A Grid-Overlay Finite Difference Method for Solving the Fractional Laplacian on Arbitrary Bounded Domains

Core Concepts

A grid-overlay finite difference method is proposed for the numerical approximation of the fractional Laplacian on arbitrary bounded domains. The method combines the advantages of uniform-grid finite difference approximation and unstructured meshes to efficiently handle complex geometries and enable mesh adaptation.

Abstract

The content presents a grid-overlay finite difference (GoFD) method for solving the fractional Laplacian problem on arbitrary bounded domains. The key aspects are: The method uses an unstructured simplicial mesh Th that fits or approximately fits the domain Ω, and an overlaying uniform grid TFD. A uniform-grid finite difference (FD) approximation h^(-2s)_FD A_FD of the fractional Laplacian is constructed on TFD, leveraging efficient matrix-vector multiplication via fast Fourier transform. The GoFD approximation of the fractional Laplacian on Th is defined as h^(-2s)_FD A_h, where A_h = D^(-1)_h (I^FD_h)^T A_FD I^FD_h. The transfer matrix I^FD_h represents data transfer from Th to TFD. Theoretical analysis shows that A_h is similar to a symmetric and positive definite matrix if I^FD_h has full column rank and positive column sums. This is guaranteed if the grid spacing h_FD is smaller than or equal to a bound proportional to the minimum element height of Th. A sparse preconditioner is proposed for the iterative solution of the resulting linear system for the homogeneous Dirichlet problem. The method can readily be incorporated with existing mesh adaptation strategies, such as the MMPDE moving mesh method. Numerical examples in 1D, 2D, and 3D demonstrate the convergence behavior and effectiveness of the sparse preconditioning and mesh adaptation.

Stats

The fractional Laplacian problem is defined as: p-Δ)^s u = f, in Ω u = 0, in Ω^c where Ω is a bounded domain in R^d, Ω^c = R^d\Ω, and s ∈ (0, 1) is the fractional order. The grid-overlay finite difference (GoFD) method approximates the solution u on an unstructured simplicial mesh Th that fits or approximately fits Ω.

Quotes

"The method takes full advantages of both uniform-grid finite difference approximation in efficient matrix-vector multiplication via the fast Fourier transform and unstructured meshes for complex geometries and mesh adaptation." "It is shown that its stiffness matrix is similar to a symmetric and positive definite matrix and thus invertible if the data transfer has full column rank and positive column sums." "Numerical examples demonstrate that the new method has similar convergence behavior as existing finite difference and finite element methods and that the sparse preconditioning is effective."

Key Insights Distilled From

A grid-overlay finite difference method for the fractional Laplacian on arbitrary bounded domains

by Weizhang Hua... at arxiv.org 04-08-2024

https://arxiv.org/pdf/2307.14437.pdf

Deeper Inquiries

How can the GoFD method be extended to handle more general boundary conditions beyond the homogeneous Dirichlet case

To extend the GoFD method to handle more general boundary conditions beyond the homogeneous Dirichlet case, we can incorporate the boundary conditions into the data transfer matrix and the stiffness matrix. For example, for Neumann boundary conditions, we can modify the data transfer matrix to account for the normal derivative of the solution at the boundary. This modification would ensure that the boundary conditions are satisfied in the discretization process. Additionally, for mixed boundary conditions, a combination of different types of boundary conditions can be implemented in the data transfer matrix and the stiffness matrix to handle the mixed conditions appropriately.

What are the potential challenges and limitations of the GoFD method when dealing with highly anisotropic or non-smooth domains

When dealing with highly anisotropic or non-smooth domains, the GoFD method may face challenges and limitations. Highly anisotropic domains can lead to significant variations in element sizes, which can affect the stability and accuracy of the method. In such cases, careful mesh refinement strategies and adaptive meshing techniques may be required to ensure the method performs well. Non-smooth domains with irregular boundaries or discontinuities can also pose challenges for the method, as the piecewise linear interpolation may not capture the geometry accurately. Special treatment or modifications may be needed to handle these non-smooth features effectively.

Can the ideas behind the GoFD method be applied to other types of non-local operators beyond the fractional Laplacian

The ideas behind the GoFD method, such as using an unstructured mesh with an overlay grid for efficient matrix-vector multiplication, can be applied to other types of non-local operators beyond the fractional Laplacian. For example, operators like the fractional diffusion equation, fractional advection-diffusion equation, or other fractional partial differential equations can benefit from a similar approach. By adapting the data transfer and interpolation techniques to suit the specific operator and its properties, the GoFD method can be extended to provide numerical approximations for a wide range of non-local operators. The key lies in understanding the characteristics of the operator and tailoring the method accordingly to ensure accuracy and efficiency in the numerical computations.