Vertex Singularities for the 3-D Laplace Equation

Although singular points in 2-D domains have been extensively investigated, the vertex singularities in 3-D domains have received scant attention due to their complexity. To the best of our knowledge, numerical methods for the investigation of vertices of conical notches, specifically the exponents of the singularity, were first introduced in [23]. Stephan and Whiteman [170] and Beagles and Whiteman [25] investigated analytically several vertices for the Laplace equation in 3-D, mainly with homogeneous Dirichlet boundary conditions, and analyzed a finite element method for the computation of eigenvalues by discretizing the Laplace-Beltrami equation (error estimates provided but no numerical results).