用自适应MG法,FMG法求解Helmholtz方程

本文针对Ｖ循环、Ｗ循环和多重网格法中最优光滑次数及循环体个数难以确定的缺点，以Ｈｅｌｍｈｏｌｔｚ方程为例给出自适应的多重网格算法和自适应的完全多重网格算法。     

等几何分析的多重网格共轭梯度法

提高NURBS基函数阶数可以提高等几何分析的精度,同时也会降低多重网格迭代收敛速度.将共轭梯度法与多重网格方法相结合,提出了一种提高收敛速度的方法,该方法用共轭梯度法作为基础迭代算法,用多重网格进行预处理.对Poisson(泊松)方程分别用多重网格方法和多重网格共轭梯度法进行了求解,计算结果表明:等几何分析中采用高阶NURBS基函数处理三维问题时,多重网格共轭梯度法比多重网格法的收敛速度更快.     

MORTAR型旋转Q_1元的多重网格方法

本文讨论了mortar型旋转Q_1元的多重网格方法.证明了W循环的多重网格法是最优的,即收敛率与网格尺寸及层数无关.同时给出了一种可变的V循环多重网格算法,得到了一个条件数一致有界的预条件子.最后,数值试验验证了我们的理论结果.     

无限元多重网格算法

在求偏微分方程数值解时,往往需要解一个规模很大的代数方程组,而多重网格是一种十分有效的迭代方法.大量数值试验证明,它具有很高的收敛速度.理论分析表明,这种迭代法的收敛速度并不随网格的加密而降低,这一突出优点是其它迭代方法望尘莫及的. 在使用有限元多重网格算法时,如果区域边界的角点使解具有奇性,理论分析会遇到     

三维泊松方程的高精度多重网格解法

利用对称网格点泰勒展开式中各阶导数项明显的对称性,得到了数值求解三维泊松方程的四阶和六阶精度的紧致差分格式,其推导过程简便直接.为了克服传统迭代法在求解高维问题时计算量大、收敛速度慢的缺陷,采用了多重网格加速技术,设计了相应的多重网格算法,求解了三维泊松方程的Dirichlet边值问题.数值实验结果表明,本文所提出的高精度紧致格式达到了期望的精度并且多重网格方法的加速效果是非常显著的.     

中子输运方程的多重网格局部Fourier分析

1 引言 多重网格作为求解椭圆偏微分方程的快速有效方法而倍受欢迎.多重网格方法有两大要素:一是光滑,二是粗网格校正.     

有限体积法与多重网格法用于提高激波捕捉质量及加速收敛

多重网格技术是一种非常有效的数值计算方法，本文采用多重网格的ＦＡＳ格式进行数值实验，计算加速效果十分明显，同时，结合矢通量分裂用有限体积法，大大提高了主激波的质量。     

以对称Kaczmarz迭代为光滑化的多重网格法的收敛性

§1 引言 近年来,已证实多重网格法对于一大类微分方程系统,特别是椭圆型微分方程的数值求解是一种非常有效的方法。 多重网格法主要由光滑化、粗网格修正过程组成。为保证整个方法的有效性,要求光滑化和粗网格修正过程分别能有效地降低误差的高、低频分量。对于对称正定系统,只要粗     

求解Oseen流的交替线松弛多重网格方法

利用Riemann解的通量差分分裂法——Godunov方法对Oseen流控制方程进行离散,得到了基于一阶上迎风格式的离散方程,并给出了使用多重网格方法求解该离散方程的V-循环算法和W-循环算法的收敛性分析.通过局部Fourier分析方法,对获得的离散方程的聚对称交替线GaussSeidel松弛的光滑性质进行了研究.结果表明:使用多重网格的两层网格及三层网格算法求解具有不同Reynolds数的Oseen流,即便是在高Reynolds数情况下,聚对称交替线Gauss-Seidel松弛具有很好的光滑性质,多重网格W-循环算法收敛性比V-循环算法好.     

经济的瀑布型多重网格法(ECMG)

提出一种新的经济的瀑布型多重网格法(ECMG), 和通常的瀑布型多重网格法(CMG)的工作量相比, 新的瀑布型多重网格法在每层上的工作量 都相应的减少, 尤其是粗网格上的工作量将大量的减少. 新格式的误差和通常的 瀑布型多重网格法一样, 都具有最优精度. 最后给出数值算例 来验证所得理论的结果.     

CASCADIC MULTIGRID METHODS FOR MORTAR WILSON FINITE ELEMENT METHODS ON PLANAR LINEAR ELASTICITY

Cascadic multigrid technique for mortar Wilson finite element method of homogeneous boundary value planar linear elasticity is described and analyzed. First the mortar Wilson finite element method for planar linear elasticity will be analyzed, and the error estimate under L2 and H1 norm is optimal. Then a cascadic multigrid method for the mortar finite element discrete problem is described. Suitable grid transfer operator and smoother are developed which lead to an optimal cascadic multigrid method. Finally, the computational results are presented.     

