Skip to main content
SpringerLink
Log in

Solving Interface Problems of the Helmholtz Equation by Immersed Finite Element Methods

  • Original Paper
  • Published:
Communications on Applied Mathematics and Computation Aims and scope Submit manuscript

Abstract

This article reports our explorations for solving interface problems of the Helmholtz equation by immersed finite elements (IFE) on interface independent meshes. Two IFE methods are investigated: the partially penalized IFE (PPIFE) and discontinuous Galerkin IFE (DGIFE) methods. Optimal convergence rates are observed for these IFE methods once the mesh size is smaller than the optimal mesh size which is mainly dictated by the wave number. Numerical experiments also suggest that higher degree IFE methods are advantageous because of their larger optimal mesh size and higher convergence rates.

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

Similar content being viewed by others

References

  1. Adjerid, S., Ben-Romdhane, M., Lin, T.: Higher degree immersed finite element methods for second-order elliptic interface problems. Int. J. Numer. Anal. Model. 11(3), 541–566 (2014)

    MathSciNet  MATH  Google Scholar 

  2. Adjerid, S., Ben-Romdhane, M., Lin, T.: Higher degree immersed finite element spaces constructed according to the actual interface. Comput. Math. Appl. 75(6), 1868–1881 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  3. Adjerid, S., Guo, R., Lin, T.: High degree immersed finite element spaces by a least squares method. Int. J. Numer. Anal. Model. 14(4/5), 604–626 (2017)

    MathSciNet  MATH  Google Scholar 

  4. Ainsworth, M.: Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods. J. Comput. Phys. 198, 106–130 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Annavarapu, C., Hautefeuille, M., Dolbow, J.: A finite element method for crack growth without remeshing. Comput. Methods Appl. Mech. Eng. 225–228, 44–54 (2012)

    Article  MATH  Google Scholar 

  6. Aziz, A.K., Werschulz, A.: On the numerical solutions of Helmholtz’s equation by the finite element method. SIAM J. Numer. Anal. 19(5), 166–178 (1995)

    MathSciNet  Google Scholar 

  7. Babuška, I.: The finite element method for elliptic equations with discontinuous coefficients. Computing (Arch. Elektron. Rechnen) 5, 207–213 (1970)

    MathSciNet  MATH  Google Scholar 

  8. Babuška, I.M., Sauter, S.A.: Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers? Comput. Math. Appl. 34(6), 2392–2423 (1997)

    MathSciNet  MATH  Google Scholar 

  9. Barrett, J.W., Elliott, C.M.: Fitted and unfitted finite-element methods for elliptic equations with smooth interfaces. IMA J. Numer. Anal. 7(3), 283–300 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bonnet-Ben Dhia, A.S., Ciarlet Jr., P., Zwölf, C.M.: Time harmonic wave diffraction problems in materials with sign-shifting coefficients. J. Comput. Appl. Math. 234, 1912–1919 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Braess, D.: Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, 2nd edn. Cambridge University Press, Cambridge (2001). Translated from the 1992 German edition by Larry L. Schumaker

    MATH  Google Scholar 

  12. Bramble, J.H., King, J.T.: A finite element method for interface problems in domains with smooth boundaries and interfaces. Adv. Comput. Math. 6(2), 109–138 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  13. Brown, D.L.: A note on the numerical solution of the wave equation with piecewise smooth coefficients. Math. Comput. 42(166), 369–391 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  14. Burman, E., Wu, H., Zhu, L.: Linear continuous interior penalty finite element method for Helmholtz equation with high wave number: one-dimensional analysis. Numer. Methods Partial Differ. Equ. 32(5), 1378–1410 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bériot, H., Prinn, A., Gabard, G.: Efficient implementation of high-order finite elements for Helmholtz problems. Int. J. Numer. Methods Eng. 106, 213–240 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Chandler-wilde, S.N., Zhang, B.: Scattering of electromagnetic waves by rough interfaces and inhomogeneous layers. SIAM J. Math. Anal. 30(3), 559–583 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  17. Chen, Z., Zou, J.: Finite element methods and their convergence for elliptic and parabolic interface problems. Numer. Math. 79(2), 175–202 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  18. Christiansen, P.S., Krenk, S.: A recursive finite element technique for acoustic fields in pipes with absorption. J. Sound Vib. 122(1), 107–118 (1988)

