Mathematical aspects of discontinuous Galerkin methods.

*(English)*Zbl 1231.65209
MathÃ©matiques & Applications (Berlin) 69. Berlin: Springer (ISBN 978-3-642-22979-4/pbk; 978-3-642-22980-0/ebook). xvii, 384 p. (2012).

The rough contents of this remarkable book are: Ch. 1 Basic concepts, Ch. 2 Steady advection-reaction, Ch. 3 Unsteady first-order partial differential equations (PDEs), Ch. 4 PDEs with diffusion, Ch. 5 Additional topics on pure diffusion, Ch. 6 Incompressible flows, Ch. 7 Friedrichs systems and Appendix A: Implementation. The book also contains a bibiography, an author index and an index.

The aim of the book is “to provide the reader with the basic mathematical concepts to design and analyze discontinuous Galerkin methods for various model problems, starting at an introductory level and further elaborating on more advanced topics”. The authors treat fairly thoroughly steady as well as unsteady linear first and second order problems and in less detail some nonlinear problems such as Navier-Stokes system. Essentially, they provide for each considered problem basic ingredients, a weak formulation along with its well-posedness, discretization and various error estimates. For unsteady problems, along with discontinuous space discretization, various schemes to march in time are analyzed. Some useful practical implementation aspects are considered in an appendix. The bibliography contains more than 300 entries.

The aim of the book is “to provide the reader with the basic mathematical concepts to design and analyze discontinuous Galerkin methods for various model problems, starting at an introductory level and further elaborating on more advanced topics”. The authors treat fairly thoroughly steady as well as unsteady linear first and second order problems and in less detail some nonlinear problems such as Navier-Stokes system. Essentially, they provide for each considered problem basic ingredients, a weak formulation along with its well-posedness, discretization and various error estimates. For unsteady problems, along with discontinuous space discretization, various schemes to march in time are analyzed. Some useful practical implementation aspects are considered in an appendix. The bibliography contains more than 300 entries.

Reviewer: Calin Ioan Gheorghiu (Cluj-Napoca)

##### MSC:

65N30 | Finite element, Rayleigh-Ritz and Galerkin 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 |

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

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

76R50 | Diffusion |

76M12 | Finite volume methods applied to problems in fluid mechanics |

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

35Q30 | Navier-Stokes equations |

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

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

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

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