Stability and convergence of iterative methods related to viscosities for the 2D/3D steady Navier–Stokes equations

Abstract

Some finite element iterative methods related to viscosities are designed to solve numerically the steady 2D/3D Navier–Stokes equations. The two-level finite element iterative methods are designed to solve numerically the steady 2D/3D Navier–Stokes equations for a large viscosity ν such that a strong uniqueness condition holds. The two-level finite element iterative methods consist of using the Stokes, Newton and Oseen iterations of m times on a coarse mesh with mesh size H and computing the Stokes, Newton and Oseen correction of one time on a fine grid with mesh size h≪H. Moreover, the one-level Oseen finite element iterative method based on a fine mesh with a small mesh size is designed to solve numerically the steady 2D/3D Navier–Stokes equations for small viscosity ν such that a weak uniqueness condition holds. The uniform stability and convergence of these methods with respect to ν and mesh sizes h and H and iterative times m are provided.