Parallel finite element variational multiscale algorithms for incompressible flow at high Reynolds numbers

Based on two-grid discretizations, some parallel finite element variational multiscale algorithms for the steady incompressible Navier-Stokes equations at high Reynolds numbers are presented and compared. In these algorithms, a stabilized Navier-Stokes system is first solved on a coarse grid, and then corrections are calculated independently on overlapped fine grid subdomains by solving a local stabilized linear problem. The stabilization terms for the coarse and fine grid problems are based on two local Gauss integrations. Error bounds for the approximate solution are estimated. Algorithmic parameter scalings are also derived. The theoretical results show that, with suitable scalings of the algorithmic parameters, these algorithms can yield an optimal rate of convergence. Numerical results are given to verify the theoretical predictions and demonstrate the effectiveness of the proposed algorithms.