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**A posteriori error estimation in finite element analysis.**
*(English)*
Zbl 1008.65076

This book provides an introduction to the subject of a posteriori error estimation of finite element approximations of partial differential equations. It is the outgrowth of over fifteen years of work by the authors on this subject and around a decade of joint work. The book attempts to cover the mathematical underpinnings of many of the most effective methods for error estimation that are available today. Its aim is to provide a systematic treatment to the theory of error estimation accessible to contemporary engineers and applied scientists who wish to learn not only the mathematical foundations of error estimation, but also the details of their implementation on boundary value problems of continuum mechanics and physics. To make the ideas clear and understandable, the focus is primarily on model scalar elliptic problems on two-dimensional domains. However, significant generalizations to unsymmetric, indefinite problems and to representative nonlinear problems, including the Navier-Stokes equations, are also presented.

The book is organized as follows: It begins with an introductory chapter in which the basic notations and assumptions, along with relevant finite element concepts, are presented. Then it presents successively five distinct techniques of error estimation that have been developed since the late 1970s. Included are chapters on explicit error estimators, recovery methods, implicit error estimators, and the use of hierarchical bases for error estimation, as well as an entire chapter on the equilibrated residual method. Also presented are techniques for computing estimates in \(L_2\) and \(L_\infty\) and in other Sobolev norms. Chapter 7 is devoted to techniques for judging the performance of the various estimators. Chapter 8 provides an account of the estimation of errors in quantities of interest, along with a discussion of local and pollution errors. Chapter 9 is devoted to a number of extensions and applications o the theory. These include (a) nonself-adjoint and indefinite problems such as are embodied in the Stokes problem and Oseen’s problem and (b) extensions to linear elliptic systems such as those found in elasticity. Examples are given of extensions to nonlinear problems such as the problem of flow characterized by the steady-state Navier-Stokes equations for incompressible fluids.

The book is organized as follows: It begins with an introductory chapter in which the basic notations and assumptions, along with relevant finite element concepts, are presented. Then it presents successively five distinct techniques of error estimation that have been developed since the late 1970s. Included are chapters on explicit error estimators, recovery methods, implicit error estimators, and the use of hierarchical bases for error estimation, as well as an entire chapter on the equilibrated residual method. Also presented are techniques for computing estimates in \(L_2\) and \(L_\infty\) and in other Sobolev norms. Chapter 7 is devoted to techniques for judging the performance of the various estimators. Chapter 8 provides an account of the estimation of errors in quantities of interest, along with a discussion of local and pollution errors. Chapter 9 is devoted to a number of extensions and applications o the theory. These include (a) nonself-adjoint and indefinite problems such as are embodied in the Stokes problem and Oseen’s problem and (b) extensions to linear elliptic systems such as those found in elasticity. Examples are given of extensions to nonlinear problems such as the problem of flow characterized by the steady-state Navier-Stokes equations for incompressible fluids.

Reviewer: I.N.Katz (St.Louis)

### MSC:

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

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

35Q30 | Navier-Stokes equations |

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

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

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

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