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Adaptive Finite Element Methods for Differential Equations

  • Wolfgang Bangerth
  • Rolf Rannacher

Part of the Lectures in Mathematics book series (LM)

Table of contents

  1. Front Matter
    Pages i-viii
  2. Wolfgang Bangerth, Rolf Rannacher
    Pages 1-14
  3. Wolfgang Bangerth, Rolf Rannacher
    Pages 15-24
  4. Wolfgang Bangerth, Rolf Rannacher
    Pages 25-40
  5. Wolfgang Bangerth, Rolf Rannacher
    Pages 41-60
  6. Wolfgang Bangerth, Rolf Rannacher
    Pages 61-70
  7. Wolfgang Bangerth, Rolf Rannacher
    Pages 71-80
  8. Wolfgang Bangerth, Rolf Rannacher
    Pages 81-100
  9. Wolfgang Bangerth, Rolf Rannacher
    Pages 101-112
  10. Wolfgang Bangerth, Rolf Rannacher
    Pages 113-128
  11. Wolfgang Bangerth, Rolf Rannacher
    Pages 129-142
  12. Wolfgang Bangerth, Rolf Rannacher
    Pages 143-160
  13. Wolfgang Bangerth, Rolf Rannacher
    Pages 161-165
  14. Back Matter
    Pages 167-208

About this book

Introduction

These Lecture Notes discuss concepts of `self-adaptivity' in the numerical solution of differential equations, with emphasis on Galerkin finite element methods. The key issues are a posteriori error estimation and it automatic mesh adaptation. Besides the traditional approach of energy-norm error control, a new duality-based technique, the Dual Weighted Residual method for goal-oriented error estimation, is discussed in detail. This method aims at economical computation of arbitrary quantities of physical interest by properly adapting the computational mesh. This is typically required in the design cycles of technical applications. For example, the drag coefficient of a body immersed in a viscous flow is computed, then it is minimized by varying certain control parameters, and finally the stability of the resulting flow is investigated by solving an eigenvalue problem. `Goal-oriented' adaptivity is designed to achieve these tasks with minimal cost.
At the end of each chapter some exercises are posed in order to assist the interested reader in better understanding the concepts presented. Solutions and accompanying remarks are given in the Appendix. For the practical exercises, sample programs are provided via internet.

Keywords

Differential equations differential equation eigenvalue fluid mechanics mechanics numerical analysis solution

Authors and affiliations

  • Wolfgang Bangerth
    • 1
  • Rolf Rannacher
    • 2
  1. 1.TICAMThe University of Texas at AustinAustinUSA
  2. 2.Institute of Applied MathematicsUniversity of HeidelbergHeidelbergGermany

Bibliographic information