## Lectures on Numerical Methods for Non-Linear Variational ProblemsWhen 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 |

463 | |

467 | |

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### Common terms and phrases

a(uh alternating-direction methods apply approximate problem augmented Lagrangian bilinear form boundary conditions bounded domain Bristeau Chapter VII Ciarlet compute conjugate gradient algorithm conjugate gradient method consider continuous denotes Dirichlet problem discrete Stokes problem discussed in Sec dx Jn Jn elliptic equivalent error estimates exists Find u e finite element approximation finite element methods fluid follows Fortin Fourier problem functional fv dx given h dx Hilbert space Hj(Q Hl(Q hypotheses implies incompressible iterative methods Jn Jn Jn least-squares formulation Lemma linear system Lions Lipschitz continuous matrix methods for solving minimization problem Navier-Stokes equations Neumann problem nonlinear problems norm notation numerical solution obtain Periaux piecewise proof properties Proposition prove Remark resp saddle point scalar product scheme step strongly subset suppose symmetric Theorem 2.1 triangles un+1 unique solution variational formulation variational inequalities variational problem vector