Why it is Difficult to Solve Helmholtz Problems with Classical Iterative Methods

Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 83)


In contrast to the positive definite Helmholtz equation, the deceivingly similar looking indefinite Helmholtz equation is difficult to solve using classical iterative methods. Simply using a Krylov method is much less effective, especially when the wave number in the Helmholtz operator becomes large, and also algebraic preconditioners such as incomplete LU factorizations do not remedy the situation. Even more powerful preconditioners such as classical domain decomposition and multigrid methods fail to lead to a convergent method, and often behave differently from their usual behavior for positive definite problems. For example increasing the overlap in a classical Schwarz method degrades its performance, as does increasing the number of smoothing steps in multigrid. The purpose of this review paper is to explain why classical iterative methods fail to be effective for Helmholtz problems, and to show different avenues that have been taken to address this difficulty.


Coarse Grid Helmholtz Equation Multigrid Method Domain Decomposition Method Schwarz Method 
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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The authors would like to acknowledge the support of the Swiss National Science Foundation Grant number 200020-121828.


  1. 1.
    Y. Achdou and F. Nataf. Dimension-wise iterated frequency filtering decomposition. Num. Lin. Alg. and Appl., 41(5):1643–1681, 2003.MathSciNetGoogle Scholar
  2. 2.
    Y. Achdou and F. Nataf. Low frequency tangential filtering decomposition. Num. Lin. Alg. and Appl., 14:129–147, 2007.CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    M. Ainsworth. Discrete dispersion relation for hp-version finite element approximation at high wave number. SIAM J. Numer. Anal., 42(2):553–575, 2004.CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    N. Bakhvalov. Convergence of one relaxation method under natural constraints on the elliptic operator. Zh. Vychisl. Mat. Mat. Fiz., 6:861–883, 1966.zbMATHGoogle Scholar
  5. 5.
    A. Bayliss, C. Goldstein, and E. Turkel. An iterative method for the Helmholtz equation. J. Comput. Phys., 49:443–457, 1983.CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    T. Betcke, S.N., Chandler-Wilde, I. Graham, S. Langdon, and M. Lindner. Condition number estimates for combined potential operators in acoustics and their boundary element discretisation. Numerical Methods for PDEs, 27(1):31–69, 2010.Google Scholar
  7. 7.
    A. Brandt. Multigrid techniques: 1984 guide with applications to fluid dynamics. GMD Studien 55, Gesellschaft für Mathematik und Datenverarbeitung, St. Augustin, Bonn, 1984.Google Scholar
  8. 8.
    A. Brandt and I. Livshits. Wave-ray multigrid method for standing wave equations. Electron. Trans. Numer. Anal., 6:162–181, 1997.zbMATHMathSciNetGoogle Scholar
  9. 9.
    A. Brandt and S. Ta’asan. Multigrid methods for nearly singular and slightly indefinite problems. In Multigrid Methods II, pages 99–121. Springer, 1986.Google Scholar
  10. 10.
    W. L. Briggs, V. E. Henson, and S. F. McCormick. A Multigrid Tutorial. SIAM, 2000.Google Scholar
  11. 11.
    B. Buzbee, F. W. Dorr, J. A. George, and G. Golub. The direct solution of the discrete Poisson equation on irregular regions. SIAM J. Numer. Anal., 8:722–736, 1974.CrossRefMathSciNetGoogle Scholar
  12. 12.
    A. Buzdin. Tangential decomposition. Computing, 61:257–276, 1998.CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    D. Colton and R. Kress. Integral Equation Methods in Scattering Theory. Wiley, New York, 1983.zbMATHGoogle Scholar
  14. 14.
    L. Debreu and E. Blayo. On the Schwarz alternating method for ocean models on parallel computers. J. Computational Physics, 141:93–111, 1998.CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    B. Després. Méthodes de décomposition de demains pour les problèms de propagation d‘ondes en régime harmonique. PhD thesis, Université Paris IX Dauphine, 1991.Google Scholar
  16. 16.
