The Poisson equation in axisymmetric domains with conical points (Q1765449)

The author presents an analysis of homogeneous Dirichlet problems for the Poisson equation in axisymmetric domains of \(\mathbb{R}^3 \) with conical points on the rotation axis. Starting with the Fourier coefficients of the solution the regularity of the problem is investigated. A description of the asymptotic behaviour of the coefficients near the conical points shows that the singularities are related to the roots of some transcendental equations involving Legendre functions. Furthermore an a priori estimate for the hybrid Fourier-finite element solution is given.