===affil2:
Institut de Math\'{e}matiques de Bordeaux
===firstname:
James
===firstname4:

===firstname3:
Ludmil
===lastname2:
Petkov
===lastname:
Adler
===firstname5:

===affil6:

===lastname3:
Zikatanov
===email:
james.adler@tufts.edu
===lastname6:

===affil5:

===otherauths:

===lastname4:

===affil4:

===lastname7:

===affil7:

===firstname7:

===postal:
503 Boston Ave.
Tufts University
Mathematics Department
Medford, MA 02155
===firstname6:

===ABSTRACT:
This work is on the numerical approximation of incoming solutions to Maxwell's equations, whose energy decays exponentially with time (asymptotically disappearing), meaning that the leading term of the back-scattering matrix becomes negligible.  For the exterior of a sphere, such solutions are obtained by Colombini, Petkov and Rauch by specifying a maximal dissipative boundary condition on the sphere and setting appropriate initial conditions.

We consider a mixed finite element approximation of Maxwell's equations in the exterior of a polyhedron whose boundary approximates the sphere.  We use the standard Nedelec-Raviart-Thomas elements and a Crank-Nicholson scheme to approximate the electric and magnetic fields.  We set discrete initial conditions with standard interpolation, modified so that these initial conditions are divergence-free. We prove that with such initial conditions, the fully discrete approximation to the electric field is weakly divergence-free for all time.  We show numerically that the finite-element solution approximates well the asymptotically disappearing solutions constructed analytically when the mesh size becomes small.
===affil3:
The Pennsylvania State University
===lastname5:

===affilother:

===title:
Numerical Approximation of Asymptotically Disappearing Solutions of Maxwell’s Equations
===firstname2:
Vesselin