Is Minimising the Convergence Rate a Good Choice for Efficient Optimized Schwarz Preconditioning in Heterogeneous Coupling? The Stokes-Darcy Case

  • Marco DiscacciatiEmail author
  • Luca Gerardo-GiordaEmail author
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 125)


Optimized Schwarz Methods (OSM) are domain decomposition techniques based on Robin-type interface condition that have become increasingly popular in the last two decades. Ensuring convergence also on non-overlapping decompositions, OSM are naturally advocated for the heterogeneous coupling of multi-physics problems. Classical approaches optimize the coefficients in the Robin condition by minimizing the effective convergence rate of the resulting iterative algorithm. However, when OSM are used as preconditioners for Krylov solvers of the resulting interface problem, such parameter optimization does not necessarily guarantee the fastest convergence. This drawback is already known for homogeneous decomposition, but in the case of heterogeneous decomposition, the poor performance of the classical optimization approach becomes utterly evident. In this paper, we highlight this drawback for the Stokes/Darcy problem and propose a more effective optimization procedure.



The second author was partly supported by the Basque government through the BERC 2014–2017, the Spanish Ministry of Economics and Competitiveness MINECO through the BCAM Severo Ochoa excellence accreditation SEV-2013-0323 and the Plan Estatal de Investigación. Desarollo e Innovación Orientada a los Retos de la Sociedad under Grant BELEMET (MTM2015-69992-R).


  1. 1.
    G. Beavers, D. Joseph, Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30, 197–207 (1967)CrossRefGoogle Scholar
  2. 2.
    M. Discacciati, L. Gerardo-Giorda, Optimized Schwarz methods for the Stokes-Darcy coupling. IMA J. Numer. Anal. (2018, to appear).
  3. 3.
    M. Discacciati, A. Quarteroni, Navier-Stokes/Darcy coupling: modeling, analysis, and numerical approximation. Rev. Mat. Complut. 22, 315–426 (2009)MathSciNetCrossRefGoogle Scholar
  4. 4.
    V. Dolean, M.J. Gander, L. Gerardo-Giorda, Optimized Schwarz methods for Maxwell’s equations. SIAM J. Sci. Comput. 31, 2193–2213 (2009)MathSciNetCrossRefGoogle Scholar
  5. 5.
    M.J. Gander, Optimized Schwarz methods. SIAM J. Numer. Anal. 44(2), 699–731 (2006)MathSciNetCrossRefGoogle Scholar
  6. 6.
    M.J. Gander, F. Magoulès, F. Nataf, Optimized Schwarz methods without overlap for the Helmholtz equation. SIAM J. Sci. Comput. 21, 38–60 (2002)MathSciNetCrossRefGoogle Scholar
  7. 7.
    L. Gerardo-Giorda, M. Perego, Optimized Schwarz methods for the Bidomain system in electrocardiology. M2AN 75, 583–608 (2013)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesLoughborough UniversityLoughboroughUK
  2. 2.BCAM - Basque Center for Applied MathematicsBilbaoSpain

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