Lectures on Numerical Methods for Non-Linear Variational Problems

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Springer Science & Business Media, Jan 22, 2008 - Mathematics - 496 pages
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When Herb Keller suggested, more than two years ago, that we update our lectures held at the Tata Institute of Fundamental Research in 1977, and then have it published in the collection Springer Series in Computational Physics, we thought, at first, that it would be an easy task. Actually, we realized very quickly that it would be more complicated than what it seemed at first glance, for several reasons: 1. The first version of Numerical Methods for Nonlinear Variational Problems was, in fact, part of a set of monographs on numerical mat- matics published, in a short span of time, by the Tata Institute of Fun- mental Research in its well-known series Lectures on Mathematics and Physics; as might be expected, the first version systematically used the material of the above monographs, this being particularly true for Lectures on the Finite Element Method by P. G. Ciarlet and Lectures on Optimization—Theory and Algorithms by J. Cea. This second version had to be more self-contained. This necessity led to some minor additions in Chapters I-IV of the original version, and to the introduction of a chapter (namely, Chapter Y of this book) on relaxation methods, since these methods play an important role in various parts of this book.
 

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Contents

Generalities on Elliptic Variational Inequalities and on Their Approximation 1 Introduction 2 Functional Context
1
3 Existence and Uniqueness Results for EVI of the First Kind
3
4 Existence and Uniqueness Results for EVI of the Second Kind
5
5 Internal Approximation of EVI of the First Kind
8
6 Internal Approximation of EVI of the Second Kind
12
7 Penalty Solution of Elliptic Variational Inequalities of the First Kind
15
8 References
26
Application of the Finite Element Method to the Approximation of Some SecondOrder EVI 1 Introduction 2 An Example of EVI of the First Kind T...
27
4 Convergence of ALG 1
171
5 Convergence of ALG 2
179
6 Applications
183
7 General Comments
194
LeastSquares Solution of Nonlinear Problems Application to Nonlinear Problems in Fluid Dynamics 1 Introduction Synopsis 2 LeastSquares Solution...
195
3 LeastSquares Solution of a Nonlinear Dirichlet Model Problem
198
4 Transonic Flow Calculations by LeastSquares and Finite Element Methods
211
Numerical Solution of the NavierStokes Equations for Incompressible Viscous Fluids by LeastSquares and Finite Element Methods
244

The ElastoPlastic Torsion Problem
41
A Simplified Signorini Problem
56
A Simplified Friction Problem
68
The Flow of a Viscous Plastic Fluid in a Pipe
78
7 On Some Useful Formulae
96
On the Approximation of Parabolic Variational Inequalities 1 Introduction References 2 Formulation and Statement of the Main Results
98
3 Numerical Schemes for Parabolic Linear Equations
99
4 Approximation of PVI of the First Kind
101
5 Approximation of PVI of the Second Kind
103
TimeDependent Flow of a Bingham Fluid in a Cylindrical Pipe
104
Applications of Elliptic Variational Inequality Methods to the Solution of Some Nonlinear Elliptic Equations 1 Introduction 2 Theoretical and Numer...
110
3 A Subsonic Flow Problem
134
Relaxation Methods and Applications1 1 Generalities 2 Some Basic Results of Convex Analysis
140
FiniteDimensional Case
142
4 Block Relaxation Methods
151
Application
152
6 Solution of Systems of Nonlinear Equations by Relaxation Methods
163
DecompositionCoordination Methods by Augmented Lagrangian Applications1 1 Introduction
166
2 Properties of P and of the Saddle Points of if and J?P
168
3 Description of the Algorithms
170
6 Further Comments on Chapter VII and Conclusion
318
A Brief Introduction to Linear Variational Problems 1 Introduction 2 A Family of Linear Variational Problems
321
3 Internal Approximation of Problem P
326
4 Application to the Solution of Elliptic Problems for Partial Differential Operators
330
Conclusion
397
A Finite Element Method with Upwinding for SecondOrder Problems with Large FirstOrder Terms 1 Introduction 2 The Model Problem
399
4 A Finite Element Approximation with Upwinding
400
6 Numerical Experiments
404
7 Concluding Comments
414
Some Complements on the NavierStokes Equations and Their Numerical Treatment 1 Introduction 2 Finite Element Approximation of the Boundary ...
415
3 Some Comments On the Numerical Treatment of the Nonlinear Term V
416
4 Further Comments on the Boundary Conditions
417
5 Decomposition Properties of the Continuous and Discrete Stokes Problems of Sec 4 Application to Their Numerical Solution
425
6 Further Comments
430
Some Illustrations from an Industrial Application
431
Bibliography
434
Glossary of Symbols
455
Author Index
463
Subject Index
467
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About the author (2008)

Roland Glowinski is Cullen Professor of Mathematics at the University of Houston and Emeritus Professor at Laboratoire J. L. Lions, University P. and M. Curie, Paris. He is a Member of the French National Academy of Sciences and in 2004 won the Von Karman prize from the Society for Industrial and Applied Mathematics. He has written over 300 research papers and this is his 3rd book.

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