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.

MARK AINSWORTH, PhD, is Professor of Applied Mathematics at Strathclyde University, UK.

Preface

xiii

Acknowledgments

xvii

Introduction

1

(18)

A Posteriori Error Estimation: The Setting

1

(1)

Status and Scope

2

(2)

Finite Element Nomenclature

4

(11)

Sobolev Spaces

5

(2)

Inverse Estimates

7

(2)

Finite Element Partitions

9

(1)

Finite Element Spaces on Triangles

10

(1)

Finite Element Spaces on Quadrilaterals

11

(1)

Properties of Lagrange Basis Functions

12

(1)

Finite Element Interpolation

12

(1)

Patches of Elements

13

(1)

Regularized Approximation Operators

14

(1)

Model Problem

15

(1)

Properties of A Posteriori Error Estimators

16

(2)

Bibliographical Remarks

18

(1)

Explicit A Posteriori Estimators

19

(24)

Introduction

19

(1)

A Simple A Posteriori Error Estimate

20

(3)

Efficiency of Estimator

23

(9)

Bubble Functions

23

(5)

Bounds on the Residuals

28

(3)

Proof of Two-Sided Bounds on the Error

31

(1)

A Simple Explicit Least Squares Error Estimator

32

(2)

Estimates for the Pointwise Error

34

(8)

Regularized Point Load

35

(3)

Regularized Green's Function

38

(1)

Two-Sided Bounds on the Pointwise Error

39

(3)

Bibliographical Remarks

42

(1)

Implicit A Posteriori Estimators

43

(22)

Introduction

43

(1)

The Subdomain Residual Method

44

(6)

Formulation of Subdomain Residual Problem

45

(1)

Preliminaries

46

(1)

Equivalence of Estimator

47

(2)

Treatment of Residual Problems

49

(1)

The Element Residual Method

50

(6)

Formulation of Local Residual Problem

50

(2)

Solvability of the Local Problems

52

(2)

The Classical Element Residual Method

54

(1)

Relationship with Explicit Error Estimators

54

(1)

Efficiency and Reliability of the Estimator

55

(1)

The Influence and Selection of Subspaces

56

(7)

Exact Solution of Element Residual Problem

56

(3)

Analysis and Selection of Approximate Subspaces

59

(3)

Conclusions

62

(1)

Bibliographical Remarks

63

(2)

Recovery-Based Error Estimators

65

(20)

Examples of Recovery-Based Estimators

66

(6)

An Error Estimator for a Model Problem in One Dimension

67

(2)

An Error Estimator for Bilinear Finite Element Approximation

69

(3)

Recovery Operators

72

(3)

Approximation Properties of Recovery Operators

73

(2)

The Superconvergence Property

75

(1)

Application to A Posteriori Error Estimation

76

(1)

Construction of Recovery Operators

77

(2)

The Zienkiewicz-Zhu Patch Recovery Technique

79

(3)

Linear Approximation on Triangular Elements

79

(2)

Quadratic Approximation on Triangular Elements

81

(1)

Patch Recovery for Quadrilateral Elements

82

(1)

A Cautionary Tale

82

(1)

Bibliographical Remarks

83

(2)

Estimators, Indicators, and Hierarchic Bases

85

(26)

Introduction

85

(3)

Saturation Assumption

88

(1)

Analysis of Estimator

89

(1)

Error Estimation Using a Reduced Subspace

90

(4)

The Strengthened Cauchy-Schwarz Inequality

94

(4)

Examples

98

(2)

Multilevel Error Indicators

100

(9)

Bibliographical Remarks

109

(2)

The Equilibrated Residual Method

111

(34)

Introduction

111

(1)

The Equilibrated Residual Method

112

(4)

The Equilibrated Flux Conditions

116

(1)

Equilibrated Fluxes on Regular Partitions

117

(11)

First-Order Equilibration Condition

118

(1)

The Form of the Boundary Fluxes

118

(2)

Equilibration Conditions in Terms of the Moments

120

(1)

Local Patch Problems for the Flux Moments

120

(3)

Procedure for Resolution of Patch Problems

123

(4)

Summary

127

(1)

Efficiency of the Estimator

128

(5)

Stability of the Equilibrated Fluxes

128

(3)

Proof of Efficiency of the Estimator

131

(2)

Equilibrated Fluxes on Partitions Containing Hanging Nodes

133

(6)

First-Order Equilibration

133

(1)

Flux Moments for Unconstrained Nodes

134

(3)

Flux Moments with Respect to Constrained Nodes

137

(1)

Recovery of Actual Fluxes

137

(2)

Equilibrated Fluxes for Higher-Order Elements

139

(4)

The Form of the Boundary Fluxes

141

(1)

Determination of the Flux Moments

141

(2)

Bibliographical Remarks

143

(2)

Methodology for the Comparison of Estimators

145

(44)

Introduction

145

(1)

Overview of the Technique

146

(3)

Approximation over an Interior Subdomain

149

(8)

Translation Invariant Meshes

149

(3)

Lower Bounds on the Error

152

(1)

Interior Estimates

153

(4)

Asymptotic Finite Element Approximation

157

(8)

Periodic Finite Element Projection on Reference Cell

157

(1)

Periodic Finite Element Projection on a Physical Cell

158

(1)

Periodic Extension on a Subdomain

159

(1)

Asymptotic Finite Element Approximation

160

(5)

Stability of Estimators

165

(9)

Verification of Stability Condition for Explicit Estimator

166

(2)

Verification of Stability Condition for Implicit Estimators

168

(1)

Verification of Stability Condition for Recovery-Based Estimator

169

(1)

Elementary Consequences of the Stability Condition

170

(2)

Evaluation of Effectivity Index in the Asymptotic Limit

172

(2)

An Application of the Theory

174

(13)

Computation of Asymptotic Finite Element Solution

174

(4)

Evaluation of the Error in Asymptotic Finite Element Approximation

178

(2)

Computation of Limits on the Asymptotic Effectivity Index for Zienkiewicz-Zhu Patch Recovery Estimator

180

(4)

Application to Equilibrated Residual Method

184

(1)

Application to Implicit Element Residual Method

184

(3)

Bibliographical Remarks

187

(2)

Estimation of the Errors in Quantities of Interest

189

(18)

Introduction

189

(2)

Estimates for the Error in Quantities of Interest

191

(2)

Upper and Lower Bounds on the Errors

193

(4)

Goal-Oriented Adaptive Refinement

197

(1)

Example of Goal-Oriented Adaptivity

198

(4)

Adaptivity Based on Control of Global Error in Energy

198

(1)

Goal-Oriented Adaptivity Based on Pointwise Quantities of Interest

198

(4)

Local and Pollution Errors

202

(3)

Bibliographical Remarks

205

(2)

Some Extensions

207

(22)

Introduction

207

(1)

Stokes and Oseen's Equations

208

(11)

A Posteriori Error Analysis

211

(7)

Summary

218

(1)

Incompressible Navier-Stokes Equations

219

(3)

Extensions to Nonlinear Problems

222

(5)

A Class of Nonlinear Problems

222

(2)

A Posteriori Error Estimation

224

(1)

Estimation of the Residual

225

(2)

Bibliographical Remarks

227

(2)

References

229

(10)

Index

239

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