*(English)*Zbl 0585.65077

The present work is an extended version of a previous text on finite element approximation of the Navier-Stokes equations [Lect. Notes Math. 749 (1979; Zbl 0413.65081)]. It is concerned with theory and application of finite element methods to the stationary Navier-Stokes equations for incompressible fluid flows. The authors restrict themselves to inner problems and provide a fairly comprehensive and exhaustive up to date treatment of this field. Emphasis is given on the mathematical foundation of finite element methods. As to implementation and numerical results the authors refer to the complementary work of *F. Thomasset* [Implementation of finite element methods for Navier-Stokes equations (1981; Zbl 0475.76036)] and of *R. Peyret* and *T. Taylor* [Computational methods for fluid flow (1983; Zbl 0514.76001)].

The book is devided into four chapters and a technical appendix: Chapter I is devoted to theoretical aspects of the Stokes equations including a study of related function spaces and variational formulations of the underlying problem. Chapter II is devoted to finite element approximations to the Stokes problem in the original variables (velocity and pressure). Here, the most popular mixed schemes are presented. In chapter III various finite element methods are presented that rest upon other variables such as stream function, vector potential and vorticity. Finally, chapter IV is concerned with theory and approximation of the full Navier-Stokes equations where the results of the two preceding chapters are systematically extended to the nonlinear equations.

##### MSC:

65Z05 | Applications of numerical analysis to physics |

65-02 | Research monographs (numerical analysis) |

65N30 | Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE) |

76-02 | Research monographs (fluid mechanics) |

35A15 | Variational methods (PDE) |

35G20 | General theory of nonlinear higher-order PDE |

35Q30 | Stokes and Navier-Stokes equations |

46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |

65N15 | Error bounds (BVP of PDE) |

76D05 | Navier-Stokes equations (fluid dynamics) |

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

76M99 | Basic methods in fluid mechanics |