## Abstract

This paper is mainly devoted to a comparative study of two iterative least-squares finite element schemes for solving the stationary incompressible Navier-Stokes equations with velocity boundary condition. Introducing vorticity as an additional unknown variable, we recast the Navier-Stokes problem into a first-order quasilinear velocity-vorticity-pressure system. Two Picard-type iterative least-squares finite element schemes are proposed to approximate the solution to the nonlinear first-order problem. In each iteration, we adopt the usual L^{2} least-squares scheme or a weighted L^{2} least-squares scheme to solve the corresponding Oseen problem and provide error estimates. We concentrate on two-dimensional model problems using continuous piecewise polynomial finite elements on uniform meshes for both iterative least-squares schemes. Numerical evidences show that the iterative L^{2} least-squares scheme is somewhat suitable for low Reynolds number flow problems, whereas for flows with relatively higher Reynolds numbers the iterative weighted L^{2} least-squares scheme seems to be better than the iterative L^{2} least-squares scheme. Numerical simulations of the two-dimensional driven cavity flow are presented to demonstrate the effectiveness of the iterative least-squares finite element approach.

Original language | English |
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Pages (from-to) | 436-461 |

Number of pages | 26 |

Journal | Numerical Functional Analysis and Optimization |

Volume | 30 |

Issue number | 5-6 |

DOIs | |

State | Published - May 2009 |

## Keywords

- Driven cavity flows
- Finite element methods
- Iterative methods
- Least squares
- Navier-Stokes equations
- Oseen equations