  This work is on the numerical approximation of incoming solutions to
Maxwell's equations with dissipative boundary conditions whose energy decays
exponentially with time. Such solutions are called asymptotically disappearing
(ADS) and they play an importarnt role in inverse back-scatering problems. The
existence of ADS is a difficult mathematical problem. For the exterior of a
sphere, such solutions have been constructed analytically by Colombini, Petkov
and Rauch [7] by specifying appropriate initial conditions. However, for
general domains of practical interest (such as Lipschitz polyhedra), the
existence of such solutions is not evident.
  This paper considers a finite-element approximation of Maxwell's equations in
the exterior of a polyhedron, whose boundary approximates the sphere. Standard
Nedelec-Raviart-Thomas elements are used with a Crank-Nicholson scheme to
approximate the electric and magnetic fields. Discrete initial conditions
interpolating the ones chosen in [7] are modified so that they are (weakly)
divergence-free. We prove that with such initial conditions, the approximation
to the electric field is weakly divergence-free for all time. Finally, we show
numerically that the finite-element approximations of the ADS also decay
exponentially with time when the mesh size and the time step become small.
