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

共 47 条

[1]

Acosta G., Bersetche F. M., Borthagaray J. P., A short FE implementation for a 2D homogeneous Dirichlet problem of a fractional Laplacian, Comput. Math. Appl, 74, pp. 784-816, (2017)

[2]

Acosta G., Borthagaray J. P., A fractional Laplace equation: Regularity of solutions and finite element approximations, SIAM J. Numer. Anal, 55, pp. 472-495, (2017)

[3]

Ainsworth M., Glusa C., Aspects of an adaptive finite element method for the fractional Laplacian: a priori and a posteriori error estimates, efficient implementation and multigrid solver, Comput. Methods Appl. Mech. Engrg, 327, pp. 4-35, (2017)

[4]

Ainsworth M., Glusa C., Towards an efficient finite element method for the integral fractional Laplacian on polygonal domains, Contemporary Computational Mathematics–A Celebration of the 80th Birthday of Ian Sloan, 1, 2, pp. 17-57, (2018)

[5]

Andrews G. E., Askey R., Roy R., Special Functions, volume 71 of Encyclopedia of Mathematics and its Applications, (1999)

[6]

Antil H., Brown T., Khatri R., Onwunta A., Verma D., Warma M., Chapter 3 – Optimal control, numerics, and applications of fractional PDEs, Handbook of Numerical Analysis, 23, pp. 87-114, (2022)

[7]

Antil H., Dondl P., Striet L., Approximation of integral fractional Laplacian and fractional PDEs via sinc-basis, SIAM J. Sci. Comput, 43, pp. A2897-A2922, (2021)

[8]

Barnett A. H., Aliasing error of the exp(β<sup>?</sup>1-z<sup>2</sup>) kernel in the nonuniform fast Fourier transform, Appl. Comput. Harmon. Anal, 51, pp. 1-16, (2021)

[9]

Barnett A. H., Magland J. F., af Klinteberg L., A parallel non-uniform fast Fourier transform library based on an “exponential of semicircle” kernel, SIAM J. Sci. Comput, 41, pp. C479-C504, (2019)

[10]

Bonito A., Lei W., Approximation of the spectral fractional powers of the Laplace-Beltrami operator, Numer. Math. Theor. Meth. Appl, 15, pp. 1193-1218, (2022)

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