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Multi-trace boundary integral equations

by X. Claeys and R. Hiptmair and C. Jerez-Hanckes

(Report number 2012-20)

Abstract
We consider the scattering of acoustic or electromagnetic waves at a penetrable object composed of different homogeneous materials. This problem can be recast as a first-kind boundary integral equation posed on the interface trace spaces through what we call a single trace boundary integral equation formulation (STF). Its Ritz-Galerkin discretization by means of low-order piecewise polynomial boundary elements on fine interface triangulations leads to ill-conditioned linear systems of equations, which defy efficient iterative solution. Powerful preconditioners for discrete boundary integral equations are provided by the policy of operator preconditioning provided that the underlying trace spaces support a duality pairing with L2 pivot space. This condition is not met by the STF. As a remedy we have proposed two variants of new multi-trace boundary integral equations (MTF); whereas the STF features unique Cauchy traces on material domain interfaces as unknowns, the multi-trace approach tears apart the traces so that local traces are recovered. Local trace spaces are in duality with respect to the L2 -pairing, and, thus, operator preconditioning becomes available for MTF.

Keywords: Helmholtz equation, Maxwell’s equation, transmission problems, boundary integral equations, PMCHWT, operator preconditioning, boundary elements, multi-trace formulations.

@Techreport{CHJ12_463,
  author = {X. Claeys and R. Hiptmair and C. Jerez-Hanckes},
  title = {Multi-trace boundary integral equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2012-20},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2012/2012-20.pdf },
  year = {2012}
}

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