On the Evaluation of the 4-D Reaction Integral for the Scalar Potential in Galerkin’s Method of Moments

Despite the great progress made in the evaluation of the 4-D reaction integrals arising in the method of moments applied to the surface integral equation formulations, there are still some aspects that deserve to be explored further. The focus in this paper lies in the evaluation of the 4-D reaction integral for the scalar potential. We demonstrate that, unless this integral is computed to machine precision in the whole impedance matrix, it is imperative to maintain the trianglewise balance in the evaluation of the scalar potential part. A small imbalance, even at very small error, results in a loss of degrees of freedom (DOFs) for the charge distribution, violating the null-space property of the divergence operator and yielding potentially spurious solutions. The effect, although present, may go unnoticed at the conventional frequency regimes, where the usual mesh sizes can be applied. However, it becomes more pronounced as we approach the quasi-static limit, eventually rendering spurious solutions at the frequencies higher than the classical low-frequency breakdown would.