We present a simple solver of the fractional Laplacian based on the hypersingular integral representation. Through singularity subtraction, we obtain a regularized integrand that is amenable to the trapezoidal rule with equispaced nodes. The resulting quadrature scheme gives a discrete operator on a regular grid that is translation-invariant and thus can be applied quickly with the fast Fourier transform. For discretizations of problems related to space-fractional diffusion on bounded domains, we observe that the underlying linear system can be efficiently solved via preconditioned Krylov methods with a preconditioner based on the finite-difference (non-fractional) Laplacian. We show numerical results illustrating the error of our simple scheme as well the efficiency of our preconditioning approach, both for the elliptic (steady-state) fractional diffusion problem and the time-dependent problem.
Notable schemes for discretizing the fractional Laplacian based on different ideas include work based on the Caffarelli-Silvestre extension @cite_15 @cite_41 @cite_13 , finite-element-based approaches @cite_48 @cite_17 @cite_8 @cite_43 @cite_30 , and work on spectral approaches @cite_19 @cite_11 @cite_14 . General references for fractional Laplacians on bounded domains include, e.g., Ros-Oton @cite_10 , D'Elia and Gunzburger @cite_38 , Felsinger @cite_4 , and @cite_0 .
@cite_30 @cite_38 @cite_11 @cite_14 @cite_4 @cite_8 @cite_41 @cite_48 @cite_10 @cite_0 @cite_43 @cite_19 @cite_15 @cite_13 @cite_17
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The fractional Laplacian in R^d has multiple equivalent characterizations. Moreover, in bounded domains, boundary conditions must be incorporated in these characterizations in mathematically distinct ways, and there is currently no consensus in the literature as to which definition of the fractional Laplacian in bounded domains is most appropriate for a given application. The Riesz (or integral) definition, for example, admits a nonlocal boundary condition, where the value of a function u(x) must be prescribed on the entire exterior of the domain in order to compute its fractional Laplacian. In contrast, the spectral definition requires only the standard local boundary condition. These differences, among others, lead us to ask the question: "What is the fractional Laplacian?" We compare several commonly used definitions of the fractional Laplacian (the Riesz, spectral, directional, and horizon-based nonlocal definitions), and we use a quantitative approach to identify their practical differences. In this work, we provide a quantitative assessment of new numerical methods as well as available state-of-the-art methods for discretizing the fractional Laplacian, and we present new results on the differences in features, regularity, and boundary behaviors of solutions to equations posed with these different definitions. We present stochastic interpretations and demonstrate the equivalence between some recent formulations. Through our efforts, we aim to further engage the research community in open problems and assist practitioners in identifying the most appropriate definition and computational approach to use for their mathematical models in addressing anomalous transport in diverse applications. This paper presents regularity results and associated high order numerical methods for one-dimensional fractional-Laplacian boundary-value problems. On the basis of a factorization of solutions as a product of a certain edge-singular weight ω times a "regular" unknown, a characterization of the regularity of solutions is obtained in terms of the smoothness of the corresponding right-hand sides. In particular, for right-hand sides which are analytic in a Bernstein ellipse, analyticity in the same Bernstein ellipse is obtained for the regular'' unknown. Moreover, a sharp Sobolev regularity result is presented which completely characterizes the co-domain of the fractional-Laplacian operator in terms of certain weighted Sobolev spaces introduced in (Babuska and Guo, SIAM J. Numer. Anal. 2002). The present theoretical treatment relies on a full eigendecomposition for a certain weighted integral operator in terms of the Gegenbauer polynomial basis. The proposed Gegenbauer-based Nystrom numerical method for the fractional-Laplacian Dirichlet problem, further, is significantly more accurate and efficient than other algorithms considered previously. The sharp error estimates presented in this paper indicate that the proposed algorithm is spectrally accurate, with convergence rates that only depend on the smoothness of the right-hand side. In particular, convergence is exponentially fast (resp. faster than any power of the mesh-size) for analytic (resp. infinitely smooth) right-hand sides. The properties of the algorithm are illustrated with a variety of numerical results. The operator square root of the Laplacian (−△) 1 2 can be obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to the Neumann condition. In this paper we obtain similar characterizations for general fractional powers of the Laplacian and other integro-differential operators. From those characterizations we derive some properties of these integro-differential equations from purely local arguments in the extension problems. We propose a new nonconforming finite element algorithm to approximate the solution to the elliptic problem involving the fractional Laplacian. We first derive an integral representation of the bilinear form corresponding to the variational problem. The numerical approximation of the action of the corresponding stiffness matrix consists of three steps: (i) apply a sinc quadrature scheme to approximate the integral representation by a finite sum where each term involves the solution of an elliptic partial differential equation defined on the entire space, (ii) truncate each elliptic problem to a bounded domain, (iii) use the finite element method for the space approximation on each truncated domain. The consistency error analysis for the three steps is discussed together with the numerical implementation of the entire algorithm. The results of computations are given illustrating the error behavior in terms of the mesh size of the physical domain, the domain truncation parameter and the quadrature spacing parameter.