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Second Kind Boundary Integral Equation for Multi-Subdomain Diffusion Problems

by X. Claeys and R. Hiptmair and E. Spindler

(Report number 2016-44)

Abstract
We consider isotropic scalar diffusion boundary value problems on \(\mathbb{R}^d\), whose diffusion coefficients are piecewise constant with respect to a partition of space into Lipschitz subdomains. We allow so-called material junctions where three or more subdomains may abut. We derive a boundary integral equation of the second kind posed on the skeleton of the subdomain partition that involves, as unknown, only one trace function at each point of each interface. We prove the well-posedness of the corresponding boundary integral equations. We also report numerical tests for Galerkin boundary element discretisations, in which the new approach proves to be highly competitive compared to the well-established first kind direct single-trace boundary integral formulation. In particular, GMRES seems to enjoy fast convergence independent of the mesh resolution for the discrete second kind BIE.

Keywords: Second-order transmission problems, boundary integral equations, second kind single integral equations, boundary element methods

@Techreport{CHS16_681,
  author = {X. Claeys and R. Hiptmair and E. Spindler},
  title = {Second Kind Boundary Integral Equation for Multi-Subdomain Diffusion Problems},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2016-44},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2016/2016-44.pdf },
  year = {2016}
}

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