Numerical solution of nonlinear elliptic problems via preconditioning operators. Theory and applications.

*(English)*Zbl 1030.65117
Advances in Computation: Theory and Practice. 11. Huntington, NY: Nova Science Publishers. xviii, 402 p. (2002).

The modelization of a great part of problems arising in mechanics, physics, chemical and biological phenomena…leads to nonlinear elliptic equations. Generally, the approximation of their solutions combines a discretization method (finite difference, finite element…) and an iterative linearization method (Newton, conjugate gradient, augmented Lagrangian…). Of course, the aim of such methods is to obtain a numerical solution which is a good approximation of a solution of the continuous problem, both with high accuracy and through a reasonable computing time. To give an efficient answer to this challenge, many researchers have developed preconditioning methods which improve the conditioning of the involved matrices and allow to reduce drastically the number of iterations. Besides the great number of papers devoted to various aspects of preconditioning, this book is, to our knowledge, the first monography entirely dedicated to the study of preconditioning operators. The monography is organized in three parts and ten chapters followed by an extended and up to date bibliography (298 references) and a medium size index.

The aim of the first part is to “motivate” the reader: Chapter 1 details some classical examples of nonlinear elliptic equations which appear in the modelling of the following problems: elasto-plastic torsion of rods, nonlinear Maxwell equations in connection with electromagnetic field theory, nonlinear elasticity, elasto-plastic bending of clamped plates, semilinear equations, flow models and non-potential problems. With these equations, they associate corresponding weak formulations. Chapter 2 focuses on the importance of conditioning properties and preconditioning for linear algebraic systems and it includes a brief discussion of implementation. Chapter 3 summarizes some basic properties of linear elliptic boundary value problems in the context of their numerical solutions : linear operators in Hilbert spaces, existence of weak solutions and associated regularity results, numerical solution of linear elliptic problems (basic methods and fast solvers), background of conditioning properties and associate efficient preconditioning approach. Chapter 4 contains a brief summary on iterations for nonlinear algebraic systems and emphasizes the importance of finding good preconditionners.

Part II develops the theoretical background that is necessary for the study of nonlinear elliptic problems into consideration. They enter in the class of the equations F(u) = b set in a real Hilbert space H with monotone potential operators F. Corresponding results are detailed in Chapter 5 and include preconditioning in Hilbert spaces. In particular, the authors underline that this concept of preconditioning gives a general framework to discuss iterative methods. Chapter 6 deals with the discussion of existence and uniqueness for the class of nonlinear elliptic problems into consideration. It gives a starting point for the study of convergent iterations. The last part of the book, i.e. Part III, is concerned with the iterative solution of nonlinear elliptic boundary value problems. Firstly, Chapter 7 discusses iterative methods which allow to construct sequences of functions on the continuous level, converging to the exact solution of a nonlinear elliptic boundary value problem. The aim of such a theoretical sequence is, through a projection to the discretization subspace, to lead to a convenient preconditioned iterative sequence for the discretized problem. Chapter 8 summarizes some general properties of the derived preconditioning methods for the discretized problems. And then, from the preconditioning operator background, it gives a list of various preconditionners which rely upon spectral equivalence. Chapter 9 discusses the associated algorithms and focuses on convergence results for the numerical methods based on preconditioning operators. Due to Sobolev space background, special attention is devoted to the approximation by finite element methods. Finally, Chapter 10 gives appropriate algorithms for the model problems considered in Chapter 1.

In conclusion, it is worth to emphasize that the great development of scientific computation leads to the modelization of problems more and more nonlinear. Their efficient numerical solutions require solvers both convergent and very fast. This monography gives a very original and pertinent answer to this important challenge by combining most advanced results of functional analysis for nonlinear elliptic problems with the up to date preconditioning techniques. Besides the strong foundations of this study of preconditioning operators, the monography benefits of an excellent and attractive presentation that should motivate a large spectrum of researchers from the applied mathematics community, and also from the engineering one, who looks for new methods in order to make their algorithms more and more performant.

The aim of the first part is to “motivate” the reader: Chapter 1 details some classical examples of nonlinear elliptic equations which appear in the modelling of the following problems: elasto-plastic torsion of rods, nonlinear Maxwell equations in connection with electromagnetic field theory, nonlinear elasticity, elasto-plastic bending of clamped plates, semilinear equations, flow models and non-potential problems. With these equations, they associate corresponding weak formulations. Chapter 2 focuses on the importance of conditioning properties and preconditioning for linear algebraic systems and it includes a brief discussion of implementation. Chapter 3 summarizes some basic properties of linear elliptic boundary value problems in the context of their numerical solutions : linear operators in Hilbert spaces, existence of weak solutions and associated regularity results, numerical solution of linear elliptic problems (basic methods and fast solvers), background of conditioning properties and associate efficient preconditioning approach. Chapter 4 contains a brief summary on iterations for nonlinear algebraic systems and emphasizes the importance of finding good preconditionners.

Part II develops the theoretical background that is necessary for the study of nonlinear elliptic problems into consideration. They enter in the class of the equations F(u) = b set in a real Hilbert space H with monotone potential operators F. Corresponding results are detailed in Chapter 5 and include preconditioning in Hilbert spaces. In particular, the authors underline that this concept of preconditioning gives a general framework to discuss iterative methods. Chapter 6 deals with the discussion of existence and uniqueness for the class of nonlinear elliptic problems into consideration. It gives a starting point for the study of convergent iterations. The last part of the book, i.e. Part III, is concerned with the iterative solution of nonlinear elliptic boundary value problems. Firstly, Chapter 7 discusses iterative methods which allow to construct sequences of functions on the continuous level, converging to the exact solution of a nonlinear elliptic boundary value problem. The aim of such a theoretical sequence is, through a projection to the discretization subspace, to lead to a convenient preconditioned iterative sequence for the discretized problem. Chapter 8 summarizes some general properties of the derived preconditioning methods for the discretized problems. And then, from the preconditioning operator background, it gives a list of various preconditionners which rely upon spectral equivalence. Chapter 9 discusses the associated algorithms and focuses on convergence results for the numerical methods based on preconditioning operators. Due to Sobolev space background, special attention is devoted to the approximation by finite element methods. Finally, Chapter 10 gives appropriate algorithms for the model problems considered in Chapter 1.

In conclusion, it is worth to emphasize that the great development of scientific computation leads to the modelization of problems more and more nonlinear. Their efficient numerical solutions require solvers both convergent and very fast. This monography gives a very original and pertinent answer to this important challenge by combining most advanced results of functional analysis for nonlinear elliptic problems with the up to date preconditioning techniques. Besides the strong foundations of this study of preconditioning operators, the monography benefits of an excellent and attractive presentation that should motivate a large spectrum of researchers from the applied mathematics community, and also from the engineering one, who looks for new methods in order to make their algorithms more and more performant.

Reviewer: Michel Bernadou (Paris La Defense)

##### MSC:

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

76M10 | Finite element methods applied to problems in fluid mechanics |

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

65-00 | General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to numerical analysis |

74B20 | Nonlinear elasticity |

35J65 | Nonlinear boundary value problems for linear elliptic equations |

65F35 | Numerical computation of matrix norms, conditioning, scaling |

65H10 | Numerical computation of solutions to systems of equations |

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