两点边值问题网格方程病态机理和预处理

Ill-condition mechanism and preconditioning principle of the grid equation of two-point boundary value problems

研究两点边值问题的不同网格方程的病态机理和预处理原理.基于结构分析的思想,通过定义并研究这些网格方程的病态结构、病态因子、去病因子,说明不同的网格方程有类似的病态结构、有相同的病态因子和与之对应的去病因子;将去病因子作为预条件子的重要组成部分,并对预处理的结果进行定量分析.结果表明,该预条件子的使用,几乎不增加迭代的计算量,预处理后的条件数接近1;去病因子是通用且最优的预条件子.

The ill conditioned mechanism and preconditioning principle of different grid equations for two-point boundary value problems were studied.Based on the idea of structural analysis,by defining and studying the ill-conditioned structure,ill-conditioned factors and cure factor of these grid equations,it is shown that different grid equations have similar ill-conditioned structures,the same ill-conditioned factors and the corresponding cure factor.The cure factor are regarded as the important components of the preconditioners.Quantitative analysis of the preconditioning results is made,it shows that the use of the preconditioner almost does not increase the amount of iterative calculation,and the number of conditions after preconditioning is close to 1,and the cure factor is a universal and optimal preconditioner.