Lectures on Numerical Methods for Non-Linear Variational Problems

By: R. Glowinski

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Paperback | 22 January 2008

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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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Some Preliminary Comments	p. xiv
Generalities on Elliptic Variational Inequalities and on Their Approximation	p. 1
Introduction	p. 1
Functional Context	p. 1
Existence and Uniqueness Results for EVI of the First Kind	p. 3
Existence and Uniqueness Results for EVI of the Second Kind	p. 5
Internal Approximation of EVI of the First Kind	p. 8
Internal Approximation of EVI of the Second Kind	p. 12
Penalty Solution of Elliptic Variational Inequalities of the First Kind	p. 15
References	p. 26
Application of the Finite Element Method to the Approximation of Some Second-Order EVI	p. 27
Introduction	p. 27
An Example of EVI of the First Kind: The Obstacle Problem	p. 27
A Second Example of EVI of the First Kind: The Elasto-Plastic Torsion Problem	p. 41
A Third Example of EVI of the First Kind: A Simplified Signorini Problem	p. 56
An Example of EVI of the Second Kind: A Simplified Friction Problem	p. 68
A Second Example of EVI of the Second Kind: The Flow of a Viscous Plastic Fluid in a Pipe	p. 78
On Some Useful Formulae	p. 96
On the Approximation of Parabolic Variational Inequalities	p. 98
Introduction: References	p. 98
Formulation and Statement of the Main Results	p. 98
Numerical Schemes for Parabolic Linear Equations	p. 99
Approximation of PVI of the First Kind	p. 101
Approximation of PVI of the Second Kind	p. 103
Application to a Specific Example: Time-Dependent Flow of a Bingham Fluid in a Cylindrical Pipe	p. 104
Applications of Elliptic Variational Inequality Methods to the Solution of Some Nonlinear Elliptic Equations	p. 110
Introduction	p. 110
Theoretical and Numerical Analysis of Some Mildly Nonlinear Elliptic Equations	p. 110
A Subsonic Flow Problem	p. 134
Relaxation Methods and Applications	p. 140
Generalities	p. 140
Some Basic Results of Convex Analysis	p. 140
Relaxation Methods for Convex Functionals: Finite-Dimensional Case	p. 142
Block Relaxation Methods	p. 151
Constrained Minimization of Quadratic Functionals in Hilbert Spaces by Under and Over-Relaxation Methods: Application	p. 152
Solution of Systems of Nonlinear Equations by Relaxation Methods	p. 163
Decomposition-Coordination Methods by Augmented Lagrangian: Applications	p. 166
Introduction	p. 166
Properties of (P) and of the Saddle Points of <$>{scr L}<$> and <$>{scr L}_r<$>	p. 168
Description of the Algorithms	p. 170
Convergence of ALG 1	p. 171
Convergence of ALG 2	p. 179
Applications	p. 183
General Comments	p. 194
Least-Squares Solution of Nonlinear Problems: Application to Nonlinear Problems in Fluid Dynamics	p. 195
Introduction: Synopsis	p. 195
Least-Squares Solution of Finite-Dimensional Systems of Equations	p. 195
Least-Squares Solution of a Nonlinear Dirichlet Model Problem	p. 198
Transonic Flow Calculations by Least-Squares and Finite Element Methods	p. 211
Numerical Solution of the Navier-Stokes Equations for Incompressible Viscous Fluids by Least-Squares and Finite Element Methods	p. 244
Further Comments on Chapter VII and Conclusion	p. 318
A Brief Introduction to Linear Variational Problems	p. 321
Introduction	p. 321
A Family of Linear Variational Problems	p. 321
Internal Approximation of Problem (P)	p. 326
Application to the Solution of Elliptic Problems for Partial Differential Operators	p. 330
Further Comments: Conclusion	p. 397
A Finite Element Method with Upwinding for Second-Order Problems with Large First- Order Terms	p. 399
Introduction	p. 399
The Model Problem	p. 399
A Centered Finite Element Approximation	p. 400
A Finite Element Approximation with Upwinding	p. 400
On the Solution of the Linear System Obtained by Upwinding	p. 404
Numerical Experiments	p. 404
Concluding Comments	p. 414
Some Complements on the Navier-Stokes Equations and Their Numerical Treatment	p. 415
Introduction	p. 415
Finite Element Approximation of the Boundary Condition u = g on ¿ if g &neq; 0	p. 415
Some Comments On the Numerical Treatment of the Nonlinear Term (u · ∇)u	p. 416
Further Comments on the Boundary Conditions	p. 417
Decomposition Properties of the Continuous and Discrete Stokes Problems of Sec. 4. Application to Their Numerical Solution	p. 425
Further Comments	p. 430
Some Illustrations from an Industrial Application	p. 431
Bibliography	p. 435
Glossary of Symbols	p. 455
Author Index	p. 463
Subject Index	p. 467
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