Numerical models for differential problems. Translated from the 5th Italian edition by Silvia Quarteroni.
2nd ed.

*(English)*Zbl 1285.65054
MS&A. Modeling, Simulation and Applications 8. Milano: Springer (ISBN 978-88-470-5521-6/hbk; 978-88-470-5522-3/ebook). xix, 656 p. (2014).

The rough contents of this book are as follows: Chapter 1 – A brief survey of partial differential equations, Chapter 2 – 2 Elements of functional analysis, Chapter 3 – Elliptic equations, Chapter 4 – The Galerkin finite element method for elliptic problems, Chapter 5 – Parabolic equations, Chapter 6 – Generation of 1D and 2D grids, Chapter 7 – Algorithms for solutions of linear systems, Chapter 8 – Elements of finite element programming, Chapter 9 – The finite volume method, Chapter 10 – Spectral methods, Chapter 11 – Discontinuous element methods (DG and mortar), Chapter 12 – Diffusion-transport-reaction equations, Chapter 13 – Finite differences for hyperbolic equations, Chapter 14 – Finite elements and spectral methods for hyperbolic equations, Chapter 15 – Nonlinear hyperbolic problems, Chapter 16 – Navier-Stokes equations, Chapter 17 – Optimal control of partial differential equations, Chapter 18 – Domain decomposition methods and Chapter 19 – Reduced basis approximation for parameterized partial differential equations.

The book offers a vast and proper perspective of the main numerical methods in solving various problems attached to linear as well as nonlinear elliptic, parabolic and hyperbolic equations. A particular attention is paid to diffusion-transport-reaction equations and to the Navier-Stokes system. The Galerkin finite element method is the principal scheme in the discretization of the spatial derivatives. Spectral and finite volume methods are also used in this respect. Finite difference schemes are currently used for time marching. The methods are formulated in their natural framework and then analyzed concerning their convergence and stability properties. It is fairly important to underline that whenever the analysis of one-dimensional problems is interesting in itself or relevant for partial differential equations, it is thoroughly provided. This aspect can seriously facilitate a possible learning process. In the same line of argument we notice that the monograph provides a large set of substantial exercises which covers both the theoretical and numerical topics.

The book offers a vast and proper perspective of the main numerical methods in solving various problems attached to linear as well as nonlinear elliptic, parabolic and hyperbolic equations. A particular attention is paid to diffusion-transport-reaction equations and to the Navier-Stokes system. The Galerkin finite element method is the principal scheme in the discretization of the spatial derivatives. Spectral and finite volume methods are also used in this respect. Finite difference schemes are currently used for time marching. The methods are formulated in their natural framework and then analyzed concerning their convergence and stability properties. It is fairly important to underline that whenever the analysis of one-dimensional problems is interesting in itself or relevant for partial differential equations, it is thoroughly provided. This aspect can seriously facilitate a possible learning process. In the same line of argument we notice that the monograph provides a large set of substantial exercises which covers both the theoretical and numerical topics.

Reviewer: Calin Ioan Gheorghiu (Cluj-Napoca)

##### MSC:

65Mxx | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |

65Nxx | Numerical methods for partial differential equations, boundary value problems |

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

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

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

76M22 | Spectral methods applied to problems in fluid mechanics |

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

65K10 | Numerical optimization and variational techniques |