Analysis of the [L2,L2,L2] least-squares finite element method for incompressible oseen-type problems

Ching L. Chang, Suh Yuh Yang

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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 [L2, L2, L2] 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 L2 norms of the residual equations over an appropriate product space. We first prove that the homogeneous least-squares functionals are coercive in the H1 × L2 × L2 norm for the velocity, vorticity, and pressure, but only continuous in the H1 × H1 × H1 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 H1 norm for velocity and suboptimal rate of convergence in the L2 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 [L2, L2, L2] least-squares finite element approach.

Original languageEnglish
Pages (from-to)402-424
Number of pages23
JournalInternational Journal of Numerical Analysis and Modeling
Volume4
Issue number3-4
StatePublished - 2007

Keywords

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

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