Finite elements and fast iterative solvers. With applications in incompressible fluid dynamics.
2nd ed.

*(English)*Zbl 1304.76002
Numerical Mathematics and Scientific Computation. Oxford: Oxford University Press (ISBN 978-0-19-967879-2/hbk; 978-0-19-967880-8/pbk). xiv, 479 p. (2014).

For the review of the first edition see [(2005; Zbl 1083.76001)].

The authors present efficient solutions of Poisson, Stokes, convection-diffusion and Navier-Stokes partial differential equations (PDEs) that arise when modelling incompressible fluid flows. The theoretical results (concerning weak formulation, the Galerkin finite element method, theory of errors, the conjugate gradient method, matrix properties, preconditioning methods, splitting operators, multigrid methods) are developed in a constructive style and illustrated by numerical experiments. Error estimation and iterative solution of linear systems of algebraic equations are placed into the context of solving the particular discrete PDEs for model problems. The computational examples and exercises are performed using the incompressible flow iterative software IFISS which can be run using MATLAB or GNU Octave. The latest version of the software package is available in source code on a web site.

The material, except for some mathematical details, is self-contained, and the book is targeted to researchers and advanced undergraduate and graduate students in engineering, numerical analysis, applied mathematics and interdisciplinary scientific computing. The integration of the numerical methods from linear algebra (in particular, Krylov subspace methods and multilevel preconditioners) into the context of solving particular discrete PDEs enables the authors to prove some convergence results. Due to this approach, the book is intended to play a bridge-building role between the PDE and linear algebra research communities.

In the second edition of the book, two new chapters have been added concerning the unsteady Navier-Stokes equations and the models of buoyancy-driven flows. The chapter devoted to the discrete steady Navier-Stokes equations has also been substantially revised.

The authors present efficient solutions of Poisson, Stokes, convection-diffusion and Navier-Stokes partial differential equations (PDEs) that arise when modelling incompressible fluid flows. The theoretical results (concerning weak formulation, the Galerkin finite element method, theory of errors, the conjugate gradient method, matrix properties, preconditioning methods, splitting operators, multigrid methods) are developed in a constructive style and illustrated by numerical experiments. Error estimation and iterative solution of linear systems of algebraic equations are placed into the context of solving the particular discrete PDEs for model problems. The computational examples and exercises are performed using the incompressible flow iterative software IFISS which can be run using MATLAB or GNU Octave. The latest version of the software package is available in source code on a web site.

The material, except for some mathematical details, is self-contained, and the book is targeted to researchers and advanced undergraduate and graduate students in engineering, numerical analysis, applied mathematics and interdisciplinary scientific computing. The integration of the numerical methods from linear algebra (in particular, Krylov subspace methods and multilevel preconditioners) into the context of solving particular discrete PDEs enables the authors to prove some convergence results. Due to this approach, the book is intended to play a bridge-building role between the PDE and linear algebra research communities.

In the second edition of the book, two new chapters have been added concerning the unsteady Navier-Stokes equations and the models of buoyancy-driven flows. The chapter devoted to the discrete steady Navier-Stokes equations has also been substantially revised.

Reviewer: Adrian Carabineanu (Bucureşti)

##### MSC:

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

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

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

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

76R99 | Diffusion and convection |

65F10 | Iterative numerical methods for linear systems |