Numerical solution of partial differential equations by the finite element method.

*(English)*Zbl 0628.65098
Cambridge etc.: Cambridge University Press. 278 p. (Orig. Studentlitteratur, Lund, Sweden) (1987).

“The purpose of this book is to give an easily accessible introduction to the finite element method as a general method for the numerical solution of partial differential equations in mechanics and physics covering all the three main types of equations namely elliptic, parabolic and hyperbolic equations.”

The author keeps the mathematical framework as simple as possible but generously illustrated by a great deal of explanatory examples. The work emphasizes the numerical aspects of the finite element method and many applications not only to linear problems but also to some nonlinear problems (compressible flow, incompressible Navier-Stokes equations, a nonlinear parabolic problem etc.) are considered.

Besides classical aspects of the finite element method, related to elliptic equations, the book treats some parabolic and hyperbolic problems using recent results of the author based on a discontinuous Galerkin and streamline diffusion type finite element method. In particular, finite elements for the time discretization are used as well. All proposed methods are accompanied by error estimates and very detailed discussions. The work also contains a chapter on the boundary element method connected to elliptic problems (finite element methods for Fredholm equations of the first and second kind are exposed). Two special chapters are devoted to a mixed finite element method to respectively, to curved elements and to the effect of numerical integration (quadrature formulas) on the accuracy of the finite element method.

The most important and the most original part of book seems to be that related to hyperbolic problems (time-dependent convection-diffusion problems, Friedrichs’ systems etc.). Actually the whole work opens new possibilities in proper understanding of the mathematical structure and features of the finite element method. It represents an important achievement in the literature on the finite element method.

The author keeps the mathematical framework as simple as possible but generously illustrated by a great deal of explanatory examples. The work emphasizes the numerical aspects of the finite element method and many applications not only to linear problems but also to some nonlinear problems (compressible flow, incompressible Navier-Stokes equations, a nonlinear parabolic problem etc.) are considered.

Besides classical aspects of the finite element method, related to elliptic equations, the book treats some parabolic and hyperbolic problems using recent results of the author based on a discontinuous Galerkin and streamline diffusion type finite element method. In particular, finite elements for the time discretization are used as well. All proposed methods are accompanied by error estimates and very detailed discussions. The work also contains a chapter on the boundary element method connected to elliptic problems (finite element methods for Fredholm equations of the first and second kind are exposed). Two special chapters are devoted to a mixed finite element method to respectively, to curved elements and to the effect of numerical integration (quadrature formulas) on the accuracy of the finite element method.

The most important and the most original part of book seems to be that related to hyperbolic problems (time-dependent convection-diffusion problems, Friedrichs’ systems etc.). Actually the whole work opens new possibilities in proper understanding of the mathematical structure and features of the finite element method. It represents an important achievement in the literature on the finite element method.

Reviewer: C.-I.Gheorghiu

##### MSC:

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

65-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis |

65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |

65K05 | Numerical mathematical programming methods |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

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

65N15 | Error bounds for boundary value problems involving PDEs |

65F05 | Direct numerical methods for linear systems and matrix inversion |

65F10 | Iterative numerical methods for linear systems |

65N40 | Method of lines for boundary value problems involving PDEs |

35K05 | Heat equation |

35L20 | Initial-boundary value problems for second-order hyperbolic equations |

35C15 | Integral representations of solutions to PDEs |

65R20 | Numerical methods for integral equations |

90C25 | Convex programming |

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