自适应多重网格法的应用

本文运用自应并行多重网格法求解了轴向大扰动,径向小扰动的跨音速方程。其计算结果表明该方法能够大大提高计算效率。     

求解线性互补问题的乘性Schwarz算法的收敛速度估计

In this paper, we consider multiplicative Schwarz algorithm for solving linear complementarity problems. Monotone convergence is obtained. under suitable conditions, we get the convergence independent of mesh size h. We also prove the finite termination property of the algorithm for the active constraints in noridegenerate case.     

CONVERGENCE RATE OF A GENERALIZED ADDITIVE SCHWARZ ALGORITHM

The convergence rate of a generalized additive Schwarz algorithm for solving boundaryvalue problems of elliptic partial differential equations is studied.A quantitative analysisof the convergence rate is given for the model Dirichlet problem.It will be shown that agreater acceleration of the algorithm can be obtained by choosing the parameter suitably.Some numerical tests are also presented in this paper.     

一类非线性椭圆问题的schwarz算法

其中考虑下述泛函最小问题:求u∈W01,α(Ω),使F(u)=     

障碍问题有限元逼近的并行Schwarz算法

1．引言设o为R‘中有界多角形区域，V为Sobolev空INH‘（fi）的子空间，叶，·）为VxV上连续强制对称双线性型，，f∈V．为简单计，设V中元素满足齐次边界条件．考虑变分不等式：求u∈K，使其中或者且w，4eH‘mnC‘m．设V‘cHI（m为V的有限元逼近，其结点参数集包含结点上的函数值．考虑（1），（2）或者（l），（3）的有限元逼近问题：求。。EK。，使aa…a；v－…三周v－uh），Vve仇；（4其中a。（·，·）为VxV上连续强制对称双线性型，fhe（V‘）”，而Kh＝（hE叶：hZ一于所有结点｝，（5或者As一扣E＊‘：。三V三中于所有结…     

A comparison of some domain decomposition and ILU preconditioned iterative methods for nonsymmetric elliptic problems

In recent years, competitive domain-decomposed preconditioned iterative techniques of Krylov-Schwarz type have been developed for nonsymmetric linear elliptic systems. Such systems arise when convection-diffusion-reaction problems from computational fluid dynamics or heat and mass transfer are linearized for iterative solution. Through domain decomposition, a large problem is divided into many smaller problems whose requirements for coordination can be controlled to allow effective solution on parallel machines. A central question is how to choose these small problems and how to arrange the order of their solution. Different specifications of decomposition and solution order lead to a plethora of algorithms possessing complementary advantages and disadvantages. In this report we compare several methods, including the additive Schwarz algorithm, the classical multiplicative Schwarz algorithm, an accelerated multiplicative Schwarz algorithm, the tile algorithm, the CGK algorithm, the CSPD algorithm, and also the popular global ILU-family of preconditioners, on some nonsymmetric or indefinite elliptic model problems discretized by finite difference methods. The preconditioned problems are solved by the unrestarted GMRES method. A version of the accelerated multiplicative Schwarz method is a consistently good performer.     

A hybrid domain decomposition method based on aggregation

A new two‐level black‐box preconditioner based on the hybrid domain decomposition technique is proposed and studied. The preconditioner is a combination of an additive Schwarz preconditioner and a special smoother. The smoother removes dependence of the condition number on the number of subdomains and variations of the diffusion coefficient and leaves minor sensitivity to the problem size. The algorithm is parallel and pure algebraic which makes it a convenient framework for the construction parallel black‐box preconditioners on unstructured meshes. Copyright © 2004 John Wiley & Sons, Ltd.     

一种新的局部时间积分的区域分解波形松弛算法

提出一种新的区域分解波形松弛算法, 使得可以在不同的子域采用不同的时间步长来并行求解线性抛物方程的初边值问题. 与传统的区域分解波形松弛算法相比, 该算法可以通过预条件子来加快收敛速度, 并且对内存的需求大大降低. 给出了局部时间步长一种具体的实现方法, 证明了离散解的存在唯一性, 并在时间连续水平分析了预条件系统. 数值实验显示了新算法的有效性.     

