A Comparative Study on Nonlocal Diffusion Operators Related to the Fractional Laplacian
In this paper, we study four nonlocal diffusion operators, including the fractional Laplacian, spectral fractional Laplacian, regional fractional Laplacian, and peridynamic operator. These operators represent the infinitesimal generators of different stochastic processes, and especially their differences on a bounded domain are significant. We provide extensive numerical experiments to understand and compare their differences. We find that these four operators collapse to the classical Laplace operator as α 2. The eigenvalues and eigenfunctions of these four operators are different, and the k-th (for k σ N) eigenvalue of the spectral fractional Laplacian is always larger than those of the fractional Laplacian and regional fractional Laplacian. For any ff 2 (0; 2), the peridynamic operator can provide a good approximation to the fractional Laplacian, if the horizon size δ is suffciently large. We find that the solution of the peridynamic model converges to that of the fractional Laplacian model at a rate of O(δ-a). In contrast, although the regional fractional Laplacian can be used to approximate the fractional Laplacian as α 2, it generally provides inconsistent result from that of the fractional Laplacian if α ≤ 2. Moreover, some conjectures are made from our numerical results, which could contribute to the mathematics analysis on these operators.
S. Duo et al., "A Comparative Study on Nonlocal Diffusion Operators Related to the Fractional Laplacian," Discrete and Continuous Dynamical Systems - Series B, vol. 24, no. 1, pp. 231 - 256, American Institute of Mathematical Sciences (AIMS), Jan 2019.
The definitive version is available at https://doi.org/10.3934/dcdsb.2018110
Mathematics and Statistics
Center for High Performance Computing Research
Keywords and Phrases
Extended Homogeneous Dirichlet Boundary Condition; Fractional Laplacian; Fractional Poisson Equation; Peridynamic Operator; Regional Fractional Laplacian; Spectral Fractional Laplacian
International Standard Serial Number (ISSN)
Article - Journal
© 2019 American Institute of Mathematical Sciences (AIMS), All rights reserved.
01 Jan 2019