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.

Finite Element Iterative Methods for the 3D Steady Navier--Stokes Equations

“…For the Stokes iterative finite element method, the stability result is and the convergence result is under the weak stability and convergence condition ; for the Newton iterative finite element method, the stability result is and the convergence result is under the strong stability and convergence condition . Compared with the results of [ 37 , 38 ], we obtain better stability and convergence results of the finite element iterative solution of of the 3D steady Navier–Stokes equations under the weak stability and convergence condition.…”

“…Furthermore, in order to overcome the difficulties mentioned above in solving the 3D steady Navier–Stokes equations, Xu and He [ 37 ] and He [ 38 ] used the finite element pair , satisfying the discrete inf-sup condition in a 2D/3D domain , which overcomes the difficulty of divergence free constraint, using the Stokes, Newton and Oseen iterative finite element methods to overcome the difficulty of nonlinearity of the steady Navier–Stokes equations in the 2D/3D space. However, in [ 37 , 38 ], they provided some poor stability and convergence results under the strong stability and convergence conditions. For the Stokes iterative finite element method, the stability result is and the convergence result is under the strong stability and convergence condition .…”

Stability and convergence of two‐grid Crank‐Nicolson extrapolation scheme for the time‐dependent natural convection equations

“…Therefore, in this paper, we consider the Crank‐Nicolson extrapolation scheme for the natural convection problem, our work is extension and supplement the previous works and provide some new stability and convergence results for the numerical solutions. At the same time, based on He, He and Li, and He et al, for the 3D Navier‐Stokes equations, we also consider the Crank‐Nicolson extrapolation scheme for 3D time‐dependent natural convection problem.…”

Some iterative finite element methods for steady Navier–Stokes equations with different viscosities

“…Recently, He's paper [19] provides the uniform stability and convergence of these methods with respect to r and mesh sizes h and H and iterative times m are provided.…”

“…Then, the error ðu À u l ; p À p l Þ satisfies the following uniform bound: In this section, we recall some iterative methods and their stability and error estimates for the 2D/3D steady Navier-Stokes equations provided in [19].…”