一类非线性对流扩散方程两重网格特征有限元方法及误差估计

主要研究了一类非线性对流扩散方程的全离散特征有限元方法的两重网格算法及其误差估计.首先在网格步长为H的粗网格上计算一个较小的非线性问题,然后利用一阶牛顿迭代和粗网格解将网格步长为h的细网格上的非线性问题转化为线性问题求解.由于非线性问题的求解仅在粗网格上进行,该两重网格算法可以节省大量的计算工作量,同时具有较高的精度,证明了该两重网格算法L~2模先验误差估计结果为O(△t+h~2+H~(4-d/2)),其中d为空间维数.     

求解半线性抛物问题的两种二重网格算法

用线性方法对半线性抛物问题进行求解。方法依赖粗、细二重网格,针对粗解在细网格上的修正提出了两种算法,算法1是乘积倍的增长精度而算法2是平方倍的增长精度,而且重复算法1、2的最后几步可以任意阶地逼近细网格上的非线性解。数值算例验证了算法的可行性和有效性。     

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。     

非定常对流扩散方程的局部和并行有限元算法

对基于两重网格的非定常对流扩散方程的局部和并行有限元算法进行了研究.算法的理论依据是两重网格的思想,解的低频分量可以用一个整体的粗网格空间来逼近,高频分量可以用局部和并行的细网格空间来逼近.因此,这种局部和并行算法仅仅涉及一个粗网格上的整体逼近和细网格上的局部校正.得到了算法的误差估计,一些数值例子验证了算法的有效性.     

基于二重网格的定常Navier-Stokes方程的局部和并行有限元算法

对二维定常的不可压缩的Navier-Stokes方程的局部和并行算法进行了研究．给出的算法是多重网格和区域分解相结合的算法,它是基于两个有限元空间:粗网格上的函数空间和子区域的细网格上的函数空间．局部算法是在粗网格上求一个非线性问题,然后在细网格上求一个线性问题,并舍掉内部边界附近的误差相对较大的解．最后,基于局部算法,通过有重叠的区域分解而构造了并行算法,并且做了算法的误差分析,得到了比标准有限元方法更好的误差估计,也对算法做了数值试验,数值结果通过比较验证了本算法的高效性和合理性．     

二维非线性对流扩散方程特征有限元的两重网络算法

针对二维非线性对流扩散方程,构造了特征有限元两重网格算法．该算法只需要在粗网格上进行非线性迭代运算,而在所需要求解的细网格上进行一次线性运算即可．对于非线性对流占优扩散方程,不仅可以消除因对流占优项引起的数值振荡现象,还可以加快收敛速度、提高计算效率．误差估计表明只要选取粗细网格步长满足一定的关系式,就可以使两重网格解与有限元解保持同样的计算精度．算例显示:两重网格算法比特征有限元算法的收敛速度明显加快．     

非匹配网格上Stokes-Darcy 问题非协调元离散的两水平和多水平方法

本文讨论了非匹配网格上Stokes-Darcy 问题的两种低阶非协调元方法, 给出了误差估计, 对耦合的非协调元离散问题, 通过粗网格求得的界面条件, 我们提出了一个解耦的两水平算法. 并且我们将两水平方法推广到多水平情形, 其只需在一个很粗的网格上解一耦合问题, 然后在逐步加细的网格上求解解耦的问题, 理论分析和数值试验都说明方法的高效性.     

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．     

大Reynolds数Navier-Stokes方程带回溯技巧的两水平方法

本文考虑了一种求解大Reynolds数定常Navier-Stokes方程带回溯(backtracking)技巧的两水平有限元方法.其基本思想是,首先在一粗网格上求解带有亚格子模型稳定项的Navier-Stokes方程,然后在细网格上求解一个亚格子模型稳定化的线性Newton问题,最后又回到粗网格上求解线性化的校正问题.通过适当的稳定化参数和粗细网格尺寸的选取,本文的算法能取得最优渐近收敛阶.数值实验检验了理论分析的正确性和算法的有效性.     

二阶椭圆问题带单位分解技巧的两重网格方法

标准的两重网格方法是一种求解二阶椭圆问题的局部并行方法,其计算所得数值解在整个求解区域上并不连续使用单位分解技术,将各个子区域上的局部解粘合在一起,从而得到全局连续解,并证明此解在H1范数意义下最优.更进一步,可以证明通过在粗网格上修正,能够改善其L2误差.数值例子验证了理论的正确性.     

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

In this article we consider a two-level finite element Galerkin method using mixed finite elements for the two-dimensional nonstationary incompressible Navier-Stokes equations. The method yields a $H^1$-optimal velocity approximation and a $L_2$-optimal pressure approximation. The two-level finite element Galerkin method involves solving one small, nonlinear Navier-Stokes problem on the coarse mesh with mesh size $H$, one linear Stokes problem on the fine mesh with mesh size $h << H$. The algorithm we study produces an approximate solution with the optimal, asymptotic in $h$, accuracy.     

