This book develops the basic mathematical theory of the finite element method, the most widely used technique for engineering design and analysis. This expanded second edition contains new chapters on additive Schwarz preconditioners and adaptive meshes. New exercises have also been added throughout. The book will be useful to mathematicians as well as engineers and physical scientists. It can be used for a course that provides an introduction to basic functional analysis, approximation theory, and numerical analysis, while building upon and applying basic techniques of real variable theory. Different course paths can be chosen, allowing the book to be used for courses designed for students with different interests.

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This book develops the basic mathematical theory of the finite element method, the most widely used technique for engineering design and analysis.

The third edition contains four new sections: the BDDC domain decomposition preconditioner, convergence analysis of an adaptive algorithm, interior penalty methods and Poincara\'e-Friedrichs inequalities for piecewise W^1_p functions. New exercises have also been added throughout.

The initial chapter provides an introducton to the entire subject, developed in the one-dimensional case. Four subsequent chapters develop the basic theory in the multidimensional case, and a fifth chapter presents basic applications of this theory. Subsequent chapters provide an introduction to:

- multigrid methods and domain decomposition methods

- mixed methods with applications to elasticity and fluid mechanics

- iterated penalty and augmented Lagrangian methods

- variational "crimes" including nonconforming and isoparametric methods, numerical integration and interior penalty methods

- error estimates in the maximum norm with applications to nonlinear problems

- error estimators, adaptive meshes and convergence analysis of an adaptive algorithm

- Banach-space operator-interpolation techniques

The book has proved useful to mathematicians as well as engineers and physical scientists. It can be used for a course that provides an introduction to basic functional analysis, approximation theory and numerical analysis, while building upon and applying basic techniques of real variable theory. It can also be used for courses that emphasize physical applications or algorithmic efficiency.

Reviews of earlier editions: "This book represents an important contribution to the mathematical literature of finite elements. It is both a well-done text and a good reference." (Mathematical Reviews, 1995)

"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."

(Zentralblatt, 2002)

From the reviews of the second edition:

*S.C. Brenner and L.R. Scott*

*The Mathematical Theory of Finite Element Methods*

*"[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."*— ZENTRALBLATT MATH

"The authors have continued ... 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 ... this book is a valuable source, and provides an accessible route to the journal literature. ... In summary, then, this is an excellent ... introduction to key mathematical topics in the finite element method, and at the same time a valuable reference and source for workers in this area." (Batamanathan D. Reddy, Zentralblatt MATH, Vol. 1012, 2003)

"This book is devoted to the mathematical theory of finite element method and is the second edition of the book from 1994. ... The book can be used as a basis for graduate-level courses for students in applied mathematics, physics, engineering sciences and other fields ... . The numerous and interesting exercises round off and complete each chapter. ... The book can be highly recommended to everyone who is teaching or researching in the field of numerical solution of partial differential equations." (I. P. Gavrilyuk, Zeitschrift für Analysis und ihre Anwendungen, Vol. 22 (1), 2003)

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ISBN 13: 9780387954516

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