A posteriori error estimation in finite element analysis:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Wiley
2000
|
| Series: | Pure and applied mathematics
A Wiley-Interscience publication |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVII, 240 S. graph. Darst. |
| ISBN: | 047129411X |
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MARC
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| 246 | 1 | 3 | |a A posteriori error estimation in finite element analysis |
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| 300 | |a XVII, 240 S. |b graph. Darst. | ||
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Record in the Search Index
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| adam_text | Contents
Preface
xiii
Acknowledgments
xvii
1
Introduction
1
1.1
A Posteriori Error Estimation: The Setting
1
1.2
Status and Scope
2
1.3
Finite Element Nomenclature
Ą
1.3.1
Sobolev Spaces
5
1.3.2
Inverse Estimates
7
1.3.3
Finite Element Partitions
9
1.3-4
Finite Element Spaces on Triangles
10
1.3.5
Finite Element Spaces on Quadrilaterals
11
1.3.6
Properties of
Lagrange
Basis Functions
12
1.3.7
Finite Element Interpolation
12
1.3.8
Patches of Elements
13
1.3.9
Regularized Approximation Operators
1Ą
1.4 Model Problem
15
1.5
Properties of A Posteriori Error Estimators
16
1.6
Bibliographical Remarks
18
viii CONTENTS
2
Explicit A Posteriori Estimators
19
2.1
Introduction
19
2.2
A Simple A Posteriori Error Estimate
20
2.3
Efficiency of Estimator
23
2.3.1
Bubble Functions
23
2.3.2
Bounds on the Residuals
28
2.3.3
Proof of Two-Sided Bounds on the Error
31
2.Ą
A Simple Explicit Least Squares Error Estimator
32
2.5
Estimates for the Pointwise Error
3Ą
2.5.1
Regularized Point Load
35
2.5.2
Regularized Green s Function
38
2.5.3
Two-Sided Bounds on the Pointwise Error
39
2.6
Bibliographical Remarks
Ą2
3
Implicit A Posteriori Estimators
A3
3.1
Introduction
A3
3.2
The
Subdomain
Residual Method
44
3.2.1
Formulation of
Subdomain
Residual
Problem
45
3.2.2
Preliminaries
46
3.2.3
Equivalence of Estimator
47
3.2.4
Treatment of Residual Problems
4$
3.3
The Element Residual Method
50
3.3.1
Formulation of Local Residual Problem
50
3.3.2
Solvability of the Local Problems
52
3.3.3
The Classical Element Residual Method
54
3.3.4
Relationship with Explicit Error
Estimators
54
3.3.5
Efficiency and Reliability of the Estimator
55
3.4
The Influence and Selection of Subspaces
56
3.4-1
Exact Solution of Element Residual
Problem
56
3.4-2
Analysis and Selection of Approximate
Subspaces
59
3.4-3
Conclusions
62
3.5
Bibliographical Remarks
63
4
Recovery-Based Error Estimators
65
4-1
Examples of Recovery-Based Estimators
66
CONTENTS ¡x
4.1.1
An
Error Estimator for a Model Problem
in One Dimension
67
4-1.2
An Error Estimator for Bilinear Finite
Element Approximation
69
4-2
Recovery Operators
72
4.2.I Approximation Properties of Recovery
Operators
73
4·3
The
Superconvergence
Property
75
4-4
Application to A Posteriori Error Estimation
76
4-5
Construction of Recovery Operators
77
4-6
The Zienkiewicz-Zhu Patch Recovery Technique
79
4-6.1
Linear Approximation on Triangular
Elements
79
4-6.2
Quadratic Approximation on Triangular
Elements
81
4.6.З
Patch Recovery for Quadrilateral
Elements
82
4.7
A Cautionary Tale
82
4.8
Bibliographical Remarks
83
Estimators, Indicators, and Hierarchic Bases
85
5.1
Introduction
85
5.2
Saturation Assumption
88
5.3
Analysis of Estimator
89
5.4
Error Estimation Using a Reduced Subspace
90
5.5
The Strengthened Cauchy-Schwarz Inequality
94
5.6
Examples
98
5.7
Multilevel Error Indicators
100
5.8
Bibliographical Remarks
109
The Equilibrated Residual Method 111
6.1
Introduction 111
6.2
The Equilibrated Residual Method
112
6.3
The Equilibrated Flux Conditions
116
6.4
Equilibrated Fluxes on Regular Partitions
117
6.4.1 First-Order Equilibration Condition
118
6.4.2
The Form of the Boundary Fluxes
118
6.4.3
Equilibration Conditions in Terms of the
Moments
120
χ
CONTENTS
6.4.4
Local Patch Problems for the Flux
Moments
120
6.4-5
Procedure for Resolution of Patch
Problems
123
6.4.6
Summary
127
6.5
Efficiency of the Estimator
128
6.5.1
Stability of the Equilibrated Fluxes
128
6.5.2
Proof of Efficiency of the Estimator
131
6.6
Equilibrated Fluxes on Partitions Containing
Hanging Nodes
133
6.6.1
First-Order Equilibration
133
6.6.2
Flux Moments for Unconstrained Nodes
134
6.6.3
Flux Moments with Respect to Constrained
Nodes
137
6.6.4
Recovery of Actual Fluxes
137
6.7
Equilibrated Fluxes for Higher-Order Elements
139
6.7.1
The Form of the Boundary Fluxes
ЦІ
6.7.2
Determination of the Flux Moments
ЦІ
6.8
Bibliographical Remarks
ЦЗ
7
Methodology for the Comparison of Estimators
145
7.1
Introduction
Ц5
7.2
Overview of the Technique
Ц6
7.3
Approximation over an Interior
Subdomain
1Ą9
7.3.1
Translation Invariant Meshes
Ц9
7.3.2
Lower Bounds on the Error
152
7.3.3
Interior Estimates
153
7.4
Asymptotic Finite Element Approximation
