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

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

Year:    2025

Author:    Weizhang Huang, Jinye Shen

Communications in Computational Physics, Vol. 37 (2025), Iss. 1 : pp. 1–29

Abstract

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.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/cicp.OA-2024-0079

Communications in Computational Physics, Vol. 37 (2025), Iss. 1 : pp. 1–29

Published online:    2025-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    29

Keywords:    Fractional Laplacian finite difference approximation stiffness matrix preconditioning overlay grid.

Author Details

Weizhang Huang

Jinye Shen