  One way of improving the behavior of finite element schemes for classical,
time-dependent Maxwell's equations, is to render them from their hyperbolic
character to elliptic form. This paper is devoted to the study of the
stabilized linear finite element method for the time harmonic Maxwell's
equations in a dual form obtained through the Laplace transformation in time.
The model problem is for the particular case of the dielectric permittivity
function which is assumed to be constant in a boundary neighborhood. For the
stabilized model a coercivity relation is derived that guarantee's the
existence of a unique solution for the iscrete problem. The convergence is
addressed both in a priori and a posteriori settings. In the a priori error
estimates we confirm the theoretical convergence of the scheme in a L2-based,
gradient dependent, triple norm. The order of convergence is
  O(h) in weighted Sobolev space and hence optimal. We also derive, similar,
optimal a posteriori error estimates controlled by a certain, weighted, norm of
the residual of the computed solution. The posteriori approach is used for
constructing adaptive algorithms for the computational purposes. Further,
assuming a sufficiently regular solution for the dual problem, we reach the
same convergence of
  O(h). Finally, through implementing several numerical examples, we validate
the robustness of the proposed scheme.
