Skip to main content
SpringerLink
Log in

A Pure Source Transfer Domain Decomposition Method for Helmholtz Equations in Unbounded Domain

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

We propose a pure source transfer domain decomposition method (PSTDDM) for solving the truncated perfectly matched layer (PML) approximation in bounded domain of Helmholtz scattering problem. The method is a modification of the STDDM proposed by Chen and Xiang (SIAM J Numer Anal 51:2331–2356, 2013). After decomposing the domain into N non-overlapping layers, the STDDM is composed of two series steps of sources transfers and wave expansions, where \(N-1\) truncated PML problems on two adjacent layers and \(N-2\) truncated half-space PML problems are solved successively. While the PSTDDM consists merely of two parallel source transfer steps in two opposite directions, and in each step \(N-1\) truncated PML problems on two adjacent layers are solved successively. One benefit of such a modification is that the truncated PML problems on two adjacent layers can be further solved by the PSTDDM along directions parallel to the layers. And therefore, we obtain a block-wise PSTDDM on the decomposition composed of \(N^2\) squares, which reduces the size of subdomain problems and is more suitable for large-scale problems. Convergences of both the layer-wise PSTDDM and the block-wise PSTDDM are proved for the case of constant wave number. Numerical examples are included to show that the PSTDDM gives good approximations to the discrete Helmholtz equations with constant wave numbers and can be used as an efficient preconditioner in the preconditioned GMRES method for solving the discrete Helmholtz equations with constant and heterogeneous wave numbers.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Data Availibility

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

  1. Ainsworth, M.: Discrete dispersion relation for hp-version finite element approximation at high wave number. SIAM J. Numer. Anal. 42, 553–575 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Babuška, I., Ihlenburg, F., Paik, E., Sauter, S.: A generalized finite element method for solving the Helmholtz equation in two dimensions with minimal pollution. Comput. Methods Appl. Mech. Eng. 128, 325–359 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  3. Babuška, I., Sauter, S.: Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers? SIAM Rev. 42, 451–484 (2000)

    MathSciNet  MATH  Google Scholar 

  4. Banjai, L., Hackbusch, W.: Hierarchical matrix techniques for low-and high-frequency Helmholtz problems. IMA J. Numer. Anal. 28, 46–79 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bebendorf, M., Kuske, C., Venn, R.: Wideband nested cross approximation for Helmholtz problems. Numer. Math. 130, 1–34 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bérenger, J.-P.: A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114, 185–200 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bérenger, J.-P.: A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 127, 363–379 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  8. Börm, S.: Directional \(H^2\)-matrix compression for high-frequency problems. Numer. Linear Algebra Appl. 24, e2112 (2017)

    Article  MATH  MathSciNet  Google Scholar 

  9. Börm, S., Melenk, J.M.: Approximation of the high-frequency Helmholtz kernel by nested directional interpolation: error analysis. Numer. Math. 137, 1–34 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bramble, J., Pasciak, J.: Analysis of a Cartesian PML approximation to acoustic scattering problems in \(\mathbb{R}^2\) and \(\mathbb{R}^3\). J. Comput. Math. 247, 209–230 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Brenner, S., Scott, L.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, Berlin (2008)

    Book  MATH  Google Scholar 

  12. Candès, E., Demanet, L., Ying, L.: A fast butterfly algorithm for the computation of Fourier integral operators. Multiscale Model. Simul. 7, 1727–1750 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cessenat, O., Despres, B.: Using plane waves as base functions for solving time harmonic equations with the ultra weak variational formulation. J. Comput. Acoust. 11, 227–238 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chandler-Wilde, S., Monk, P.: Wave-number-explicit bounds in time-harmonic scattering. SIAM J. Math. Anal. 39, 1428–1455 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Chen, H., Wu, H., Xu, X.: Multilevel preconditioner with stable coarse grid corrections for the Helmholtz equation. SIAM J. Sci. Comput. 37, A221–A244 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Chen, W., Weedom, W.: A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates. Microwave Opt. Tech. Lett. 7, 599–604 (1994)

    Article  Google Scholar 

  17. Chen, Z., Wu, H.: An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures. SIAM J. Numer. Anal. 41, 799–826 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Chen, Z., Wu, X.: An adaptive uniaxial perfectly matched layer technique for time-harmonic scattering problems. Numer. Math. Theory Methods Appl. 1, 113–137 (2008)

