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**Numerical methods for nonlinear variational problems.**
*(English)*
Zbl 0536.65054

Springer Series in Computational Physics. New York etc.: Springer-Verlag. xv, 493 p. DM 158.00; $ 61.30 (1984).

This book is an improved and up-dated version of a monograph with the same title (1980; Zbl 0456.65035). It provides a study of approximate methods for nonlinear variational problems of elliptic (EVI) and parabolic (PVI) type. The book contains seven chapters and three appendices.

By comparison with the previous version, Section 7 in Chapter I, Chapter V and all appendices are new and Chapter VII is developed and entirely rewritten. Existence and uniqueness results and Ritz-Galerkin methods for EVI are presented in Chap. I. Sec. 7 in this chapter deals with the penalty solution for EVI.

In Chap. II, the finite element method is used to solve second-order elliptic problems arising from mechanics (the obstacle problem, the elasto-plastic torsion problem, a simplified Signorini problem, a simplified friction problem, the flow of a viscous, plastic fluid in a pipe). The finite dimensional problems obtained are treated by iterative schemes (relaxation methods, duality methods and Uzawa’s algorithm).

Chap. III is an introduction to the approximation of PVI by Galerkin methods and iterative schemes; an example concerning Bingham fluids is given in detail. Chap. IV shows the use of EVI methods in solving a family of mildly nonlinear elliptic equations and a nonlinear elliptic equation modelling the subsonic flow of a perfect compressible fluid. Various gradient type methods, Newton’s method and relaxation methods are used to handle the discrete problems.

Since relaxation methods are often involved in the text of this book, they are presented in Chap. V together with applications to constrained minimization of quadratic functionals in Hilbert spaces and to systems of nonlinear equations.

The iterative solution to some variational problems with a very specific structure, allowing the use of decomposition-coordination methods via augmented Lagrangians, is discussed in Chap. VI.

Chap. VII deals with the solution of nonlinear problems in fluid dynamics by a combination of least-squares, conjugate gradient and finite element method. The solution of nonlinear equations in \(R^n\) by least-squares methods is discussed first. A Dirichlet model problem is solved and comments about the pseudo-arc-length-continuation methods are given. Next, the steady transonic potential flow of an inviscid compressible fluid and the steady and unsteady Navier-Stokes equations for incompressible viscous flow in the pressure-velocity formulation are treated. Many numerical results are given.

Appendix I, which contains about 80 pages, provides a useful presentation of linear variational problems with applications to the solution of elliptic problems for partial differential operators. Examples are taken from geophysics and the mechanics of continuous media. A finite element method with upwinding for elliptic boundary-value problems with large first-order terms is described in Appendix II. Appendix III contains various information and results for the practical solution of the Navier-Stokes equations, being a complement to Chap. VII. Finally, an industrial application, namely the aerodynamical performances of an aircraft, is mentioned.

Well structured, well written, provided with interesting remarks and references in the text and with about 350 final references with 82 figures, many of them showing computer results, this book is a very useful one, from both theoretical and practical point of view, in numerical mathematics.

By comparison with the previous version, Section 7 in Chapter I, Chapter V and all appendices are new and Chapter VII is developed and entirely rewritten. Existence and uniqueness results and Ritz-Galerkin methods for EVI are presented in Chap. I. Sec. 7 in this chapter deals with the penalty solution for EVI.

In Chap. II, the finite element method is used to solve second-order elliptic problems arising from mechanics (the obstacle problem, the elasto-plastic torsion problem, a simplified Signorini problem, a simplified friction problem, the flow of a viscous, plastic fluid in a pipe). The finite dimensional problems obtained are treated by iterative schemes (relaxation methods, duality methods and Uzawa’s algorithm).

Chap. III is an introduction to the approximation of PVI by Galerkin methods and iterative schemes; an example concerning Bingham fluids is given in detail. Chap. IV shows the use of EVI methods in solving a family of mildly nonlinear elliptic equations and a nonlinear elliptic equation modelling the subsonic flow of a perfect compressible fluid. Various gradient type methods, Newton’s method and relaxation methods are used to handle the discrete problems.

Since relaxation methods are often involved in the text of this book, they are presented in Chap. V together with applications to constrained minimization of quadratic functionals in Hilbert spaces and to systems of nonlinear equations.

The iterative solution to some variational problems with a very specific structure, allowing the use of decomposition-coordination methods via augmented Lagrangians, is discussed in Chap. VI.

Chap. VII deals with the solution of nonlinear problems in fluid dynamics by a combination of least-squares, conjugate gradient and finite element method. The solution of nonlinear equations in \(R^n\) by least-squares methods is discussed first. A Dirichlet model problem is solved and comments about the pseudo-arc-length-continuation methods are given. Next, the steady transonic potential flow of an inviscid compressible fluid and the steady and unsteady Navier-Stokes equations for incompressible viscous flow in the pressure-velocity formulation are treated. Many numerical results are given.

Appendix I, which contains about 80 pages, provides a useful presentation of linear variational problems with applications to the solution of elliptic problems for partial differential operators. Examples are taken from geophysics and the mechanics of continuous media. A finite element method with upwinding for elliptic boundary-value problems with large first-order terms is described in Appendix II. Appendix III contains various information and results for the practical solution of the Navier-Stokes equations, being a complement to Chap. VII. Finally, an industrial application, namely the aerodynamical performances of an aircraft, is mentioned.

Well structured, well written, provided with interesting remarks and references in the text and with about 350 final references with 82 figures, many of them showing computer results, this book is a very useful one, from both theoretical and practical point of view, in numerical mathematics.

Reviewer: Viorel Arnăutu (Iaşi)

### MSC:

65K10 | Numerical optimization and variational techniques |

49J40 | Variational inequalities |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

49M15 | Newton-type methods |

35J20 | Variational methods for second-order elliptic equations |

35Q99 | Partial differential equations of mathematical physics and other areas of application |

90C52 | Methods of reduced gradient type |

74S05 | Finite element methods applied to problems in solid mechanics |

76G25 | General aerodynamics and subsonic flows |

76H05 | Transonic flows |

76D05 | Navier-Stokes equations for incompressible viscous fluids |