  We present a Calder\'on preconditioner for the electric field integral
equation (EFIE), which does not require a barycentric refinement of the mesh
and which yields a Hermitian, positive definite (HPD) system matrix allowing
for the usage of the conjugate gradient (CG) solver. The resulting discrete
equation system is immune to the low-frequency and the dense-discretization
breakdown and, in contrast to existing Calder\'on preconditioners, no second
discretization of the EFIE operator with Buffa-Christiansen (BC) functions is
necessary. This preconditioner is obtained by leveraging on spectral
equivalences between (scalar) integral operators, namely the single layer and
the hypersingular operator known from electrostatics, on the one hand, and the
Laplace-Beltrami operator on the other hand. Since our approach incorporates
Helmholtz projectors, there is no search for global loops necessary and thus
our method remains stable on multiply connected geometries. The numerical
results demonstrate the effectiveness of this approach for both canonical and
realistic (multi-scale) problems.
