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Finite element methods (Part 1). (English) Zbl 0712.65091
Handbook of Numerical Analysis 2. Amsterdam etc.: North-Holland (ISBN 0-444-70365-9). ix, 928 p. (1991).

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[For Vol. I, Part 1 (1990) see Zbl 0689.65001.]
This volume is entirely devoted to the finite element methods (FEM). It contains six articles which survey some major aspects of this field; a second volume will be issued later.
Firstly, J. T. Oden briefly introduces (12 pages) FEM and underlines their interests, early history and mathematical theory.
Next, P. G. Ciarlet thoroughly details (330 pages) FEM for linear elliptic problems. After a short introduction of elliptic boundary value problems the author describes the main aspects of the FEM. The basic error estimates are mathematically analyzed for second-order problems: conforming and nonconforming methods, effect of numerical integration, problems posed over curved domains. This article ends by the approximation of fourth-order problems. The whole paper is written in the same spirit as the well-known book from P. G. Ciarlet [The finite element method for elliptic problems, Studies in Mathematics and its Applications 4 (1978; Zbl 0383.65058)].
The third article (170 pages) from L. B. Wahlbin concerns the local behavior in FEM. This subject is mainly related to the approximation of problems which do not have smooth solutions. Several basic techniques for one-space dimension problems are presented in the introduction. Next the local estimates for second-order elliptic problems and some examples are considered. Subsequently the author discusses singularly perturbed elliptic to elliptic and convection-dominated model problems. This paper ends with an example of time localized behavior and with results on superconvergence.
The fourth paper (120 pages) from J. E. Roberts and J. M. Thomas is devoted to mixed and hybrid methods. Basic required mathematical results on interpolation and approximation are given in chapters 2 and 3 while the basic aspects of mixed and hybrid methods are clearly and thoroughly detailed in chapters 4 and 5. The authors conclude with some extensions and variations.
The fifth article (150 pages) from I. Babuska and J. Osborn is concerned with the approximation of eigenvalue problems. After some examples and general considerations, they detail the abstract spectral approximation results and next, they apply these abstract results to several representative problems.
The last paper (140 pages) from H. Fujita and T. Suzuki is devoted to evolution problems. They start with records on the use of FEM to solve elliptic boundary value problems. Next they study the FEM for initial value problems along the line of semigroup of operators. Then, they extend the error estimates to the temporary inhomogeneous case by using the methods of Helfrich and of energy. The discretization of hyperbolic equations is briefly considered.
All these contributions give a very nice volume: the contents are of very high quality and the presentation is high level.
Reviewer: M.Bernadou

65-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to numerical analysis
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
65-03 History of numerical analysis
65N15 Error bounds for boundary value problems involving PDEs
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
35J25 Boundary value problems for second-order elliptic equations
65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs