Asymptotic and numerical analysis for Holland and Simpson's thin wire formalism

In the context of simulation of electromagnetic propagation, the thin wire formalism of Holland and Simpson allows one to deal with scattering by perfectly conducting thin wires by coupling a standard FDTD method with a discrete 1D wave equation ruling the current inside the wires. This method can be very accurate, but it involves a fitting parameter that requires careful calibration. We propose a consistency analysis and derive a formula for the calibration of this parameter in the case of a simplified 2D analogue of the method of Holland and Simpson. Our proof relies on the observation that this method is actually a hidden version of the singular function method well known in the context of elliptic equations in domains with a singular boundary.