Ainsworth M., Oden J. Tinsley A Posterori Error Estimation in Finite Element Analysis

John Wiley & Sons, Inc., 2000. 264 p. ISBN: 047129411XThis 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
mplicit 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 elementsA Posteriori Error Estimation: The Setting
Status and Scope
Finite Element Nomenclature
Sobolev Spaces
Inverse Estimates
Finite Element Partitions
Finite Element Spaces on Triangles
Finite Element Spaces on Quadrilaterals
Properties of Lagrange Basis Functions
Finite Element Interpolation
Patches of Elements
Regularized Approximation Operators
Model Problem
Properties of A Posteriori Error Estimators
Bibliographical Remarks
Explicit A Posteriori EstimatorsA Simple A Posteriori Error Estimate
Efficiency of Estimator
Bubble Functions
Bounds on the Residuals
Proof of Two-Sided Bounds on the Error
A Simple Explicit Least Squares Error Estimator
Estimates for the Pointwise Error
Regularized Point Load
Regularized Green's Function
Two-Sided Bounds on the Pointwise Error
Bibliographical Remarks
Implicit A Posteriori EstimatorsThe Subdomain Residual Method
Formulation of Subdomain Residual Problem
Preliminaries
Equivalence of Estimator
Treatment of Residual Problems
The Element Residual Method
Formulation of Local Residual Problem
Solvability of the Local Problems
The Classical Element Residual Method
Relationship with Explicit Error Estimators
Efficiency and Reliability of the Estimator
The Influence and Selection of Subspaces
Exact Solution of Element Residual Problem
Analysis and Selection of Approximate Subspaces
Conclusions
Bibliographical Remarks
Recovery-Based Error Estimators
Examples of Recovery-Based Estimators
An Error Estimator for a Model Problem in One Dimension
An Error Estimator for Bilinear Finite Element Approximation
Recovery Operators
Approximation Properties of Recovery Operators
The Superconvergence Property
Application to A Posteriori Error Estimation
Construction of Recovery Operators
The Zienkiewicz-Zhu Patch Recovery Technique
Linear Approximation on Triangular Elements
Quadratic Approximation on Triangular Elements
Patch Recovery for Quadrilateral Elements
A Cautionary Tale
Bibliographical Remarks
Estimators, Indicators, and Hierarchic BasesSaturation Assumption
Analysis of Estimator
Error Estimation Using a Reduced Subspace
The Strengthened Cauchy-Schwarz Inequality
Examples
Multilevel Error Indicators
Bibliographical Remarks
The Equilibrated Residual MethodThe Equilibrated Residual Method
The Equilibrated Flux Conditions
Equilibrated Fluxes on Regular Partitions
First-Order Equilibration Condition
The Form of the Boundary Fluxes
Equilibration Conditions in Terms of the Moments
Local Patch Problems for the Flux Moments
Procedure for Resolution of Patch ProblemsEfficiency of the Estimator
Stability of the Equilibrated Fluxes
Proof of Efficiency of the Estimator
Equilibrated Fluxes on Partitions Containing Hanging Nodes
First- Order Equilibration
Flux Moments for Unconstrained Nodes
Flux Moments with Respect to Constrained Nodes
Recovery of Actual Fluxes
Equilibrated Fluxes for Higher-Order Elements
The Form of the Boundary Fluxes
Determination of the Flux Moments
Bibliographical Remarks
Methodology for the Comparison of EstimatorsOverview of the Technique
Approximation over an Interior Subdomain
Translation Invariant Meshes
Lower Bounds on the Error
Interior Estimates
Asymptotic Finite Element Approximation
Periodic Finite Element Projection on Reference Cell
Periodic Finite Element Projection on a Physical Cell
Periodic Extension on a Subdomain
Asymptotic Finite Element Approximation
Stability of Estimators
Verification of Stability Condition for Explicit Estimator
Verification of Stability Condition for Implicit Estimators
Verification of Stability Condition for Recovery-Based Estimator
Elementary Consequences of the Stability Condition
Evaluation of Effectivity Index in the Asymptotic Limit
An Application of the Theory
Computation of Asymptotic Finite Element Solution
Evaluation of the Error in Asymptotic Finite Element Approximation
Computation of Limits on the Asymptotic Effectivity Index for Zienkiewicz-Zhu Patch Recovery Estimator
Application to Equilibrated Residual Method
Application to Implicit Element Residual Method
Bibliographical Remarks
Estimation of the Errors in Quantities of InterestEstimates for the Error in Quantities of Interest
Upper and Lower Bounds on the Errors
Goal-Oriented Adaptive Refinement
Example of Goal-Oriented Adaptivity
Adaptivity Based on Control of Global Error in Energy
Goal-Oriented Adaptivity Based on Pointwise Quantities of Interest
Local and Pollution Errors
Bibliographical Remarks
Some ExtensionsStokes and Oseen's Equations
A Posteriori Error AnalysisIncompressible Navier-Stokes Equations
Extensions to Nonlinear Problems
A Class of Nonlinear Problems
A Posteriori Error Estimation
Estimation of the Residual
Bibliographical Remarks