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Quotient-Space Boundary Element Methods for Scattering at Complex Screens

by X. Claeys and L. Giacomel and R. Hiptmair and C. Urzua-Torres

(Report number 2020-11)

Abstract
A complex screen is an arrangement of panels that may not be even locally orientable because of junction lines. A comprehensive trace space framework for first-kind variational boundary integral equations on complex screens has been established in \([\) X. Claeys and R. Hiptmair, Integral equations on multi-screens, Integral Equations and Operator Theory, 77 (2013), pp. 167-197\(]\) for the Helmholtz equation, and in \([\) X. Claeys and R. Hiptmair, Integral equations for electromagnetic scattering at multi-screens, Integral Equations and Operator Theory, 84 (2016), pp. 33-68\(]\) for Maxwell's equations in frequency domain. The gist is a quotient space perspective that allows to make sense of jumps of traces as factor spaces of multi-trace spaces modulo single-trace spaces without relying on orientation. This paves the way for formulating first-kind boundary integral equations in weak form posed on energy trace spaces. In this article we extend that idea to the Galerkin boundary element (BE) discretization of first-kind boundary integral equations. Instead of trying to approximate jumps directly, the new quotient space boundary element method employs a Galerkin BE approach in multi-trace boundary element spaces. This spawns discrete boundary integral equations with large null spaces comprised of single-trace functions. Yet, since the right-hand-sides of the linear systems of equations are consistent, Krylov subspace iterative solvers like GMRES are not affected by the presence of a kernel and still converge to a solution. This is strikingly confirmed by numerical tests.

Keywords: Complex screens, Galerkin Boundary Element Method, Quotient Space Boundary Element Method

@Techreport{CGHU20_884,
  author = {X. Claeys and L. Giacomel and R. Hiptmair and C. Urzua-Torres},
  title = {Quotient-Space Boundary Element Methods for Scattering at Complex Screens},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2020-11},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2020/2020-11.pdf },
  year = {2020}
}

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