A fast algorithm with error bounds for Quadrature by Expansion

Research output: Contribution to journalArticle

Abstract

Quadrature by Expansion (QBX) is a quadrature method for approximating the value of the singular integrals encountered in the evaluation of layer potentials. It exploits the smoothness of the layer potential by forming locally-valid expansions which are then evaluated to compute the near or on-surface value of the potential. Recent work towards coupling of a Fast Multipole Method (FMM) to QBX yielded a first step towards the rapid evaluation of such integrals (and the solution of related integral equations), albeit with only empirically understood error behavior. In this paper, we improve upon this approach with a modified algorithm for which we give a comprehensive analysis of error and cost in the case of the Laplace equation in two dimensions. For the same levels of (user-specified) accuracy, the new algorithm empirically has cost-per-accuracy comparable to prior approaches. We provide experimental results to demonstrate scalability and numerical accuracy.

Original languageEnglish (US)
Pages (from-to)135-162
Number of pages28
JournalJournal of Computational Physics
Volume374
DOIs
StatePublished - Dec 1 2018

Keywords

  • Fast algorithms
  • Fast multipole method
  • Integral equations
  • Quadrature
  • Singular integrals

ASJC Scopus subject areas

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

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