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**Multilevel finite element approximation. Theory and applications.**
*(English)*
Zbl 0830.65107

Teubner Skripten zur Numerik. Stuttgart: B. G. Teubner. 160 p. (1994).

As is mentioned in the introduction, the author makes an attempt to collect some specific information from approximation theory and function spaces, by presenting it in a form understandable for specialists working on large scale computational methods for partial differential equations. He presents recent developments in the field of multilevel-multigrid methods and includes an introduction of the wavelet concept.

These notes reflect the present research interests of the author, extended by the idea of other specialists working in the field. They are based on lectures given in 1993 by the author at the Institutes of Mathematics and Informatics at the Technical University of Munich.

The author’s main aim is to survey the approximation-theoretical background of stable splittings of Sobolev spaces with respect to multilevel finite element schemes.

One can say that this book represents a collection of research notes rather than a textbook or a monograph.

The book consists of four chapters. The chapter headings are as follows: (i) Finite element approximation; (ii) Function spaces; (iii) Applications to multilevel methods; (iv) Error estimates and adaptivity. It closes with a bibliography, extended on 20 pages, and a subject index.

This book will be a useful source of information for mathematicians and engineers using numerical methods in general and finite element, multigrid and wavelet methods in particular.

These notes reflect the present research interests of the author, extended by the idea of other specialists working in the field. They are based on lectures given in 1993 by the author at the Institutes of Mathematics and Informatics at the Technical University of Munich.

The author’s main aim is to survey the approximation-theoretical background of stable splittings of Sobolev spaces with respect to multilevel finite element schemes.

One can say that this book represents a collection of research notes rather than a textbook or a monograph.

The book consists of four chapters. The chapter headings are as follows: (i) Finite element approximation; (ii) Function spaces; (iii) Applications to multilevel methods; (iv) Error estimates and adaptivity. It closes with a bibliography, extended on 20 pages, and a subject index.

This book will be a useful source of information for mathematicians and engineers using numerical methods in general and finite element, multigrid and wavelet methods in particular.

Reviewer: D.D.Stancu (Cluj-Napoca)

### MSC:

65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |

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

65M55 | Multigrid methods; domain decomposition 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 |

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

46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |