A posteriori error estimation techniques for finite element methods.

*(English)*Zbl 1279.65127
Numerical Mathematics and Scientific Computation. Oxford: Oxford University Press (ISBN 978-0-19-967942-3/hbk). xx, 393 p. (2013).

Self-adaptive discretization methods are now an indispensable tool for the numerical solution of partial differential equations that arise from physical and technical applications. The aim is to obtain a numerical solution within a prescribed tolerance using a minimal amount of work. The main tools in achieving this goal are a posteriori error estimates which give global and local information on the error of the numerical solution and which can easily be computed from the given numerical solution and the data of the differential equation.

This book reviews the most frequently used a posteriori error estimation techniques and applies them to a broad class of linear and nonlinear elliptic and parabolic equations. Although there are various approaches to adaptivity and a posteriori error estimation, they are all based on a few common principles. The main aim of the book is to elaborate these basic principles and to give guidelines for developing adaptive schemes for new problems.

In Chapter 1, the most frequently used a posteriori error estimates are presented within the framework of a simple model problem: the linear or bi-linear conforming finite element discretisation of the two-dimensional Poisson equation with mixed Dirichlet and Neumann boundary conditions. In Chapter 2, it is shown how to use a posteriori error estimates for adaptive mesh-refinement and how to implement a simple adaptive discretisation scheme. Chapters 1 and 2 are quite elementary and present various error indicators and their use for mesh adaptation in the framework of a simple model problem. The basic principles are introduced using a minimal amount of notations and techniques providing a complete overview for the non-specialist.

Chapter 3 collects the technical prerequisites for the a posteriori error estimates of Chapters 4–6. Chapters 4–6 are more advanced and present a posteriori error estimates within a general framework using the technical tools collected in Chapter 3. Chapter 4 is devoted to a posteriori error estimates for linear elliptic equations. In Chapter 5, the results of Chapter 4 are extended to nonlinear elliptic equations. Chapter 6 is concerned with a posteriori error estimates for linear and nonlinear parabolic equations. Most sections close with a bibliographical remark which indicates the historical development and hints at further results.

This book reviews the most frequently used a posteriori error estimation techniques and applies them to a broad class of linear and nonlinear elliptic and parabolic equations. Although there are various approaches to adaptivity and a posteriori error estimation, they are all based on a few common principles. The main aim of the book is to elaborate these basic principles and to give guidelines for developing adaptive schemes for new problems.

In Chapter 1, the most frequently used a posteriori error estimates are presented within the framework of a simple model problem: the linear or bi-linear conforming finite element discretisation of the two-dimensional Poisson equation with mixed Dirichlet and Neumann boundary conditions. In Chapter 2, it is shown how to use a posteriori error estimates for adaptive mesh-refinement and how to implement a simple adaptive discretisation scheme. Chapters 1 and 2 are quite elementary and present various error indicators and their use for mesh adaptation in the framework of a simple model problem. The basic principles are introduced using a minimal amount of notations and techniques providing a complete overview for the non-specialist.

Chapter 3 collects the technical prerequisites for the a posteriori error estimates of Chapters 4–6. Chapters 4–6 are more advanced and present a posteriori error estimates within a general framework using the technical tools collected in Chapter 3. Chapter 4 is devoted to a posteriori error estimates for linear elliptic equations. In Chapter 5, the results of Chapter 4 are extended to nonlinear elliptic equations. Chapter 6 is concerned with a posteriori error estimates for linear and nonlinear parabolic equations. Most sections close with a bibliographical remark which indicates the historical development and hints at further results.

Reviewer: Nina Shokina (Freiburg)

##### MSC:

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

65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |

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

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

65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

65M08 | Finite volume methods for initial value and initial-boundary value problems involving PDEs |

35J25 | Boundary value problems for second-order elliptic equations |

35J60 | Nonlinear elliptic equations |

35K20 | Initial-boundary value problems for second-order parabolic equations |

35K55 | Nonlinear parabolic equations |

65M50 | Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs |

65N08 | Finite volume methods for boundary value problems involving PDEs |