解Stokes特征值问题的一种两水平稳定化有限元方法

基于局部Gauss积分,研究了解Stokes特征值问题的一种两水平稳定化有限元方法．该方法涉及在网格步长为H的粗网格上解一个Stokes特征值问题,在网格步长为h=O(H2)的细网格上解一个Stokes问题．这样使其能够仍旧保持最优的逼近精度,求得的解和一般的稳定化有限元解具有相同的收敛阶,即直接在网格步长为h的细网格上解一个Stokes特征值问题．因此,该方法能够节省大量的计算时间．数值试验验证了理论结果．     

求解定常不可压Stokes方程的两层罚函数方法

借助于两套有限元网格空间提出了一种求解定常不可压Stokes方程的两层罚函数方法.该方法只需要求解粗网格空间上的Stokes方程和细网格空间上的两个易于求解的罚参数方程（离散后的线性方程组具有相同的对称正定系数矩阵）.收敛性分析表明粗网格空间相对于细网格空间可以选择很小，并且罚参数的选取只与粗网格步长和问题的正则性有关.因此罚参数不必选择很小仍能够得到最优解.最后通过数值算例验证了上述理论结果，并且数值对比可知两层罚函数方法对于求解定常不可压Stokes方程具有很好的效果.     

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.     

Numerical approximation of the Smagorinsky model by a two-level method

A two-level method for discretizing the Smagorinsky model for the numerical simulation of turbulent flows is proposed. In the two-level algorithm, the solution to the fully nonlinear coarse mesh problem is utilized in a single-step linear fine mesh problem. When modeling parameters are chosen appropriately, the error in the two-level algorithm is comparable to the error in solving the fully nonlinear problem on the fine mesh. We provide an a priori error estimate for the two-level method, which yields appropriate scalings between the coarse and fine mesh-sizes (H and h, respectively), and the radius of the spatial filter used in the Smagorinsky model (δ). In addition, we provide an algorithm in which a coarse mesh correction is performed to further enhance the accuracy. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Two-level stabilized nonconforming finite element method for the Stokes equations

In this article, we present a new two-level stabilized nonconforming finite elements method for the two dimensional Stokes problem. This method is based on a local Gauss integration technique and the mixed nonconforming finite element of the NCP ₁ — P ₁ pair (nonconforming linear element for the velocity, conforming linear element for the pressure). The two-level stabilized finite element method involves solving a small stabilized Stokes problem on a coarse mesh with mesh size H and a large stabilized Stokes problem on a fine mesh size h = H/3. Numerical results are presented to show the convergence performance of this combined algorithm.     

A two-level finite element method for the Navier–Stokes equations based on a new projection

We consider a fully discrete two-level approximation for the time-dependent Navier–Stokes equations in two dimension based on a time-dependent projection. By defining this new projection, the iteration between large and small eddy components can be reflected by its associated space splitting. Hence, we can get a weakly coupled system of large and small eddy components. This two-level method applies the finite element method in space and Crank–Nicolson scheme in time. Moreover,the analysis and some numerical examples are shown that the proposed two-level scheme can reach the same accuracy as the classical one-level Crank–Nicolson method with a very fine mesh size h by choosing a proper coarse mesh size H. However, the two-level method will involve much less work.     

Two-level multiscale finite element methods for the steady navier-stokes problem

In this article, on the basis of two-level discretizations and multiscale finite element method, two kinds of finite element algorithms for steady Navier-Stokes problem are presented and discussed. The main technique is first to use a standard finite element discretization on a coarse mesh to approximate low frequencies, then to apply the simple and Newton scheme to linearize discretizations on a fine grid. At this process, multiscale finite element method as a stabilized method deals with the lowest equal-order finite element pairs not satisfying the inf-sup condition. Under the uniqueness condition, error analyses for both algorithms are given. Numerical results are reported to demonstrate the effectiveness of the simple and Newton scheme.     

TWO-LEVEL MULTISCALE FINITE ELEMENT METHODS FOR THE STEADY NAVIER-STOKES PROBLEM

In this article, on the basis of two-level discretizations and multiscale finite element method, two kinds of finite element algorithms for steady Navier-Stokes problem are presented and discussed. The main technique is first to use a standard finite element discretization on a coarse mesh to approximate low frequencies, then to apply the simple and Newton scheme to linearize discretizations on a fine grid. At this process, multiscale finite element method as a stabilized method deals with the lowest equal-order finite element pairs not satisfying the inf-sup condition. Under the uniqueness condition, error analyses for both algorithms are given. Numerical results are reported to demonstrate the effectiveness of the simple and Newton scheme.