解泊松方程的快速预处理共轭梯度法

A Fast Preconditioned Conjugate Gradient Method for Solving Poisson's Equation

为了满足电子技术中电磁问题求解器的工程需求 ,通过分析泊松方程均匀差分离散所得模型问题的矩阵结构 ,提出了共轭梯度法的三角阵预处理器 .在用数值试验考察了其参数的特性后 ,给出了参数的经验估计方法 .实现了带参数的三角预处理器共轭梯度法求解器 .实例表明 ,该算法比常规共轭梯度法和超松弛法具有更低的计算复杂度 ,而它们存储复杂度相同 .不仅所实现的求解器具有实用价值 ,而且所给出的预处理构造技术具有进一步发展的余地 .

In order to meet the requirements of solver for electromagnetic problems needed in advanced electronic engineering, the triangular preconditioner for conjugate gradient method (CGM) is presented in this paper by use of the matrix structure obtained from the uniform discretization of Poisson's equation with finite difference method. The empirical formula for estimation of the preconditioner factor is given and tested after the properties of the preconditioner and the factor are explored. The solver of the triangular preconditioned CGM (TPCGM) is implemented. And the numerical results verify that the novel algorithm possesses lower computational complexity than conventional CGM and the SOR method while it has the same storage complexity as the other methods. Besides the validity of the solver, the technique to construct the preconditioner has the value to be further developed.