In the finite difference approximation of the fractional Laplacian the stiffness matrix is typically dense and needs to be approximated numerically. The effect of the accuracy in approximating the stiffness matrix on the accuracy in the whole computation is analyzed and shown to be significant. Four such approximations are discussed. While they are shown to work well with the recently developed grid-over finite difference method (GoFD) for the numerical solution of boundary value problems of the fractional Laplacian, they differ in accuracy, economics to compute, performance of preconditioning, and asymptotic decay away from the diagonal line. In addition, two preconditioners based on sparse and circulant matrices are discussed for the iterative solution of linear systems associated with the stiffness matrix. Numerical results in two and three dimensions are presented.

翻译：在分数阶拉普拉斯算子的有限差分近似中，刚度矩阵通常为稠密矩阵，需要进行数值逼近。本文分析了刚度矩阵逼近精度对整体计算精度的影响，结果表明该影响十分显著。文中讨论了四种此类逼近方法。尽管这些方法在近期发展的用于分数阶拉普拉斯算子边值问题数值求解的网格覆盖有限差分法（GoFD）中均表现良好，但它们在精度、计算经济性、预处理性能以及非对角线元素的渐近衰减特性方面存在差异。此外，针对刚度矩阵相关线性系统的迭代求解，本文讨论了基于稀疏矩阵与循环矩阵的两种预处理方法。文中给出了二维与三维数值算例。