Advances in Computational Mathematics

, Volume 43, Issue 1, pp 195–234 | Cite as

Error estimation for quadrature by expansion in layer potential evaluation

  • Ludvig af Klinteberg
  • Anna-Karin Tornberg
Open Access


In boundary integral methods it is often necessary to evaluate layer potentials on or close to the boundary, where the underlying integral is difficult to evaluate numerically. Quadrature by expansion (QBX) is a new method for dealing with such integrals, and it is based on forming a local expansion of the layer potential close to the boundary. In doing so, one introduces a new quadrature error due to nearly singular integration in the evaluation of expansion coefficients. Using a method based on contour integration and calculus of residues, the quadrature error of nearly singular integrals can be accurately estimated. This makes it possible to derive accurate estimates for the quadrature errors related to QBX, when applied to layer potentials in two and three dimensions. As examples we derive estimates for the Laplace and Helmholtz single layer potentials. These results can be used for parameter selection in practical applications.


Nearly singular Quadrature Layer potential Error estimate 

Mathematics Subject Classification (2010)

65D30 65D32 65G99 


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Copyright information

© The Author(s) 2016

Authors and Affiliations

  1. 1.Numerical Analysis, Department of MathematicsKTH Royal Institute of TechnologyStockholmSweden

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