Highly accurate operator factorization methods for the integral fractional Laplacian and its generalization

by   Yixuan Wu, et al.

In this paper, we propose a new class of operator factorization methods to discretize the integral fractional Laplacian (-Δ)^α/2 for α∈ (0, 2). The main advantage of our method is to easily increase numerical accuracy by using high-degree Lagrange basis functions, but remain the scheme structure and computer implementation unchanged. Moreover, our discretization of the fractional Laplacian results in a symmetric (multilevel) Toeplitz differentiation matrix, which not only saves memory cost in simulations but enables efficient computations via the fast Fourier transforms. The performance of our method in both approximating the fractional Laplacian and solving the fractional Poisson problems was detailedly examined. It shows that our method has an optimal accuracy of 𝒪(h^2) for constant or linear basis functions, while 𝒪(h^4) if quadratic basis functions are used, with h a small mesh size. Note that this accuracy holds for any α∈ (0, 2) and can be further increased if higher-degree basis functions are used. If the solution of fractional Poisson problem satisfies u ∈ C^m, l(Ω̅) for m ∈ℕ and 0 < l < 1, then our method has an accuracy of 𝒪(h^min{m+l, 2}) for constant and linear basis functions, while 𝒪(h^min{m+l, 4}) for quadratic basis functions. Additionally, our method can be readily applied to study generalized fractional Laplacians with a symmetric kernel function, and numerical study on the tempered fractional Poisson problem demonstrates its efficiency.


page 1

page 2

page 3

page 4


Fast Fourier-like Mapped Chebyshev Spectral-Galerkin Methods for PDEs with Integral Fractional Laplacian in Unbounded Domains

In this paper, we propose a fast spectral-Galerkin method for solving PD...

A universal solution scheme for fractional and classical PDEs

We propose a unified meshless method to solve classical and fractional P...

Finite difference method for inhomogeneous fractional Dirichlet problem

We make the split of the integral fractional Laplacian as (-Δ)^s u=(-Δ)(...

Numerical Approximation of the Fractional Laplacian on R Using Orthogonal Families

In this paper, using well-known complex variable techniques, we compute ...

A unified meshfree pseudospectral method for solving both classical and fractional PDEs

In this paper, we propose a meshfree method based on the Gaussian radial...

Approximation of Integral Fractional Laplacian and Fractional PDEs via sinc-Basis

Fueled by many applications in random processes, imaging science, geophy...

Radial basis function collocation method for decoupled fractional Laplacian wave equations

Decoupled fractional Laplacian wave equation can describe the seismic wa...