A fast algorithm for Quadrature by Expansion in three dimensions

Research output: Contribution to journalArticlepeer-review


This paper presents an accelerated quadrature scheme for the evaluation of layer potentials in three dimensions. Our scheme combines a generic, high order quadrature method for singular kernels called Quadrature by Expansion (QBX) with a modified version of the Fast Multipole Method (FMM). Our scheme extends a recently developed formulation of the FMM for QBX in two dimensions, which, in that setting, achieves mathematically rigorous error and running time bounds. In addition to generalization to three dimensions, we highlight some algorithmic and mathematical opportunities for improved performance and stability. Lastly, we give numerical evidence supporting the accuracy, performance, and scalability of the algorithm through a series of experiments involving the Laplace and Helmholtz equations.

Original languageEnglish (US)
Pages (from-to)655-689
Number of pages35
JournalJournal of Computational Physics
StatePublished - Jul 1 2019
Externally publishedYes


  • Fast algorithms
  • Fast multipole method
  • Integral equations
  • Quadrature
  • Singular integrals
  • Three dimensional problems

ASJC Scopus subject areas

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


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