Accuracy of the approximate velocity of the steady incompressible Navier-Stokes equations computed by the standard mixed finite element methods is often affected by the pressure. In order to circumvent or weaken the influence of pressure on the accuracy of the computed velocity, by combining grad-div stabilized method with two-level finite element method, this paper presents a kind of two-level grad-div stabilized finite element methods for solving the steady incompressible Navier-Stokes equations numerically. The basic idea of the methods is to first solve a grad-div stabilized nonlinear Navier-Stokes problems on a coarse grid, and then solve, respectively, Stokes-linearized, Newton-linearized and Oseen-linearized Navier-Stokes problem with grad-div stabilization on a fine grid. Numerical examples are given to verify the high efficiency of the two-level grad-div stabilized finite element methods.

Based on two-grid discretizations and domain decomposition techniques, this paper presents three parallel finite element algorithms for numerically solving the steady Navier-Stokes equations with damping term. The basic idea of the present algorithms is to first solve a fully nonlinear problem on a coarse grid to get a coarse grid solution, then solve Stokes, Oseen, and Newton linearized residual problems in parallel in overlapping local fine grid subdomains, and finally update the approximate solution in non-overlapping fine grid subdomains. The effectiveness of the proposed algorithms is demonstrated by some numerical examples.

In numerical solution of unsteady Navier-Stokes equations with standard finite element method, errors of computed velocity are usually affected by pressure errors, where smaller viscosity coefficients lead to greater velocity errors. To improve pressurerobustness, in this paper, we introduce a grad-div stabilization term to improve accuracy of approximate solutions. We present parallel two-level grad-div stabilized finite element algorithms for unsteady Navier-Stokes equations, where implicit Euler scheme and Galerkin finite element methods are used for temporal and spatial discretizations, respectively. At each time step, firstly we solve a nonlinear Navier-Stokes problem with grad-div stabilization on a coarse grid, and then linearized and grad-div stabilized problems are solved with Stokes, Oseen and Newton iterations on overlapping fine grid subdomains in a parallel manner to correct the coarse grid solution. Finally, numerical experiments are given to verify correctness of theoretical predictions and demonstrate effectiveness of the algorithms.