## 有限元方法: 精度及其改善Distributed by Elsevier Science on behalf of Science Press. This book discusses the accuracy of various finite element approximations and how to improve them, with the help of extrapolations and super convergence's post-processing technique. The discussion is based on asymptotic expansions for finite elements and finally reduces to the technique of integration by parts, embedding theorems and norm equivalence lemmas. The book is also devoted to explaining the origin of theorems. * Masterly exposition of the accuracy and improvement of finite element methods, highlighting the postprocessing * Emphasis on understanding of higher knowledge * Accessible to students, engaging for experts and professionals * Written by leading Chinese mathematicians, available internationally for the first time |

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### Contents

Eulers Algorithm and Finite Element Method | 3 |

Function Spaces and Norm Equivalence Lemmas | 23 |

From K to Eigenvalue Computation of PDEs | 65 |

Appendix 1 | 110 |

Appendix 2 | 120 |

Appendix 3 | 132 |

Appendix 4 | 143 |

First Part Bibliography | 149 |

Expansion of Integrals on Rectangular Elements | 165 |

Expansion of Integrals on Triangle Elements | 226 |

Quasisuperconvergence and Quasiexpansion | 252 |

Postprocessing | 298 |

Second Part Bibliography | 317 |

### Common terms and phrases

adjacent elements ah(u BH lemma bilinear interpolation boundary broken-line common edge Consider the functional construct postprocessing interpolation convergence defined derivatives direct computation shows domain dxdy ECHL edge-integral eigenfunction EQ\ot error estimate error expansion Example expansion lemma extrapolation finite difference method finite element interpolation finite element method finite element solution finite element space fishbone four small elements function space fundamental formula G Vh give a proof gives the lemma Green formula h2 f Hq(Q interpolation function inverse inequality isosceles right triangle L2-norm Legendre polynomial Lin Q line integrals mixed element space nonconforming elements nonconforming error norm equivalence lemma piecewise Poisson equation polygon polynomial Proof Let prove quasi-superconvergence Ratio(2 real superconvergence reference element right triangle mesh rotated bilinear element satisfies scaling argument gives Schwartz inequality Sobolev embedding theorem Suppose that Vh uniform convergence uniform rectangular mesh vector vertices Vh,o weak equation wxvxdxdy zero