The spectrum of the MFIE and Calderon preconditioned EFIE for scattering by two-dimensional non-smooth objects

The mixed MFIE and Caldeŕon preconditioned EFIE both can be used to accurately model the scattering of time-harmonic electromagnetic waves by two-dimensional perfect electrical conductors. In the case those conductors are bounded by smooth surfaces, the spectra of the linear systems are clustered around a single non-zero finite value. This configuration is optimal for the iterative solution of these systems by iterative algorithms. Regrettably, it has been demonstrated that this optimal configuration is lost when the methods are applied to scattering by non-smooth surfaces. In this case, the spectrum tends to spread out, negatively influencing the number of iterations required for iterative solvers to converge. In this contribution, this spreading out of the spectrum is studied quantitatively. It is shown that even though the spectrum spreads out, it remains bounded away from zero and oriented along the negative real axis. It can be concluded that iterative solution remains an option, even for non-smooth geometries. In the case the geometry is so complicated that the spectrum is bounded away from zero by only a very small distance, further preconditioning may be required. Here, a quasi-block diagonal preconditioner is introduced that will compress the spectrum. It is explained how this preconditioner can be applied efficiently as expansion in a Neumann series.