    Article  Google Scholar 

  19. Chu, C.-C., Graham, I.G., Hou, T.-Y.: A new multiscale finite element method for high-contrast elliptic interface problems. Math. Comput. 79(272), 1915–1955 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Clough, R.W., Tocher, J.L.: Finite element stiffness matrices for analysis of plate bending. In: Przemieniecki, J.S., Bader,R.M., Bozich, W.F., Johnson, J.R., Mykytow, W.J. (eds.) Matrix Methods in Structural Mechanics, The Proceedings of the Conference held at Wright-Parrterson Air Force Base, Ohio, 26–28, October, 1965, pp. 515–545, Washington, 1966. Air Force Flight Dynamics Laboratory

  21. Douglas Jr., J., Sheen, D., Santos, J.E.: Approximation of scalar waves in the space-frequency domain. Math. Models Methods Appl. Sci. 4(4), 509–531 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  22. 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(2), 782–804 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Farhat, C., Harari, I., Hetmaniuk, U.: A discontinuous Galerkin method with lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime. Comput. Methods Appl. Mech. Eng. 192, 1389–1419 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  24. Feng, X., Wu, H.: Discontinuous Galerkin methods for the Helmholtz equation with large wave number. SIAM J. Numer. Anal. 47(4), 2872–2896 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gittelson, G., 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 

  26. He, X., Lin, T., Lin, Y.: Approximation capability of a bilinear immersed finite element space. Numer. Methods Partial Differ. Equ. 24(5), 1265–1300 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. He, X., Lin, T., Lin, Y.: Interior penalty bilinear IFE discontinuous Galerkin methods for elliptic equations with discontinuous coefficient. J. Syst. Sci. Complex. 23(3), 467–483 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. He, X., Lin, T., Lin, Y.: Immersed finite element methods for elliptic interface problems with non-homogeneous jump conditions. Int. J. Numer. Anal. Model. 8(2), 284–301 (2011)

    MathSciNet  MATH  Google Scholar 

  29. He, X., Lin, T., Lin, Y.: The convergence of the bilinear and linear immersed finite element solutions to interface problems. Numer. Methods Partial Differ. Equ. 28(1), 312–330 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  30. He, X., Lin, T., Lin, Y.: A selective immersed discontinuous Galerkin method for elliptic interface problems. Math. Methods Appl. Sci. 37(7), 983–1002 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  31. He, X.: Bilinear immersed finite elements for interface problems. PhD thesis, Virginia Polytechnic Institute and State University (2009)

  32. Hou, T.Y., Wu, X.: A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134(1), 169–189 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  33. 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), 9–37 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  34. Ihlenburg, F., Babuška, I.: Finite element solutions of the Helmholtz equation with high wave number part II: the h-p version of the FEM. SIAM J. Numer. Anal. 34(1), 315–358 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  35. Jensen, F.B., Kuperman, W.A., Porter, M.B., Schmidt, H.: Computational Ocean Acoustics. Springer, Berlin (1995)

    MATH  Google Scholar 

  36. Klenow, B., Nisewonger, A., Batra, R.C., Brown, A.: Reflection and transmission of plane waves at an interface between two fluids. Comput. Fluids 36, 1298–1306 (2007)

    Article  MATH  Google Scholar 

  37. Kreiss, H., Petersson, N.A.: An embedded boundary method for the wave equation with discontinuous coefficients. SIAM J. Sci. Comput. 28(6), 2054–2074 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  38. Lam, C.Y., Shu, C.-W.: A phase-based interior penalty discontinuous Galerkin method for the Helmholtz equation with spatially varying wavenumber. Comput. Methods Appl. Mech. Eng. 318, 456–473 (2017)

    Article  MathSciNet  Google Scholar 

  39. LeVeque, R.J., Li, Z.L.: The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal. 31(4), 1019–1044 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  40. Li, Z., Ito, K.: The Immersed Interface Method: Numerical Solutions of PDEs Involving Interfaces and Irregular Domains. Frontiers in Applied Mathematics, vol. 33. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2006)