    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. Comp., 23:1290–1314, 2001.CrossRefMathSciNetGoogle Scholar
  17. 17.
    B. Engquist and L. Ying. Sweeping preconditioner for the Helmholtz equation: Hierarchical matrix representation. Preprint, 2010.Google Scholar
  18. 18.
    B. Engquist and L. Ying. Sweeping preconditioner for the Helmholtz equation: Moving perfectly matched layers. Preprint, 2010.Google Scholar
  19. 19.
    B. Engquist and L. Ying. Fast algorithms for high-frequency wave propagation. this volume, page 127, 2011.Google Scholar
  20. 20.
    Y. Erlangga. Advances in iterative methods and preconditioners for the Helmholtz equation. Archives Comput. Methods in Engin., 15:37–66, 2008.CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Y. Erlangga, C. Vuik, and C. Oosterlee. On a class of preconditioners for solving the Helmholtz equation. Applied Numerical Mathematics, 50:409–425, 2004.CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    O. G. Ernst. Fast Numerical Solution of Exterior Helmholtz Problems with Radiation Boundary Condition by Imbedding. PhD thesis, Stanford University, 1994.Google Scholar
  23. 23.
    O. G. Ernst. A finite element capacitance matrix method for exterior Helmholtz problems. Numer. Math., 75:175–204, 1996.CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    O. G. Ernst. Residual-minimizing Krylov subspace methods for stabilized discretizations of convection-diffusion equations. SIAM J. Matrix Anal. Appl., 22(4):1079–1101, 2000.CrossRefMathSciNetGoogle Scholar
  25. 25.
    C. Farhat, P. Avery, R. Tesaur, and J. Li. FETI-DPH: a dual-primal domain decomposition method for acoustic scattering. Journal of Computational Acoustics, 13(3):499–524, 2005.CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    C. Farhat, P. Avery, R. Tezaur, and J. Li. FETI-DPH: a dual-primal domain decomposition method for accoustic scattering. Journal of Computational Acoustics, 13:499–524, 2005.CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    C. Farhat, A. Macedo, and R. Tezaur. FETI-H: a scalable domain decomposition method for high frequency exterior Helmholtz problem. In C.-H. Lai, P. Bjørstad, M. Cross, and O. Widlund, editors, Eleventh International Conference on Domain Decomposition Method, pages 231–241. DDM.ORG, 1999.Google Scholar
  28. 28.
    C. Farhat and F.-X. Roux. A method of Finite Element Tearing and Interconnecting and its parallel solution algorithm. Int. J. Numer. Meth. Engrg., 32:1205–1227, 1991.CrossRefzbMATHGoogle Scholar
  29. 29.
    R. Federenko. A relaxation method for solving elliptic difference equations. USSR Comput. Math. and Math. Phys., 1(5):1092–1096, 1961.Google Scholar
  30. 30.
    R. W. Freund and N. M. Nachtigal. QMR: a quasi-minimal residual method for non-Hermitian linear systems. Numer. Math., 60:315–339, 1991.CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    M. J. Gander. Optimized Schwarz methods. SIAM J. Numer. Anal., 44(2):699–731, 2006.CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    M. J. Gander. Schwarz methods over the course of time. Electronic Transactions on Numerical Analysis (ETNA), 31:228–255, 2008.Google Scholar
  33. 33.
    M. J. Gander, L. Halpern, and F. Magoulès. An optimized Schwarz method with two-sided robin transmission conditions for the Helmholtz equation. Int. J. for Num. Meth. in Fluids, 55(2):163–175, 2007.CrossRefzbMATHGoogle Scholar
  34. 34.
    M. J. Gander, L. Halpern, and F. Nataf. Optimized Schwarz methods. In T. Chan, T. Kako, H. Kawarada, and O. Pironneau, editors, Twelfth International Conference on Domain Decomposition Methods, Chiba, Japan, pages 15–28, Bergen, 2001. Domain Decomposition Press.Google Scholar
  35. 35.
