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Volume 15, Issue 4
Efficient Hermite Spectral-Galerkin Methods for Nonlocal Diffusion Equations in Unbounded Domains

Huiyuan Li, Ruiqing Liu & Li-Lian Wang

DOI: 10.4208/nmtma.OA-2022-0007s

Numer. Math. Theor. Meth. Appl., 15 (2022), pp. 1009-1040.

Published online: 2022-10

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  • Abstract

In this paper, we develop an efficient Hermite spectral-Galerkin method for nonlocal diffusion equations in unbounded domains. We show that the use of the Hermite basis can de-convolute the troublesome convolutional operations involved in the nonlocal Laplacian. As a result, the “stiffness” matrix can be fast computed and assembled via the four-point stable recursive algorithm with $\mathcal{O}(N^2)$ arithmetic operations. Moreover, the singular factor in a typical kernel function can be fully absorbed by the basis. With the aid of Fourier analysis, we can prove the convergence of the scheme. We demonstrate that the recursive computation of the entries of the stiffness matrix can be extended to the two-dimensional nonlocal Laplacian using the isotropic Hermite functions as basis functions. We provide ample numerical results to illustrate the accuracy and efficiency of the proposed algorithms.

  • Keywords

Nonlocal diffusion equation, spectral-Galerkin, Hermite functions, correlation/convolution, recurrence algorithm.

  • AMS Subject Headings

65N35, 65N25, 33C45, 65M70

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-15-1009, author = {Li , HuiyuanLiu , Ruiqing and Wang , Li-Lian}, title = {Efficient Hermite Spectral-Galerkin Methods for Nonlocal Diffusion Equations in Unbounded Domains}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2022}, volume = {15}, number = {4}, pages = {1009--1040}, abstract = {

In this paper, we develop an efficient Hermite spectral-Galerkin method for nonlocal diffusion equations in unbounded domains. We show that the use of the Hermite basis can de-convolute the troublesome convolutional operations involved in the nonlocal Laplacian. As a result, the “stiffness” matrix can be fast computed and assembled via the four-point stable recursive algorithm with $\mathcal{O}(N^2)$ arithmetic operations. Moreover, the singular factor in a typical kernel function can be fully absorbed by the basis. With the aid of Fourier analysis, we can prove the convergence of the scheme. We demonstrate that the recursive computation of the entries of the stiffness matrix can be extended to the two-dimensional nonlocal Laplacian using the isotropic Hermite functions as basis functions. We provide ample numerical results to illustrate the accuracy and efficiency of the proposed algorithms.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2022-0007s}, url = {http://global-sci.org/intro/article_detail/nmtma/21088.html} }
TY - JOUR T1 - Efficient Hermite Spectral-Galerkin Methods for Nonlocal Diffusion Equations in Unbounded Domains AU - Li , Huiyuan AU - Liu , Ruiqing AU - Wang , Li-Lian JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 1009 EP - 1040 PY - 2022 DA - 2022/10 SN - 15 DO - http://doi.org/10.4208/nmtma.OA-2022-0007s UR - https://global-sci.org/intro/article_detail/nmtma/21088.html KW - Nonlocal diffusion equation, spectral-Galerkin, Hermite functions, correlation/convolution, recurrence algorithm. AB -

In this paper, we develop an efficient Hermite spectral-Galerkin method for nonlocal diffusion equations in unbounded domains. We show that the use of the Hermite basis can de-convolute the troublesome convolutional operations involved in the nonlocal Laplacian. As a result, the “stiffness” matrix can be fast computed and assembled via the four-point stable recursive algorithm with $\mathcal{O}(N^2)$ arithmetic operations. Moreover, the singular factor in a typical kernel function can be fully absorbed by the basis. With the aid of Fourier analysis, we can prove the convergence of the scheme. We demonstrate that the recursive computation of the entries of the stiffness matrix can be extended to the two-dimensional nonlocal Laplacian using the isotropic Hermite functions as basis functions. We provide ample numerical results to illustrate the accuracy and efficiency of the proposed algorithms.

Li , HuiyuanLiu , Ruiqing and Wang , Li-Lian. (2022). Efficient Hermite Spectral-Galerkin Methods for Nonlocal Diffusion Equations in Unbounded Domains. Numerical Mathematics: Theory, Methods and Applications. 15 (4). 1009-1040. doi:10.4208/nmtma.OA-2022-0007s
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