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Volume 37, Issue 1
Approximating and Preconditioning the Stiffness Matrix in the GoFD Approximation of the Fractional Laplacian

Weizhang Huang & Jinye Shen

DOI: 10.4208/cicp.OA-2024-0079

Commun. Comput. Phys., 37 (2025), pp. 1-29.

Published online: 2025-01

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  • 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.

  • Keywords

Fractional Laplacian, finite difference approximation, stiffness matrix, preconditioning, overlay grid.

  • AMS Subject Headings

65N06, 35R11

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-37-1, author = {Huang , Weizhang and Shen , Jinye}, title = {Approximating and Preconditioning the Stiffness Matrix in the GoFD Approximation of the Fractional Laplacian}, journal = {Communications in Computational Physics}, year = {2025}, volume = {37}, number = {1}, pages = {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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2024-0079}, url = {http://global-sci.org/intro/article_detail/cicp/23779.html} }
TY - JOUR T1 - Approximating and Preconditioning the Stiffness Matrix in the GoFD Approximation of the Fractional Laplacian AU - Huang , Weizhang AU - Shen , Jinye JO - Communications in Computational Physics VL - 1 SP - 1 EP - 29 PY - 2025 DA - 2025/01 SN - 37 DO - http://doi.org/10.4208/cicp.OA-2024-0079 UR - https://global-sci.org/intro/article_detail/cicp/23779.html KW - Fractional Laplacian, finite difference approximation, stiffness matrix, preconditioning, overlay grid. AB -

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.

Huang , Weizhang and Shen , Jinye. (2025). Approximating and Preconditioning the Stiffness Matrix in the GoFD Approximation of the Fractional Laplacian. Communications in Computational Physics. 37 (1). 1-29. doi:10.4208/cicp.OA-2024-0079
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