Accurate numerical methods for two and three dimensional integral fractional Laplacian with applications

Abstract

In this paper, we propose accurate and efficient finite difference methods to discretize the two- and three-dimensional fractional Laplacian $(-Δ)^{\fracα{2}}$ ($0 < α< 2$) in hypersingular integral form. The proposed finite difference methods provide a fractional analogue of the central difference schemes to the fractional Laplacian, and as $α\to 2^-$, they collapse to the central difference schemes of the classical Laplace operator $-Δ$. We prove that our methods are consistent if $u \in C^{\lfloorα\rfloor, α-\lfloorα\rfloor+ε}({\mathbb R}^d)$, and the local truncation error is ${\mathcal O}(h^ε)$, with $ε> 0$ a small constant and $\lfloor \cdot \rfloor$ denoting the floor function. If $u \in C^{2+\lfloorα\rfloor, α-\lfloorα\rfloor+ε}({\mathbb R}^d)$, they can achieve the second order of accuracy for any $α\in (0, 2)$. These results hold for any dimension $d \ge 1$ and thus improve the existing error estimates for the finite difference method of the one-dimensional fractional Laplacian. Extensive numerical experiments are provided and confirm our analytical results. We then apply our method to solve the fractional Poisson problems and the fractional Allen-Cahn equations. Numerical simulations suggest that to achieve the second order of accuracy, the solution of the fractional Poisson problem should {\it at most} satisfy $u \in C^{1,1}({\mathbb R}^d)$. One merit of our methods is that they yield a multilevel Toeplitz stiffness matrix, an appealing property for the development of fast algorithms via the fast Fourier transform (FFT). Our studies of the two- and three-dimensional fractional Allen-Cahn equations demonstrate the efficiency of our methods in solving the high-dimensional fractional problems.