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.

翻译：针对任意有界区域上分数阶拉普拉斯算子的齐次狄利克雷边值问题，近期提出了一种称为网格覆盖有限差分法（GoFD）的数值求解方法。该方法兼具有限差分法和有限元法的优势，包括通过快速傅里叶变换实现高效计算、适用于复杂区域以及支持网格自适应。本文旨在研究无网格框架下的GoFD，其关键在于构建从给定点云到均匀网格的数据传输矩阵。为此提出两种方法：一种基于移动最小二乘拟合，另一种基于Delaunay三角剖分和分段线性插值。针对凸/凸区域及多种点云类型的数值实验表明，两种方法均能获得可比结果。随着点云密度增加，所提无网格GoFD的收敛阶与基于非结构化网格的GoFD及有限元逼近相当。此外，数值结果证明该方法对点位置随机扰动具有鲁棒性。