Vertex singularities associated with conical points for the 3D Laplace equation

T. Zaltzman, Z. Yosibash

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


The solution u to the Laplace equation in the neighborhood of a vertex in a three-dimensional domain may be described by an asymptotic series in terms of spherical coordinates u = ∑i Aiρ ν ifi(θ,φ). For conical vertices, we derive explicit analytical expressions for the eigenpairs νi and f i(θ,δ), which are required as benchmark solutions for the verification of numerical methods. Thereafter, we extend the modified Steklov eigen-formulation for the computation of vertex eigenpairs using p/spectral finite element methods and demonstrate its accuracy and high efficiency by comparing the numerically computed eigenpairs to the analytical ones. Vertices at the intersection of a crack front and a free surface are also considered and numerical eigenpairs are provided. The numerical examples demonstrate the efficiency, robustness, and high accuracy of the proposed method, hence its potential extension to elasticity problems.

Original languageEnglish
Pages (from-to)662-679
Number of pages18
JournalNumerical Methods for Partial Differential Equations
Issue number3
StatePublished - 1 May 2011


  • Laplace equation
  • Steklov method
  • vertex singularities

ASJC Scopus subject areas

  • Analysis
  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics


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