Semi-discrete Schwarz waveform relaxation algorithms for reaction diffusion equations

For time dependent problems, the Schwarz waveform relaxation (SWR) algorithm can be analyzed both at the continuous and semi-discrete level. For consistent space discretizations, one would naturally expect that the semi-discrete algorithm performs as predicted by the continuous analysis. We show in this paper for the reaction diffusion equation that this is not always the case. We consider two space discretization methods—the 2nd-order central finite difference method and the 4th-order compact finite difference method, and for each method we show that the semi-discrete SWR algorithm with Dirichlet transmission condition performs as predicted by the continuous analysis. However, for Robin transmission condition the semi-discrete SWR algorithm performs worse than predicted by the continuous analysis. For each type of transmission conditions, we show that the convergence factors of the semi-discrete SWR algorithm using the two space discretization methods are (almost) equal. Numerical results are presented to validate our conclusions.     

热传导方程的一类新型重叠型并行区域分解有限差分算法

0引言随着大规模科学工程计算的发展和计算精度要求的提高,区域分解和并行计算的发展越来越受到人们的重视.区域分解方法把复杂或大型的问题分解成若干重叠或非重叠子区域上的子问题,再在子区域上利用各种算法求解子问题.借助于区域分解,各子区域之间的计算可以并行,这引起了人们的研究兴趣和极大的应用前景.重叠型区域分解法的原始思想来源于Schwarz交替法.近年来建立在Schwarz交替法基础上的区域分解法在理论分析和实际应用中取得令人注目的发展,已成为一种有效的迭代方法.经典的Schwarz交替法本质上是串行的.随着并行计算的发展,出现了多种可完全并行化的Schwarz算法     

MONOTONIC ITERATIVE ALGORITHMS FOR A QUASICOMPLEMENTARITY PROBLEM

1. IntroductionConsider the following QCP: find u E Re such thatwhere A, B: R" -+ Re are operators, f E R". The quasivariational inequality which is equivalent to (1) sounds: to find u E Re such that u 2 Ba andIf Ba = c E R" for ally v E R" then QCP(1) reduces into CP. (1) appears in mathematicalprogramming (see, for example, [2] and the references therein), also comes from the discretization of QCP in mathematical physics and control theory (see [1]). For various generalization,see…     

Parallel Smoothed Aggregation Multilevel Schwarz Preconditioned Newton-Krylov Algorithms for Poisson-Boltzmann Problems

We study a multilevel Schwarz preconditioned Newton-Krylov algorithm to solve the Poisson-Boltzmann equation with applications in multi-particle colloidal simulation. The smoothed aggregation-type coarse mesh space is introduced in collaboration with the one-level Schwarz method as a composite preconditioner for accelerating the convergence of a Krylov subspace method for solving the Jacobian system at each Newton step. The important feature of the proposed solution algorithm is that the geometric mesh information needed for constructing the multilevel preconditioner is the same as the one-level Schwarz method on the fine mesh. Other components, such as the definition of the coarse mesh, all the mesh transfer operators, and the coarse mesh problem, are taken care of by the Trillinos/ML packages of the Sandia National Laboratories in the United States. After algorithmic parameter tuning, we show that the proposed smoothed aggregation multilevel Newton-Krylov-Schwarz (NKS) algorithm numerically outperforms than smoothed aggregation multigrid method and one-level version of the NKS algorithm with satisfactory parallel performances up to a few thousand cores. Besides, we investigate how the electrostatic forces between particles for the separation distance depend on the radius of spherical colloidal particles and valence ratios of cation and anion in a cubic system.     

带间断系数椭圆问题的P1非协调四边形元的加性 Schwarz 方法

本文讨论了带间断系数的二阶椭网问题的P1非协调四边形元的加性Schwarz方法.通过分析加性Schwaxz预处理后系统的特征值分布,我们证明了除少数小特征值外,其余所有特征值都有正的关于间断系数和网格尺寸拟一致的上下界.数值试验验证了我们的结论.     

