The mimetic finite difference method for elliptic problems.

*(English)*Zbl 1286.65141
MS&A. Modeling, Simulation and Applications 11. Cham: Springer (ISBN 978-3-319-02662-6/hbk; 978-3-319-02663-3/ebook). xvi, 392 p. (2014).

This book deals with theoretical and computational issues of the mimetic finite difference method for a large classes of multidimensional elliptic problems. They include but are not limited to diffusion, advection-diffusion, Stokes, elasticity, magnetostatics, plate bending problems.

The monograph is structured into three parts, each containing four chapters.

Part one is entitled ‘Foundation’ and contains the historical development of the method, the motivation as well as unique features of the method. Part two is entitled ‘Mimetic discretization of basic partial differential equations’. It describes the implementation of the method to steady-state diffusion equations, Maxwell’s equations and steady Stokes equations. Part three is entitled ‘Further developments’ and emphasizes the application of the mimetic finite difference method in addressing challenging problems from structural mechanics, convection-diffusion problems or obstacle problems.

The research monograph is a useful source for scientists and engineers interested in computational treatment for various mathematical models arising in real life. It also proves to be a valuable research monograph for graduate students in Applied Mathematics or Computational Physics.

The monograph is structured into three parts, each containing four chapters.

Part one is entitled ‘Foundation’ and contains the historical development of the method, the motivation as well as unique features of the method. Part two is entitled ‘Mimetic discretization of basic partial differential equations’. It describes the implementation of the method to steady-state diffusion equations, Maxwell’s equations and steady Stokes equations. Part three is entitled ‘Further developments’ and emphasizes the application of the mimetic finite difference method in addressing challenging problems from structural mechanics, convection-diffusion problems or obstacle problems.

The research monograph is a useful source for scientists and engineers interested in computational treatment for various mathematical models arising in real life. It also proves to be a valuable research monograph for graduate students in Applied Mathematics or Computational Physics.

Reviewer: Marius Ghergu (Dublin)

##### MSC:

65N06 | Finite difference methods for boundary value problems involving PDEs |

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

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

35Q30 | Navier-Stokes equations |

35Q61 | Maxwell equations |

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

74S20 | Finite difference methods applied to problems in solid mechanics |

76M20 | Finite difference methods applied to problems in fluid mechanics |