Efficient Hermite Spectral-Galerkin Methods for Nonlocal Diffusion Equations in Unbounded Domains
Efficient Hermite Spectral-Galerkin Methods for Nonlocal Diffusion Equations in Unbounded Domains
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DOI：
10.4208/nmtma.oa-2022-0007s
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发表时间：
2022-06
期刊：
影响因子：
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通讯作者：
Hui-yuan Li;Ruiqing Liu null;Lilian Wang
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文献类型：
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作者：
Hui-yuan Li;Ruiqing Liu null;Lilian Wang
本文提出了一种求解无界域上非局部扩散方程的有效Hermite谱-Galerkin方法。我们证明了使用Hermite基可以对非局部拉普拉斯算子中麻烦的卷积运算进行去卷积。结果表明，用O(N2)次运算量的四点稳定递推算法可以快速计算和组装“刚度”矩阵。此外，该基能充分吸收典型核函数中的奇异因子。借助于傅立叶分析，我们可以证明该格式的收敛。我们证明了用各向同性Hermite函数作为基函数，可以将刚度矩阵项的递推计算推广到二维非局部拉普拉斯函数。我们给出了大量的数值结果来说明所提出的算法的精度和效率。
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 O(N2) 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.