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 - <p style="text-align: justify;">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.</p>