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The mathematical theory of finite element methods. (English) Zbl 0804.65101
Texts in Applied Mathematics. 15. New York: Springer-Verlag. xii, 294 p. (1994).
The aim of this text is to provide a detailed introduction to the mathematical theory of the finite element method. In order to achieve this goal it is essential that a variety of mathematical tools, mainly from functional analysis and function spaces, be mastered; these tools are also given a treatment in the text.
The book is divided into two parts: the first part, comprising Chapters 0 through 5, cover the background mathematics and basic theory of finite elements. The second part comprises further 7 chapters, each of which builds on the material in the first part, and is devoted to a specific advanced topic.
While the authors claim that the only prerequisite is a course in real variables, and that “even this has not been necessary for well-prepared engineers and scientists \(\dots\)”, an examination of the book gives the impression that this may well be too optimistic a claim: it is true that all of the necessary topics from functional analysis are covered, but these are often given a very brief treatment (for example, normed spaces, Lebesgue integration theory and Sobolev spaces up to trace theorems and dual spaces in Chapter 1, which has around 20 pages), which affords little time or opportunity for students to assimilate the material to the extent that they would feel confident in applying the results later. This comment is made notwithstanding the extensive sets of exercises at the end of each chapter; these range from simple to challenging, yet one is left with the feeling that a more extensive introduction to functional analysis would have been a better idea.
Nevertheless, the chapters covering the basic material are written in a competent, readable and well-organized manner, with the material arranged in such a way as to be read with the minimum of paging back and forth in search of relevant results. The introductory Chapter 0 is a good idea: it gives an overview of the essential ideas which are further developed in subsequent chapters. The treatment of elliptic variational problems and the basic theory of finite elements follow a conventional route – these topics are covered in Chapters 2 and 3 – though, to their credit, the authors give more attention to nonsymmetric variational problems than one is accustomed to seeing in texts at this level.
Finite element interpolation theory, covered in Chapter 4, takes a route a little different from the original theory in that it is developed by using an appropriate generalization of Taylors theorem to find polynomial approximations of functions in Sobolev spaces. Some important results such as the Bramble-Hilbert lemma and Friedrich’s inequality are obtained along the way. This basic theory is then applied to variational problems in Chapter 5, in which a careful exposition of the theory, in the context of Poisson’s equation, is developed.
The chapters on more advanced topics deal with multigrid methods, max- norm estimates, “variational crimes”, linear elasticity, mixed methods, and there is a final chapter on operator-interpolation theory. Much of the material in these chapters is based on fairly recent research, though the authors have taken care to make the treatment accessible to a students readership. For example, minimal smoothness assumptions on functions are relaxed at times to simplify the theory. The material in these later chapters reflects the authors’ particular interests, but this is quite acceptable given the broad range of topics from which it is possible to choose.
A comment about the notation used in Chapter 9 (on elasticity) is in order. The authors choose to make use of undertildes to denote vector- and matrix- valued functions or operators. This is an extremely unsightly notation: the undertilde is conventionally used in handwritten material to indicate quantities that in printed form would be boldface. Indeed, it is the printer’s way of denoting a quantity to be rendered boldface. The author’s use of the undertilde is therefore unconventional, and should have been replaced by boldface quantities.
The above occasional criticisms aside, this is a well-written book. A great deal of material is covered, and students who have taken the trouble to master at least some of the advanced material in the later chapters would be well-placed to embark on research in the area. The book would work even better as a course text if computational and programming aspects of finite elements were to be integrated into the course work, or if a course on computational aspects of finite elements were offered in tandem.

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition 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