A three-step subgrid stabilized Oseen iterative method for Navier–Stokes type variational inequality

Abstract

This work is devoted to developing a two-grid subgrid stabilized Oseen iterative finite element method for the convection dominated Navier–Stokes problem with friction boundary conditions whose weak form is the variational inequality of the second kind. This method inherits the best algorithmic advantages of each and involves three steps. Specifically, in the first step, a nonlinear Navier–Stokes type variational inequality is solved by applying m steps Oseen iterations to approximate the low frequencies of the solution on a coarse grid (H mesh), and then in the second and third steps, two subgrid stabilized and Newton-linearized Navier–Stokes type variational inequalities are solved to attain the high frequencies on a fine grid (h mesh). The developed method is easily implementable since there have identical stiffness matrices with only various right-hand sides in the second and third steps. Error bounds of the approximate velocity and pressure with regard to the mesh sizes h and H, stabilization parameters αH and αh and iteration step m are estimated, and new termination condition for nonlinear iterations is derived. The scalings both for the coarse grid and fine grid sizes as well as fine grid stabilization parameter obtained by our present method are improved in comparison with the developed method that only stays in the second step. Performance and effectiveness of the developed method are illustrated through a series of numerical experiments, which show that the accuracy of the calculated velocity and pressure of each step is higher than that of the previous step.