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Second kind boundary integral equation for multi-subdomain diffusion problems

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Abstract

We consider isotropic scalar diffusion boundary value problems 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.

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Acknowledgments

The work of X.Claeys received support from the ANR research grant ANR-15-CE23-0017-01. E. Spindler was supported by SNF grant 200021_137873/1.

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Authors and Affiliations

  1. Sorbonne Universités, UPMC Université Paris 6 Laboratoire Jacques-Louis Lions, CNRS UMR 7598, Paris, France

    X. Claeys

  2. INRIA Paris, EPC Alpines, Paris, France

    X. Claeys

  3. Seminar of Applied Mathematics, ETH Zurich, Zürich, Switzerland

    R. Hiptmair & E. Spindler

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  1. X. Claeys
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  2. R. Hiptmair
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  3. E. Spindler
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Correspondence to X. Claeys.

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Communicated by: Leslie Greengard

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Claeys, X., Hiptmair, R. & Spindler, E. Second kind boundary integral equation for multi-subdomain diffusion problems. Adv Comput Math 43, 1075–1101 (2017). https://doi.org/10.1007/s10444-017-9517-0

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