## Abstract

In this paper, we study the L^{2} least-squares finite element approximations to the Oseen problem for the stationary incompressible Navier-Stokes equations with the velocity boundary condition. The Oseen problem is first recast into the velocity-vorticity-pressure first-order system formulation by introducing the vorticity variable. We then derive some a priori estimates for the first-order system problem and identify the dependence of the estimates on the Reynolds number. Such a priori estimates play the crucial roles in the error analysis for least-squares approximations to the incompressible velocity-vorticity-pressure Oseen problem. It is proved that, with respect to the order of approximation for smooth exact solutions, the L^{2} least-squares method exhibits an optimal rate of convergence in the H^{1} norm for velocity and a suboptimal rate of convergence in the L^{2} norm for vorticity and pressure. Numerical results that confirm this analysis are given. Furthermore, in order to maintain the coercivity and continuity of the homogeneous least-squares functional that are destroyed by large Reynolds numbers, a weighted least-squares energy functional is proposed and analyzed. Numerical experiments in two dimensions are presented, which demonstrate the effectiveness of the weighted least-squares approach. Finally, approximate solutions of the incompressible velocity-vorticity-pressure Navier-Stokes problem with various Reynolds numbers are also given by solving a sequence of Oseen problems arising from a Picard-type iteration scheme. Numerical evidences show that, except for large Reynolds numbers, the convergence rates of the weighted least-squares approximations for the Navier-Stokes problem are similar to that for the Oseen problem.

Original language | English |
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Pages (from-to) | 211-232 |

Number of pages | 22 |

Journal | Journal of Computational and Applied Mathematics |

Volume | 182 |

Issue number | 1 |

DOIs | |

State | Published - 1 Oct 2005 |

## Keywords

- Driven cavity flows
- Least-squares finite element methods
- Navier-stokes equations
- Oseen equations
- Velocity-vorticity-pressure formulation