## Abstract

In this paper we analyze several first-order systems of Oseen-type equations that are obtained from the time-dependent incompressible Navier-Stokes equations after introducing the additional vorticity and possibly total pressure variables, time-discretizing the time derivative and linearizing the nonlinear terms. We apply the [L^{2}, L^{2}, L^{2}] least-squares finite element scheme to approximate the solutions of these Oseen-type equations assuming homogeneous velocity boundary conditions. All of the associated least-squares energy functionals are defined to be the sum of squared L^{2} norms of the residual equations over an appropriate product space. We first prove that the homogeneous least-squares functionals are coercive in the H^{1} × L^{2} × L^{2} norm for the velocity, vorticity, and pressure, but only continuous in the H^{1} × H^{1} × H^{1} norm for these variables. Although equivalence between the homogeneous least-squares functionals and one of the above two product norms is not achieved, by using these a priori estimates and additional finite element analysis we are nevertheless able to prove that the least-squares method produces an optimal rate of convergence in the H^{1} norm for velocity and suboptimal rate of convergence in the L^{2} norm for vorticity and pressure. Numerical experiments with various Reynolds numbers that support the theoretical error estimates are presented. In addition, numerical solutions to the time-dependent incompressible Navier-Stokes problem are given to demonstrate the accuracy of the semi-discrete [L^{2}, L^{2}, L^{2}] least-squares finite element approach.

Original language | English |
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Pages (from-to) | 402-424 |

Number of pages | 23 |

Journal | International Journal of Numerical Analysis and Modeling |

Volume | 4 |

Issue number | 3-4 |

State | Published - 2007 |

## Keywords

- Finite element methods
- Least squares
- Navier-Stokes equations
- Oseen-type equations