Fourier Spectral Method for Nonlocal Equations on Bounded Domains

TL;DR

This paper develops efficient spectral solvers for nonlocal equations on bounded domains by leveraging Fourier transforms and the 2D-FC algorithm, enabling fast solutions and exploring solution regularity through numerical experiments.

Contribution

It extends existing spectral methods by integrating 2D-FC for bounded domains, improving efficiency and accuracy in solving nonlocal PDEs.

Findings

Spectral solvers achieve high accuracy and efficiency.

Numerical experiments reveal solution regularity issues.

Method extends to 2D nonlocal Poisson and diffusion equations.

Abstract

This work introduces efficient and accurate spectral solvers for nonlocal equations on bounded domains. These spectral solvers exploit the fact that integration in the nonlocal formulation transforms into multiplication in Fourier space and that nonlocality is decoupled from the grid size, allowing fast and accurate solutions to the nonlocal problems. Our approach extends the spectral solvers developed by Alali and Albin (2020) for periodic domains by incorporating the two-dimensional Fourier continuation (2D-FC) algorithm introduced by Bruno and Paul (2022). We evaluate the performance of the proposed methods on two-dimensional nonlocal Poisson and nonlocal diffusion equations defined on bounded domains. While the regularity of solutions to these equations in bounded settings remains an open problem, we conduct numerical experiments to explore this issue, particularly focusing on studying discontinuities.