The surface integral equation (SIE) method, discretized with the method of moments, is a well-established methodology for the scattering analysis of subwavelength plasmonic nanoparticles. SIEs are usually discretized with low-order basis functions that preserve the normal continuity of the surface currents across the edges arising in the meshed boundary, such as Rao-Wilton-Glisson (RWG) functions. However, the plasmonic enhancement modeling on sharp-edged particles is an extremely challenging task, especially due to the singular fields exerted at sharp corners, exposing a slow (or no) convergence in the computation of the scattering and absorption spectra. In this paper, we propose an alternative discretization strategy based on a discontinuous basis function set in conjunction with a volumetric-tetrahedral testing scheme. We demonstrate the potential of the proposed discretization scheme by studying scattering and absorption spectra of three canonical plasmonic polyhedra, i.e., a hexahedral, an octahedral, and a tetrahedral silver inclusion. The results expose an improved accuracy and faster convergence in both far-field and near-field regions when compared to the standard RWG implementation. The proposed discretization scheme can offer faster and more accurate routes towards the exploration and design of the plasmonic resonant spectrum of sharp-edged nanoparticles and nanoantennas.