The Multilevel Krylov-Multigrid Method for the Helmholtz Equation Preconditioned by the Shifted Laplacian

  • Yogi A. ErlanggaEmail author
  • Luis García Ramos
  • Reinhard Nabben
Part of the Geosystems Mathematics book series (GSMA)


This chapter discusses a multilevel Krylov method (MK-method) for solving the Helmholtz equation preconditioned by the shifted Laplacian preconditioner, resulting in the so-called multilevel Krylov-multigrid (MKMG) method. This method was first presented in Erlangga and Nabben (E. Trans. Numer. Anal. 31:403–424, 2008). By combining the MK method with the shifted Laplacian preconditioner, it is expected that the issues related to indefiniteness and small eigenvalues of the Helmholtz matrix can be resolved simultaneously, leading to an equivalent system, whose matrix is spectrally favorable for fast convergence of a Krylov method. The eigenvalues of the preconditioned system preconditioned by the ideal MKMG operator lie on (or inside) the same circles known from the shifted Laplace preconditioning. But they are much better clustered. Here, we distinguish between the so-called ideal MKMG and the practical MKMG method. Numerical results for the practical MKMG presented here suggest that it is indeed possible to achieve an almost gridsize- and wavenumber-independent convergence, provided that the coarse-grid system in the ideal MK method is properly and accurately approximated.


FGMRES Helmholtz equation Multilevel Krylov methods Shifted-Laplace preconditioner 


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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Yogi A. Erlangga
    • 1
    Email author
  • Luis García Ramos
    • 2
  • Reinhard Nabben
    • 2
  1. 1.Mathematics DepartmentNazarbayev UniversityAstanaKazakhstan
  2. 2.Institut für MathematikTU BerlinBerlinGermany

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