二维边值问题九点差分方程的病态因子与通用预条件子

Ill-conditioned factors and general preconditioners for 9-point difference equations of 2D boundary value problems

用有限差分方法求解二维Poisson方程边值问题时,基于九点差分格式形成的网格方程,具有病态、稀疏的特点;与五点差分格式形成的网格方程也有类似的病态结构;对应的去病因子是通用的,可用于五点差分方程的预处理,仍然是最优预条件子,几乎不增加迭代的计算量;预处理后方程组的主体保持正定对称,条件数接近常数.

In the process of solving the boundary value problem of 2-D Poisson equation by finite difference method,the mesh equation based on 9-point difference scheme is ill-conditioned and sparse;its ill-conditioned structure is similar to the mesh equation formed by the 5-point difference scheme;the corresponding cure factor is universal and can be used for the pre-conditioning of 5-point difference equation,and it is still the optimum preconditioner,which hardly increases the computational complexity of iteration.After pretreatment,the main body of the equations remains positive definite symmetry,and the number of conditions is close to a constant.