The finite element method for elliptic problems

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The Finite Element Method for Elliptic Problems is the only book available that analyzes in depth the mathematical foundations of the finite element method. It is a valuable reference and introduction to current research on the numerical analysis of the finite element method, as well as a working textbook for graduate courses in numerical analysis. It includes many useful figures, and there are many exercises of varying difficulty.

Although nearly 25 years have passed since this book was first published, the majority of its content remains up-to-date. Chapters 1 through 6, which cover the basic error estimates for elliptic problems, are still the best available sources for material on this topic. The material covered in Chapters 7 and 8, however, has undergone considerable progress in terms of new applications of the finite element method; therefore, the author provides, in the Preface to the Classics Edition, a bibliography of recent texts that complement the classic material in these chapters.

Audience This book is particularly useful to graduate students, researchers, and engineers using finite element methods. The reader should have knowledge of analysis and functional analysis, particularly Hilbert spaces, Sobolev spaces, and differential calculus in normed vector spaces. Other than these basics, the book is mathematically self-contained.

Author(s): Philippe G. Ciarlet
Series: Classics in applied mathematics 40
Edition: 2nd
Publisher: Society for Industrial and Applied Mathematics
Year: 2002

Language: English
Pages: 559
City: Philadelphia, PA

The Finite Element Method for Elliptic Problems......Page 1
TABLE OF CONTENTS......Page 10
PREFACE TO THE CLASSICS EDITION......Page 16
PREFACE......Page 20
GENERAL PLAN AND INTERDEPENDENCE TABLE......Page 27
CHAPTER 1 ELLIPTIC BOUNDARY VALUE PROBLEMS......Page 30
1.1. Abstract problems......Page 31
1.2. Examples of elliptic boundary value problems......Page 39
CHAPTER 2 INTRODUCTION TO THE FINITE ELEMENT METHOD......Page 65
2.1. Basic aspects of the finite element method......Page 66
2.2. Examples of finite elements and finite element spaces......Page 72
2.3. General properties of finite elements and finite element spaces......Page 107
2.4. General considerations on convergence......Page 132
CHAPTER 3 CONFORMING FINITE ELEMENT METHODS FOR SECOND-ORDER PROBLEMS......Page 139
3.1. Interpolation theory in Sobolev spaces......Page 141
3.2. Application to second-order problems over polygonal domains......Page 160
3.3. Uniform convergence......Page 176
CHAPTER 4 OTHER FINITE ELEMENT METHODS FOR SECOND-ORDER PROBLEMS......Page 203
4.1. The effect of numerical integration......Page 207
4.2. A nonconforming method......Page 236
4.3. Isoparametric finite elements......Page 253
4.4 Application to second-order problems over curved domains......Page 277
Additional bibliography and comments......Page 305
CHAPTER 5 APPLICATION OF THE FINITE ELEMENT METHOD......Page 316
5.1. The obstacle problem......Page 318
5.2. The minimal surface problem......Page 330
5.3. Nonlinear problems of monotone type......Page 341
Additional bibliography and comments......Page 359
CHAPTER 6 FINITE ELEMENT METHODS FOR THE PLATE PROBLEM......Page 362
6.1. Conforming methods......Page 363
6.2. Nonconfonning methods......Page 391
CHAPTER 7 A MIXED FINITE ELEMENT METHOD......Page 410
7.1. A mixed finite element method for the biharmonic problem......Page 412
7.2. Solution of the discrete problem by duality techniques......Page 424
Additional bibliography and comments......Page 436
CHAPTER 8 FINITE ELEMENT METHODS FOR SHELLS......Page 454
8.1. The shell problem......Page 455
8.2. Conforming methods......Page 468
8.3. A nonconforming method for the arch problem......Page 480
Bibliography and comments......Page 495
EPILOGUE:Some "real-life" finite element model examples......Page 498
BIBLIOGRAPHY......Page 510
GLOSSARY OF SYMBOLS......Page 541
INDEX......Page 550