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**On the approximation of the unsteady Navier-Stokes equations by finite element projection methods.**
*(English)*
Zbl 0914.76051

Summary: This paper provides an analysis of a fractional-step projection method to compute incompressible viscous flows by means of finite element approximations. The analysis is based on the idea that the appropriate functional setting for projection methods must accommodate two different spaces for representing the velocity fields, calculated respectively in the viscous and the incompressible half steps of the method. Such a theoretical distinction leads to a finite element projection method with a Poisson equation for the incremental pressure unknown and to a very practical implementation of the method with only the intermediate velocity appearing in the numerical algorithm. Error estimates in finite time are given. An extension of the method to a problem with unconventional boundary conditions is also considered to illustrate the flexibility of the proposed method.

### MSC:

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

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

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

35Q30 | Navier-Stokes equations |

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |