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The mathematical theory of finite element methods. 2nd ed. (English) Zbl 1012.65115
Texts in Applied Mathematics. 15. Berlin: Springer. xv, 361 p. (2002).
The first edition of this book (1994) was reviewed in Zbl 0804.65101, by this reviewer. The sentiments expressed in that review remain relevant to this second edition. For example, it is still this reviewer’s opinion that the content and pace of presentation is at a relatively advanced level for the intended readership. On the other hand, works on the finite element method exist in abundance, and even texts dealing with mathematical aspects of the method are plentiful nowadays. So the key question is then the following: have the authors of a new work on the subject added anything new by way of presentation, insights, or topics?
In the case of the work under review the response is resoundingly affirmative. This was in fact so in the first edition, in which the topics covered included some for which no treatment at this level was to be found. Good examples were the chapters on max-norm estimates and operator interpolation theory, the latter being particularly welcome given the difficulty that exists in finding accessible and concise treatments. The authors have continued along this path in the second edition, adding chapters on additive Schwarz preconditioners with applications to domain decomposition methods, and on a posteriori estimators and adaptivity. For researchers in finite elements and graduate students who are embarking on research in the area, this book is a valuable source, and provides an accessible route to the journal literature. In fact, the authors have drawn extensively on recent results in the literature, by themselves and others, in their choice of material and its presentation. Thus, overall, one is the beneficiary of a novel point of view, at this level.
In summary, then, this is an excellent, though demanding, introduction to key mathematical topics in the finite element method, and at the same time a valuable reference and source for workers in the area.

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
74S05 Finite element methods applied to problems in solid mechanics
74B05 Classical linear elasticity
35Jxx Elliptic equations and elliptic systems
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
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