On using charge as an additional unknown in the EFIE-hd to improve mesh stability

It has been recently observed that a new electric field integral equation (EFIE) based formulation that relies on the Helmholtz decomposition (HD) (EFIE-hd) of the current can overcome the low frequency breakdown problem of the EFIE. It has been demonstrated that the EFIE-hd is frequency stable and provides accurate solutions for the electric and magnetic fields at high and low frequencies. While the resulting EFIE-hd is frequency stable, it is not stable with respect to mesh refinement. The purpose of this work is to obtain an improved formulation that is also stable with mesh refinement by augmenting the original EFIE-hd with the continuity equation (referred to as EFIE-hdc) and including charge as additional set of unknowns with appropriate diagonal scaling.