A Two-Level Defect–correction Method for Navier–stokes Equations

Abstract

A two-level defect-correction method for the steady-state Navier-Stokes equations with a high Reynolds number is considered in this paper. The defect step is accomplished in a coarse-level subspace H m by solving the standard Galerkin equation with an artificial viscosity parameter σ as a stability factor, and the correction step is performed in a fine-level subspace H M by solving a linear equation. H 1 error estimates are derived for this two-level defect-correction method. Moreover, some numerical examples are presented to show that the two-level defect-correction method can reach the same accuracy as the standard Galerkin method in fine-level subspace H M . However, the two-level method will involve much less work than the one-level method.2000 Mathematics subject classification: primary 65N35; secondary 76D05, 35Q30.

Error analysis of a fully discrete finite element variational multiscale method for time‐dependent incompressible Navier–Stokes equations

“…Moreover, we assume that the velocity space X h satisfies that 13) which indicates that the method proposed in this article is only suitable for those elements pair where the finite element space for velocity is restricted to (P 2 ) d . We introduce the following spaces…”

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

“…Recently, two-level strategy has been studied for steady semi-linear elliptic equations and nonlinear PDEs by Xu [35,36], and two-level strategy or multi-level strategy has been studied for the steady Navier-Stokes equations by Layton [24], Layton and Tobiska [28], Layton and Leferink [26,27] and Layton, Lee and Peterson [25] and Girault and Lions [7] and He et al [12,14,19] and Liu and Hou [29], and two level discretization of flows of electrically conducting, incompressible fluids has been provided by Ervin, Layton and Maubach in [6]. Moreover, a combination of two-level methods and iterative methods for solving the 2D/3D steady Navier-Stokes equations is provided by He et al [20].…”

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

“…Recently, a two-level strategy has been studied for steady semi-linear elliptic equations and nonlinear PDEs by Xu [33,34], and two-level strategy or multi-level strategy has been studied for the steady Navier-Stokes equations by Layton [21], Layton and Tobiska [26], Layton and Lenferink [23,24] and Layton, Lee and Peterson [25] and Girault and Lions [4] and He et al [11,14,15] and Liu and Hou [27], and two level discretization of flows of electrically conducting, incompressible fluids has been provided by Ervin, Layton and Maubach in [3]. Moreover, a combination of two-level methods and iterative methods for solving the 2D/3D steady Navier-Stokes equations is provided by He et al [17,18].…”