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Preconditioning the EFIE on Screens

by R. Hiptmair and C. Urzua-Torres

(Report number 2019-44)

Abstract
We consider the electric field integral equation (EFIE) modeling the scattering of time-harmonic electromagnetic waves at a perfectly conducting screen. When discretizing the EFIE by means of low-order Galerkin boundary methods (BEM), one obtains linear systems that are ill-conditioned on fine meshes and for low wave numbers \(k\). This makes iterative solvers perform poorly and entails the use of preconditioning. In order to construct optimal preconditioners for the EFIE on screens, the authors recently derived compact equivalent inverses of the EFIE operator on simple Lipschitz screens in [R. Hiptmair and C. Urzúa-Torres, Compact Equivalent Inverse of the Electric Field Integral Operator on Screens, Report 2018-46, Seminar for Applied Mathematics, ETH Zürich, 2018]. This paper elaborates how to use this result to build an optimal operator preconditioner for the EFIE on screens that can be discretized in a stable fashion. Furthermore, the stability of the preconditioner relies only on the stability of the discrete L2 duality pairing for scalar functions, instead of the vectorial one. Therefore, this novel approach not only offers h-independent and \(k\)-robust condition numbers, but it is also easier to implement and accommodates non-uniform meshes without additional computational effort.

Keywords: Electric field integral equation; Screens; Operator preconditioning

@Techreport{HU19_848,
  author = {R. Hiptmair and C. Urzua-Torres},
  title = {Preconditioning the EFIE on Screens},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2019-44},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2019/2019-44.pdf },
  year = {2019}
}

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