We show that the mass matrix derived from finite elements can be effectively used as a preconditioner for iteratively solving the linear system arising from finite-difference discretization of the Poisson equation, using the conjugate gradient method. We derive analytically the condition number of the preconditioned operator. Theoretical analysis shows that the ratio of the condition number of the Laplacian to the preconditioned operator is $8/3$ in one dimension, $9/2$ in two dimensions, and $2^9/3^4 \approx 6.3$ in three dimensions. From this it follows that the expected iteration count for achieving a fixed reduction of the norm of the residual is smaller than a half of the number of the iterations of unpreconditioned CG in 2D and 3D. The scheme is easy to implement, and numerical experiments show its efficiency.

翻译：我们表明,从有限元素中得出的质量矩阵可有效用作利用同梯度法,迭代解决Poisson等式的有限差异离散产生的线性系统的前提条件,我们从分析中得出先决条件操作者的条件号。理论分析表明,拉普拉西亚人的条件号与先决条件操作者的条件号的比率一个维数是8/3美元,两个维数是9/2美元,三个维数是2-9/3/4\ approx 6.3美元。从这一点来看,实现固定减少残余物规范的预期迭代数小于2D和3D中未附加条件的CG的迭代数的一半。 这个办法易于实施,数字实验显示了其效率。