Approximating and Preconditioning the Stiffness Matrix in the GoFD Approximation of the Fractional Laplacian

被引：0

作者：

Huang, Weizhang ^{[1
]}

Shen, Jinye ^{[2
]}

机构：

[1] Univ Kansas, Dept Math, Lawrence, KS USA

[2] Southwestern Univ Finance & Econ, Sch Math, Chengdu, Peoples R China

来源：

COMMUNICATIONS IN COMPUTATIONAL PHYSICS | 2025年 / 37卷 / 01期

基金：

中国国家自然科学基金;

关键词：

Fractional Laplacian; finite difference approximation; stiffness matrix; precondition-; ing; overlay grid; FINITE-DIFFERENCE METHOD; DIFFUSION; REGULARITY; EQUATIONS; ERROR;

D O I：

10.4208/cicp.OA-2024-0079

中图分类号：

O4 [物理学];

学科分类号：

0702 ;

摘要：

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.

引用

页码：1 / 29

页数：29

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