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Novel Multi-Trace Boundary Integral Equations for Transmission Boundary Value Problems

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

(Report number 2014-05)

Abstract
We consider scalar 2nd-order transmission problems in the exterior of a bounded domain \(\Omega_{Z}\subset\mathbb{R}^{d}\). The coefficients are assumed to be piecewise constant with respect to a partition of \(\mathbb{R}^{d}\setminus\overline{\Omega}_{Z}\) into subdomains. Dirichlet boundary conditions are imposed on \(\partial\Omega_{Z}\). We recast the transmission problems into two novel well-posed multi-trace boundary integral equations. Their unknowns are functions on the product of subdomain boundaries. Compared to conventional single-trace formulations they offer the big benefit of being amenable to operator preconditioning. We outline the analysis of the new formulations, give the details of operator preconditioning applied to them, and, for one type of a multi-trace formulation, report numerical tests confirming the efficacy of operator preconditioning.

Keywords: Multi-trace boundary integral equations; boundary element methods; 1st-kind integral equations; operator preconditioning; domain decomposition

@Techreport{CHJP14_555,
  author = {X. Claeys and R. Hiptmair and C. Jerez-Hanckes and S. Pintarelli},
  title = {Novel Multi-Trace Boundary Integral Equations for Transmission Boundary Value Problems},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2014-05},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2014/2014-05.pdf },
  year = {2014}
}

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