Based on two-grid discretization and domain decomposition, three finite element parallel algorithms for unsteady Navier-Stokes equations are proposed. The key idea of the algorithms is to solve nonlinear problem firstly by Oseen iteration method on a coarse grid, and then to solve Oseen, Newton or Stokes problem in parallel on a fine grid to correct the coarse grid solution at each time step, respectively. Conforming finite element pairs are used for spatial discretization and backward Euler scheme for temporal discretization. Numerical results are shown to verify effectiveness of the algorithms.

Based on two-grid discretizations,three two-level subgrid stabilized finite element algorithms for stationary Navier-Stokes equations at high Reynolds numbers are proposed and compared. Basic idea of the algorithms is to solve a fully nonlinear Navier-Stokes problem with a subgrid stabilization term on a coarse grid,and then solve a subgrid stabilized linear fine grid problem based on one step of Newton,Oseen or Stokes iterations for Navier-Stokes equations.It shows that with suitable stabilization parameters and coarse and fine grid sizes,those algorithms yield an optimal convergence rate. Finally, numerial results are given to show efficiency of the algorithms.

Based on fully overlapping domain decomposition,three parallel finite element algorithms for time-dependent Navier-Stokes equations are proposed.Basic idea of algorithms is to discretize spatial space with fully overlapping domain decomposition technique,and then to solve ordinary differential equations with respect to time independently in backward Euler scheme on overlapped subdomains.The nonlinear convective term is dealt with semi-and fully-implicit schemes,respectively.In these algorithms,each subproblem is a global problem with vast majority of degrees of freedom associated with a particular subdomain that is responsible for,which allows algorithms to be implemented easily with low communication costs.Numerical test illustrates efficiency and good parallel performance of the algorithms.