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The mathematical theory of finite element methods. 3rd ed. (English) Zbl 1135.65042
Texts in Applied Mathematics 15. New York, NY: Springer (ISBN 978-0-387-75933-3/hbk). xvii, 397 p. (2008).
The rough contents of this book are as follows: Ch. 0, Basic concepts; Ch. 1, Sobolev spaces; Ch. 2, Variational formulation of elliptic boundary value problems; Ch. 3, The construction of finite element space; Ch. 4, Polynomial approximation theory in Sobolev spaces; Ch. 5, \(n\)-dimensional variational problems; Ch. 6, Finite element multigrid methods; Ch. 7, Additive Schwarz preconditioners, Ch. 8, Max-norm estimates, Ch. 9, Adaptive meshes; Ch. 10, Variational crimes, Ch. 11, Application to planar elasticity; Ch. 12, Mixed methods; Ch. 13, Iterative techniques for mixed methods; Ch. 14, Applications of operator-interpolation theory; References and Index.
The book represents an excelent survey of the deep mathematical roots of finite element methods as well as of some of the newest and most formal results concerning these methods. With respect to the second edition [The mathematical theory of finite element methods. 2nd ed., Berlin: Springer (2002; Zbl 1012.65115)], the authors add four new sections and operate some improvements throughout the text. The approach remains very clear and precise with emphasis on the constructive aspects of the methods. A significant number of examples and exercises improve considerably the accessability of the text. The authors also point out different ways the book could be used in various courses.
A comment seems to be suitable here. Except the first chapter, which is fairly useful and accessible to a large variety of students, the rest of the book requires quite subtle knowledge of some sophisticated parts of mathematics. However, this impressive book remains a fairly valuable reference and source for researchers (mainly mathematicians) in the topic.

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
46N40 Applications of functional analysis in numerical analysis
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
74S05 Finite element methods applied to problems in solid mechanics
76M10 Finite element methods applied to problems in fluid mechanics
35J25 Boundary value problems for second-order elliptic equations
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
65F35 Numerical computation of matrix norms, conditioning, scaling
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