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**Finite element methods and Navier-Stokes equations.**
*(English)*
Zbl 0649.65059

The finite element method (FEM) is one of the most commonly used methods for solving partial differential equations (PDEs). It makes use of the computer and is very general in the sense that it can be applied to both steady-state and transient, linear and nonlinear problems in geometries of arbitrary space dimension. The FEM is in fact a method which transforms a PDE into a system of linear (algebraic) equations.

The following aspects may be identified in the study of a physical phenomenon:

(i) engineering-mathematical sciences to formulate the problem correctly in terms of PDE’s;

(ii) numerical methods to construct and to solve the system of algebraic equations; applied numerical functional analysis to give error estimates and convergence proofs;

(iii) informatics and programming to perform the calculations efficiently on the computer.

In this book we pay attention to all three of the aspects, considering in particular the finite element analysis of the Navier-Stokes equations, for which we restrict ourselves to the study of incompressible fluids. Applications include all kinds of internal fluid flows in complex geometries.

Part I (Chapters 1 to 5) serves as an introduction to the finite element analysis of PDEs of elliptic type. In part II (Chapters 6 to 10) three approaches to finite element analysis of the Navier-Stokes equations will be studied, including time dependent terms. In part III (Chapters 11 to 14) a more mathematical approach will be followed to give error estimates and convergence proofs. Finally in part IV (Chapters 15 to 18) we overview some current areas of research.

The following aspects may be identified in the study of a physical phenomenon:

(i) engineering-mathematical sciences to formulate the problem correctly in terms of PDE’s;

(ii) numerical methods to construct and to solve the system of algebraic equations; applied numerical functional analysis to give error estimates and convergence proofs;

(iii) informatics and programming to perform the calculations efficiently on the computer.

In this book we pay attention to all three of the aspects, considering in particular the finite element analysis of the Navier-Stokes equations, for which we restrict ourselves to the study of incompressible fluids. Applications include all kinds of internal fluid flows in complex geometries.

Part I (Chapters 1 to 5) serves as an introduction to the finite element analysis of PDEs of elliptic type. In part II (Chapters 6 to 10) three approaches to finite element analysis of the Navier-Stokes equations will be studied, including time dependent terms. In part III (Chapters 11 to 14) a more mathematical approach will be followed to give error estimates and convergence proofs. Finally in part IV (Chapters 15 to 18) we overview some current areas of research.

### MSC:

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

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

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

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

35Q30 | Navier-Stokes equations |

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

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

65F10 | Iterative numerical methods for linear systems |