Stationary iteration methods for solving 3D electromagnetic scattering problems

• Stationary iterations constitute most efficient approach for numerical solution of VSIEs describing wave scattering by dielectric structures.
• A finite algorithm of finding optimal iteration parameter for arbitrary spectrum localization has been constructed.
• Sufficient conditions imposed on the domain of the spectrum localization have been obtained providing convergence of iterations.
• We have shown that if the operator spectrum is a circle on the complex plane, all iteration parameters are equal to GSI optimal iteration parameter.
• We have confirmed numerically that greater dimensions lead to increasing the memory volume rather than to improvement of convergence.

Generalized Chebyshev iteration (GCI) applied for solving linear equations with nonselfadjoint operators is considered. Sufficient conditions providing the convergence of iterations imposed on the domain of localization of the spectrum on the complex plane are obtained. A minimax problem for the determination of optimal complex iteration parameters is formulated. An algorithm of finding an optimal iteration parameter in the case of arbitrary location of the operator spectrum on the complex plane is constructed for the generalized simple iteration method. The results are applied to numerical solution of volume singular integral equations (VSIEs) associated with the problems of the mathematical theory of wave diffraction by 3D dielectric bodies. In particular, the domain of the spectrum location is described explicitly for low-frequency scattering problems and in the general case. The obtained results are discussed and recommendations concerning their applications are given.