  We investigate in a $2$D setting the scattering of time-harmonic
electromagnetic waves by a plasmonic device, represented as a non dissipative
bounded and penetrable obstacle with a negative permittivity. Using the
$\textrm{T}$-coercivity approach, we first prove that the problem is well-posed
in the classical framework $H^1_{\text{loc}} $ if the negative permittivity
does not lie in some critical interval whose definition depends on the shape of
the device. When the latter has corners, for values inside the critical
interval, unusual strong singularities for the electromagnetic field can
appear. In that case, well-posedness is obtained by imposing a radiation
condition at the corners to select the outgoing black-hole plasmonic wave, that
is the one which carries energy towards the corners. A simple and systematic
criterion is given to define what is the outgoing solution. Finally, we propose
an original numerical method based on the use of Perfectly Matched Layers at
the corners. We emphasize that it is necessary to design an $\textit{ad hoc}$
technique because the field is too singular to be captured with standard finite
element methods.
