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 $(-\Delta {)}^{\frac{\alpha }{2}}$ ($0<\alpha <2$) in hypersingular integral form. The proposed finite difference methods provide a fractional analogue of the central difference schemes to the fractional Laplacian, and as $\alpha \to 2^{-}$, they collapse to the central difference schemes of the classical Laplace operator $-\Delta$. We prove that our methods are consistent if $u\in C^{\lfloor \alpha \rfloor ,\alpha -\lfloor \alpha \rfloor +\epsilon }({\mathbb{R}}^d)$, and the local truncation error is $\mathcal{O}(h^{\epsilon })$, with $\epsilon >0$ a small constant and $\lfloor \cdot \rfloor$ denoting the floor function. If $u\in C^{2+\lfloor \alpha \rfloor ,\alpha -\lfloor \alpha \rfloor +\epsilon }({\mathbb{R}}^d)$, they can achieve the second order of accuracy for any $\alpha \in (0,2)$. 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.

arXiv comment24 pages, 6 figures, and 6 tables

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• Numerical Methods for a Diffusive Class Nonlocal Operators
• A monotone discretization for integral fractional Laplacian on bounded Lipschitz domains: Pointwise error estimates under Hölder regularity
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