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On the Minimal Shift in the Shifted Laplacian Preconditioner for Multigrid to Work

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Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE,volume 104)

Abstract

Over the past years, the shifted Laplacian has been advocated as a way of making multigrid work for the indefinite Helmholtz equation. The idea is to use a shift into the complex plane of the wave number in the operator, and then to use the shifted operator as a preconditioner for a Krylov method. The hope is that due to the shift, it becomes possible to use standard multigrid to invert the preconditioner, and if the shift is not too big, it is still an effective preconditioner for the Helmholtz equation with a real wave number. There are however two conflicting requirements here: the shift should be not too large for the shifted preconditioner to be a good preconditioner, and it should be large enough for multigrid to work. It was rigorously proved last year that the preconditioner is good if the shift is at most of the size of the wavenumber. We prove here rigorously that if the shift is less than the size of the wavenumber squared, multigrid will not work. It is therefore not possible to solve the shifted Laplace preconditioner with multigrid in the regime where it is a good preconditioner.

Keywords

  • Multigrid
  • Good Preconditioner
  • Real Wave Number
  • Helmholtz Equation
  • Krylov Methods

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. T. Airaksinen, E. Heikkola, A. Pennanen, J. Toivanen, An algebraic multigrid based shifted-laplacian preconditioner for the Helmholtz equation. J. Comput. Phys. 226(1), 1196–1210 (2007)

    MathSciNet  CrossRef  MATH  Google Scholar 

  2. A. Bayliss, C. Goldstein, E. Turkel, An iterative method for the Helmholtz equation. J. Comput. Phys. 49, 443–457 (1983)

    MathSciNet  CrossRef  MATH  Google Scholar 

  3. A. Brandt, O.E. Livne, in Multigrid Techniques, 1984 Guide with Applications to Fluid Dynamics, Revised Edition. Classics in Applied Mathematics, vol. 67 (SIAM, Philadelphia, 2011)

    Google Scholar 

  4. S. Cools, W. Vanroose, Local Fourier analysis of the complex shifted Laplacian preconditioner for Helmholtz problems. Numerical Linear Algebra with Applications 19(2), 232–252 (2013)

    MathSciNet  MATH  Google Scholar 

  5. Y. Erlangga, Advances in iterative methods and preconditioners for the Helmholtz equation. Arch. Comput. Meth. Eng. 15, 37–66 (2008)

    MathSciNet  CrossRef  MATH  Google Scholar 

  6. Y. Erlangga, C. Vuik, C. Oosterlee, On a class of preconditioners for solving the Helmholtz equation. Appl. Numer. Math. 50, 409–425 (2004)

    MathSciNet  CrossRef  MATH  Google Scholar 

  7. O. Ernst, M. Gander, Why it is Difficult to Solve Helmholtz Problems with Classical Iterative Methods, in Numerical Analysis of Multiscale Problems, ed. by I. Graham, T. Hou, O. Lakkis, R. Scheichl (Springer, Berlin, 2012), pp. 325–363

    CrossRef  Google Scholar 

  8. M. Gander, O. Ernst, Multigrid Methods for Helmholtz Problems: A Convergent Scheme in 1d Using Standard Components, in Direct and Inverse Problems in Wave Propagation and Applications (De Gruyter, Boston, 2013), pp. 135–186

    MATH  Google Scholar 

  9. M. Gander, I.G. Graham, E.A. Spence, How should one choose the shift for the shifted laplacian to be a good preconditioner for the Helmholtz equation? Numer. Math. (2015). doi:10.1007/s00211-015-0700-2

    Google Scholar 

  10. M.V. Gijzen, Y. Erlangga, C. Vuik, Spectral analysis of the discrete Helmholtz operator preconditioned with a shifted Laplacian. SIAM J. Sci. Comput. 29(5), 1942–1958 (2007)

    MathSciNet  CrossRef  MATH  Google Scholar 

  11. W. Hackbusch, Multi-Grid Methods and Applications (Springer, Berlin, 1985)

    CrossRef  MATH  Google Scholar 

  12. U. Trottenberg, C.C.W. Oosterlee, A. Schüller, Multigrid (Academic Press, New York, 2001)

    MATH  Google Scholar 

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Correspondence to Martin J. Gander .

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Cocquet, PH., Gander, M.J. (2016). On the Minimal Shift in the Shifted Laplacian Preconditioner for Multigrid to Work. In: Dickopf, T., Gander, M., Halpern, L., Krause, R., Pavarino, L. (eds) Domain Decomposition Methods in Science and Engineering XXII. Lecture Notes in Computational Science and Engineering, vol 104. Springer, Cham. https://doi.org/10.1007/978-3-319-18827-0_12

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