三维预处理技术初探

本文应用高阶近似LU分解法对三维椭圆型方程的正规七点差分矩阵给出了预处理阵,并对典型问题给出了预处理共轭梯度法的计算结果。文中所采用的分阶方论以及阶矩阵等概念,适用于一般的非奇异对角优势稀疏矩阵。对不同网格点数的计算结果表明,三维预处理共轭梯度法仍具有超线性的收敛速率,在高阶情形下超线性的特征尤为突出,由于非零对角线数增长过快,高阶的三维预处理方法不宜采用,零阶方法和一阶方法是值得推荐的。

A PRELIMINARY STUDY OF 3-D PRECONDITIONING METHOD

A method of high order approximate LU decomposition is adopted to give the preconditioner for the coefficient matrix of 3-D elliptic partial differential equation when the normal 7-point scheme is used. Numerical results of the preconditioned conjugate gradient (PCG) solving a tipycal problem are given.The concepts of the order, the order matrix and the P-order influence areas given in this paper can be applied to a general sparse non-singular and diagonally dominant linear system of equations. The results of different meshsize show that 3-D PCG has a superlinear convergence rate. In the case of higher orders the superlinearity of the PCG convergence rate is stronger even more. Since the number of non-zero diagonals in the preconditioner increases rapidly when the order of the PCG method increases, it is not beneficial to use the preconditioner of order P if P is larger than one. The preeonditioners of order zero and order one are recommended.