Discontinuous Galerkin method. Analysis and applications to compressible flow.

*(English)*Zbl 1401.76003
Springer Series in Computational Mathematics 48. Cham: Springer (ISBN 978-3-319-19266-6/hbk; 978-3-319-19267-3/ebook). xiv, 572 p. (2015).

The rough contents of this important monograph are as follows: Ch. 1 Introduction, Part I Analysis of the Discontinuous Galerkin Method, Ch. 2 DGM for Elliptic Problems, Ch. 3 Methods Based on a Mixed Formulation, Ch. 4 DGM for Convection-Diffusion Problems, Ch. 5 Space-Time Discretization by Multistep Methods, Ch. 6 Space-Time Discretization Galerkin Method, Ch. 7 Generalization of the DGM, Part II Applications of the Discontinuous Galerkin Method, Ch. 8 Inviscid Compressible Flow, Ch. 9 Viscous Compressible Flow, Ch. 10 Fluid-Structure Interaction, References and an Index. The References contain 287 works almost all having a mathematical structure. An important number of them belong to mathematicians of Czech extraction.

The book is devoted to the foundation of DGM and its theoretical analysis (Part I) as well as to a large range of solutions obtained by this method (Part II). The first part categorically dominates the second one with respect to the length, richness of ideas and mathematical results and, last but not least, with respect to the numerous and illuminating examples of initial and/or boundary value problems. These examples are effectively carried out by the authors and are fairly useful. Along with the variational and discrete formulations of the problems as well as with the rigorous error analysis, these examples could provide some graduate courses in numerical analysis of partial differential equations.

The second part is concerned with numerical solutions to some challenging problems from inviscid and viscous compressible flows.

However, the numerical results obtained by authors in solving model problems, mainly in solving various boundary value problems attached to the Laplace equation, as well as the numerical results carried out corresponding to certain compressible fluid mechanics problems, are compared with those obtained by finite elements and finite volume methods. Unfortunately, any evaluation of DGM in the broader context of spectral element method is avoided.

The book is devoted to the foundation of DGM and its theoretical analysis (Part I) as well as to a large range of solutions obtained by this method (Part II). The first part categorically dominates the second one with respect to the length, richness of ideas and mathematical results and, last but not least, with respect to the numerous and illuminating examples of initial and/or boundary value problems. These examples are effectively carried out by the authors and are fairly useful. Along with the variational and discrete formulations of the problems as well as with the rigorous error analysis, these examples could provide some graduate courses in numerical analysis of partial differential equations.

The second part is concerned with numerical solutions to some challenging problems from inviscid and viscous compressible flows.

However, the numerical results obtained by authors in solving model problems, mainly in solving various boundary value problems attached to the Laplace equation, as well as the numerical results carried out corresponding to certain compressible fluid mechanics problems, are compared with those obtained by finite elements and finite volume methods. Unfortunately, any evaluation of DGM in the broader context of spectral element method is avoided.

Reviewer: Calin Ioan Gheorghiu (Cluj-Napoca)

##### MSC:

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |

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

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

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

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

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

76M30 | Variational methods applied to problems in fluid mechanics |