A Posteriori Error Estimation Techniques For Finite Element Methods (numerical Mathematics And Scientific Computation)

A posteriori error estimation techniques are fundamental to the efficient numerical solution of PDEs arising in physical and technical applications. This book gives a unified approach to these techniques and guides graduate students, researchers, and practitioners towards understanding, applying and developing self-adaptive discretization methods. Abstract: A posteriori error estimation techniques are fundamental to the efficient numerical solution of PDEs arising in physical and technical applications. This book gives a unified approach to these techniques and guides graduate students, researchers, and practitioners towards understanding, applying and developing self-adaptive discretization methods. Read more... Cover 1 Contents 20 1 A Simple Model Problem 22 1.1 Motivation and Overview 22 1.2 The Model Problem and its Discretisation 25 1.3 Notations and Auxiliary Results 26 1.4 Residual Estimates 31 1.5 A Vertex-Oriented Residual Error Indicator 38 1.6 Edge Residuals 41 1.7 Auxiliary Local Problems 46 1.8 A Hierarchical Approach 52 1.9 Gradient Recovery 57 1.10 Equilibrated Residuals 62 1.11 Dual Weighted Residuals 66 1.12 The Hyper-Circle Method 69 1.13 Efficiency and Asymptotic Exactness 74 1.14 Convergence of the Adaptive Process I 79 1.15 Summary and Outlook 83 2 Implementation 85 2.1 Mesh-Refinement 85 2.2 Mesh-Coarsening 90 2.3 Mesh-Smoothing 91 2.4 Data Structures 95 2.5 Numerical Examples 97 3 Auxiliary Results 100 3.1 Function Spaces 100 3.2 Finite Element Meshes and Spaces 102 3.3 Trace Inequalities 108 3.4 Poincaré and Friedrichs’ Inequalities 112 3.5 Interpolation Error Estimates 129 3.6 Inverse Estimates 133 3.7 Decomposition of Affine Functions in L[sup(p)] (0, 1; Y*) 151 3.8 Estimation of Residuals 153 4 Linear Elliptic Equations 172 4.1 Abstract Linear Problems 172 4.2 The Model Problem Revisited 178 4.3 Reaction–Diffusion Equations 180 4.4 Convection–Diffusion Equations 184 4.5 Anisotropic Meshes 198 4.6 Non-Smooth Coefficients 212 4.7 Eigenvalue Problems 226 4.8 Mixed Formulation of the Poisson Equation 229 4.9 The Equations of Linear Elasticity 244 4.10 The Stokes Equations 258 4.11 The Bi-harmonic Equation 269 4.12 Non–Conforming Discretisations 282 4.13 Convergence of the Adaptive Process II 285 5 Nonlinear Elliptic Equations 302 5.1 Abstract Nonlinear Problems 302 5.2 Quasilinear Equations of Second Order 311 5.3 Eigenvalue Problems Revisited 320 5.4 The Stationary Navier–Stokes Equations 322 6 Parabolic Equations 330 6.1 The Heat Equation 330 6.2 Time-Dependent Convection–Diffusion Equations 338 6.3 Linear Parabolic Equations of Second Order 347 6.4 The Method of Characteristics 350 6.5 The Time-Dependent Stokes Equations 356 6.6 Nonlinear Parabolic Equations of Second Order 368 6.7 Finite Volume Methods 381 6.8 Convergence of the Adaptive Process III 383 References 394 List of Symbols 408 Index 411 A 411 B 411 C 411 D 411 E 412 F 412 G 412 H 412 I 412 J 412 K 412 L 412 M 412 N 413 O 413 P 413 Q 413 R 413 S 413 T 414 U 414 V 414 W 414 Y 414 Z 414 Self-adaptive discretization methods are now an indispensable tool for the numerical solution of partial differential equations that arise from physical and technical applications. The aim is to obtain a numerical solution within a prescribed tolerance using a minimal amount of work. The main tools in achieving this goal are a posteriori error estimates which give global and local information on the error of the numerical solution and which can easily be computed from the given numerical solution and the data of the differential equation. This book reviews the most frequently used a posteriori error estimation techniques and applies them to a broad class of linear and nonlinear elliptic and parabolic equations. Although there are various approaches to adaptivity and a posteriori error estimation, they are all based on a few common principles. The main aim of the book is to elaborate these basic principles and to give guidelines for developing adaptive schemes for new problems. Chapters 1 and 2 are quite elementary and present various error indicators and their use for mesh adaptation in the framework of a simple model problem. The basic principles are introduced using a minimal amount of notations and techniques providing a complete overview for the non-specialist. Chapters 4-6 on the other hand are more advanced and present a posteriori error estimates within a general framework using the technical tools collected in Chapter 3. Most sections close with a bibliographical remark which indicates the historical development and hints at further results.