Numerically stable analytical computation of double surface integrals over almost parallel flat supports for time and frequency domain Galerkin discretization of wave scattering problems.

In this contribution, we consider recently proposed analytical methods for the exact evaluation of double surface integrals present in the Galerkin discretization of acoustic and electromagnetic scattering, in both frequency and time domain cases. Either in the frequency or time domain, it is possible to obtain a closed form of the double surface integrals of the impedance matrix. However, the result of this evaluation is affected by numerical instabilities for certain geometrical configurations of the integration domain. We analyze the cause of such instabilities. In particular, we focus on the case of almost parallel domains of integration and illustrate why it is necessary to take care of this configuration when dealing with large-scale problems. Although past works discussed this problem in the frequency domain, we show that time domain analytical methods require a detailed analysis for analogous configurations. We propose a strategy that allows to stabilize the calculation. The method is based on the analysis of the explicit dependence of the formula on the relative angle between the two domains of integration. It furnishes a stable result for small angles by avoiding the computation of differences between big numbers in the formula.