Abstract:

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.

Key words: unsteady Navier-Stokes equations, grad-div stabilization, two-level method, parallel algorithm, finite element method