Unconditionally Optimal Error Estimates of the Bilinear-Constant Scheme for Time-Dependent Navier-Stokes Equations

In this paper,the unconditional error estimates are presented for the time-dependent Navier-Stokes equations by the bilinear-constant scheme.The corresponding optimal error estimates for the velocity and the pressure are derived unconditionally,while the previous works require certain time-step restrictions.The analysis is based on an iterated time-discrete system,with which the error function is split into a temporal error and a spatial error.The τ-independent(τ is the time stepsize)error estimate between the numerical solution and the solution of the time-discrete system is proven by a rigorous analysis,which implies that the numerical solution in L^∞-norm is bounded.Thus optimal error estimates can be obtained in a traditional way.Numerical results are provided to confirm the theoretical analysis.     

Error estimates for an operator-splitting method for Navier-Stokes equations: Second-order schemes

We present in this paper an error analysis of a fractional-step method for the approximation of the unsteady incompressible Navier-Stokes equations. Under mild regularity assumptions on the continuous solution, we obtain second-order error estimates in the time step size, both for velocity and pressure. Numerical results in agreement with the error analysis are also presented.     

非定常不可压Navier-Stokes方程两重网格方法的稳定性和L2误差估计

讨论了二维非定常不可压Navier-Stokes方程的两重网格方法．此方法包括在粗网格上求解一个非线性问题,在细网格上求解一个Stokes问题．采用一种新的全离散(时间离散用Crank-Nicolson格式,空间离散用混合有限元方法)格式数值求解N-S方程．证明了该全离散格式的稳定性．给出了L2误差估计．对比标准有限元方法,在保持同样精度的前提下,TGM能节省大量的计算量．     

On Some Model Equations of Euler and Navier-Stokes Equations

The author proposes a two-dimensional generalization of Constantin-Lax-Majda model. Some results about singular solutions are given. This model might be the first step toward the singular solutions of the Euler equations. Along the same line(vorticity formulation), the author presents some further model equations. He possibly models various aspects of difficulties related with the singular solutions of the Euler and Navier-Stokes equations. Some discussions on the possible connection between tur...     

可压缩Navier-Stokes方程的压力梯度局部投影间断有限元法

将压力梯度投影与间断有限元法相结合,对可压缩线性化N-S方程提出了一种稳定化间断有限元格式．证明了此格式在速度和压力有限元空间无需满足B-B型条件的情况下,解的存在性和唯一性,以及相应的误差估计．     

Stokes方程的投影／方向分裂谱方法

我们提出和分析了一种求解Stokes方程的数值方法．新方法基于空间上的Legendre谱离散,时间上则采用投影／方向分裂格式．更确切地说,时间离散的出发点是旋度形式的压力校正投影法,在此基础上进一步应用方向分裂法,把速度和压力方程分裂为一系列一维的椭圆型子问题．然后生成的这些一维子问题用Legendre谱方法进行空间离散．另外,我们证明了全离散格式的稳定性．一些数值实验验证了收敛性和方法的有效性．     

定常Navier—Stokes方程的Petrov—Galerkin方法

研究了定常Navier－Stokes方程的四种Petrov－Galerkin有限元方法：PG1，PG2，SD和GLS．它们都是稳定的，避免了经典混合方法中必要的Babuska－Brezzi条件．给出了各种方法有限元解的存在性、唯一性和唯一解的误差估计．     

Reduced Finite Element Discretizations of the Stokes and Navier-Stokes Equations

ABSTRACT

If finite element spaces for the velocity and pressure do not satisfy the Babu?ka-Brezzi condition, a stable conforming discretization of the Stokes or Navier-Stokes equations can be obtained by enriching the velocity space by suitable functions. Writing any function from the enriched space as a sum of a function from the original space and a function from the supplementary space, the discretization will contain a number of additional terms compared with a conforming discretization for the original pair of spaces. We show that not all these terms are necessary for the solvability of the discrete problem and for optimal convergence properties of the discrete solutions, which is useful for saving computer memory and for establishing a connection to stabilized methods.     

The Viscosity Splitting Solutions of the Navier-Stokes Equations

The solutions of the initial boundary value problems or the Navier-Stokes equations are constructed by means of a viscosity splitting scheme. Convergence results are proved.     

A TWO-LEVEL FINITE ELEMENT GALERKIN METHOD FOR THE NONSTATIONARY NAVIER-STOKES EQUATIONS II： TIME DISCRETIZATION

In this article we consider the fully discrete two-level finite element Galerkin method for the two-dimensional nonstationary incompressible Navier-Stokes equations. This method consists in dealing with the fully discrete nonlinear Navier-Stokes problem on a coarse mesh with width $H$ and the fully discrete linear generalized Stokes problem on a fine mesh with width $h << H$. Our results show that if we choose $H=O(h^{1/2}$) this method is as the same stability and convergence as the fully discrete standard finite element Galerkin method which needs dealing with the fully discrete nonlinear Navier-Stokes problem on a fine mesh with width $h$. However, our method is cheaper than the standard fully discrete finite element Galerkin method.     

