We have identified the sources of the different problems plaguing the EFIE at low frequencies in both the frequency and the TD, as well as their traditional cures. Despite their apparent effectiveness, these techniques have been shown to have a limited applicability because they introduce their own set of problems which include the high computational burden of the LS decomposition and its effect on the high-refinement conditioning of the FD-EFIE and the numerical instabilities introduced by the treatment of the TD-EFIE. Techniques leveraging qH projectors, immune from the aforementioned side-effects, have been introduced to address the different aspects of the low-frequency breakdown of the FD formulation and of the large time step breakdown of its TD counterpart. In case of the FD, using projectors allows the same re-scaling of the solenoidal and non-solenoidal parts of the RWG space as traditional LS, but it has the added benefits of not requiring identification of the global loops of the structure as well as not introducing any further high-refinement ill-conditioning. In the TD case, the projectors are still used to separate the loop and star parts of the discretized space, but this separation is used to apply the correct derivative and integrative terms to the different parts of the operators. Coupled with an adequate mixed time-discretization scheme, this technique fully addresses the low-frequency limitations of the TD-EFIE. Along with presenting these purely theoretical concepts, we have provided implemen-tation related hints, allowing the techniques presented in this chapter to be reliably and readily implemented into existing solvers. Finally, while we have addressed their low-frequency breakdown, both EFIE formulations still suffer from a high-refinement breakdown. While in standard low-frequency scenarios, a curing of low-frequency issues may suffice, for more pathological cases techniques addressing both break-downs may be required. Strategies based on qH projectors and Calderon identities have recently been introduced for the frequency and TD formulations [23, 40] and should be used in this case.

Boundary integral equation methods for analyzing electromagnetic scattering phenomena typically suffer from several of the following shortcomings: 1) ill-conditioning when the frequency is low; 2) ill-conditioning when the discretization density is high; 3) ill-conditioning when the structure contains global loops (which are computationally expensive to detect); 4) incorrect solution at low frequencies due to a loss of significant digits; and 5) the presence of spurious resonances. In this article, quasi-Helmholtz projectors are leveraged to obtain magnetic field integral equation (MFIE) that is immune to drawbacks 1)-4). Moreover, when this new MFIE is combined with a regularized electric field integral equation (EFIE), a new quasi-Helmholtz projector-combined field integral equation (CFIE) is obtained that also is immune to 5). The numerical results corroborate the theory and show the practical impact of the newly proposed formulations.