Introduction to adaptive methods for differential equations.

*(English)*Zbl 0829.65122
Iserles, A. (ed.), Acta Numerica 1995. Cambridge: Cambridge University Press. 105-158 (1995).

The authors of this paper are well-known specialists in adaptive finite element methods for elliptic and parabolic problems. Herein, they present a general framework for the design and analysis of computational methods for differential equations.

The errors which appear between a physical process and its mathematical representation includes various components like modelling, data and computational errors. After some general considerations about modelling and data errors, this paper presents the main lines and results on the state of the art of adaptive computational methods. This presentation is illustrated by well-chosen examples: elliptic, parabolic and hyperbolic one-dimensional linear or nonlinear problems as well as elliptic and parabolic two-dimensional problems.

These results are implemented in the software Femlab which is developed by the authors and which is publicly available through the Internet. To conclude, this is a very nice and attractive introduction to these strategical adaptive methods: with no doubt, the concerned reader will find herein a great motivation to read the coming book and to use Femlab software.

For the entire collection see [Zbl 0817.00007].

The errors which appear between a physical process and its mathematical representation includes various components like modelling, data and computational errors. After some general considerations about modelling and data errors, this paper presents the main lines and results on the state of the art of adaptive computational methods. This presentation is illustrated by well-chosen examples: elliptic, parabolic and hyperbolic one-dimensional linear or nonlinear problems as well as elliptic and parabolic two-dimensional problems.

These results are implemented in the software Femlab which is developed by the authors and which is publicly available through the Internet. To conclude, this is a very nice and attractive introduction to these strategical adaptive methods: with no doubt, the concerned reader will find herein a great motivation to read the coming book and to use Femlab software.

For the entire collection see [Zbl 0817.00007].

Reviewer: M.Bernadou (Le Chesnay)

##### MSC:

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

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

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

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

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

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

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