Superconvergence Analysis and a Posteriori Error Estimation in Finite Element Methods – Series in Information and Computational Science 40

【Introduction】

The purpose of this book is to give an essentially self-contained presentation of the mathematical theory underlying the global superconvergence analysis and the recovery type a posteriori error estimates. Since, as mentioned in the beginning, the literature and research on superconvergence and a posteriori error estimation are huge, it is impossible to include all relevant material in this book. This book tries to summarize most of the research results on global superconvergence analysis completed by the author and her colleagues, especially by Professor Q. Lin’s group. Instead of using the Green function theory as the theoretical basis as in most other earlier books on superconvergence, our global superconvergence analysis is based on the so-called integral identity technique. For a posteriori error estimates, this book focuses on the recovery type a posteriori error estimate, which is more close to the superconvergence analysis in its theoretical analysis.

【Main Contents】

Chapter 1 Basic Framework
1.1 Preliminaries
1.2 Model problem
1.3 Integral identity
1.4 Global superconvergence analysis
1.5 Brief summary and notes
Chapter 2 Integral identities
2.1 Bilinear rectangular element
2.2 General results for bilinear elements
2.3 Rectangular Lagrange element of order р
2.4 Rectangular finite elements with derivative degree of freedom
2.5 Rectangular mixed finite elements
2.6 Summary of integral identities
Chapter 3 Superconvergence Analysis
3.1 Elliptic partial differential equations
3.2 Nonconforming finite elements
3.3 Evolution partial differential equation
3.4 Mixed finite elements
3.5 Integral equations
3.6 Optimal control problems
3.7 Optimal control problems
3.8 Summary of superconvergence analysis
Chapter 4 More discussions on high accuracy analysis
Chapter 5 A posteriori error estimates