A mesh-free pseudospectral approach to estimating the fractional Laplacian via radial basis functions

JOURNAL OF COMPUTATIONAL PHYSICS

卷:390

Article

Rosenfeld, Joel A.¹  Rosenfeld, Spencer A.²  Dixon, Warren E.³ 

[1] Vanderbilt Univ, Dept Elect Engn & Comp Sci, 221 Kirkland Hall, Nashville, TN 37235 USA

[2] Univ Florida, Dept Phys, Gainesville, FL 32611 USA

[3] Univ Florida, Dept Mech & Aerosp Engn, Nonlinear Controls & Robot NCR Lab, Gainesville, FL USA

关键词: Mesh-free methods;    Fractional Laplacian;    Wendland RBFs;    Fractional calculus;    Pseudospectral methods;    Fractional Poissonequation;   

DOI  :  10.1016/j.jcp.2019.02.015

来源: Elsevier

【 摘 要 】

This paper investigates the use of radial basis function (RBF) interpolants to estimate a function's fractional Laplacian of a given order through a mesh-free pseudospectral method. The mesh-free approach yields an algorithm that can be implemented in high dimensional settings without adjustment. Moreover, the fractional Laplacian is defined in terms of the Fourier transform, and the symmetry of RBFs can be exploited to simplify the estimation problem. Convergence rates are established for RBFs when the function whose fractional Laplacian to be estimated is compactly supported. Further results demonstrate convergence when a function is in the native space for a Wendland RBF (i.e. a Sobolev space) and satisfies a certain L-1 condition. Numerical experiments demonstrate the developed method by estimating the fractional Laplacian of several functions and by solving a fractional Poisson equation with extended Dirichlet condition in one and two dimensions. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free