Posteriori Error Estimation in Finite Element Analysis

MARK AINSWORTH, PhD, is Professor of Applied Mathematics atStrathclyde University, UK.J. TINSLEY ODEN, PhD, is Director of the Texas Institute forComputational and Applied Mathematics at the University of Texas,Austin.

• Preface xiiiAcknowledgments xvii1 Introduction 11.1 A Posteriori Error Estimation: The Setting 11.2 Status and Scope 21.3 Finite Element Nomenclature 41.3.1 Sobolev Spaces 51.3.2 Inverse Estimates 71.3.3 Finite Element Partitions 91.3.4 Finite Element Spaces on Triangles 101.3.5 Finite Element Spaces on Quadrilaterals 111.3.6 Properties of Lagrange Basis Functions 121.3.7 Finite Element Interpolation 121.3.8 Patches of Elements 131.3.9 Regularized Approximation Operators 141.4 Model Problem 151.5 Properties of A Posteriori Error Estimators 161.6 Bibliographical Remarks 182 Explicit A Posteriori Estimators 192.1 Introduction 192.2 A Simple A Posteriori Error Estimate 202.3 Efficiency of Estimator 232.3.1 Bubble Functions 232.3.2 Bounds on the Residuals 282.3.3 Proof of Two-Sided Bounds on the Error 312.4 A Simple Explicit Least Squares Error Estimator 322.5 Estimates for the Pointwise Error 342.5.1 Regularized Point Load 352.5.2 Regularized Green's Function 382.5.3 Two-Sided Bounds on the Pointwise Error 392.6 Bibliographical Remarks 4%3 Implicit A Posteriori Estimators 433.1 Introduction 433.2 The Subdomain Residual Method 443.2.1 Formulation of Subdomain Residual Problem 453.2.2 Preliminaries 463.2.3 Equivalence of Estimator ^73.2.4 Treatment of Residual Problems 493.3 The Element Residual Method 503.3.1 Formulation of Local Residual Problem 503.3.2 Solvability of the Local Problems 523.3.3 The Classical Element Residual Method 543.3.4 Relationship with Explicit Error Estimators 543.3.5 Efficiency and Reliability of the Estimator 553.4 The Influence and Selection of Subspaces 563.4.1 Exact Solution of Element Residual Problem 563.4.2 Analysis and Selection of Approximate Subspaces 593.4.3 Conclusions 623.5 Bibliographical Remarks 634 Recovery-Based Error Estimators 654.1 Examples of Recovery-Based Estimators 664.1.1 An Error Estimator for a Model Problem in One Dimension 614.1.2 An Error Estimator for Bilinear Finite Element Approximation 694.2 Recovery Operators 724.2.1 Approximation Properties of Recovery Operators 734.3 The Superconvergence Property 754.4 Application to A Posteriori Error Estimation 764.5 Construction of Recovery Operators 774.6 The Zienkiewicz-Zhu Patch Recovery Technique 794.6.1 Linear Approximation on Triangular Elements 794.6.2 Quadratic Approximation on Triangular Elements 814.6.3 Patch Recovery for Quadrilateral Elements 824.7 A Cautionary Tale 824.8 Bibliographical Remarks 835 Estimators, Indicators, and Hierarchic Bases 855.1 Introduction 855.2 Saturation Assumption 885.3 Analysis of Estimator 895.4 Error Estimation Using a Reduced Subspace 905.5 The Strengthened Cauchy-Schwarz Inequality 945.6 Examples 985.7 Multilevel Error Indicators 1005.8 Bibliographical Remarks 1096 The Equilibrated Residual Method 1116.1 Introduction 1116.2 The Equilibrated Residual Method 1126.3 The Equilibrated Flux Conditions 1166.4 Equilibrated Fluxes on Regular Partitions 1176.4.1 First-Order Equilibration Condition 1186.4.2 The Form of the Boundary Fluxes 1186.4.3 Equilibration Conditions in Terms of the Moments 1206.4.4 Local Patch Problems for the Flux Moments 1206.4.5 Procedure for Resolution of Patch Problems 1236.4.6 Summary 1276.5 Efficiency of the Estimator 1286.5.1 Stability of the Equilibrated Fluxes 1286.5.2 Proof of Efficiency of the Estimator 1316.6 Equilibrated Fluxes on Partitions Containing Hanging Nodes 1336.6.1 First-Order Equilibration 1336.6.2 Flux Moments for Unconstrained Nodes 1346.6.3 Flux Moments with Respect to Constrained Nodes 1376.6.4 Recovery of Actual Fluxes 1376.7 Equilibrated Fluxes for Higher-Order Elements 1396.7.1 The Form of the Boundary Fluxes 1416.7.2 Determination of the Flux Moments 1416.8 Bibliographical Remarks 1437 Methodology for the Comparison of Estimators 1457.1 Introduction 1457.2 Overview of the Technique 1467.3 Approximation over an Interior Subdomain 1497.3.1 Translation Invariant Meshes 1497.3.2 Lower Bounds on the Error 1527.3.3 Interior Estimates 1537.4 Asymptotic Finite Element Approximation 1577.4.1 Periodic Finite Element Projection on Reference Cell 1577.4.2 Periodic Finite Element Projection on a Physical Cell 1587.4.3 Periodic Extension on a Subdomain 1597.4.4 Asymptotic Finite Element Approximation 1607.5 Stability of Estimators 1657.5.1 Verification of Stability Condition for Explicit Estimator 1667.5.2 Verification of Stability Condition for Implicit Estimators 1687.5.3 Verification of Stability Condition for Recovery-Based Estimator 1697.5.4 Elementary Consequences of the Stability Condition 1707.5.5 Evaluation of Effectivity Index in the Asymptotic Limit 1727.6 An Application of the Theory 1747.6.1 Computation of Asymptotic Finite Element Solution 1747.6.2 Evaluation of the Error in Asymptotic Finite Element Approximation 1787.6.3 Computation of Limits on the Asymptotic Effectivity Index for Zienkiewicz-Zhu Patch Recovery Estimator 1807.6.4 Application to Equilibrated Residual Method 1847.6.5 Application to Implicit Element Residual Method 1847.7 Bibliographical Remarks 1878 Estimation of the Errors in Quantities of Interest 1898.1 Introduction 1898.2 Estimates for the Error in Quantities of Interest 1918.3 Upper and Lower Bounds on the Errors 1938.4 Goal-Oriented Adaptive Refinement 1978.5 Example of Goal-Oriented Adaptivity 1988.5.1 Adaptivity Based on Control of Global Error in Energy 1988.5.2 Goal-Oriented Adaptivity Based on Pointwise Quantities of Interest 1988.6 Local and Pollution Errors 2028.7 Bibliographical Remarks 2059 Some Extensions 2079.1 Introduction 2079.2 Stokes and Oseen's Equations 2089.2.1 A Posteriori Error Analysis 2119.2.2 Summary 2189.3 Incompressible Navier-Stokes Equations 2199.4 Extensions to Nonlinear Problems 2229.4.1 A Class of Nonlinear Problems 2229.4.2 A Posteriori Error Estimation 2249.4.3 Estimation of the Residual 2259.5 Bibliographical Remarks 227References 229Index 239