Regulated kernel for the electric field integral equation

We provide a theoretical justification for the filtered kernel of Herrmann and Strain (1993), and show how to extend the method to three-dimensional problems. The solution error for the moment method consists of two components: approximation error, or the error incurred by projecting the exact solution into the space spanned by the basis functions, and sampling error due to the aliasing of high order eigenfunctions associated with the singularity of the kernel. For the surface current, the approximation error is dominant. Specular scattering cross-sections of smooth scatterers, on the other hand, are only sensitive to the sampling error. Thus, if spectral content beyond the Nyquist frequency of the mesh is filtered out, the cross section error can in principle be significantly reduced. In addition, the regulated kernel is smooth, and hence can be accurately integrated using lower order quadrature rules, reducing the computational cost of near-neighbor matrix elements, and simplifying the implementation of self-term integrals.