k-循环矩阵的最佳SOR方法

本文研究 K-循环矩阵的SOR迭代,提出一种确定最佳松弛因子的方法，应用和改进了Young和Eidson等人的结果，同时给出了计算实例．     

Exact SOR convergence regions for a general class ofp-cyclic matrices

Linear systems whose associated block Jacobi iteration matrixB is weakly cyclic generated by the cyclic permutation = (₁,₂,..., _p ) in the spirit of Li and Varga are considered. Regions of convergence for the corresponding blockp-cyclic SOR method are derived and the exact convergence domains for real spectra, (B ^p ), of the same sign are obtained. Moreover, analytical expressions for two special cases forp = 5 are given and numerical results are presented confirming the theory developed. The tools used for this work are mainly from complex analysis and extensive use of (asteroidal) hypocycloids in the complex plane is made to produce our results.This work was supported in part by AFOSR grant F49620-92-J-0069 and NSF grant 9202536-CCR.     

A refined mixed finite-element method for the stationary Navier Stokes equations with mixed boundary conditions

Serge Nicaise ^{This paper is concerned with the mixed formulation of the Navier–Stokesequations with mixed boundary conditions in 2D polygonal domainsand its numerical approximation. We first describe the regularityof any solution. The problem is then approximated by a mixedfinite-element method where the strain tensor and the antisymmetricgradient tensor, quantities of practical importance, are introducedas new unknowns. An existence result for the finite-elementsolution and convergence results are proved near a nonsingularsolution. Quasi-optimal error estimates are finally presented.}     

具有完全并行度的SOR算法

1　引　　言Jacobi和 SOR迭代是求解线性方程组的两类基本的迭代方法 .并行计算机的出现使人们能立刻注意到它们在拥有并行处理性能上的显著差别 .Jacobi迭代因其各个分量的修正相互独立而具有十分明显的内在并行计算特性 .SOR则完全不同 ,其中诸分量的计算是逐个相关的 .由此而导致一般认为 SOR不适合并行处理 ,其内在并行性远不如 Jacobi迭代[1 ] [2 ] .由于 SOR多用于有限差分或有限元方法导致的大型稀疏方程组求解 ,因此 ,利用系数矩阵零元素或非零元素的特殊分布 ,采用红 -黑或多色排序成为实现 SOR并行处理的有效途径 .然而 ,…     

On preconditioned Uzawa methods and SOR methods for saddle-point problems

This paper studies convergence analysis of a preconditioned inexact Uzawa method for nondifferentiable saddle-point problems. The SOR-Newton method and the SOR-BFGS method are special cases of this method. We relax the Bramble-Pasciak-Vassilev condition on preconditioners for convergence of the inexact Uzawa method for linear saddle-point problems. The relaxed condition is used to determine the relaxation parameters in the SOR-Newton method and the SOR-BFGS method. Furthermore, we study global convergence of the multistep inexact Uzawa method for nondifferentiable saddle-point problems.     

A fully discrete stabilized finite-element method for the time-dependent Navier-Stokes problem

A fully discrete stabilized finite-element method is presentedfor the two-dimensional time-dependent Navier–Stokes problem.The spatial discretization is based on a finite-element spacepair (X_h, M_h) for the approximation of the velocity and thepressure, constructed by using the Q₁ – P₀ quadrilateralelement or the P₁ – P₀ triangular element; the time discretizationis based on the Euler semi-implicit scheme. It is shown thatthe proposed fully discrete stabilized finite-element methodresults in the optimal order error bounds for the velocity andthe pressure.     

Semiconvergence of block SOR method for singular linear systems with p-cyclic matrices

In this paper, we discuss semiconvergence of the block SOR method for solving singular linear systems with p-cyclic matrices. Some sufficient conditions for the semiconvergence of the block SOR method for solving a general p-cyclic singular system are proved.     

A posteriori error estimates for H 1-Galerkin mixed finite-element method for parabolic problems

The purpose of this article is to derive a posteriori error estimates for the H ¹-Galerkin mixed finite element method for parabolic problems. We study both semidiscrete and fully discrete a posteriori error analyses using standard energy argument. A fully discrete a posteriori error analysis based on the backward Euler method is analysed and upper bounds for the errors are derived. The estimators yield upper bounds for the errors which are global in space and time. Our analysis is based on residual approach and the estimators are free from edge residuals.     

