数学物理方程离散特征值问题的几何网格因式分解算法

ON FACTORIZATION ALGORITHM WITH GEOMETRIC PREPROCESSING FOR DISCRETE EIGEN-PROBLEMS IN MATHEMATICAL-PHYSICS

本文提出求解数学物理方程大型离散特征值问题的几何网格预变换块因式分解算法(简称GPA算法)通过长期研究我们发现:结构化网格矩阵G满足幂等方程G^(m)=I_(N),(m<N=dim(G)),故可在实数域或复数范围内进行因式分解;且G与有限元刚度矩阵A之间乘法存在互易性:A·G=G·A,利用G的几何不变性可把N阶大型矩阵A正交分解为m一块对角块矩阵异步并行是我们算法的计算数学基础。本文以正三角形、方形、平行六边形及正十七边形等结构化网格为例,特别是详细分析了六边形上的离散特征值异步并行算法及程序实现细节.文后附有若干2-3万阶量级离散矩阵特征值的桌面电脑数值计算例子(正三角形与方形网格,串行加速比分别为3-4倍),符合本文算法分析得出的“几何网格预处理的并行度与正多边形边数成正比”的结论.这类几何网格因式分解算法原则上可推广到三维乃至高维数学物理方程离散特征值计算问题,也可用于大型线性方程组的高效并行求解.

A geometric asynchronous parallel algorithm for solving large-scale discrete mathematicalphysical system with structured mesh is presented.By using the intrinsic geometry symmetry,geometric matrix G and the stiff matrix A satisfies reciprocal operator A·G=G.A,which G satisfies G^(m)=I,large scale system solvers can be replaced to block-solver as a.pretreatment.In this paper,we restrict ourselves to arbitrarily regular polygon mesh,such as triangle,square,hexagon,as well as heptadecagon.Mathematical analysis and numerical tests are included.The pretreatment algorithm has been extended to 3-D structured grid,such as cube,dodecahedron,icosahedron.