A Simple Solver for the Fractional Laplacian in Multiple Dimensions
Victor Minden, Lexing Ying
Abstract
We present a simple discretization scheme for the hypersingular integral representa- tion of the fractional Laplace operator and solver for the corresponding fractional Laplacian problem. Through singularity subtraction, we obtain a regularized integrand that is amenable to the trape- zoidal rule with equispaced nodes, assuming a high degree of regularity in the underlying function (i.e., $u \in C^6({R}^d)$). 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 (nonfractional) 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.