Stationary iteration methods for solving 3D electromagnetic scattering problems

TL;DR

This paper develops and analyzes stationary iteration methods, including GCI, for solving 3D electromagnetic scattering problems, focusing on convergence conditions, optimal parameters, and applications to volume integral equations.

Contribution

It introduces new convergence criteria, formulates a minimax problem for optimal iteration parameters, and applies these methods to 3D wave diffraction problems.

Findings

Convergence conditions depend on spectrum localization in the complex plane.

An algorithm for optimal iteration parameters is constructed.

Results are applied to low-frequency scattering and VSIEs.

Abstract

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.