Navier-Stokes方程的非奇异解分支的谱Galerkin逼近

No error estimate of the spectral Galerkin approximation for the steady-state Navier-Stokes equations was given without assuming that the data of the externalforce field and the boundary conditions are small enough. In this paper, under the condition that the solutions of the Navier-Stokes equations are nonsingular,we proved the existence and convergence of the spectral Galerkin approximation solutions and gave the error estimate. At last, this approximation method wasapplied to simulate the spherical Couette flow.     

关于二维Navier-Stokes方程非线性Galerkin校正的注记

本文在对二维Navier-Stokes方程及其近似惯性流形领域估计的基础上，讨论了非线性Galerkin方法校正量的一种可能的最优选取。     

对称破坏分歧点的分裂迭代方法

构造了求解对称破坏分歧点的扩充系统，采用分裂分块迭代方法逼近对称破坏分歧点，并对2.Box Brusselator反应模型进行了数值模拟．     

三维不可压Navier-Stokes方程的Fourier-Chebyshev谱方法

对三维不可压Navier－Stokes方程构造了Fourier-Chebyshev谱格式，并严格证明了该格式的广义稳定性和收敛性，效植结果表明这一方法有较高的精度．     

定常的Navier-Stokes方程的非线性Galerkin/Petrov最小二乘混合元法

给出定常的Navier-Stokes方程的一种非线性Galerkin/Petrov最小二乘混合元法，该方法是将余量形式的Petrov最小二乘方法与非线性Galerkin混合元结合起来，使得速度和压力的混合元空间无需满足离散的Babuska-Brezzi稳定性条件，从而使得它们的有限元空间可以任意选择。并证明该方法的解的存在唯一性和收敛性。     

Galerkin方法的一种简单后处理过程

基于近似惯性流形思想,以流函数形式定常Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程为例,给出了一种简单的后处理Ｇａｌｅｒｋｉｎ方法。其主要思想是利用近似惯性流形概念和对真解的一种新的分解,构造高低频分量间的近似作用规律。文中证明了这种简单的后处理Ｇａｌｅｒｋｉｎ方法可以较小的代价获得较经典Ｇａｌｅｒｋｉｎ方法高得多的精度。     

Couette-Taylor流的谱Galerkin逼近

利用谱方法对轴对称的旋转圆柱间的Couette-Taulor流进行数值模拟．首先给出Navier-Stokes方程流函数形式,利用Couette流把边界条件齐次化．其次给出Stokes算子的特征函数的解析表达式,证明其正交性,并对特征值进行估计．最后利用Stokes算子的特征函数作为逼近子空间的基函数,给出谱Galerkin逼近方程的表达式．证明了Navier-Stokes方程非奇异解的谱Galerkin逼近的存在性、唯一性和收敛性,给出了解谱Galerkin逼近的误差估计,并展示了数值计算结果．     

三维区域中的Navier-Stokes方程的初边值问题

本文在初边值适当小的假设下，建立了任意三维区域中Navier－Stokes方程初边值问题整体强解的存在性定理．     

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．     

简单高阶对称破坏分歧点的研究

In this paper,the bifurcation problem g(x, λ) = 0 with symmetry con-dition is studied. An extended system is introduced for computing the simple high order symmetry-breaking bifurcation point. Numerical example is given. Key words Bifurcation problem, Symmetry-breaking bifurcation point, High order Singularities     

二维N－S方程的Fourier非线性Galerkin方法

本文对周期边界条件Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程，证明了其Ｆｏｕｒｉｅｒ非线性Ｇａｌｅｒｋｉｎ逼近解的存在唯一性，同时给出了逼近解的误差估计．     

Numerical Approximation of Transcritical Simple Bifurcation Point of the Navier-Stokes Equations

The extended system of nondegenerate simple bifurcation point of the Navier-Stokes equations is constructed in this paper, due to its derivative has a block lower triangular form, we design a Newton-like method, using the extended system and splitting iterative technique to compute transcritical nondegenerate simple bifurcation point, we not only reduces computational complexity, but also obtain quadratic convergence of algorithm.     

Uniform stability of spectral nonlinear Galerkin methods

This article provides a stability analysis for the backward Euler schemes of time discretization applied to the spatially discrete spectral standard and nonlinear Galerkin approximations of the nonstationary Navier‐Stokes equations with some appropriate assumption of the data (λ, u₀, f). If the backward Euler scheme with the semi‐implicit nonlinear terms is used, the spectral standard and nonlinear Galerkin methods are uniform stable under the time step constraint Δt ≤ (2/λλ₁). Moreover, if the backward Euler scheme with the explicit nonlinear terms is used, the spectral standard and nonlinear Galerkin methods are uniform stable under the time step constraints Δt = O(λ) and Δt = O(λ), respectively, where λ ≤ λ, which shows that the restriction on the time step of the spectral nonlinear Galerkin method is less than that of the spectral standard Galerkin method. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

一维Navier-Stokes方程组整体经典解

本文作者考虑了一维Navier-Stokes方程组Cauchy问题,得到了其经典解的整体光滑可解性。     

加罚Navier—Stokes方程的最佳非线性Galerkin算法

该文提出了求解二维加罚Navier-Stokes方程的最佳非线性Galerkin算法．这个算法在于在粗网格有限元空间上求解一非线性子问题，在细网格增量有限元空间Wh上求解一线性子问题．如果线性有限元被使用及，则该算法具有和有限元Galerkin算法同阶的收敛速度．然而该文提出的算法可以节省可观的计算时间．     

Navier-Stokes方程恰当弱解的存在性

本文假定外力f(x,t)∈Ｌ^1(0,T;L^2(Ω）,初始速度u0∈Ｈ,证明了三维Ｎavier-Stokes方程恰当弱解的存在性。     

Pointwise Error Estimate for Spectral Galerkin Approximations of Micropolar Equations

The goal of this article is to present pointwise time error estimates in suitable Hilbert spaces by considering spectral Galerkin approximations of the micropolar fluid model for strong solutions. In fact, we use the properties of the Stokes and Lamé operators for prove the pointwise convergence rate in the H²-norm for the ordinary velocity and microrotational velocity and the pointwise convergence rate in the L²-norm for the time-derivative of both velocities. The novelty of our method is that we do not impose any compatibility conditions in the initial data.     

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.     

发展型Navier-Stokes方程基于压力稳定化方法的谱逼近格式

本文采用压力稳定化方法近似模拟不可压缩条件,进而构造了发展型非周期NavierStokes方程的全离散Legendre谱逼近计算格式,严格分析了格式的广义稳定性与收敛性．本文建立的逼近结果也适用其它非周期问题．