On two iterative least-squares finite element schemes for the incompressible navier-stokes problem

Mei Chun Chen, Po Wen Hsieh, Chun Ting Li, Yun Tsz Wang, Suh Yuh Yang

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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 L2 least-squares scheme or a weighted L2 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 L2 least-squares scheme is somewhat suitable for low Reynolds number flow problems, whereas for flows with relatively higher Reynolds numbers the iterative weighted L2 least-squares scheme seems to be better than the iterative L2 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 languageEnglish
Pages (from-to)436-461
Number of pages26
JournalNumerical Functional Analysis and Optimization
Issue number5-6
StatePublished - May 2009


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


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