Numerical approximation of asymptotically disappearing solutions of maxwell's equations

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 important role in inverse back-scattering 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 [Proc. Amer. Math. Soc., 139 (2011), pp. 2163-2173] 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 Nédélec-Raviart- Thomas elements are used with a Crank-Nicolson scheme to approximate the electric and magnetic fields. Discrete initial conditions interpolating the ones chosen by Colombini, Petkov, and Rauch 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 approximates this exponential decay (quadratically) with time when the mesh size and the time step become small.