Two-Level Preconditioners for the Helmholtz Equation

  • Marcella BonazzoliEmail author
  • Victorita DoleanEmail author
  • Ivan G. GrahamEmail author
  • Euan A. SpenceEmail author
  • Pierre-Henri TournierEmail author
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 125)


In this paper we compare numerically two different coarse space definitions for two-level domain decomposition preconditioners for the Helmholtz equation, both in two and three dimensions. While we solve the pure Helmholtz problem without absorption, the preconditioners are built from problems with absorption. In the first method, the coarse space is based on the discretization of the problem with absorption on a coarse mesh, with diameter constrained by the wavenumber. In the second method, the coarse space is built by solving local eigenproblems involving the Dirichlet-to-Neumann (DtN) operator.



This work has been supported in part by the French National Research Agency (ANR), project MEDIMAX, ANR-13-MONU-0012.


  1. 1.
    L. Conen, V. Dolean, R. Krause, F. Nataf, A coarse space for heterogeneous Helmholtz problems based on the Dirichlet-to-Neumann operator. J. Comput. Appl. Math. 271, 83–99 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    I.G. Graham, E.A. Spence, E. Vainikko, Recent results on domain decomposition preconditioning for the high-frequency Helmholtz equation using absorption, in Modern Solvers for Helmholtz Problems. Geosystems Mathematics (Springer, Cham, 2017), pp. 3–26Google Scholar
  3. 3.
    I.G. Graham, E.A. Spence, E. Vainikko, Domain decomposition preconditioning for high-frequency Helmholtz problems with absorption. Math. Comput. 86(307), 2089–2127 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    J.-H. Kimn, M. Sarkis, Restricted overlapping balancing domain decomposition methods and restricted coarse problems for the Helmholtz problem. Comput. Methods Appl. Mech. Eng. 196(8), 1507–1514 (2007)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Université Côte d’Azur, CNRS, LJADNiceFrance
  2. 2.University of StrathclydeGlasgowUK
  3. 3.University of BathBathUK
  4. 4.UPMC Univ Paris 06LJLLParisFrance

Personalised recommendations