Skip to main content
SpringerLink
Log in

A lowest-order virtual element method for the Helmholtz transmission eigenvalue problem

  • Published:
Calcolo Aims and scope Submit manuscript

Abstract

In this paper, we introduce a \(C^{0}\) virtual element method for the Helmholtz transmission eigenvalue problem, which is a fourth-order non-selfadjoint eigenvalue problem. We consider the mixed formulation of the eigenvalue problem discretized by the lowest-order virtual elements. This discrete scheme is based on a conforming \(H^{1}(\varOmega )\times H^{1}(\varOmega )\) discrete formulation, which makes use of lower regular virtual element spaces. However, the discrete scheme is a non-classical mixed method due to the non-selfadjointness, then we cannot use the framework of classical eigenvalue problem directly. We employ the spectral theory of compact operator to prove the spectral approximation. Finally, some numerical results show that numerical eigenvalues obtained by the proposed numerical scheme can achieve the optimal convergence order.

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

Similar content being viewed by others

References

  1. Ahmad, B., Alsaedi, A., Brezzi, F., Marini, L., Russo, A.: Equivalent projections for virtual element methods. Comput. Math. Appl. 66(3), 376–391 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. An, J., Shen, J.: A spectral-element method for transmission eigenvalue problems. J. Sci. Comput. 57, 670–688 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Antonietti, P., Beirão Da Veiga, L., Scacchi, S., Verani, M.: A \({C}^{1}\) virtual element method for the Cahn-Hilliard equation with polygonal meshes. SIAM J. Numer. Anal. 54(1), 36–56 (2016)

    MATH  Google Scholar 

  4. Bab\(\check{u}\)ska, I., Osborn, J.: Eigenvalue problems. In: Handbook of Numerical Analysis, vol. II, pp. 641–787. North-Holland, Amsterdam (1991)

  5. Beirão Da Veiga, L., Manzini, G.: A virtual element menthod with arbitrary regularity. IMA J. Numer. Anal. 34(2), 759–781 (2013)

    Article  MATH  Google Scholar 

  6. Beirão Da Veiga, L., Mora, D., Rivera, G., Rodríguez, R.: A virtual element method for the acoustic vibration problem. Numer. Math. 136(3), 725–763 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  7. Beirão da Veiga L, Brezzi F, Cangiani A, Manzini G, Marini LD, Russo A Basic principles of virtual element methods. In: Mathematical Models and Methods in Applied Sciences, vol. 1, pp. 199–214 (2013)

  8. Beirão Da Veiga, L., Lipnikov, K., Manzini, G.: Arbitrary-order nodal mimetic discretizations of elliptic problems on polygonal meshes. SIAM J. Numer. Anal. 49(5), 1737–1760 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Beirão Da Veiga, L., Brezzi, F., Marini, L., Russo, A.: H(div) and H(curl)-conforming VEM. Numer. Math. 133(2), 303–332 (2015)

    MATH  Google Scholar 

  10. Beirão Da Veiga, L., Lovadina, C., Mora, D.: A virtual element method for elastic and inelastic problems on polytope meshes. Comput. Methods Appl. Mech. Eng. 295, 327–346 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Beirão Da Veiga, L., Brezzi, F., Marini, L., Russo, A.: Mixed virtual element methods for general second order elliptic problems on polygonal meshes. ESAIM Math. Model. Numer. Anal. 50(3), 727–747 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Beirão Da Veiga, L., Brezzi, F., Marini, L., Russo, A.: Virtual element methods for general second-order elliptic problems on polygonal meshes. Math. Models Methods Appl. Sci. 26(4), 729–750 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Beirão Da Veiga, L., Dassi, F., Russo, A.: High-order virtual element method on polyhedral meshes. Comput. Math. Appl. 74(5), 1110–1122 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Beirão Da Veiga, L., Lovadina, C., Vacca, G.: Divergence free virtual elements for the Stokes problem on polygonal meshes. ESAIM Math. Model. Numer. Anal. 51(2), 509–535 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bellis, C., Cakoni, F., Guzina, B.: Nature of the tranmission eigenvalue spectrum for elastic bodies. IMA J. Appl. Math. 78, 895–923 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Benedetto, M., Berrone, S., Borio, A., Pieraccini, S., Scialò, S.: A hybrid mortar virtual element method for discrete fracture network simulations. J. Comput. Phys. 306, 148–166 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Benedetto, M., Berrone, S., Borio, A., Pieraccini, S., Scialò, S.: Order preserving SUPG stabilization for the virtual element formulation of advection-diffusion problems. Comput. Methods Appl. Mech. Eng. 311, 18–40 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Boffi, D., Gardini, F., Gastaldi, L.: Approximation of PDE eigenvalue problems involving parameter dependent matrices. arXiv:2001.01304v1 (2020)

