  A so-called grid-overlay finite difference method (GoFD) was proposed
recently for the numerical solution of homogeneous Dirichlet boundary value
problems of the fractional Laplacian on arbitrary bounded domains. It was shown
to have advantages of both finite difference and finite element methods,
including its efficient implementation through the fast Fourier transform and
ability to work for complex domains and with mesh adaptation. The purpose of
this work is to study GoFD in a meshfree setting, a key to which is to
construct the data transfer matrix from a given point cloud to a uniform grid.
Two approaches are proposed, one based on the moving least squares fitting and
the other based on the Delaunay triangulation and piecewise linear
interpolation. Numerical results obtained for examples with convex and concave
domains and various types of point clouds are presented. They show that both
approaches lead to comparable results. Moreover, the resulting meshfree GoFD
converges at a similar order as GoFD with unstructured meshes and finite
element approximation as the number of points in the cloud increases.
Furthermore, numerical results show that the method is robust to random
perturbations in the location of the points.
