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Meshfree methods for the fractional Laplacian and its applications

Abstract

Recently, the fractional Laplacian has received great attention in modeling complex phenomena that involve long-range interactions. However, its nonlocality introduces considerable challenges in both mathematical analysis and numerical simulations. In this talk, I will introduce the recently developed meshfree methods based on the radial basis function to solve problems with the fractional Laplacian. The proposed methods take advantage of the Laplacian of radial basis functions so as to accommodate the discretization of classical and fractional Laplacians in a single scheme and also avoid large computational cost for numerical evaluation of the fractional derivatives. Moreover, our methods are simple and easy to handle complex geometry and local refinements, and their computer program implementation remains the same for any dimension d. Numerical accuracy of these methods will be discussed, and some applications of nonlocal problems with the fractional Laplacian will also be demonstrated.