  19. Boffi, D.: Finite element approximation of eigenvalue problems. Acta Numer. 19, 1–120 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Brenner, S., Scott, R.: The Mathematical Theory of Finte Element Methods (Texts in Applied Mathematics), vol. 15. Springer, New York (2008)

    Google Scholar 

  21. Brezzi, F., Falk, R., Marini, L.: Basic principles of mixed virtual element methods. ESAIM Math. Model. Numer. Anal. 48(4), 1227–1240 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  22. Cáceres, E., Gatica, G.: A mixed virtual element method for the pseudostress-velocity formulation of the Stokes problem. IMA J. Numer. Anal. 37, 296–331 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. Cáceres, E., Gatica, G., Sequeira, F.: A mixed virtual element method for the Brinkman problem. Math. Models Methods Appl. Sci. 27, 707–743 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  24. Cáceres, E., Gatica, G., Sequeira, F.: A mixed virtual element method for quasi-Newtonian Stokes flows. SIAM J. Numer. Anal. 56, 317–343 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  25. Cakoni, F., Haddar, H.: On the existence of transmission eigenvalues in an inhomogeneous medium. Appl. Anal. 88(4), 475–493 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Cakoni, F., Gintides, D., Haddar, H.: The existence of an infinite discrete set of transmission eigenvalues. SIAM J. Math. Anal. 42, 237–255 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Cakoni, F., Monk, P., Sun, J.: Error analysis for the finite element approximation of transmission eigenvalues. Comput. Methods Appl. Math. 14(4), 419–427 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  28. Camaño, J., Rodríguez, R., Venegas, P.: Convergence of a lowest-order finite element method for the transmission eigenvalue problem. Calcolo 55, 33 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  29. Cangiani, A., Gardini, F., Manzini, G.: Convergence of the mimetic finite difference method for eigenvalue problems in mixed form. Comput. Methods Appl. Mech. Eng. 200, 1150–1160 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. Cangiani, A., Georgoulis, E., Pryer, T., Sutton, O.: A posteriori error estimates for the virtual element method. Numer. Math. 137(4), 857–893 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  31. Čertík, O., Gardini, F., Manzini, G., Mascotto, L., Vacca, G.: The virtual element method for eigenvalue problems with potential terms on polytopic meshes. Appl. Math. 63, 333–365 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  32. Ciarlet, P., Raviart, P.: A mixed finite element method for the biharmonic equation. In: Mathematical Aspects of Finite Elements in Partial Differential Equations, pp. 125–145. Academic Press, New York (1974)

  33. Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory, 3rd edn. Springer, New York (2013)

    Book  MATH  Google Scholar 

  34. Colton, D., Monk, P., Sun, J.: Analytical and computational methods for transmission eigenvalues. Inverse Prob. 26(4), 045011 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  35. Descloux, J., Nassif, N., Rappaz, J.: On spectral approximation. Part 1: The problem of convergence. RAIRO Anal. Numer. 12, 97–112 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  36. Gardini, F., Vacca, G.: Virtual element method for second order elliptic eigenvalue problems. IMA J. Numer. Anal. 38(4), 2026–2054 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  37. Gardini, F., Manzini, G., Vacca, G.: The nonconforming virtual element method for eigenvalue problems. ESAIM Math. Model. Numer. Anal. 53(3), 749–774 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  38. Han, J., Yang, Y., Bi, H.: A new multigrid finite element method for the transmission eigenvalue problems. Appl. Math. Comput. 292, 96–106 (2017)

    MathSciNet  MATH  Google Scholar 

  39. Huang, T., Huang, W., Lin, W.: A robust numerical algorithm for computing Maxwell’s tranmission eigenvalue problems. SIAM J. Sci. Comput. 37, A2403–A2423 (2015)

