The paper studies the meshfree solution of homogeneous Dirichlet problems of the fractional Laplacian using the grid-overlay finite difference method (GoFD). GoFD combines the advantages of finite difference and finite element methods, allowing efficient implementation through fast Fourier transform while being able to handle complex domains and mesh adaptation.
The key to the success of GoFD in a meshfree setting is the construction of the transfer matrix from the given point cloud to the uniform grid. Two approaches are proposed:
Moving least squares fitting with inverse distance weighting: For each grid point, the nearest n points in the point cloud are used to construct a local linear polynomial approximation. The coefficients of this approximation are used to define the entries of the transfer matrix.
Delaunay triangulation and piecewise linear interpolation: The point cloud is first partitioned into a constrained Delaunay triangulation. The transfer matrix is then constructed based on piecewise linear interpolation on this triangulation.
Numerical results are presented for examples with convex and concave domains, as well as various types of point clouds. The results show that both approaches lead to comparable solution accuracy, with the error behaving like Op¯hminp1,s+0.5qq for quasi-uniform point clouds and second-order for adaptive point clouds, where ¯h = 1/√Nv and Nv is the number of points in the cloud. The method is also shown to be robust with respect to random perturbations in the point locations.
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by Jinye Shen,B... at arxiv.org 04-09-2024
https://arxiv.org/pdf/2404.04407.pdfDeeper Inquiries