  The aim of this article is to investigate the well-posedness, stability and
convergence of solutions to the time-dependent Maxwell's equations for electric
field in conductive media in continuous and discrete settings. The situation we
consider would represent a physical problem where a subdomain is emerged in a
homogeneous medium, characterized by constant dielectric permittivity and
conductivity functions. It is well known that in these homogeneous regions the
solution to the Maxwell's equations also solves the wave equation which makes
calculations very efficient. In this way our problem can be considered as a
coupling problem for which we derive stability and convergence analysis. A
number of numerical examples validate theoretical convergence rates of the
proposed stabilized explicit finite element scheme.