    Article  MATH  Google Scholar 

  40. Ishihara, K.: A mixed finite element method for the biharmonic eigenvalue problems of plate bending. Publ. RIMS Kyoto Univ. 14, 399–414 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  41. Ji, X., Sun, J., Turner, T.: Algorithm 922: a mixed finite element method for Helmholtz transmission eigenvalues. ACM Trans. Math. Softw. 38(4), 1–8 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  42. Ji, X., Xi, Y., Xie, H.: Nonconforming finite element method for the transmission eigenvalue problem. Adv. Appl. Math. Mech. 9(1), 92–103 (2017)

    Article  MathSciNet  Google Scholar 

  43. Kirsch, A.: The denseness of the far field patterns for the transmission problem. IMA J. Appl. Math. 37, 213–226 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  44. Kirsch, A.: On the existence of transmission eigenvalues. Inverse Probl. Imaging 3(2), 155–172 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  45. Liu, X., Chen, Z.: A virtual element method for the Cahn-Hilliard problem in mixed form. Appl. Math. Lett. 87, 115–124 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  46. Liu, X., He, Z., Chen, Z.: A fully discrete virtual element scheme for the Cahn-Hilliard equation in mixed form. Comput. Phys. Commun. 246, 106870 (2020)

    Article  MathSciNet  Google Scholar 

  47. Meng, J., Mei, L.: The matrix domain and the spectra of a generalized difference operator. J. Math. Anal. Appl. 470, 1095–1107 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  48. Meng, J., Zhang, Y., Mei, L.: A virtual element method for the Laplacian eigenvalue problem in mixed form. Appl. Numer. Math. 156, 1–13 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  49. Monzón, G.: A virtual element method for a biharmonic Steklov eigenvalue problem. Adv. Pure Appl. Math. 10(4), 325–337 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  50. Mora, D., Velásquez, I.: A virtual element method for the transmission eigenvalue problem. Math. Models Methods Appl. Sci. 28(14), 2803–2831 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  51. Mora, D., Velásquez, I.: Virtual element for the buckling problem of Kirchhoff-Love plates. Comput. Methods Appl. Mech. Eng. 360, 112687 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  52. Mora, D., Rivera, G., Rodríguez, R.: A virtual element method for the Steklov eigenvalue. Math. Models Methods Appl. Sci. 25(8), 1421–1445 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  53. Mora, D., Rivera, G., Rodríguez, R.: A posteriori error estimates for a virtual element menthod for the Steklov eigenvalue. Comput. Math. Appl. 74(9), 2172–2190 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  54. Mora, D., Rivera, G., Velásquez, I.: A virtual element method for the vibration problem of Kirchhoff plates. ESAIM Math. Model. Numer. Anal. 52, 1437–1456 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  55. Srivastava, P., Kumar, S.: Fine spectrum of the generalized difference operator \(\triangle _{uv}\) over the sequence space \(\ell _{1}\). Appl. Math. Comput. 218, 6407–6414 (2012)

    MathSciNet  MATH  Google Scholar 

  56. Sun, J.: Estimation of transmission eigenvalues and the index of refraction from Cauchy date. Inverse Probl. 27, 015009 (2011)

    Article  MATH  Google Scholar 

  57. Yang, Y., Bi, H., Li, H., Han, J.: Mixed methods for the Helmholtz transmission eigenvalues. SIAM J. Sci. Comput. 38(3), A1383–A1403 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  58. Yang, Y., Bi, H., Li, H., Han, J.: A \({C}^{0}\) IPG method and its error estimates for the Helmholtz transmission eigenvalue problem. J. Comput. Appl. Math. 326, 71–86 (2017)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We wish to thank the referee for his/her constructive comments and suggestions. The work is supported by the Science Challenge Project (No. TZ2016002) and the Fundamental Research Funds for the Central Universities (No. xzy022019040).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, Shaanxi, People’s Republic of China

    Jian Meng & Liquan Mei

  2. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710129, Shaanxi, People’s Republic of China

    Gang Wang

Authors
  1. Jian Meng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Liquan Mei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Liquan Mei.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Meng, J., Wang, G. & Mei, L. A lowest-order virtual element method for the Helmholtz transmission eigenvalue problem. Calcolo 58, 2 (2021). https://doi.org/10.1007/s10092-020-00391-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10092-020-00391-5

Keywords

Mathematics Subject Classification

Search

Navigation