An up-to-date, one-stop reference–complete with applications
This volume presents the most up-to-date information available on a posteriori error estimation for finite element approximation in mechanics and mathematics. It emphasizes methods for elliptic boundary value problems and includes applications to incompressible flow and nonlinear problems.
Recent years have seen an explosion in the study of a posteriori error estimators due to their remarkable influence on improving both accuracy and reliability in scientific computing. In an effort to provide an accessible source, the authors have sought to present key ideas and common principles on a sound mathematical footing.
Topics covered in this timely reference include:
- Implicit and explicit a posteriori error estimators
- Recovery-based error estimators
- Estimators, indicators, and hierarchic bases
- The equilibrated residual method
- Methodology for the comparison of estimators
- Estimation of errors in quantities of interest
A Posteriori Error Estimation in Finite Element Analysis is a lucid and convenient resource for researchers in almost any field of finite element methods, and for applied mathematicians and engineers who have an interest in error estimation and/or finite elements.
Author(s): Mark Ainsworth, J. Tinsley Oden
Edition: 1st
Publisher: Wiley-Interscience
Year: 2000
Language: English
Pages: 264
Front Cover......Page 1
Title Page......Page 4
Copyright......Page 5
Dedication......Page 6
Contents ......Page 8
Preface ......Page 14
Acknowledgments ......Page 18
1.1 A Posteriori Error Estimation: The Setting ......Page 22
1.2 Status and Scope ......Page 23
1.3 Finite Element Nomenclature ......Page 25
1.3.1 Sobolev Spaces ......Page 26
1.3.2 Inverse Estimates ......Page 28
1.3.3 Finite Element Partitions ......Page 30
1.3.4 Finite Element Spaces on Triangles ......Page 31
1.3.5 Finite Element Spaces on Quadrilaterals ......Page 32
1.3.7 Finite Element Interpolation ......Page 33
1.3.8 Patches of Elements ......Page 34
1.3.9 Regularized Approximation Operators ......Page 35
1.4 Model Problem ......Page 36
1.5 Properties of A Posteriori Error Estimators ......Page 37
1.6 Bibliographical Remarks ......Page 39
2.1 Introduction ......Page 40
2.2 A Simple A Posteriori Error Estimate ......Page 41
2.3.1 Bubble Functions ......Page 44
2.3.2 Bounds on the Residuals ......Page 49
2.3.3 Proof of Two-Sided Bounds on the Error ......Page 52
2.4 A Simple Explicit Least Squares Error Estimator ......Page 53
2.5 Estimates for the Pointwise Error ......Page 55
2.5.1 Regularized Point Load ......Page 56
2.5.2 Regularized Green's Function ......Page 59
2.5.3 Two-Sided Bounds on the Pointwise Error ......Page 60
2.6 Bibliographical Remarks ......Page 63
3.1 Introduction ......Page 64
3.2 The Subdomain Residual Method ......Page 65
3.2.1 Formulation of Subdomain Residual Problem ......Page 66
3.2.2 Preliminaries ......Page 67
3.2.3 Equivalence of Estimator ......Page 68
3.2.4 Treatment of Residual Problems ......Page 70
3.3.1 Formulation of Local Residual Problem ......Page 71
3.3.2 Solvability of the Local Problems ......Page 73
3.3.4 Relationship with Explicit Error Estimators ......Page 75
3.3.5 Efficiency and Reliability of the Estimator ......Page 76
3.4.1 Exact Solution of Element Residual Problem ......Page 77
3.4.2 Analysis and Selection of Approximate Subspaces ......Page 80
3.4.3 Conclusions ......Page 83
3.5 Bibliographical Remarks ......Page 84
4 Recovery-Based Error Estimators ......Page 86
4.1 Examples of Recovery-Based Estimators ......Page 87
4.1.1 An Error Estimator for a Model Problem in One Dimension ......Page 88
4.1.2 An Error Estimator for Bilinear Finite Element Approximation ......Page 90
4.2 Recovery Operators ......Page 93
4.2.1 Approximation Properties of Recovery Operators ......Page 94
