Harmonic density interpolation methods for high-order evaluation of Laplace layer potentials in 2D and 3D

Carlos Pérez-Arancibia*, Luiz M. Faria, Catalin Turc

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

16 Citations (Scopus)


We present an effective harmonic density interpolation method for the numerical evaluation of singular and nearly singular Laplace boundary integral operators and layer potentials in two and three spatial dimensions. The method relies on the use of Green's third identity and local Taylor-like interpolations of density functions in terms of harmonic polynomials. The proposed technique effectively regularizes the singularities present in boundary integral operators and layer potentials, and recasts the latter in terms of integrands that are bounded or even more regular, depending on the order of the density interpolation. The resulting boundary integrals can then be easily, accurately, and inexpensively evaluated by means of standard quadrature rules. A variety of numerical examples demonstrate the effectiveness of the technique when used in conjunction with the classical trapezoidal rule (to integrate over smooth curves) in two-dimensions, and with a Chebyshev-type quadrature rule (to integrate over surfaces given as unions of non-overlapping quadrilateral patches) in three-dimensions.
Original languageEnglish
Pages (from-to)411-434
Number of pages24
JournalJournal of computational physics
Early online date3 Oct 2018
Publication statusPublished - 1 Jan 2019
Externally publishedYes


  • Boundary integral operators
  • Harmonic polynomials
  • Laplace equation
  • Layer potentials
  • Nyström method
  • Taylor interpolation


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