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

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 A_iρ ^{ν i}f_i(θ,φ). For conical vertices, we derive explicit analytical expressions for the eigenpairs ν_i 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.