    Book  MATH  Google Scholar 

  41. Li, Z., Lin, T., Lin, Y., Rogers, R.C.: An immersed finite element space and its approximation capability. Numer. Methods Partial Differ. Equ. 20(3), 338–367 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  42. Li, Z., Lin, T., Wu, X.: New Cartesian grid methods for interface problems using the finite element formulation. Numer. Math. 96(1), 61–98 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  43. Lin, T., Lin, Y., Rogers, R., Lynne Ryan, M.: A Rectangular Immersed Finite Element Space for Interface Problems. Scientific Computing and Applications (Kananaskis, AB, 2000). Advances in Computation: Theory and Practice, vol. 7, pp. 107–114. Nova Sci. Publ, Huntington (2001)

    MATH  Google Scholar 

  44. Lin, T., Lin, Y., Zhang, X.: Partially penalized immersed finite element methods for elliptic interface problems. SIAM J. Numer. Anal. 53(2), 1121–1144 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  45. Lin, T., Yang, Q., Zhang, X.: A priori error estimates for some discontinuous Galerkin immersed finite element methods. J. Sci. Comput. 65, 875–894 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  47. Perugia, I.: A note on the discontinuous Galerkin approximation of the Helmholtz equation. Lecture notes. ETH, Zürich (2006)

  48. Romdhane, M.B.: Higher-degree immersed finite elements for second-order elliptic interface problems. PhD thesis, Virginia Polytechnic Institute and State University (2011)

  49. Semblat, J.F., Brioist, J.J.: Efficiency of higher order finite elements for the analysis of seismic wave propagation. J. Sound Vib. 231(2), 460–467 (2000)

    Article  Google Scholar 

  50. Speck, F.O.: Sommerfeld diffraction problems with first and second kind boundary conditions. SIAM J. Math. Anal. 20(2), 396–407 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  51. Suater, S.A., Warnke, R.: Composite finite elements for elliptic boudnary value problems with discontinuous coefficients. Computing 77, 29–55 (2006)

    Article  MathSciNet  Google Scholar 

  52. Wang, K., Wong, Y.: Pollution-free finite difference schemes for non-homogeneous Helmholtz equation. Int. J. Numer. Anal. Model. 11(4), 787–815 (2014)

    MathSciNet  Google Scholar 

  53. Xu, J.: Estimate of the convergence rate of the finite element solutions to elliptic equation of second order with discontinuous coefficients. Nat. Sci. J. Xiangtan Univ. 1, 1–5 (1982)

    Google Scholar 

  54. Zhang, J.: Wave propagation across fluid-solid interfaces: a grid method approach. Geophys. J. Int. 159, 240–252 (2004)

    Article  Google Scholar 

  55. Zhang, S.M., Li, Z.: An augmented IIM for Helmholtz/Poisson equations on irregular domains in complex space. Int. J. Numer. Anal. Model. 13(1), 166–178 (2016)

    MathSciNet  MATH  Google Scholar 

  56. Zhang, X.: Nonconforming immersed finite element methods for interface problems. PhD thesis, Virginia Polytechnic Institute and State University (2013)

  57. Zou, Z., Aquino, W., Harari, I.: Nitsche's method for Helmholtz problems with embedded interfaces. Int. J. Numer. Meth. Eng. 110, 618–636 (2017)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This research was partially supported by GRF B-Q56D of HKSAR and Polyu G-UA7V.

Author information

Authors and Affiliations

  1. Department of Mathematics, Virginia Tech, Blacksburg, VA, 24061, USA

    Tao Lin & Qiao Zhuang

  2. Department of Applied Mathematics, Hong Kong Polytechnic University, Kowloon, Hong Kong, China

    Yanping Lin

Authors
  1. Tao Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yanping Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Qiao Zhuang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yanping Lin.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lin, T., Lin, Y. & Zhuang, Q. Solving Interface Problems of the Helmholtz Equation by Immersed Finite Element Methods. Commun. Appl. Math. Comput. 1, 187–206 (2019). https://doi.org/10.1007/s42967-019-0002-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s42967-019-0002-2

Keywords

Mathematics Subject Classification

Search

Navigation