Fast algorithms for Quadrature by Expansion I: Globally valid expansions

Manas Rachh, Andreas Klöckner, Michael O'Neil

Research output: Contribution to journalArticlepeer-review


The use of integral equation methods for the efficient numerical solution of PDE boundary value problems requires two main tools: quadrature rules for the evaluation of layer potential integral operators with singular kernels, and fast algorithms for solving the resulting dense linear systems. Classically, these tools were developed separately. In this work, we present a unified numerical scheme based on coupling Quadrature by Expansion, a recent quadrature method, to a customized Fast Multipole Method (FMM) for the Helmholtz equation in two dimensions. The method allows the evaluation of layer potentials in linear-time complexity, anywhere in space, with a uniform, user-chosen level of accuracy as a black-box computational method. Providing this capability requires geometric and algorithmic considerations beyond the needs of standard FMMs as well as careful consideration of the accuracy of multipole translations. We illustrate the speed and accuracy of our method with various numerical examples.

Original languageEnglish (US)
Pages (from-to)706-731
Number of pages26
JournalJournal of Computational Physics
StatePublished - Sep 15 2017


  • Fast multipole method
  • High-order accuracy
  • Integral equations
  • Layer potentials
  • Quadrature
  • Singular integrals

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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