边值问题离散系统数值稳定性的新度量

对于用有限元法或差分法求解微分方程边值问题,流行着这样的观点:当网格剖分出现小角度的三角形或窄长的矩形时,离散系统的数值稳定性就差.这种观点的根据是由于把系数矩阵条件数作为稳定性度量.

A NEW MEASURE OF NUMERICAL STABILITY FOR THE DISCRETIZED SYSTEM OF ELLIPTIC BOUNDARY VALUE PROBLEMS

The condition number of the discretization matrix of an elliptic boundary value problemdepends on the shape and the size of the grid. It becomes large when the grid has some dis-torted elements, and tends to infinity when h tends to zero. As situation in general linearalgebra, this condition number is commonly taken as a measure for the numerical stability ofthe discretized system. But, it is pointed out in this paper that the condition number of a coef-ficient matrix does not give a reasonable guide in this case, and a new measure, which is in-dependent of the grid size and the grid shape, is defined.