Electrical engineering

Preconditioner

Mathematics

Elliptic operator

Laplace transform

Mathematical optimization

Engineering

Laplace operator

Mathematical analysis

Solver

Discretization

Applied mathematics

Bounded function

Quadrature (astronomy)

Linear system

SIAM Journal on Scientific Computing

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.

A Simple Solver for the Fractional Laplacian in Multiple Dimensions