The Mathematical Theory of Finite Element Methods
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The Mathematical Theory of Finite Element Methods

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This is the third and yet further updated edition of a highly regarded mathematical text. Brenner develops the basic mathematical theory of the finite element method, the most widely used technique for engineering design and analysis. Her volume formalizes basic tools that are commonly used by researchers in the field but not previously published. The book is ideal for 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. This new edition is substantially updated with additional exercises throughout and new chapters on Additive Schwarz Preconditioners and Adaptive Meshes.
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Product details

  • Paperback | 400 pages
  • 155 x 235 x 21.84mm | 640g
  • New York, NY, United States
  • English
  • Revised
  • Softcover reprint of hardcover 3rd ed. 2008
  • XVIII, 400 p.
  • 1441926119
  • 9781441926111
  • 1,736,420

Back cover copy

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)
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Table of contents

Preface(3rdEd).- Preface(2ndEd).- Preface(1stED).- Basic Concepts.- Sobolev Spaces.- Variational Formulation of Elliptic Boundary Value Problems.- The Construction of a Finite Element of Space.- Polynomial Approximation Theory in Sobolev Spaces.- n-Dimensional Variational Problems.- Finite Element Multigrid Methods.- Additive Schwarz Preconditioners.- Max-norm Estimates.- Adaptive Meshes.- Variational Crimes.- Applications to Planar Elasticity.- Mixed Methods.- Iterative Techniques for Mixed Methods.- Applications of Operator-Interpolation Theory.- References.- Index.
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Review Text

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."

ZENTRALBLATT MATH

From the reviews of the third edition:

"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. ... The approach remains very clear and precise ... . 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. ... valuable reference and source for researchers (mainly mathematicians) in the topic." (Calin Ioan Gheorghiu, Zentralblatt MATH, Vol. 1135 (13), 2008)
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Review quote

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."

ZENTRALBLATT MATH

From the reviews of the third edition:



"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. ... The approach remains very clear and precise ... . 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. ... valuable reference and source for researchers (mainly mathematicians) in the topic." (Calin Ioan Gheorghiu, Zentralblatt MATH, Vol. 1135 (13), 2008)
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