Multitrace Formulations and Dirichlet-Neumann Algorithms

  • Victorita DoleanEmail author
  • Martin J. Gander
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 104)


Multitrace formulations (MTF) for boundary integral equations (BIE) were developed over the last few years in [1, 2, 4] for the simulation of electromagnetic problems in piecewise constant media, see also [3] for associated boundary integral methods. The MTFs are naturally adapted to the developments of new block preconditioners, as indicated in [5], but very little is known so far about such associated iterative solvers. The goal of our presentation is to give an elementary introduction to MTFs, and also to establish a natural connection with the more classical Dirichlet-Neumann algorithms that are well understood in the domain decomposition literature, see for example [6, 7]. We present for a model problem a convergence analysis for a naturally arising block iterative method associated with the MTF, and also first numerical results to illustrate what performance one can expect from such an iterative solver.


  1. 1.
    X. Claeys, R. Hiptmair, Electromagnetic scattering at composite objects: a novel multi-trace boundary integral formulation. ESAIM Math. Model. Numer. Anal. 46(6), 1421–1445 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    X. Claeys, R. Hiptmair, Multi-trace boundary integral formulation for acoustic scattering by composite structures. Commun. Pure Appl. Math. 66(8), 1163–1201 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    X. Claeys, R. Hiptmair, E. Spindler, A second-kind Galerkin boundary element method for scattering at composite objects, in Technical Report 2013-13 (revised), Seminar for Applied Mathematics, ETH Zürich (2013)Google Scholar
  4. 4.
    R. Hiptmair, C. Jerez-Hanckes, Multiple traces boundary integral formulation for Helmholtz transmission problems. Adv. Comput. Math. 37(1), 39–91 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    R. Hiptmair, C. Jerez-Hanckes, J. Lee, Z. Peng, Domain decomposition for boundary integral equations via local multi-trace formulations, in Technical Report 2013-08 (revised), Seminar for Applied Mathematics, ETH Zürich (2013)Google Scholar
  6. 6.
    A. Quarteroni, A. Valli, Domain Decomposition Methods for Partial Differential Equations (Oxford Science Publications, Oxford, 1999)zbMATHGoogle Scholar
  7. 7.
    A. Toselli, O. Widlund, Domain Decomposition Methods: Algorithms and Theory. Springer Series in Computational Mathematics, vol. 34 (Springer, Berlin, 2004)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.University of StrathclydeGlasgowUK
  2. 2.University of GenevaGenevaSwitzerland

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