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**Adaptive finite element methods for differential equations.**
*(English)*
Zbl 1020.65058

Lectures in Mathematics, ETH Zürich. Basel: Birkhäuser. viii, 207 p. EUR 22.00/net; sFr 35.00 (2003).

Adaptivity as a rule improves significantly discretization techniques since it selects the elements to be refined according to their influence upon the goal of the problem. In case of the solution of a differential equation this could be an error bound, but the concept of specified objectives allows a significantly greater variety of adaptations, e.g. to objective functionals of optimal control problems with differential equations as state equations. The dual weighted residual method turns out as the method of choice considered throughout the book because it enables excellently adaptation to various goal functionals. The material of the book covers a lecture series of the second author at ETH Zürich. The lecture notes condense research results mainly achieved at IWR Heidelberg over a longer period, partially published recently in comprehensive articles.

The book is structured into the following 12 chapters:

1. Introduction, 2. An ordinary differential equation model case, 3. A partial differential equation model case, 4. Practical aspects, 5. The limits of theoretical analysis, 6. An abstract approach to nonlinear problems, 7. Eigenvalue problems, 8. Optimization problems, 9. Time-dependent problems, 10. Applications in structural mechanics, 11. Applications in fluid mechanics, 12. Miscellaneous and open problems.

The authors guide the interested reader from basic principles of a-posteriori estimates to a wide range of applications of the developed techniques for goal oriented grid marking and refinement. The basic idea of dual weighted residuals is presented in chapter 1 and its easy presentation helps later to understand following applications of this principle. Beside a few academic examples to underline the basics the main emphasis of the authors is toward real life applications. To support these applications also different problems of technical aspects of the implementations are discussed. Numerous computer results demonstrate the efficiency of the proposed adaptive grid refinements. In addition, exercises at the end of the chapters completed by solutions at the end of the book support the reading and understanding.

Overall the book can be recommended graduate students in applied mathematics as well as researchers interested in principles of error control in state of the art finite element implementations including those for optimal control problems.

The book is structured into the following 12 chapters:

1. Introduction, 2. An ordinary differential equation model case, 3. A partial differential equation model case, 4. Practical aspects, 5. The limits of theoretical analysis, 6. An abstract approach to nonlinear problems, 7. Eigenvalue problems, 8. Optimization problems, 9. Time-dependent problems, 10. Applications in structural mechanics, 11. Applications in fluid mechanics, 12. Miscellaneous and open problems.

The authors guide the interested reader from basic principles of a-posteriori estimates to a wide range of applications of the developed techniques for goal oriented grid marking and refinement. The basic idea of dual weighted residuals is presented in chapter 1 and its easy presentation helps later to understand following applications of this principle. Beside a few academic examples to underline the basics the main emphasis of the authors is toward real life applications. To support these applications also different problems of technical aspects of the implementations are discussed. Numerous computer results demonstrate the efficiency of the proposed adaptive grid refinements. In addition, exercises at the end of the chapters completed by solutions at the end of the book support the reading and understanding.

Overall the book can be recommended graduate students in applied mathematics as well as researchers interested in principles of error control in state of the art finite element implementations including those for optimal control problems.

Reviewer: Christian Grossmann (Dresden)

### MSC:

65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |

49M15 | Newton-type methods |

65L70 | Error bounds for numerical methods for ordinary differential equations |

65K10 | Numerical optimization and variational techniques |

49J15 | Existence theories for optimal control problems involving ordinary differential equations |

49J20 | Existence theories for optimal control problems involving partial differential equations |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

65N15 | Error bounds for boundary value problems involving PDEs |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |