Numerical Methods for the Fractional Laplacian: a Finite Difference-quadrature Approach

2013/11/29 by Yanghong Huang, Adam M. Oberman, Huang, Yanghong +1 · 4 citations
Mathematics · #Differential Equations and Numerical Methods #FOS: Mathematics #Fractional Differential Equations Solutions #Nonlinear Partial Differential Equations #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.1311.7691

openalex publication_date 2013/11/29 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28

Abstract

The fractional Laplacian (-Δ)^α/2 is a non-local operator which depends on the parameter α and recovers the usual Laplacian as α→ 2. A numerical method for the fractional Laplacian is proposed, based on the singular integral representation for the operator. The method combines finite difference with numerical quadrature, to obtain a discrete convolution operator with positive weights. The accuracy of the method is shown to be O(h^3-α). Convergence of the method is proven. The treatment of far field boundary conditions using an asymptotic approximation to the integral is used to obtain an accurate method. Numerical experiments on known exact solutions validate the predicted convergence rates. Computational examples include exponentially and algebraically decaying solution with varying regularity. The generalization to nonlinear equations involving the operator is discussed: the obstacle problem for the fractional Laplacian is computed.

Citations

• Regularity of the obstacle problem for a fractional power of the laplace operator
• Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian

Cited by

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• A note on spectral Monte-Carlo method for fractional Poisson equation on high-dimensional ball
• Numerical Approximation of the logarithmic Laplacian via sinc-basis
• A preconditioned boundary value method for advection-diffusion equations with half Laplacian via spectrum doubling

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