Debye Sources and the Numerical Solution of the Time Harmonic Maxwell Equations
This work provides a robust numerical method for electromagnetic scattering problems, addressing known bottlenecks like spurious resonances and low-frequency breakdown, which is important for computational electromagnetics.
The paper introduces a new representation for outgoing solutions to time harmonic Maxwell equations, leading to a Fredholm integral equation of the second kind for scattering from perfect conductors that avoids spurious resonances and low-frequency breakdown. It also proves the existence of non-trivial families of time harmonic solutions with vanishing normal components on non-simply connected boundaries, termed k-Neumann fields.
In this paper, we develop a new representation for outgoing solutions to the time harmonic Maxwell equations in unbounded domains in $\bbR^3.$ This representation leads to a Fredholm integral equation of the second kind for solving the problem of scattering from a perfect conductor, which does not suffer from spurious resonances or low frequency breakdown, although it requires the inversion of the scalar surface Laplacian on the domain boundary. In the course of our analysis, we give a new proof of the existence of non-trivial families of time harmonic solutions with vanishing normal components that arise when the boundary of the domain is not simply connected. We refer to these as $k$-Neumann fields, since they generalize, to non-zero wave numbers, the classical harmonic Neumann fields. The existence of $k$-harmonic fields was established earlier by Kress.