On a Low-Frequency and Refinement Stable PMCHWT Integral Equation Leveraging the Quasi-Helmholtz Projectors

Abstract

Classical Poggio-Miller-Chan-Harrington-Wu-Tsai (PMCHWT) formulations for modeling radiation and scattering from penetrable objects suffer from ill-conditioning when the frequency is low or when the mesh density is high. The most effective techniques to solve these problems, unfortunately, either require the explicit detection of the so-called global loops of the structure, or suffer from numerical cancellation at extremely low frequency. In this contribution, a novel regularization method for the PMCHWT equation is proposed, which is based on the quasi-Helmholtz projectors. This method not only solves both the low frequency and the dense mesh ill-conditioning problems of the PMCHWT, but it is immune from low-frequency numerical cancellations and it does not require the detection of global loops. This is obtained by projecting the range space of the PMCHWT operator onto a dual basis, by rescaling the resulting quasi-Helmholtz components, by replicating the strategy in the dual space, and finally, by combining the primal and the dual equations in a Calder�n-like fashion. Implementation-related treatments and details alternate the theoretical developments in order to maximize impact and practical applicability of the approach. Finally, numerical results corroborate the theory and show the effectiveness of the new schemes in real case scenarios.