Weighted least-squares finite element methods for the linearized Navier–Stokes equations

Abstract

We implemented weighted least-squares finite element methods for the linearized Navier-Stokes equations based on the velocity–pressure–stress and the velocity–vorticity–pressure formulations. The least-squares functionals involve the L²-norms of the residuals of each equation multiplied by the appropriate weighting functions. The weights included a mass conservation constant, a mesh-dependent weight, a nonlinear weighting function, and Reynolds numbers. A feature of this approach is that the linearized system creates a symmetric and positive-definite linear algebra problem at each Newton iteration. We can prove that least-squares approximations converge with the linearized version solutions of the Navier–Stokes equations at the optimal convergence rate. Model problems considered in this study were the flow past a planar channel and 4-to-1 contraction problems. We presented approximate solutions of the Navier–Stokes problems by solving a sequence of the linearized Navier–Stokes problems arising from Newton iterations, revealing the convergence rates of the proposed schemes, and investigated Reynolds number effects.

Keywords:

• least-squares finite element methods
• Navier–Stokes equations
• nonlinear weighting function
• Reynolds number

2010 AMS Subject Classifications:

• 65N30
• 76M10

Funding

This work was supported in part by the National Science Council of Taiwan under contract no. 100-2115-M-160-001.