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

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

Abstract

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.

Keywords

FGMRES Helmholtz equation Multilevel Krylov methods Shifted-Laplace preconditioner 

References

  1. 1.
    S. Abarbanel and D. Gottlieb. A mathematical analysis of the PML method. J. Comput. Phys., 134:357–363, 1997.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    S. Abarbanel and D. Gottlieb. On the construction and analysis of absorbing layers in CEM. Appl. Numer. Math., 27:331–340, 1998.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    I. Babuska, F. Ihlenburg, T. S. Trouboulis, and S. K. Gangaraj. A posteriori error estimation for finite element solutions of Helmholtz’s equation. Part II: Estimation of the pollution error. Internat. J. Numer. Methods Engrg., 40:3883–3900, 1997.Google Scholar
  4. 4.
    A. Bayliss, C. I. Goldstein, and E. Turkel. An iterative method for Helmholtz equation. J. Comput. Phys., 49:443–457, 1983.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    A. Bayliss, C. I. Goldstein, and E. Turkel. On accuracy conditions for the numerical computation of waves. J. Comput. Phys., 59:396–404, 1985.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    J. P. Berenger. A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys., 114:185–200, 1994.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    J. P. Berenger. Three-dimensional perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys., 127:363–379, 1996.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    A. Brandt and I. Livshits. Wave-ray multigrid method for standing wave equations. Electronic Transactions on Numerical Analysis, 6:161–181, 1997.MathSciNetzbMATHGoogle Scholar
  9. 9.
    M. Eiermann, O. G. Ernst, and O. Schneider. Analysis of acceleration strategies for restarted minimal residual methods. J. Comput. Appl. Math., 123:261–292, 2000.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    H. C. Elman, O. G. Ernst, and D. P. O’Leary. A multigrid method enhanced by Krylov subspace iteration for discrete Helmholtz equations. SIAM J. Sci. Comput., 22:1291–1315, 2001.CrossRefzbMATHGoogle Scholar
  11. 11.
    B. Engquist and A. Majda. Absorbing boundary conditions for the numerical simulation of waves. Math. Comp., 31:629–651, 1977.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    B. Engquist and L. Ying. Sweeping preconditioner for the Helmholtz equation: moving perfectly matched layers. Multiscale Model. Simul., 9:686–710, 2011.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Y. A. Erlangga and R. Nabben. Deflation and balancing preconditioners for Krylov subspace methods applied to nonsymmetric matrices. SIAM J. Matrix Anal. Appl., 30:684–699, 2008.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Y. A. Erlangga and R. Nabben. Multilevel projection-based nested Krylov iteration for boundary value problems. SIAM J. Sci. Comput., 30:1572–1595, 2008.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Y. A. Erlangga and R. Nabben. On a multilevel Krylov method for the Helmholtz equation preconditioned by shifted Laplacian. E. Trans. Numer. Anal., 31:403–424, 2008.MathSciNetzbMATHGoogle Scholar
  16. 16.
    Y. A. Erlangga and R. Nabben. Algebraic multilevel Krylov methods. SIAM J. Sci. Comput., 31:3417–3437, 2009.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Y. A. Erlangga and R. Nabben. On the convergence of two-level Krylov methods for singular symmetric systems. 2015.Google Scholar
  18. 18.