    MathSciNet  MATH  Google Scholar 

  19. Chen, Z., Xiang, X.: A source transfer domain decomposition method for Helmholtz equations in unbounded domain. SIAM J. Numer. Anal. 51, 2331–2356 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Chen, Z., Zheng, W.: Convergence of the uniaxial perfectly matched layer method for time-harmonic scattering problems in two-layered media. SIAM J. Numer. Anal. 48, 2158–2185 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Chevalier, P., Nataf, F.: Symmetrized method with optimized second-order conditions for the Helmholtz equation. Contemp. Math. 218, 400–407 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  22. Chew, W., Jin, J., Michielssen, E.: Complex coordinate stretching as a generalized absorbing boundary condition. Microwave Opt. Technol. Lett. 15, 363–369 (1997)

    Article  Google Scholar 

  23. Deraemaeker, A., Babuška, I., Bouillard, P.: Dispersion and pollution of the FEM solution for the Helmholtz equation in one, two and three dimensions. Int. J. Numer. Methods Eng. 46, 471–499 (1999)

    Article  MATH  Google Scholar 

  24. Du, Y., Wu, H.: Preasymptotic error analysis of higher order FEM and CIP-FEM for Helmholtz equation with high wave number. SIAM J. Numer. Anal. 53, 782–804 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Engquist, B., Ying, L.: Sweeping preconditioner for the Helmholtz equation: moving perfectly matched layers. Multiscale Model. Simul. 9, 686–710 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Eslaminia, M., Guddati, M.N.: A double-sweeping preconditioner for the Helmholtz equation. J. Comput. Phys. 314, 800–823 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. Feng, X., Wu, H.: Discontinuous Galerkin methods for the Helmholtz equation with large wave numbers. SIAM J. Numer. Anal. 47, 2872–2896 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. Feng, X., Wu, H.: \(hp\)-discontinuous Galerkin methods for the Helmholtz equation with large wave number. Math. Comput. 80, 1997–2024 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Gander, M.J., et al.: Schwarz methods over the course of time. Electron. Trans. Numer. Anal. 31, 228–255 (2008)

    MathSciNet  MATH  Google Scholar 

  30. Gander, M.J., Magoules, F., Nataf, F.: Optimized schwarz methods without overlap for the Helmholtz equation. SIAM J. Sci. Comput. 24, 38–60 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  31. Gander, M.J., Zhang, H.: A class of iterative solvers for the Helmholtz equation: factorizations, sweeping preconditioners, source transfer, single layer potentials, polarized traces, and optimized Schwarz methods. SIAM Rev. 61, 3–76 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  32. Gittelson, C., Hiptmair, R., Perugia, I.: Plane wave discontinuous Galerkin methods: analysis of the h-version. ESAIM Math. Model. Numer. Anal. 43, 297–331 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  33. Hackbusch, W.: A sparse matrix arithmetic based on \(mathcal H \)-matrices. Part I: Introduction to \(mathcal H \)-matrices. Computing 62, 89–108 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  34. Hackbusch, W., Khoromskij, B.N., Hackbusch, W., Khoromskij, B.N.: A sparse \(mathcal H \)-matrix arithmetic. Part II: Application to multidimensional problems. Computing 64, 21–47 (2000)

