Periodic Fast Multipole Method

TL;DR

This paper introduces a novel periodic boundary condition scheme for unit cells applicable to various PDEs, which is aspect ratio insensitive, easily integrated with fast multipole methods, and accelerable with NUFFT for large cells.

Contribution

The proposed scheme extends to multiple equations, provides explicit low-rank representations, and is compatible with adaptive FMMs, improving efficiency and flexibility over traditional lattice sum methods.

Findings

Scheme is insensitive to unit cell aspect ratio.

Coupling with NUFFT accelerates computations for large cells.

Performance demonstrated through numerical examples.

Abstract

A new scheme is presented for imposing periodic boundary conditions on unit cells with arbitrary source distributions. We restrict our attention here to the Poisson, modified Helmholtz, Stokes and modified Stokes equations. The approach extends to the oscillatory equations of mathematical physics, including the Helmholtz and Maxwell equations, but we will address these in a companion paper, since the nature of the problem is somewhat different and includes the consideration of quasiperiodic boundary conditions and resonances. Unlike lattice sum-based methods, the scheme is insensitive to the unit cell's aspect ratio and is easily coupled to adaptive fast multipole methods (FMMs). Our analysis relies on classical "plane-wave" representations of the fundamental solution, and yields an explicit low-rank representation of the field due to all image sources beyond the first layer of neighboring unit cells. When the aspect ratio of the unit cell is large, our scheme can be coupled with the nonuniform fast Fourier transform (NUFFT) to accelerate the evaluation of the induced field. Its performance is illustrated with several numerial examples.