A fast method for imposing periodic boundary conditions on arbitrarily-shaped lattices in two dimensions

Ruqi Pei, Travis Askham, Leslie Greengard, Shidong Jiang

Research output: Contribution to journalArticlepeer-review


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 numerical examples.

Original languageEnglish (US)
Article number111792
JournalJournal of Computational Physics
StatePublished - Feb 1 2023


  • Arbitrarily-shaped lattices
  • Fast multipole method
  • Low rank factorization
  • Nonuniform fast Fourier transform
  • Periodic boundary conditions
  • Plane wave representation

ASJC Scopus subject areas

  • Numerical Analysis
  • Modeling and Simulation
  • Physics and Astronomy (miscellaneous)
  • General Physics and Astronomy
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics


Dive into the research topics of 'A fast method for imposing periodic boundary conditions on arbitrarily-shaped lattices in two dimensions'. Together they form a unique fingerprint.

Cite this