  Numerical calculation of modes in dispersive and absorptive systems is
performed using the finite element method. The dispersion is tackled in the
frame of an extension of Maxwell's equations where auxiliary fields are added
to the electromagnetic field. This method is applied to multi-domain cavities
and photonic crystals including Drude and Drude-Lorentz metals. Numerical
results are compared to analytical solutions for simple cavities and to
previous results of the literature for photonic crystals, showing excellent
agreement. The advantages of the developed method lie on the versatility of the
finite element method regarding geometries, and in sparing the use of tedious
complex poles research algorithm. Hence the complex spectrum of resonances of
non-hermitian operators and dissipative systems, like two-dimensional photonic
crystal made of absorbing Drude metal, can be investigated in detail. The
method is used to reveal unexpected features of their complex band structures.