On convergence of SOR methods for nonsmooth equations

We study the choice of relaxation parameters ω for convergence of the SOR Newton method and the SOR method for the system of equations F(x)=0 in a unified framework, where F is strongly monotone, locally Lipschitz continuous but not necessarily differentiable. Applications to non‐smooth Dirichlet problems are discussed. Copyright © 2001 John Wiley & Sons, Ltd.     

SOR方法的拓展与应用

对流体润滑的压力控制方程,在有限差分法的基础上,通过对SOR超松弛因子和迭代精度的选择,采用SOR逐次超松弛迭代法对控制方程进行了数值求解.在保证方程求解精度的基础上,还具有收敛快、稳定性好,计算工作量小等特点.     

A new type of full multigrid method for the elasticity eigenvalue problem

This paper introduces a new type of full multigrid method for the elasticity eigenvalue problem. The main idea is to avoid solving large scale elasticity eigenvalue problem directly by transforming the solution of the elasticity eigenvalue problem into a series of solutions of linear boundary value problems defined on a multilevel finite element space sequence and some small scale elasticity eigenvalue problems defined on the coarsest correction space. The involved linear boundary value problems will be solved by performing some multigrid iterations. Besides, some efficient techniques such as parallel computing and adaptive mesh refinement can also be absorbed in our algorithm. The efficiency and validity of the multigrid methods are verified by several numerical experiments.     

Conservative upwind finite-element method for a simplified Keller-Segel system modelling chemotaxis

^** Email: saito{at}infsup.jp Finite-element approximation for a non-linear parabolic–ellipticsystem is considered. The system describes the aggregation ofslime moulds resulting from their chemotactic features and iscalled a simplified Keller–Segel system. Applying an upwindtechnique, first we present a finite-element scheme that satisfiesboth positivity and mass conservation properties. Consequently,if the triangulation is of acute type, our finite-element approximationpreserves the L¹ norm, which is an important property of theoriginal system. Then, under some assumptions on the regularityof a solution and on the triangulation, we establish error estimatesin L^p x W^1, with a suitable p > d, where d is the dimensionof a spatial domain. Our scheme is well suited for practicalcomputations. Some numerical examples that validate our theoreticalresults are also presented.     

External finite-element approximations of eigenfunctions in the case of multiple eigenvalues

The paper deals with the finite-element analysis of second-order elliptic eigenvalue problems when the approximate domains Ω_h are not subdomains of the original domain . The considerations are restricted to piecewise linear approximations. Special attention is devoted to the convergence of approximate eigenfunctions in the case of multiple exact eigenvalues. As yet the approximate solutions have been compared with linear combinations of exact eigenfunctions with coefficients depending on the mesh parameter h. We avoid this disadvantage.     

Residual a posteriori error estimates for a 3D mortar finite-element method: the Stokes system

We consider a non-conforming domain decomposition techniquefor the discretization of the three-dimensional Stokes equationsby the mortar finite-element method. Relying on the velocity–pressureformulation of the system, we perform the numerical analysisof residual error indicators for this problem and we prove thatthe error estimators provide upper and lower bounds for theenergy norm of the mortar finite-element solution.     

Mixed methods with dynamic finite-element spaces for miscible displacement in porous media

Miscible displacement in porous media is modeled by a nonlinear coupled system of two partial differential equations. We approximate the pressure equation, which is elliptic, and the concentration equation, which is parabolic but normally convection-dominated, by the mixed methods with dynamic finite-element spaces, i.e., different number of elements and different basis functions are adopted at different time levels; and the approximate concentration is projected onto the next finite-element space in weighted L²-norm for starting a new time step. This allows us to make local grid refinements or unrefinements and basis function improvements. Two fully discrete schemes are presented and analysed. Error estimates show that these methods have optimal convergent rate in some sense. The efficiency and capability of the dynamic finite-element method are commented for accurately solving time-dependent problems with localized phenomena, such as fronts, shocks, and boundary layers.     

一类矩阵的AOR迭代收敛性分析及其与SOR迭代的比较

1 引言 许多实际问题最后常归结为解一个或一些矩阵的线性代数方程组Ax=b （1.1）这里讨论A为（1,1）相容次序矩阵的情形。     

A comparison of two algorithms for the assignment problem

State-of-the-art computational results have shown that the shortest augmenting path (SAP) methods are more efficient than other primal-dual and primal-simplex based methods for solving the linear assignment problem on uniprocessor computers. There is, however, some controversy concerning their merits when compared with Bertsekas' auction algorithm on multiprocessor computers. In this study we investigate the performance of these competing methods on the Alliant FX/8. For each method, theoretical motivation, sources of parallelism and computational results are presented.