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. The emphases and selection of the topics reflects our (author and colleagues) involvement in the field over the past 15 years.

Chapter 1 Basic framework

1.1 Preliminaries

1.2 Model problem

1.3 Integral identity

1.4 Global superconvergence analysis

   1.4.1 Superclose analysis

   1.4.2 Global superconvergence

   1.4.3 A posteriori error estimate

1.5 Brief summary and notes

Chapter 2 Integral identities

2.1 Bilinear rectangular element

   2.1.1 Integral identity for ∫Ω(u-uI)xvx

   2.1.2 Integral identity for ∫Ω(u-uI)xvy

   2.1.3 Integralidentities for ∫Ω(u-uI)xv and ∫Ω(u-uI)v

   2.1.4 Summary

  2.2 General results for bilinear elements

   2.2.1 General elliptic bilinear form of order two

   2.2.2 Bilinear finite element on general domain

   2.2.3 Regular locally refined mesh

  2.3 Rectangular Lagrange elements of order

   2.3.1 Integral identity for ∫Ω(u-uI)xvx(p=2)

   2.3.2 Integral identity for ∫Ω(u-uI)xvx(p≥3)

   2.3.3 Integral identity for ∫Ω(u-uI)xvy

   2.3.4 Integral identity for ∫Ω(u-uI)xv and ∫Ωγ(u-uI)v

   2.3.5 General elliptic bilinear form

 2.4 Rectangular finite elements with derivative degrees of freedom

   2.4.1 Bicubic Hermite dement 

   2.4.2 Adini element

  2.5 Rectangular mixed finite elements

   2.5.1 Mixed finite element for Stokes equation

   2.5.2 Mixed finite element for elliptic equation

   2.5.3 Mixed finite element for Maxwell equation

2.6 Summary of integral identities

Chapter 3 Superconvergence Analysis

3.1 Elliptic partial differential equations

   3.1.1 Poisson’s equation

   3.1.2 Elliptic equation of order two (Lagrange element)

   3.1.3 Elliptic equation of order two (Adini element)

   3.1.4 Singular problem

   3.1.5 Coupling of different finite element spaces

   3.1.6 Elliptic equation of order four

3.2 Nonconforming finite elements

   3.2.1 Elliptic equation of order two

   3.2.2 Elliptic equation of order four

3.3 Evolution partial differential equations

   3.3.1 Parabolic equation

   3.3.2 Hyperbolic equation of order two

   3.3.3 Integral-differential equation

   3.3.4 Other evolution partial differential equations

3.4 Hyperbolic equation of order one

   3.4.1 Standard finite element scheme

   3.4.2 Streamline diffusion method

   3.4.3 Discontinuous Galerkin method

3.5 Mixed finite dements

   3.5.1 Elliptic PDE of order two

   3.5.2 Elliptic PDE of order four

   3.5.3 Stokes equation

   3.5.4 Maxwdl’s equation

3.6 Integral equations

3.7 Optimal control problems

3.8 Summary of superconvergence analysis

Chapter 4 More discussions on high accuracy analysis

4.1 Global superconvergence

   4.1.1 Interpolation post-processing

   4.1.2 Global superconvergence

   4.1.3 Summary of global superconvergence

4.2 Extrapolation

4.3 Defect correction

4.4 Local superconvergence

4.5 Ultraconvergence

4.6 Eigenvalue problems

4.7 Numerical examples

Chapter 5 A posteriori error estimates

5.1 Introduction

5.2 Residual type a posteriori error estimate

5.3 Recovery type a posteriori error estimate

5.4 Equivalence of recovery type estimator

5.5 Asymptotically exactness of recovery type estimator

5.6 Some remarks on two kinds of estimators

5.7 A posteriori error estimate for optimal control problems

   5.7.1 Model problem

   5.7.2 Residual type a posteriori error estimate

   5.7.3 Recovery type a posteriori error estimate

5.8 Numerical examples

Bibliography