Meshfree Finite Difference Solution of Fractional Laplacian Boundary Value Problems on Arbitrary Domains

核心概念

This work explores the feasibility of using the grid-overlay finite difference method (GoFD) with point clouds for the numerical solution of homogeneous Dirichlet problems of the fractional Laplacian. Two approaches are proposed to construct the transfer matrix from the point cloud to the uniform grid, one based on moving least squares fitting and the other based on Delaunay triangulation and piecewise linear interpolation.

摘要

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:

1. 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.

2. 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.

统计

The fractional Laplacian can be expressed as:
p´∆qsupxq " C2,s p.v.
ż
R2
upxq ´ upyq
| x ´ y |22s dy, C2,s " 22ssΓps 1q
πΓp1 ´ sq
The uniform-grid finite difference approximation of the fractional Laplacian can be written as:
p´∆hqsupxj, ykq "
1
h2s
FD
8
ÿ
m"´8
8
ÿ
n"´8
Apj,kq,pm,nqum,n

引用

"A so-called grid-overlay finite difference method (GoFD) was recently proposed in [17] for the numerical solution of homogeneous Dirichlet boundary value problems (BVPs) of the fractional Laplacian."
"The objective of this work is to study GoFD in a meshfree/meshless setting in two dimensions."

从中提取的关键见解

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

by Jinye Shen,B... 在 arxiv.org 04-09-2024

https://arxiv.org/pdf/2404.04407.pdf

更深入的查询

How can the proposed meshfree GoFD approach be extended to higher-dimensional problems

The proposed meshfree GoFD approach can be extended to higher-dimensional problems by generalizing the construction of the transfer matrix from a given point cloud to a uniform grid in three or more dimensions. In higher dimensions, the point cloud would represent the discretization of the domain in a meshless manner, and the transfer matrix would be constructed using techniques suitable for the specific dimensionality. For example, in three dimensions, the Delaunay triangulation approach could be adapted to create tetrahedral elements for interpolation. The key would be to maintain the efficiency and accuracy of the method while handling the increased complexity of higher-dimensional geometries.

What are the potential limitations or challenges in applying the Delaunay triangulation approach to constructing the transfer matrix for complex geometries

One potential limitation or challenge in applying the Delaunay triangulation approach to constructing the transfer matrix for complex geometries is the computational cost and complexity of the triangulation process. In complex geometries with irregular boundaries or intricate shapes, the Delaunay triangulation may result in a large number of triangles or tetrahedra, leading to increased computational overhead. Additionally, ensuring the quality of the triangulation and the accuracy of the interpolation in regions with sharp features or high curvature can be challenging. The Delaunay triangulation approach may also struggle with handling concave geometries or regions with narrow passages, where the triangulation may not capture the geometry accurately.

How can the meshfree GoFD method be combined with adaptive strategies to further improve the computational accuracy and convergence order for fractional Laplacian boundary value problems

The meshfree GoFD method can be combined with adaptive strategies to further improve computational accuracy and convergence order for fractional Laplacian boundary value problems. By incorporating adaptive mesh refinement techniques, the method can dynamically adjust the resolution of the point cloud or mesh based on the solution behavior, such as gradients or singularities. This adaptive refinement can help concentrate computational resources in regions of interest, leading to more accurate solutions with improved convergence rates. Adaptive strategies can also enhance the method's efficiency by reducing the number of points or elements in areas where the solution is smooth and increasing resolution where it is needed most. This adaptive approach can optimize the computational resources and improve the overall performance of the meshfree GoFD method.