##
**Mixed finite element methods and applications.**
*(English)*
Zbl 1277.65092

Springer Series in Computational Mathematics 44. Berlin: Springer (ISBN 978-3-642-36518-8/hbk; 978-3-642-36519-5/ebook). xiv, 685 p. (2013).

The rough contents of this monograph are as follows: Chapter 1: Variational formulations and finite element methods, Chapter 2: Function spaces and finite element approximations, Chapter 3: Algebraic aspects of saddle point problems, Chapter 4: Saddle point problems in Hilbert spaces, Chapter 5: Approximation of saddle point problems, Chapter 6: Complements: Stabilization methods, eigenvalue problems, Chapter 7: Mixed methods for elliptic problems, Chapter 8: Incompressible materials and flow problems, Chapter 9: Complements on elasticity problems, Chapter 10: Complements on plate problems and Chapter 11: Mixed finite elements for electromagnetic problems. The book also contains a short Preface, References and an Index.

The authors are leading experts in the mathematical foundation of the finite element method with outstanding publications in mixed and hybrid formulations of this method. The book is mainly built up on their results. Thus, the authors first review the basic results from functional analysis and linear algebra on which they construct the mixed finite element methods. The second-and fourth-order elliptic problems, which form the core of the work, are separately analyzed for the one-, two- and three-dimensional cases. The mixed (hybrid) finite element methods are compared with finite volume methods and various error estimates are provided. A particular attention is paid to eigenvalue problems. Mixed variational formulations are also provided for a large variety of problems from linear elasticity, incompressible Stokes flow and electromagnetism. Along with these formulations the authors introduce a fairly general and rigorous framework for the analysis of finite element approximations to problems involving incompressible type conditions. All in all, the book contains the fundamental results as well as a lot of usual and peculiar ingredients one needs to understand and efficiently practice the mixed and related finite element methods.

The authors are leading experts in the mathematical foundation of the finite element method with outstanding publications in mixed and hybrid formulations of this method. The book is mainly built up on their results. Thus, the authors first review the basic results from functional analysis and linear algebra on which they construct the mixed finite element methods. The second-and fourth-order elliptic problems, which form the core of the work, are separately analyzed for the one-, two- and three-dimensional cases. The mixed (hybrid) finite element methods are compared with finite volume methods and various error estimates are provided. A particular attention is paid to eigenvalue problems. Mixed variational formulations are also provided for a large variety of problems from linear elasticity, incompressible Stokes flow and electromagnetism. Along with these formulations the authors introduce a fairly general and rigorous framework for the analysis of finite element approximations to problems involving incompressible type conditions. All in all, the book contains the fundamental results as well as a lot of usual and peculiar ingredients one needs to understand and efficiently practice the mixed and related finite element methods.

Reviewer: Calin Ioan Gheorghiu (Cluj-Napoca)

### MSC:

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

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

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

78M10 | Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory |

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

65N25 | Numerical methods for eigenvalue problems for boundary value problems involving PDEs |

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

65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |

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

74B05 | Classical linear elasticity |

78A25 | Electromagnetic theory (general) |

76D07 | Stokes and related (Oseen, etc.) flows |

35P15 | Estimates of eigenvalues in context of PDEs |

35J40 | Boundary value problems for higher-order elliptic equations |