Meshfree finite difference solution of homogeneous Dirichlet problems of the fractional Laplacian

TL;DR

This work extends the grid-overlay finite difference (GoFD) method to a meshfree setting for homogeneous Dirichlet problems of the fractional Laplacian, enabling point-cloud representations of complex domains. It introduces two transfer-matrix constructions—moving least squares (MLS) and constrained Delaunay triangulation—to connect a uniform overlay grid with a given point cloud, allowing FFT-based efficient application of the fractional FD operator. The GoFD system remains symmetric and positive definite under a sufficient invertibility condition, and numerical results on convex, concave, and complex domains show comparable accuracy and robustness to point perturbations, with convergence consistent with GoFD on meshes. The approach preserves FFT efficiency, supports mesh adaptation, and generalizes to higher dimensions, suggesting practical utility for fractional PDEs on irregular domains.

Abstract

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.

Figures (14)

• Figure 1: A rectangular/cubic domain $\Omega_{\text{FD}} \supseteq \Omega$ and a uniform grid $\mathcal{T}_{\text{FD}}$ (in green color) on $\Omega_{\text{FD}}$ are used in the GoFD solution of fractional Laplacian BVPs. BVPs are solved on an unstructured mesh $\mathcal{T}_h$ (in blue color).
• Figure 2: Example \ref{['exam-1']}. The solution (\ref{['u-1']}) is plotted for $s = 0.25$, 0.5, and 0.75.
• Figure 3: Example \ref{['exam-1']}. Point Cloud 1 and a corresponding perturbed point cloud.
• Figure 4: Example \ref{['exam-1']}. The solution error in $L^2$ norm is plotted as a function of $N_v$ for perturbed Point Cloud 1.
• Figure 5: Example \ref{['exam-1']}. Point Cloud 2 and a corresponding perturbed point cloud.