A comparative study on nonlocal diffusion operators related to the fractional Laplacian

TL;DR

This study compares four nonlocal diffusion operators related to the fractional Laplacian, analyzing their differences, eigenvalues, and approximation properties through extensive numerical experiments.

Contribution

It provides a comprehensive numerical comparison of four nonlocal operators, highlighting their differences and approximation behaviors, especially as parameters vary.

Findings

Eigenvalues differ among the operators, with the spectral fractional Laplacian having larger eigenvalues.

All operators converge to the classical Laplacian as   2.

Peridynamic operator approximates the fractional Laplacian well for large horizon size   .

Abstract

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 \alpha \to 2. The eigenvalues and eigenfunctions of these four operators are different, and the k-th (for k \in N) eigenvalue of the spectral fractional Laplacian is always larger than those of the fractional Laplacian and regional fractional Laplacian. For any \alpha \in (0, 2), the peridynamic operator can provide a good approximation to the fractional Laplacian, if the horizon size \delta is sufficiently large. We find that the solution of the peridynamic model converges to that of the fractional Laplacian model at a rate of O(\delta^{-\alpha}). In contrast, although the regional fractional Laplacian can be used to approximate the fractional Laplacian as \alpha \to 2, it generally provides inconsistent result from that of the fractional Laplacian if \alpha \ll 2. Moreover, some conjectures are made from our numerical results, which could contribute to the mathematics analysis on these operators.