A walk-on-sphere-motivated finite-difference method for the fractional Poisson equation on a bounded d-dimensional domain

Abstract

Inspired by the idea of ‘walk-on-sphere’ algorithm, we propose a novel finite-difference framework for solving the fractional Poisson equation under the help of the Feynman-Kac representation of its solution, i.e., walk-on-sphere-motivated finite-difference scheme. By choosing suitable basis functions in interpolatory quadrature and using graded meshes, the convergence rates can achieve up to $O(h^{2})$ in arbitrary $d$-dimensional bounded Lipschitz domain satisfying the exterior ball condition, where $d>1$; while the convergence rate can reach $O(h^{10})$ in 1-dimensional bounded domain under some regularity assumptions on the source term $f$. Furthermore, we propose a strict convergence analysis and several numerical examples in different domains, including circle, L-shape, pentagram and ball, are provided to illustrate the effectiveness of the above built scheme.