    M. J. Gander, F. Magoulès, and F. Nataf. Optimized Schwarz methods without overlap for the Helmholtz equation. SIAM J. Sci. Comput., 24(1):38–60, 2002.CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    M. J. Gander, V. Martin, and J.-P. Chehab. GMRES convergence analysis for diffusion, convection diffusion and Helmholtz problems. In preparation, 2011.Google Scholar
  37. 37.
    M. J. Gander and F. Nataf. AILU: A preconditioner based on the analytic factorization of the elliptic operator. Num. Lin. Alg. and Appl., 7(7-8):543–567, 2000.MathSciNetGoogle Scholar
  38. 38.
    M. J. Gander and F. Nataf. An incomplete LU preconditioner for problems in acoustics. Journal of Computational Acoustics, 13(3):1–22, 2005.CrossRefMathSciNetGoogle Scholar
  39. 39.
    M. J. Gander and G. Wanner. From Euler, Ritz and Galerkin to modern computing. SIAM Review, 2011. to appear.Google Scholar
  40. 40.
    M. J. Gander and H. Zhang. Domain Decomposition Methods for the Helmholtz equation: A Numerical Study, submitted to Domain Decomposition Methods in Science and Engineering XX, Lecture Notes in Computational Science and Engineering, Springer-Verlag, 2012.Google Scholar
  41. 41.
    M. V. Gijzen, Y. Erlangga, and C. Vuik. Spectral analysis of the discrete Helmholtz operator preconditioned with a shifted Laplacian. SIAM J. Sci. Comput., 29(5):1942–1958, 2007.CrossRefzbMATHMathSciNetGoogle Scholar
  42. 42.
    E. Giladi and J. B. Keller. Iterative solution of elliptic problems by approximate factorization. Journal of Computational and Applied Mathematics, 85:287–313, 1997.CrossRefzbMATHMathSciNetGoogle Scholar
  43. 43.
    W. Hackbusch. Multi-Grid Methods and Applications. Springer-Verlag, 1985.Google Scholar
  44. 44.
    T. Hagstrom, R. P. Tewarson, and A. Jazcilevich. Numerical experiments on a domain decomposition algorithm for nonlinear elliptic boundary value problems. Appl. Math. Lett., 1(3), 1988.Google Scholar
  45. 45.
    E. Heikkola, J. Toivanen, and T. Rossi. A parallel fictitious domain method for the three-dimensional Helmholtz equation. SIAM J. Sci. Comput, 24(5):1567–1588, 2003.CrossRefzbMATHMathSciNetGoogle Scholar
  46. 46.
    M. A. Hyman. Non-iterative numerical solution of boundary-value problems. Appl. Sci. Res. Sec. B2, 2:325–351, 1952.Google Scholar
  47. 47.
    F. Ihlenburg and I. Babuška. Finite element solution to the Helmholtz equation with high wave number. Part I: The h-version of the FEM. Computer Methods in Applied Mechanics and Engineering, 39:9–37, 1995.Google Scholar
  48. 48.
    F. Ihlenburg and I. Babuška. Finite element solution to the Helmholtz equation with high wave number. Part II: The h-p version of the FEM. SIAM Journal on Numerical Analysis, 34:315–358, 1997.Google Scholar
  49. 49.
    J. B. Keller. Rays, waves and asymptotics. Bull. Amer. Math. Soc., 84(5):727–750, 1978.CrossRefzbMATHMathSciNetGoogle Scholar
  50. 50.
    B. Lee, T. Manteuffel, S. McCormick, and J. Ruge. First-order system least squares (FOSLS) for the Helmholtz equation. SIAM J. Sci. Comp., 21:1927–1949, 2000.CrossRefzbMATHMathSciNetGoogle Scholar
  51. 51.
    L. M. Leslie and B. J. McAveney. Comparative test of direct and iterative methods for solving Helmholtz-type equations. Mon. Wea. Rev., 101:235–239, 1973.CrossRefGoogle Scholar
  52. 52.