4.3 The Superconvergence Property ......Page 96
4.4 Application to A Posteriori Error Estimation ......Page 97
4.5 Construction of Recovery Operators ......Page 98
4.6.1 Linear Approximation on Triangular Elements ......Page 100
4.6.2 Quadratic Approximation on Triangular Elements ......Page 102
4.7 A Cautionary Tale ......Page 103
4.8 Bibliographical Remarks ......Page 104
5.1 Introduction ......Page 106
5.2 Saturation Assumption ......Page 109
5.3 Analysis of Estimator ......Page 110
5.4 Error Estimation Using a Reduced Subspace ......Page 111
5.5 The Strengthened Cauchy-Schwarz Inequality ......Page 115
5.6 Examples ......Page 119
5.7 Multilevel Error Indicators ......Page 121
5.8 Bibliographical Remarks ......Page 130
6.1 Introduction ......Page 132
6.2 The Equilibrated Residual Method ......Page 133
6.3 The Equilibrated Flux Conditions ......Page 137
6.4 Equilibrated Fluxes on Regular Partitions ......Page 138
6.4.2 The Form of the Boundary Fluxes ......Page 139
6.4.4 Local Patch Problems for the Flux Moments ......Page 141
6.4.5 Procedure for Resolution of Patch Problems ......Page 144
6.4.6 Summary ......Page 148
6.5.1 Stability of the Equilibrated Fluxes ......Page 149
6.5.2 Proof of Efficiency of the Estimator ......Page 152
6.6.1 First- Order Equilibration ......Page 154
6.6.2 Flux Moments for Unconstrained Nodes ......Page 155
6.6.4 Recovery of Actual Fluxes ......Page 158
6.7 Equilibrated Fluxes for Higher-Order Elements ......Page 160
6.7.2 Determination of the Flux Moments ......Page 162
6.8 Bibliographical Remarks ......Page 164
7.1 Introduction ......Page 166
7.2 Overview of the Technique ......Page 167
7.3.1 Translation Invariant Meshes ......Page 170
7.3.2 Lower Bounds on the Error ......Page 173
7.3.3 Interior Estimates ......Page 174
7.4.1 Periodic Finite Element Projection on Reference Cell ......Page 178
7.4.2 Periodic Finite Element Projection on a Physical Cell ......Page 179
7.4.3 Periodic Extension on a Subdomain ......Page 180
7.4.4 Asymptotic Finite Element Approximation ......Page 181
7.5 Stability of Estimators ......Page 186
7.5.1 Verification of Stability Condition for Explicit Estimator ......Page 187
7.5.2 Verification of Stability Condition for Implicit Estimators ......Page 189
7.5.3 Verification of Stability Condition for Recovery-Based Estimator ......Page 190
7.5.4 Elementary Consequences of the Stability Condition ......Page 191
7.5.5 Evaluation of Effectivity Index in the Asymptotic Limit ......Page 193
7.6.1 Computation of Asymptotic Finite Element Solution ......Page 195
7.6.2 Evaluation of the Error in Asymptotic Finite Element Approximation ......Page 199
7.6.3 Computation of Limits on the Asymptotic Effectivity Index for Zienkiewicz-Zhu Patch Recovery Estimator ......Page 201
7.6.5 Application to Implicit Element Residual Method ......Page 205
7.7 Bibliographical Remarks ......Page 208
8.1 Introduction ......Page 210
8.2 Estimates for the Error in Quantities of Interest ......Page 212
8.3 Upper and Lower Bounds on the Errors ......Page 214
8.4 Goal-Oriented Adaptive Refinement ......Page 218
8.5.2 Goal-Oriented Adaptivity Based on Pointwise Quantities of Interest ......Page 219
8.6 Local and Pollution Errors ......Page 223
8.7 Bibliographical Remarks ......Page 226
9.1 Introduction ......Page 228
9.2 Stokes and Oseen's Equations ......Page 229
9.2.1 A Posteriori Error Analysis ......Page 232
9.2.2 Summary ......Page 239
9.3 Incompressible Navier-Stokes Equations ......Page 240
9.4.1 A Class of Nonlinear Problems ......Page 243
9.4.2 A Posteriori Error Estimation ......Page 245
9.4.3 Estimation of the Residual ......Page 246
9.5 Bibliographical Remarks ......Page 248
References ......Page 250
Index ......Page 260
Series Titles......Page 262
Back Cover......Page 264
