Numerical evaluation of the eigenexpansion of the electrostatic potential near a dielectric cone

The purpose of this paper is to elaborate on some numerical aspects of the evaluation of the eigenfunction expansion for the electrostatic potential near a dielectric cone of circular cross-section. This expansion has been derived analytically by the authors in a recent article [SIAM J. Appl. Math., 70 (2010), pp. 2329-2352] with the aid of the two-interval Sturm-Liouville theory. In our effort to implement numerically the aforesaid expansion we discovered a numerical instability on the spherical surface which is centered at the vertex of the cone and passes through the point source. In this paper we present an analytical way of removing that instability. Furthermore, we propose some numerical methods for calculating the terms of the eigenfunction expansion and we present some numerical results in the form of equipotential contour plots.