    R. M. Lewis. Asymptotic theory of wave-propagation. Archive for Rational Mechanics and Analysis, 20(3):191–250, 1965.CrossRefzbMATHMathSciNetGoogle Scholar
  53. 53.
    J. Liesen and Z. Strakoš. GMRES convergence analysis for a convection-diffusion model problem. SIAM J. Sci. Comput, 26(6):1989–2009, 2005.CrossRefzbMATHMathSciNetGoogle Scholar
  54. 54.
    P.-L. Lions. On the Schwarz alternating method. III: a variant for nonoverlapping subdomains. In T. F. Chan, R. Glowinski, J. Périaux, and O. Widlund, editors, Third International Symposium on Domain Decomposition Methods for Partial Differential Equations , held in Houston, Texas, March 20-22, 1989, Philadelphia, PA, 1990. SIAM.Google Scholar
  55. 55.
    I. Livshits. Multigrid Solvers for Wave Equations. PhD thesis, Bar-Ilan University, Ramat-Gan, Israel, 1996.Google Scholar
  56. 56.
    F. Nataf. Résolution de l’équation de convection-diffusion stationaire par une factorisation parabolique. C. R. Acad. Sci., I 310(13):869–872, 1990.Google Scholar
  57. 57.
    Q. Niu, L. Grigori, P. Kumar, and F. Nataf. Modified tangential frequency filtering decomposition and its Fourier analysis. Numerische Mathematik, 116(1), 2010.Google Scholar
  58. 58.
    W. Proskurowski and O. B. Widlund. On the numerical solution of Helmholtz’s equation by the capacitance matrix method. Math. Comp., 30:433–468, 1976.zbMATHMathSciNetGoogle Scholar
  59. 59.
    T. E. Rosmond and F. D. Faulkner. Direct solution of elliptic equations by block cyclic reduction and factorization. Mon. Wea. Rev., 104:641–649, 1976.CrossRefGoogle Scholar
  60. 60.
    Y. Saad. Iterative Methods for Sparse Linear Systems. PWS Publishing Company, 1996.Google Scholar
  61. 61.
    V. Saul’ev. On solution of some boundary value problems on high performance computers by fictitious domain method. Siberian Math. J., 4:912–925, 1963 (in Russian).Google Scholar
  62. 62.
    H. A. Schwarz. Über einen Grenzübergang durch alternierendes Verfahren. Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich, 15:272–286, May 1870.Google Scholar
  63. 63.
    R. V. Southwell. Relaxation Methods in Theoretical Physics. Oxford University Press, 1946.Google Scholar
  64. 64.
    E. Stiefel. Über einige Methoden der Relaxationsrechnung. Z. Angew. Math. Phys., 3:1–33, 1952.CrossRefzbMATHMathSciNetGoogle Scholar
  65. 65.
    C. Wagner. Tangential frequency filtering decompositions for symmetric matrices. Numer. Math., 78(1):119–142, 1997.CrossRefzbMATHMathSciNetGoogle Scholar
  66. 66.
    C. Wagner. Tangential frequency filtering decompositions for unsymmetric matrices. Numer. Math., 78(1):143–163, 1997.CrossRefzbMATHMathSciNetGoogle Scholar
  67. 67.
    G. Wittum. An ILU-based smoothing correction scheme. In Parallel algorithms for partial differential equations, volume 31, pages 228–240. Notes Numer. Fluid Mech., 1991. 6th GAMM-Semin., Kiel/Ger.Google Scholar
  68. 68.
    G. Wittum. Filternde Zerlegungen. Schnelle Löser für grosse Gleichungssysteme. Teubner Skripten zur Numerik, Stuttgart, 1992.Google Scholar
  69. 69.
    E. Zauderer. Partial Differential Equations of Applied Mathematics. John Wiley & Sons, second edition, 1989.Google Scholar

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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Section of MathematicsUniversity of GenevaGeneva 4Switzerland
  2. 2.TU Bergakademie Freiberg, Institut für Numerische Mathematik und OptimierungFreibergGermany

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