    MathSciNet  MATH  Google Scholar 

  35. Ihlenburg, F., Babuška, I.: Finite element solution of the Helmholtz equation with high wave number. I. The \(h\)-version of the FEM. Comput. Math. Appl. 30, 9–37 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  36. Ihlenburg, F., Babuška, I.: Finite element solution of the Helmholtz equation with high wave number. II. The \(h\)-\(p\) version of the FEM. SIAM J. Numer. Anal. 34, 315–358 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  37. Kim, S., Pasciak, J.: Analysis of a Cartesian PML approximation to acoustic scattering problems in \(\mathbb{R}^2\). J. Math. Anal. Appl. 370, 168–186 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  38. Lassas, M., Somersalo, E.: Analysis of the PML equations in general convex geometry. Proc. R. Soc. Ed. 131, 1183–1207 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  39. Leng, W., Ju, L.: An additive overlapping domain decomposition method for the Helmholtz equation. SIAM J. Sci. Comput. 41, A1252–A1277 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  40. Li, Y., Wu, H.: FEM and CIP-FEM for Helmholtz equation with high wave number and PML truncation. SIAM J. Numer. Anal. 57, 96–126 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  41. Liu, F., Ying, L.: Additive sweeping preconditioner for the Helmholtz equation. Multiscale Model. Simul. 14, 799–822 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  42. Liu, Y., Xu, X.: An optimized Schwarz method with relaxation for the Helmholtz equation: the negative impact of overlap. ESAIM M2AN 53, 249–268 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  43. Melenk, J., Parsania, A., Sauter, S.: General DG-methods for highly indefinite Helmholtz problems. J. Sci. Comput. 57, 536–581 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  44. Melenk, J.M., Sauter, S.: Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions. Math. Comput. 79, 1871–1914 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  45. Melenk, J.M., Sauter, S.: Wavenumber explicit convergence analysis for Galerkin discretizations of the Helmholtz equation. SIAM J. Numer. Anal. 49, 1210–1243 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  46. Messner, M., Schanz, M., Darve, E.: Fast directional multilevel summation for oscillatory kernels based on Chebyshev interpolation. J. Comput. Phys. 231, 1175–1196 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  47. Michielssen, E., Boag, A.: A multilevel matrix decomposition algorithm for analyzing scattering from large structures. IEEE Trans. Antennas Propag. 44, 086–1093 (1996)

    Article  Google Scholar 

  48. Målqvist, A., Peterseim, D.: Localization of elliptic multiscale problems. Math. Comput. 83, 2583–2603 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  49. Ohlberger, M., Verfürth, B.: A new heterogeneous multiscale method for the Helmholtz equation with high contrast. Multiscale Model. Simul. 16, 385–411 (2018)

    Article  MathSciNet  Google Scholar 

  50. Peterseim, D.: Eliminating the pollution effect in Helmholtz problems by local subscale correction. Math. Comput. 86, 1005–1036 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  51. St-Cyr, A., Gander, M.J., Thomas, S.J.: Optimized multiplicative, additive, and restricted additive schwarz preconditioning. SIAM J. Sci. Comput. 29, 2402–2425 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  52. Stolk, C.C.: A rapidly converging domain decomposition method for the Helmholtz equation. J. Comput. Phys. 241, 240–252 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  53. Stolk, C.C.: An improved sweeping domain decomposition preconditioner for the Helmholtz equation. Adv. Comput. Math. 43, 45–76 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  54. Thompson, L.: A review of finite-element methods for time-harmonic acoustics. J. Acoust. Soc. Am. 119, 1315–1330 (2006)

    Article  Google Scholar 

  55. Wang, S., de Hoop, M.V., Xia, J.: On 3D modeling of seismic wave propagation via a structured parallel multifrontal direct Helmholtz solver. Geophys. Prospect. 59, 857–873 (2011)

    Article  Google Scholar 

  56. Wang, S., Maarten, V., Xia, J.: Acoustic inverse scattering via Helmholtz operator factorization and optimization. J. Comput. Phys. 229, 8445–8462 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  57. Wu, H.: Pre-asymptotic error analysis of CIP-FEM and FEM for the Helmholtz equation with high wave number. Part I: linear version. IMA J. Numer. Anal. 34, 1266–1288 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  58. Xiang, X.: Double source transfer domain decomposition method for Helmholtz problems. Commun. Comput. Phys. 26, 434–468 (2019)

    Article  MathSciNet  Google Scholar 

  59. Zepeda-Núñez, L., Demanet, L.: The method of polarized traces for the 2D Helmholtz equation. J. Comput. Phys. 308, 347–388 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  60. Zhu, L., Wu, H.: Pre-asymptotic error analysis of CIP-FEM and FEM for Helmholtz equation with high wave number. Part II: \(hp\) version. SIAM J. Numer. Anal. 51, 1828–1852 (2013)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Xiangtan University, Hunan, 411105, People’s Republic of China

    Yu Du

  2. Department of Mathematics, Nanjing University, Jiangsu, 210093, People’s Republic of China

    Haijun Wu

Authors
  1. Yu Du
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Haijun Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yu Du.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was funded by the Natural Science Foundation of China under Grants 11601026, 11525103 and 91630309, and the Hunan Provincial Natural Science Foundation of China (No. 2019JJ50572).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Du, Y., Wu, H. A Pure Source Transfer Domain Decomposition Method for Helmholtz Equations in Unbounded Domain. J Sci Comput 83, 68 (2020). https://doi.org/10.1007/s10915-020-01249-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-020-01249-2

Keywords

Mathematics Subject Classification

Search

Navigation