Overlapping Schwarz Waveform Relaxation for the Heat Equation in N Dimensions

We analyze overlapping Schwarz waveform relaxation for the heat equation in n spatial dimensions. We prove linear convergence of the algorithm on unbounded time intervals and superlinear convergence on bounded time intervals. In both cases the convergence rates are shown to depend on the size of the overlap. The linear convergence result depends also on the number of subdomains because it is limited by the classical steady state result of overlapping Schwarz for elliptic problems. However the superlinear convergence result is independent of the number of subdomains. Thus overlapping Schwarz waveform relaxation does not need a coarse space for robust convergence independent of the number of subdomains, if the algorithm is in the superlinear convergence regime. Numerical experiments confirm our analysis. We also briefly describe how our results can be extended to more general parabolic problems.     

The Lions domain decomposition algorithm on non-matching cell-centred finite volume meshes

A finite volume scheme for convection diffusion equations onnon-matching grids is presented. Sharp error estimates for H²solutions of the continuous problem are obtained. A finite volumeversion of an adaptation of the Schwarz algorithm due to P.L. Lions is then studied. For a fixed mesh, its convergencetowards the finite volume scheme on the whole domain is proven.Numerical experiments are performed to illustrate the theoreticalrate of convergence of the finite volume sequences of solutionsas the mesh is refined, together with the speed of convergenceof the Schwarz algorithm.     

一类非线性单调型方程的区域分裂法

本文考虑了一类非线性单调问题的加性Schwgrz交替法和异步平行算法,并得到了在能量模意义下的收敛性结果,最后还讨论了格式的有限元离散。     

Additive Schwarz algorithm for the nonlinear complementarity problem with M-function

In this paper, an additive Schwarz algorithm is considered for solving the finite-dimensional nonlinear complementarity problem with M-function. The monotone convergence of the algorithm is obtained with special choices of initial values. Moreover, the weighted max-norm bound is obtained for the iterative errors.     

Schwarz 算法的 Lions 框架与异步并行算法的收敛性证明

经典的Schwarz算法,早在1870年就提出了,是求解不规则椭圆型方程的交替法.本世纪苏联学者等又在变分框架下论证了收敛性.近年来以 Schwarrz算法为基础的区域分解算法,发展十分活跃,当前该方法正与并行算法,预处理,快速直接解,多水平及多层网格技术结合,成为计算数学领域内最有前途一个分支.虽然表面看 Schwarz 算法不是并行的,但康立山等打破了分解区域为两子域的     

A new coarse space for overlapping Schwarz algorithms for H(curl) problems in three dimensions with irregular subdomains

Numerical Algorithms - A new coarse space for a two-level overlapping Schwarz algorithm is presented for problems posed in three dimensions in the space H(curl, Ω). Previous studies for these...     

Schwarz methods for inequalities with contraction operators

We prove the convergence of some multiplicative and additive Schwarz methods for inequalities which contain contraction operators. The problem is stated in a reflexive Banach space and it generalizes the well-known fixed-point problem in the Hilbert spaces. Error estimation theorems are given for three multiplicative algorithms and two additive algorithms. We show that these algorithms are in fact Schwarz methods if the subspaces are associated with a decomposition of the domain. Also, for the one- and two-level methods in the finite element spaces, we write the convergence rates as functions of the overlapping and mesh parameters. They are similar with the convergence rates of these methods for linear problems. Besides the direct use of the five algorithms for the inequalities with contraction operators, we can use the above results to obtain the convergence rate of the Schwarz method for other types of inequalities or nonlinear equations. In this way, we prove the convergence and estimate the error of the one- and two-level Schwarz methods for some inequalities in Hilbert spaces which are not of the variational type, and also, for the Navier–Stokes problem. Finally, we give conditions of existence and uniqueness of the solution for all problems we consider. We point out that these conditions and the convergence conditions of the proposed algorithms are of the same type.