Overcoming low-frequency breakdown of the magnetic field integral equation

In the electromagnetics literature, significant attention has been paid to the problem of low-frequency breakdown in the electric field integral equation. By contrast, the magnetic field integral equation is well-conditioned (in simply connected domains) and can be used in the low frequency limit without modification or preconditioning. Reconstruction of the electric field, however, is subject to catastrophic cancellation unless appropriate measure are taken. In this paper, we show that solving an auxiliary (scalar) integral equation for the charge overcomes this form of low frequency breakdown, both in the near and far fields. Moreover, both the current and charge can be discretized using simple piecewise polynomial basis functions on triangulated surfaces. We also analyze an alternative formulation involving magnetic current and charge and illustrate the performance of the methods with several numerical examples.