Numerical Methods for the Fractional Laplacian: A Finite Difference-Quadrature Approach

2014/01/01 by Yanghong Huang, Adam Oberman, Adam M. Oberman · 13 citations
Mathematics · #Fractional Differential Equations Solutions #Differential Equations and Numerical Methods #Numerical methods in inverse problems

paper · doi:10.1137/140954040

Abstract

The fractional Laplacian (-Δ)^α/2 is a nonlocal 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 differences 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 solutions with varying regularity. The generalization to nonlinear equations involving the operator is discussed: the obstacle problem for the fractional Laplacian is computed.

Citations

• Analysis and Approximation of Nonlocal Diffusion Problems with Volume Constraints
• 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

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