Numerical noise in analytical evaluation of MoM integrals

In the Method of Moments (MoM), the evaluation of near-field interactions is a numerically complex task due to the singularity of the free-space Green's function. Recently, an analytical evaluation method has been proposed to evaluate these interactions. While being theoretically exact, the method deals with recursive formulas that can be ill-conditioned in specific cases when dealing with finite machine precision. In this work, we study the numerical stability of the method. We show that large errors can appear when one edge of the basis function (BF) is nearly parallel to the testing function (TF), and vice-versa. The error depends on the relative distance between the BF and the TF, their polynomial order and their relative orientation. Asymptotic behaviours are extracted from the numerical experiments for the different cases considered. We conclude that, provided that relative angles below 1 are avoided and that first-order basis and testing functions are used, the method provides accurate results for basis and testing functions separated by typically less than two or three other basis functions, making it suitable to evaluate near-field interactions.