    Y. A. Erlangga, C. W. Oosterlee, and C. Vuik. A novel multigrid-based preconditioner for the heterogeneous Helmholtz equation. SIAM J. Sci. Comput., 27:1471–1492, 2006.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Y. A. Erlangga, C. Vuik, and C. W. Oosterlee. On a class of preconditioners for solving the Helmholtz equation. Appl. Numer. Math., 50:409–425, 2004.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Y. A. Erlangga, C. Vuik, and C. W. Oosterlee. Comparison of multigrid and incomplete LU shifted-Laplace preconditioners for the inhomogeneous Helmholtz equation. Appl. Numer. Math., 56:648–666, 2006.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    J. Frank and C. Vuik. On the construction of deflation-based preconditioners. SIAM J. Sci. Comput., 23:442–462, 2001.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    M. J. Gander, I. G. Graham, and E. A. Spence. Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioning: what is the largest shift for which wavenumber-independent convergence is guaranteed? Numerische Mathematik, 131 (3):567–614, 2015.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    L. García Ramos and R. Nabben. On the spectrum of deflated matrices with applications to the deflated shifted Laplace preconditioner for the Helmholtz equation. 2016.Google Scholar
  24. 24.
    A. L. Laird and M. B. Giles. Preconditioned iterative solution of the 2D Helmholtz equation. Technical Report NA 02-12, Comp. Lab., Oxford Univ., 2002.Google Scholar
  25. 25.
    I. Livshits. A scalable multigrid method for solving indefinite Helmholtz equations with constant wave numbers. Numerical Linear Algebra with Applications, 21:177–193, 2014.MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    R. B. Morgan. A restarted GMRES method augmented with eigenvectors. SIAM J. Matrix Anal. Appl., 16:1154–1171, 1995.MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    R. Nabben and C. Vuik. A comparison of deflation and the balancing preconditioner. SIAM J. Sci. Comput., 27:1742–1759, 2006.MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    R. A. Nicolaides. Deflation of conjugate gradients with applications to boundary value problems. SIAM J. Numer. Anal., 24:355–365, 1987.MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    C. W. Oosterlee. A GMRES-based plane smoother in multigrid to solve 3D anisotropic fluid flow problems. J. Comput. Phys., 130:41–53, 1997.CrossRefzbMATHGoogle Scholar
  30. 30.
    J. Poulson, B. Engquist, S. Li, and L. Ying. A parallel sweeping preconditioner for heterogeneous 3D Helmholtz equations. SIAM J. Sci. Comput., 35(3):C194–C212, 2013.MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Y. Saad. A flexible inner-outer preconditioned GMRES algorithm. SIAM J. Sci. Comput., 14:461–469, 1993.MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Y. Saad. Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia, 2003.CrossRefzbMATHGoogle Scholar
  33. 33.
    Y. Saad and M. H. Schultz. GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput., 7:856–869, 1986.MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    A. H. Sheikh, D. Lahaye, L. García Ramos, R. Nabben, and C. Vuik. Accelerating the Shifted Laplace Preconditioner for the Helmholtz Equation by Multilevel Deflation. J. Comput. Phys., 322:473–490, 2016.MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    A. H. Sheikh, D. Lahaye, and C. Vuik. On the convergence of shifted Laplace preconditioner combined with multilevel deflation. Numerical Linear Algebra with Applications, 20(4):645–662, Apr 2013.Google Scholar
  36. 36.
    C. C. Stolk. A rapidly converging domain decomposition method for the Helmholtz equation. J. Comput. Phys., 241:240–252, 2013.CrossRefGoogle Scholar
  37. 37.
    J. Tang, R. Nabben, C. Vuik, and Y. A. Erlangga. Theoretical and numerical comparison of projection methods derived from deflation. J. Sci. Comput., 39:340–370, 2009.MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    U. Trottenberg, C. Oosterlee, and A. Schüller. Multigrid. Academic Press, New York, 2001.zbMATHGoogle Scholar
  39. 39.
    S. Tsynkov and E. Turkel. A cartesian perfectly matched layer for the Helmholtz equation. In L. Tourette and L. Harpern, editors, Absorbing Boundaries and Layers, Domain Decomposition Methods Applications to Large Scale Computation, pages 279–309. Springer, Berlin, 2001.Google Scholar
  40. 40.
    M. B. van Gijzen, Y. A. Erlangga, and C. Vuik. Spectral analysis of the discrete Helmholtz operator preconditioned with a shifted Laplacian. SIAM J. Sci. Comput., 29:1942–1952, 2006.MathSciNetCrossRefzbMATHGoogle Scholar

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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