157
7-4-1
Periodic Finite Element Projection on
Reference Cell
157
7.4-2
Periodic Finite Element Projection on a
Physical Cell
158
7.4.З
Periodic Extension on
a Subdomain
159
7Ą-4
Asymptotic Finite Element Approximation
160
7.5
Stability of Estimators
165
7.5.1
Verification of Stability Condition for
Explicit Estimator
166
7.5.2
Verification of Stability Condition for
Implicit Estimators
168
CONTENTS
x¡
7.5.3
Verification of Stability Condition for
Recovery-Based Estimator
169
7.5.Ą
Elementary Consequences of the Stability
Condition
170
7.5.5
Evaluation of Effectivity Index in the
Asymptotic Limit
172
7.6
An Application of the Theory 174
7.6.1
Computation of Asymptotic Finite
Element Solution
17Ą
7.6.2
Evaluation of the Error in Asymptotic
Finite Element Approximation
178
7.6.3
Computation of Limits on the Asymptotic
Effectivity Index for Zienkiewicz-Zhu
Patch Recovery Estimator
180
7.6.4
Application to Equilibrated Residual
Method
184
7.6.5
Application to Implicit Element Residual
Method
184
7.7
Bibliographical Remarks
187
8
Estimation of the Errors in Quantities of Interest
189
8.1
Introduction
189
8.2
Estimates for the Error in Quantities of Interest
191
8.3
Upper and Lower Bounds on the Errors
193
8.4
Goal-Oriented Adaptive Refinement
197
8.5
Example of Goal-Oriented Adaptivity
198
8.5.1
Adaptivity Based on Control of Global
Error in Energy
198
8.5.2
Goal-Oriented Adaptivity Based on
Pointwise Quantities of Interest
198
8.6
Local and Pollution Errors
202
8.7
Bibliographical Remarks
205
9
Some Extensions
207
9.1
Introduction
207
9.2
Stokes and Oseen s Equations
208
9.2.1
A Posteriori Error Analysis
211
9.2.2
Summary
218
9.3
Incompressible Navier-Stokes Equations
219
9.4
Extensions to Nonlinear Problems
222
xii CONTENTS
9.4-1
A Class of Nonlinear Problems
9.4-2
A Posteriori Error Estimation
224
9-4-3
Estimation of the Residual
225
9.5
Bibliographical Remarks
227
References
229
Index
239
|
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| id | DE-604.BV013803641 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:52:14Z |
| institution | BVB |
| isbn | 047129411X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009436613 |
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| physical | XVII, 240 S. graph. Darst. |
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| series2 | Pure and applied mathematics A Wiley-Interscience publication |
| spelling | Ainsworth, Mark 1965- Verfasser (DE-588)121031373 aut A posteriori error estimation in finite element analysis Mark Ainsworth and J. Tinsely Oden A posteriori error estimation in finite element analysis New York [u.a.] Wiley 2000 XVII, 240 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Pure and applied mathematics A Wiley-Interscience publication Fehleranalyse - Finite-Elemente-Methode Finite-Elemente-Methode (DE-588)4017233-8 gnd rswk-swf Fehleranalyse (DE-588)4016608-9 gnd rswk-swf A-posteriori-Abschätzung (DE-588)4346907-3 gnd rswk-swf Finite-Elemente-Methode (DE-588)4017233-8 s A-posteriori-Abschätzung (DE-588)4346907-3 s DE-604 Fehleranalyse (DE-588)4016608-9 s Oden, John Tinsley 1936-2023 Verfasser (DE-588)108393224 aut Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009436613&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ainsworth, Mark 1965- Oden, John Tinsley 1936-2023 A posteriori error estimation in finite element analysis Fehleranalyse - Finite-Elemente-Methode Finite-Elemente-Methode (DE-588)4017233-8 gnd Fehleranalyse (DE-588)4016608-9 gnd A-posteriori-Abschätzung (DE-588)4346907-3 gnd |
| subject_GND | (DE-588)4017233-8 (DE-588)4016608-9 (DE-588)4346907-3 |
| title | A posteriori error estimation in finite element analysis |
| title_alt | A posteriori error estimation in finite element analysis |
| title_auth | A posteriori error estimation in finite element analysis |
| title_exact_search | A posteriori error estimation in finite element analysis |
| title_full | A posteriori error estimation in finite element analysis Mark Ainsworth and J. Tinsely Oden |
| title_fullStr | A posteriori error estimation in finite element analysis Mark Ainsworth and J. Tinsely Oden |
| title_full_unstemmed | A posteriori error estimation in finite element analysis Mark Ainsworth and J. Tinsely Oden |
| title_short | A posteriori error estimation in finite element analysis |
| title_sort | a posteriori error estimation in finite element analysis |
| topic | Fehleranalyse - Finite-Elemente-Methode Finite-Elemente-Methode (DE-588)4017233-8 gnd Fehleranalyse (DE-588)4016608-9 gnd A-posteriori-Abschätzung (DE-588)4346907-3 gnd |
| topic_facet | Fehleranalyse - Finite-Elemente-Methode Finite-Elemente-Methode Fehleranalyse A-posteriori-Abschätzung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009436613&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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