A posteriori error analysis via duality theory. With applications in modeling and numerical approximations.

*(English)*Zbl 1081.65065
Advances in Mechanics and Mathematics 8. New York, NY: Springer (ISBN 0-387-23536-1/hbk). xvi, 302 p. (2005).

The subject of this book is the a posteriori error analysis for mathematical idealizations in applied boundary value problems (BVPs), especially those arising in mechanical models, and for numerical approximations of nonlinear variational problems. An error estimate is called a posteriori if the computed solution is used in evaluating its accuracy. The main mathematical tool used by the author is the duality theory of convex analysis [e.g. I. Ekeland and R. Temam, Convex analysis and variational problems, North-Holland, Amsterdam (1976; Zbl 0322.90046)].

The book is divided into six chapters. The first and the second are devoted to the mathematical background, namely functional analysis, variational formulation of BVPs, numerical analysis, elements of convex analysis and duality theory.

Chapter 3 deals with a posteriori error estimates for idealizations of linear elliptic problems on nonsmooth domains. The coefficients of a partial differential equation modeling a physical process depend on the physical properties of the material involved. They cannot be determined exactly. The goal of this chapter is to asses quantitatively the influence of the coefficient uncertainty in the concrete environment of the problem to be solved. The results obtained are applied to linear torsion problems and to heat conduction ones. A posteriori error estimates for the effect of linearization on solutions of nonlinear problems are derived in Chapter 4. Many consistent examples are studied. They belong nonlinear elasticity, heat conduction, nonlinear problems with small parameters, laminar flow of a Bingham fluid, obstacle problems.

Chapter 5 is dedicated to a posteriori error analysis for numerical methods such as the regularization method and the Kačanov iteration method. The regularization method is used to handle non-smooth terms, while the Kačanov one provides a sequence of linear problems to approximate a nonlinear one. For a practical implementation a posteriori error estimates lead, for both methods, to an error bound of the approximate solution. Examples of applications to model problems are given. Chapter 6 is devoted to elliptic variational inequalities of the second kind. A simplified friction problem is considered as model and a posteriori error estimates for the finite element method (FEM) approximation are obtained. Then the corresponding dual problem is used to get residual based error estimates and gradient recovery-based error estimates. Numerical results are obtained for the model problem and also for a more complicated frictional contact problem.

A very nice book, well structured and written, coupling mathematical theory and numerical results and tests for applied problems.

The book is divided into six chapters. The first and the second are devoted to the mathematical background, namely functional analysis, variational formulation of BVPs, numerical analysis, elements of convex analysis and duality theory.

Chapter 3 deals with a posteriori error estimates for idealizations of linear elliptic problems on nonsmooth domains. The coefficients of a partial differential equation modeling a physical process depend on the physical properties of the material involved. They cannot be determined exactly. The goal of this chapter is to asses quantitatively the influence of the coefficient uncertainty in the concrete environment of the problem to be solved. The results obtained are applied to linear torsion problems and to heat conduction ones. A posteriori error estimates for the effect of linearization on solutions of nonlinear problems are derived in Chapter 4. Many consistent examples are studied. They belong nonlinear elasticity, heat conduction, nonlinear problems with small parameters, laminar flow of a Bingham fluid, obstacle problems.

Chapter 5 is dedicated to a posteriori error analysis for numerical methods such as the regularization method and the Kačanov iteration method. The regularization method is used to handle non-smooth terms, while the Kačanov one provides a sequence of linear problems to approximate a nonlinear one. For a practical implementation a posteriori error estimates lead, for both methods, to an error bound of the approximate solution. Examples of applications to model problems are given. Chapter 6 is devoted to elliptic variational inequalities of the second kind. A simplified friction problem is considered as model and a posteriori error estimates for the finite element method (FEM) approximation are obtained. Then the corresponding dual problem is used to get residual based error estimates and gradient recovery-based error estimates. Numerical results are obtained for the model problem and also for a more complicated frictional contact problem.

A very nice book, well structured and written, coupling mathematical theory and numerical results and tests for applied problems.

Reviewer: Viorel Arnăutu (Iaşi)

##### MSC:

65K10 | Numerical optimization and variational techniques |

49M20 | Numerical methods of relaxation type |

49J20 | Existence theories for optimal control problems involving partial differential equations |

49N15 | Duality theory (optimization) |

65N15 | Error bounds for boundary value problems involving PDEs |

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

74M10 | Friction in solid mechanics |

74B20 | Nonlinear elasticity |

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

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

49J40 | Variational inequalities |

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

49M15 | Newton-type methods |