Navier-Stokes方程流函数形式两重网格算法的误差分析

对定常Navier-Stokes方程流函数形式两重网格有限元算法进行了误差分析。此方法包括在粗网格上求解一个非线性问题，在细网格上求解一个线性问题，然后再在粗网格上求解一个线性校正问题。分析了包括校正项和不包括校正项两种方法的误差，得出对于任意固定的Beynolds数，能达到最优逼近阶。     

The Decay Rates of Strong Solutions for Navier-Stokes Equations

In this paper, we deduce the estimates on decay rates of higher order derivatives about time variable and space variables for the strong solution to the Cauchy problem of the Navier-Stokes equations. The rate obtained is optimal in the sense that it coincides with that of solution to the heat equation.     

The Asymptotic Expansion of Solutions for the Navier-Stokes Equations with Large Parameters

In this paper, we decomposite the coefficient matrix of the Navier-Stokes equations with large parameter in λ power. By factor analysis and energy estimation, the long time existence and asymptotic expansion of the solution tor the system are obtained. Key words Navier-Stokes equations; expansion; approximation.     

Error Estimates for Finite-Element Navier-Stokes Solvers without Standard Inf-Sup Conditions

The authors establish error estimates for recently developed finite-element methods for incompressible viscous flow in domains with no-slip boundary conditions. The methods arise by discretization of a well-posed extended Navier-Stokes dynamics for which pressure is determined from current velocity and force fields. The methods use C^1 elements for velocity and C^0 elements for pressure. A stability estimate is proved for a related finite-element projection method close to classical time-splitting methods of Orszag, Israeli, DeVille and Karniadakis.     

Auxiliary Equations Approach for the Stochastic Unsteady Navier-Stokes Equations with Additive Random Noise

This paper presents a Martingale regularization method for the stochastic Navier-Stokes equations with additive noise. The original system is split into two equivalent parts, the linear stochastic Stokes equations with Martingale solution and the stochastic modified Navier-Stokes equations with relatively-higher regularities. Meanwhile, a fractional Laplace operator is introduced to regularize the noise term. The stability and convergence of numerical scheme for the pathwise modified Navier-Stokes equations are proved.The comparisons of non-regularized and regularized noises for the Navier-Stokes system are numerically presented to further demonstrate the efficiency of our numerical scheme.     

关于轴对称Navier-Stokes方程正则性的一个注记

建立了一个关于轴对称不可压Navier-Stokes系统的正则性准则.证明了如果局部的轴对称光滑解u满足‖ωr‖_{Lα1((0,T);Lβ1)}+‖ωθ/r‖_{Lα2((0,T);Lβ2)}<∞,其中2/α₁+3/β₁≤1+3/β₁,2/α₂+3/β₂≤2和β₁≥3, β₂>3/2,那么此强解将保持光滑性直至时刻T.     

可压缩的Navier—Stokes方程H^2强解的整体存在性

该文对于三维可压缩的Navier-Stokes方程，当其具有小扰动初值时，证明了H2（R3）强解的整体存在性．     

Zero Dissipation Limit to Rarefaction Waves for the 1-D Compressible Navier-Stokes Equations

The zero dissipation limit for the one-dimensional Navier-Stokes equations of compressible,isentropic gases in the case that the corresponding Euler equations have rarefaction wave solutions is investi...     

Penalty-Factor-Free Stabilized Nonconforming Finite Elements for Solving Stationary Navier-Stokes Equations

Two nonconforming penalty methods for the two-dimensional stationary Navier-Stokes equations are studied in this paper. These methods are based on the weakly continuous $P_1$ vector fields and the locally divergence-free (LDF) finite elements, which respectively penalize local divergence and are discontinuous across edges. These methods have no penalty factors and avoid solving the saddle-point problems. The existence and uniqueness of the velocity solution are proved, and the optimal error estimates of the energy norms and $L^2$-norms are obtained. Moreover, we propose unified pressure recovery algorithms and prove the optimal error estimates of $L^2$-norm for pressure. We design a unified iterative method for numerical experiments to verify the correctness of the theoretical analysis.     

The Stability of Navier-Stokes Equations and the Estimation of Its Attractor Dimension

In this paper, for a class of exterior force term 2s²W^'_{s,s} we analyse the existence of unstable modes of linearized Navier-Stokes Equations (NSE), and associate them with integer points in plane. Furthermore we give the lower boundary dimension estimation of the attractor of NSE. Liu discussed the condition where the exterior force term is W^'_{0,s} in (1, 2), but his method can't be extended to the condition where the exterior force term is W^'_{s_1,s_2} (s_1 ≠ 0, s_2 ≠ 0). So this paper may look as the extention of [1, 2]. The method which we give in this paper has direct application for further study of other properties of NSE (such as Hopf bifurcation). See [3].