Multilevel preconditioners for a quadrature Galerkin solution of a biharmonic problem

Efficient multilevel preconditioners are developed and analyzed for the quadrature finite element Galerkin approximation of the biharmonic Dirichlet problem. The quadrature scheme is formulated using the Bogner–Fox–Schmit rectangular element and the product two‐point Gaussian quadrature. The proposed additive and multiplicative preconditioners are uniformly spectrally equivalent to the operator of the quadrature scheme. The preconditioners are implemented by optimal algorithms, and they are used to accelerate convergence of the preconditioned conjugate gradient method. Numerical results are presented demonstrating efficiency of the preconditioners. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006     

An improved progressive preconditioning method for steady non‐cavitating and sheet‐cavitating flows

An improved progressive preconditioning method for analyzing steady inviscid and laminar flows around fully wetted and sheet‐cavitating hydrofoils is presented. The preconditioning matrix is adapted automatically from the pressure and/or velocity flow‐field by a power‐law relation. The cavitating calculations are based on a single fluid approach. In this approach, the liquid/vapour mixture is treated as a homogeneous fluid whose density is controlled by a barotropic state law. This physical model is integrated with a numerical resolution derived from the cell‐centered Jameson's finite volume algorithm. The stabilization is achieved via the second‐and fourth‐order artificial dissipation scheme. Explicit four‐step Runge–Kutta time integration is applied to achieve the steady‐state condition. Results presented in the paper focus on the pressure distribution on hydrofoils wall, velocity profiles, lift and drag forces, length of sheet cavitation, and effect of the power‐law preconditioning method on convergence speed. The results show satisfactory agreement with numerical and experimental works of others. The scheme has a progressive effect on the convergence speed. The results indicate that using the power‐law preconditioner improves the convergence rate, significantly. Copyright © 2010 John Wiley & Sons, Ltd.     

基于多重网格分层四边形单元有限元空间的预处理方法

用有限元方法解二阶问题时,常选用三角单元和四边形单元.对于基于三角形单元的有限元方程,采用多重网格分层函数的预处理方法,Yserentant,H.已作了详细的讨论.本文对基于四边形单元的有限元方程,采用类似于Yserentant,H.的多重网格分层函数的预处理方法,使得系数矩阵的条件数由原来的O((1/h)~2)缩小为O((log(1/h)~3).     

限制加性许瓦兹预条件的变形及其在二维三温能量方程中的应用

给出标准限制加性许瓦兹预条件的变形,并应用当前流行的Newton-Krylov-Schwarz方法,结合该预条件子,求解由二维三温能量方程离散得到的非线性代数方程组,减少收敛所需要的迭代次数和所需的CPU时间.数值实验表明,该方法比标准限制加性许瓦兹预条件方法收敛所需要的迭代次数和CPU时间要少.     

An overlapping additive Schwarz preconditioner for interpolation on the unit sphere with spherical radial basis functions

The problem of interpolation of scattered data on the unit sphere has many applications in geodesy and Earth science in which the sphere is taken as a model for the Earth. Spherical radial basis functions provide a convenient tool for constructing the interpolant. However, the underlying linear systems tend to be ill-conditioned. In this paper, we present an additive Schwarz preconditioner for accelerating the solution process. An estimate for the condition number of the preconditioned system will be discussed. Numerical experiments using MAGSAT satellite data will be presented.     

Solution of large‐scale weighted least‐squares problems

A sequence of least‐squares problems of the form min_y∥G^1/2(A^T y?h)∥₂, where G is an n×n positive‐definite diagonal weight matrix, and A an m×n (m?n) sparse matrix with some dense columns; has many applications in linear programming, electrical networks, elliptic boundary value problems, and structural analysis. We suggest low‐rank correction preconditioners for such problems, and a mixed solver (a combination of a direct solver and an iterative solver). The numerical results show that our technique for selecting the low‐rank correction matrix is very effective. Copyright © 2002 John Wiley & Sons, Ltd.     

Flexible incomplete Cholesky factorization with multi‐parameters to control the number of nonzero elements in preconditioners

An incomplete Cholesky (IC) factorization with multi‐parameters is presented. The marked virtue of the proposed IC factorization algorithm is to dynamically control the number of nonzero elements in each column of the IC factorization preconditioner L with the help of these involved parameters. Parameter setting strategies are also given. Numerical results show that the total computing time for both computation of the preconditioner L and iterative solution is evidently reduced for almost all test matrices. In general, these parameters can obviously enhance the effectiveness and performance of the IC factorization. Copyright © 2011 John Wiley & Sons, Ltd.     

一类求解鞍点问题的广义不精确Uzawa方法

本文提出了一类求解大型稀疏鞍点问题的新的广义不精确Uzawa算法.该方法不仅可以包含 前人的方法, 而且可以拓展出很多新方法. 理论分析给出该方法收敛的条件, 并详细的分析了其收敛性质和参数矩阵的选取方法. 通过对有限元离散的Stokes问题的数值实验表明, 新方法是行之有效的, 其收敛速度明显优于原来的算法.     

On the optimal convergence factor of the accelerated parameterized inexact Uzawa method with three parameters for augmented systems

Under the assumption that all eigenvalues of the preconditioned Schur complement are real, we present an analytical proof for obtaining the optimal convergence factor of the real accelerated parameterized inexact Uzawa (APIU) method when P=A. It is proved that the optimal convergence factor is the same as that of the generalized successive overrelaxation method, which was published at the same time, and that it can be attained only at the unique optimum point of parameters, regardless of whether m>n or m=n. In addition, we generalize the APIU method and analyze the relationship between the APIU method and 10 additional Uzawa‐like methods.