A Domain Derivative for Electromagnetic Scattering by Perfect Conductors in the Time Domain

Abstract

A domain derivative for time-dependent electromagnetic scattering from perfect conductors is established. By proceeding through the Laplace domain, frequency-dependent bounds on solutions to Maxwell's equations are derived. These bounds are used both for establishing time regularity properties of the domain derivative and for proving convergence of the proposed Runge--Kutta convolution quadrature semi-discretization in time. A full convergence analysis is also carried out for pointwise evaluations of the domain derivative, when this time discretization is combined with a Galerkin method in space. Numerical experiments validate the convergence orders in time and space. Eventually, the domain derivative is applied in an iterative shape reconstruction algorithm, in which the electric near field is measured at some receiver positions away from the perfect conductor. Numerical examples show the feasibility of this algorithm and in particular highlight its robustness, when additional noise is applied to the data.

arXiv commentCompared to the first submission, many incorrect sign errors, but also errors in frequency dependent bounds have been corrected. The higher order regularity result is now stated more precisely, which facilitates its use in Section 4. Finally, Section 5 has been reworked and includes now a numerical verification of the error bounds of Theorem 4.9