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