In this article, a locally stabilized nite element formulation of the two-dimensional Navier-Stokes problem is used. A macroelement condition which provides the stability of the Q1

A multi-level spectral Galerkin method for the two-dimensional nonstationary Navier-Stokes equations is presented. The method proposed here is a multiscale method in which the fully nonlinear Navier-Stokes equations are solved only on a low-dimensional space Hm1; subsequent approximations are generated on a succession of higher-dimensional spaces Hmj, j=2, . . . , J, by solving a linearized Navier-Stokes problem around the solution on the previous level. Error estimates depending on the kinematic viscosity 0<ν<1 are also presented for the J -level spectral Galerkin method. The optimal accuracy is achieved when mj=O (m 3/2 j−1), j=2, . . . , J. We demonstrate theoretically that the J-level spectral Galerkin method is much more efficient than the standard onelevel spectral Galerkin method on the highest-dimensional space HmJ .

In this article, the two-level stabilized finite element formulations of the two-dimensional steady Navier-Stokes problem are analyzed. A macroelement condition is introduced for constructing the local stabilized formulation of the steady Navier–Stokes problem. By satisfying this condition the stability of the Q1-P0 quadrilateral element and the P1-P0 triangular element are established. Moreover, the two-level stabilized finite element methods involve solving one small Navier–Stokes problem on a coarse mesh with mesh size H, a large Stokes problem for the simple two-level stabilized finite element method on a fine mesh with mesh size h=O (H2) or a large general Stokes problem for the Newton two-level stabilized finite element method on a fine mesh with mesh size h=O (|log h|1/2H3). The methods we study provide an approximate solution (uh, ph) with the convergence rate of same order as the usual stabilized finite element solution, which involves solving one large Navier-Stokes problem on a fine mesh with mesh size h. Hence, our methods can save a large amount of computational time.

A stabilized finite-element method for the two-dimensional stationary incompressible Navier-Stokes equations is investigated in this work. A macroelement condition is introduced for constructing the local stabilized formulation of the stationary Navier-Stokes equations. By satisfying this condition, the stability of the Q1−P0 quadrilateral element and the P1−P0 triangular element are established. Moreover, the well-posedness and the optimal error estimate of the stabilized finite-element method for the stationary Navier-Stokes equations are obtained. Finally, some numerical tests to confirm the theoretical results of the stabilized finite-element method are provided.

fully discrete penalty finite element method is presented for the two-dimensional time-dependent Navier-Stokes equations. The time discretization of the penalty Navier-Stokes equations is based on the backward Euler scheme; the spatial discretization of the time discretized penalty Navier-Stokes equations is based on a finite element space pair (Xh,Mh) which satisfies some approximate assumption. An optimal error estimate of the numerical velocity and pressure is provided for the fully discrete penalty finite element method when the parameters e, Δt and h are sufficiently small.