16:20:39Z |
| indexdate | 2024-07-09T20:51:57Z |
| institution | BVB |
| isbn | 0444513760 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015397556 |
| oclc_num | 249888589 |
| open_access_boolean | |
| owner | DE-706 DE-11 |
| owner_facet | DE-706 DE-11 |
| physical | X, 305 S. graph. Darst. |
| publishDate | 2004 |
| publishDateSearch | 2004 |
| publishDateSort | 2004 |
| publisher | Elsevier |
| record_format | marc |
| series | Studies in mathematics and its applications |
| series2 | Studies in mathematics and its applications |
| spelling | Neittaanmäki, Pekka 1951- Verfasser (DE-588)132647702 aut Reliable methods for computer simulation error control and a posteriori estimates 1. ed. Amsterdam [u.a.] Elsevier 2004 X, 305 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Studies in mathematics and its applications 33 Literaturverz. S. 281 - 299 Fehlerkorrekturcode - Computersimulation Approximation theory Error-correcting codes (Information theory) Numerical analysis A-posteriori-Abschätzung (DE-588)4346907-3 gnd rswk-swf Computersimulation (DE-588)4148259-1 gnd rswk-swf Computersimulation (DE-588)4148259-1 s DE-604 A-posteriori-Abschätzung (DE-588)4346907-3 s Repin, Sergej I. Verfasser aut Studies in mathematics and its applications 33 (DE-604)BV000000646 33 Digitalisierung UB Bamberg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015397556&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Neittaanmäki, Pekka 1951- Repin, Sergej I. Reliable methods for computer simulation error control and a posteriori estimates Studies in mathematics and its applications Fehlerkorrekturcode - Computersimulation Approximation theory Error-correcting codes (Information theory) Numerical analysis A-posteriori-Abschätzung (DE-588)4346907-3 gnd Computersimulation (DE-588)4148259-1 gnd |
| subject_GND | (DE-588)4346907-3 (DE-588)4148259-1 |
| title | Reliable methods for computer simulation error control and a posteriori estimates |
| title_auth | Reliable methods for computer simulation error control and a posteriori estimates |
| title_exact_search | Reliable methods for computer simulation error control and a posteriori estimates |
| title_exact_search_txtP | Reliable methods for computer simulation error control and a posteriori estimates |
| title_full | Reliable methods for computer simulation error control and a posteriori estimates |
| title_fullStr | Reliable methods for computer simulation error control and a posteriori estimates |
| title_full_unstemmed | Reliable methods for computer simulation error control and a posteriori estimates |
| title_short | Reliable methods for computer simulation |
| title_sort | reliable methods for computer simulation error control and a posteriori estimates |
| title_sub | error control and a posteriori estimates |
| topic | Fehlerkorrekturcode - Computersimulation Approximation theory Error-correcting codes (Information theory) Numerical analysis A-posteriori-Abschätzung (DE-588)4346907-3 gnd Computersimulation (DE-588)4148259-1 gnd |
| topic_facet | Fehlerkorrekturcode - Computersimulation Approximation theory Error-correcting codes (Information theory) Numerical analysis A-posteriori-Abschätzung Computersimulation |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015397556&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000646 |
| work_keys_str_mv | AT neittaanmakipekka reliablemethodsforcomputersimulationerrorcontrolandaposterioriestimates AT repinsergeji reliablemethodsforcomputersimulationerrorcontrolandaposterioriestimates |