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.

The asymptotic behavior and the Euler time discretization analysis are presented for the two-dimensional non-stationary Navier-Stokes problem. If the data ν and lim t→∞ f (t) satisfy a uniqueness condition corresponding to the stationary Navier-Stokes problem, we then obtain the convergence of the non-stationary Navier-Stokes problem to the stationary Navier-Stokes problem and the uniform boundedness, stability and error estimates of the Euler time discretization for the non-stationary Navier-Stokes problem.

A fully discrete modified finite element nonlinear Galerkin method is presented for the two-dimensional equation of Navier-Stokes type. The spatial discretization is based on two finite element spaces XH and Xh defined on a coarse grid with grid size H and a fine grid with grid size h << H, respectively; the time discretization is based on the Euler explicit scheme with respect to the nonlinear term.We analyze the stability and convergence rate of the method. Comparing with the andard finite element Galerkin method and the nonlinear Galerkin method, this method can admit a larger time step under the same convergence rate of same order. Hence this method can save a large amount of computational time. Finally, we provide some numerical tests on this method, the standard finite element Galerkin method, and the nonlinear Galerkin method, which are in a good agreement with the theoretical analysis.

discrete stabilized finite-element method is presented for the two-dimensional timedependent Navier-Stokes problem. The spatial discretization is based on a finite-element space pair (Xh, Mh) for the approximation of the velocity and the pressure, constructed by using the Q1-P0 quadrilateral element or the P1−P0 triangular element; the time discretization is based on the Euler semi-implicit scheme. It is shown that the proposed fully discrete stabilized finite-element method results in the optimal order error bounds for the velocity and the pressure.

A fully discrete two-level finite element method (the two-level method) is presented for solving the two-dimensional time-dependent Navier-Stokes problem. The method requires a Crank-Nicolson extrapolation solution (uH,τ0, pH,τ0) on a spatial-time coarse grid JH,τ0 and a backward Euler solution (uh,τ, ph,τ) on a space-time fine grid Jh,τ. The error estimates of optimal order of the discrete solution for the two-level method are derived. Compared with the standard Crank- Nicolson extrapolation method (the one-level method) based on a space-time fine grid Jh,τ, the two-level method is of the error estimates of the same order as the one-level method in the H1-norm for velocity and the L2-norm for pressure. However, the two-level method involves much less work than the one-level method.

In this article, we present a new fully discrete finite element nonlinear Galerkin method, which are well suited to the long time integration of the Navier-Stokes equations. Spatial discretization is based on two-grid finite element technique; time discretization is based on Euler explicit scheme with variable time step size. Moreover, we analyse the boundedness, convergence and stability condition of the finite element nonlinear Galerkin method. Our discussion shows that the time step constraints of the method depend only on the coarse grid parameter H and the time step constraints of the finite element Galerkin method depend on the fine grid parameter h << H under the same convergence accuracy.