New intergrid transfer operator in multigrid method for P1-nonconforming finite element method

For a second-order elliptic boundary value problem, We develop an intergrid transfer operator in multigrid method for the P₁-nonconforming finite element method. This intergrid transfer operator needs smaller computation than previous intergrid transfer operators. Multigrid method with this operator converges well.     

The Multigrid Method of Nonconforming Finite Elements for Solving the Biharmonic Equation

1.IntroductionSeveralaspectsofthenonconfOrmingfi11iteelementmethodhavebeendiscussedin[1-51.Inthispaper,wewillintroducethemllltigridmethodandprovethatthemultigridmethodofnonconformingfiniteelementscanattainthesameoptimalconvergenceorderaJsthenonconformingflniteelementmethodinenergy-norm.Themultigridmethodofconformingfiniteelementshasbeen.t.died[7-8].Forthemultigridmethodofnonconformingfiniteele1nents,becausethefiniteelementspacesassociatedwiththenetsarenotnest(Vk-1ctVK),itisdifficulttodefinet…     

PRECONDITIONING MIXED ELEMENT METHOD FOR NONSELFADJOINT AND INDEFINITE SECOND ORDER ELLIPTIC PROBLEMS*

In this paper, an equivalence between mixed element method and nonconforming element method for nonselfad joint and indefinite second order elliptic problems is established without using any bubble functions. It is proved that the H~1-condition number of preconditioned operator B_h~(-1)A_h is uniformly bounded and its B_h-singular values cluster in a positive finite interval, where A_h is the equivalent nonconforming element discretization of nonselfad joint and indefinite second order elliptic operator A, B_h is usual noncon forming element discretization of selfadjoint and positive definite second order elliptic operator B. Finally a simple V-cycle multigrid implementation of B_h~(-1) is given.     

双参数任意四边形非协调板元的构造及收敛性分析

本文利用双参数有限元方法[1]构造出十二参和十三参任意四边形板元，并对其收敛性进行分析证明．     

A new semi-smooth Newton multigrid method for control-constrained semi-linear elliptic PDE problems

In this paper a new multigrid algorithm is proposed to accelerate the convergence of the semi-smooth Newton method that is applied to the first order necessary optimality systems arising from a class of semi-linear control-constrained elliptic optimal control problems. Under admissible assumptions on the nonlinearity, the discretized Jacobian matrix is proved to have an uniformly bounded inverse with respect to mesh size. Different from current available approaches, a new numerical implementation that leads to a robust multigrid solver is employed to coarsen the grid operator. Numerical simulations are provided to illustrate the efficiency of the proposed method, which shows to be computationally more efficient than the full-approximation-storage multigrid in current literature. In particular, our proposed approach achieves a mesh-independent convergence and its performance is highly robust with respect to the regularization parameter.     

Optimal Multigrid Methods with New Transfer Operators Based on Finite Difference Approximations

We constructed new interpolation operator in multigrid methods, which is efficient to transfer residual error from coarse grid to fine grid. This operator used idea of solving local residual equation using the standard stencil and the skewed stencil of the centered difference approximation to the Laplacian operator. We also compared our new multigrid methods with traditional multigrid methods, and found that new method is optimal.     

九参数双参数元与广义协调元的对称列式

1.引言 最近石钟慈和陈绍春使用双参数法对九参数拟协调元和九参数广义协调元进行了研究。研究结果表明,基于函数空间     

Two‐level Fourier analysis of multigrid for higher‐order finite‐element discretizations of the Laplacian

In this paper, we employ local Fourier analysis (LFA) to analyze the convergence properties of multigrid methods for higher‐order finite‐element approximations to the Laplacian problem. We find that the classical LFA smoothing factor, where the coarse‐grid correction is assumed to be an ideal operator that annihilates the low‐frequency error components and leaves the high‐frequency components unchanged, fails to accurately predict the observed multigrid performance and, consequently, cannot be a reliable analysis tool to give good performance estimates of the two‐grid convergence factor. While two‐grid LFA still offers a reliable prediction, it leads to more complex symbols that are cumbersome to use to optimize parameters of the relaxation scheme, as is often needed for complex problems. For the purposes of this analytical optimization as well as to have simple predictive analysis, we propose a modification that is “between” two‐grid LFA and smoothing analysis, which yields reasonable predictions to help choose correct damping parameters for relaxation. This exploration may help us better understand multigrid performance for higher‐order finite element discretizations, including for Q₂‐Q₁ (Taylor‐Hood) elements for the Stokes equations. Finally, we present two‐grid and multigrid experiments, where the corrected parameter choice is shown to yield significant improvements in the resulting two‐grid and multigrid convergence factors.     

一个新非协调单元对扩散对流反应方程的应用

利用最近提出的一个新型非协调双参数单元,将流线扩散有限元方法成功地应用于对流占优的扩散对流反应方程,并且得到流线扩散模意义下的误差估计结果.