  We consider time-harmonic scalar transmission problems between dielectric and
dispersive materials with generalized Lorentz frequency laws. For certain
frequency ranges such equations involve a sign-change in their principle part.
Due to the resulting loss of coercivity properties, the numerical simulation of
such problems is demanding. Furthermore, the related eigenvalue problems are
nonlinear and give rise to additional challenges. We present a new finite
element method for both of these types of problems, which is based on a weakly
coercive reformulation of the PDE. The new scheme can handle
$C^{1,1}$-interfaces consisting piecewise of elementary geometries. Neglecting
quadrature errors, the method allows for a straightforward convergence
analysis. In our implementation we apply a simple, but nonstandard quadrature
rule to achieve negligible quadrature errors. We present computational
experiments in 2D and 3D for both source and eigenvalue problems which confirm
the stability and convergence of the new scheme.
