Skip to main content
SpringerLink
Log in

The Mixed Discontinuous Galerkin Method for Transmission Eigenvalues for Anisotropic Medium

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

Abstract

Transmission eigenvalues play an important role in the inverse scattering theory. In this paper, we study the mixed discontinuous Galerkin method for the transmission eigenvalue problem and the modified transmission eigenvalue problem for anisotropic inhomogeneous medium in \(\varOmega \subset {\mathbb {R}}^d\,(d=2,3)\). We use the \({\mathbb {T}}\)-coercivity, Gårding’s inequality, the consistency of the DG method and the compact embeddings of broken Sobolev spaces to prove that the discrete solution operator \({\mathbb {K}}_{h}\) converges pointwise to the solution operator \({\mathbb {K}}\) and \(\{{\mathbb {K}}_{h}\}\) is collectively compact. Then we employ the spectral approximation theory and the approximation property of the DG finite element space to prove the hp a priori error estimate of approximate eigenpairs.

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

Data Availability

Enquiries about data availability should be directed to the authors.

References

  1. Cakoni, F., Colton, D., Haddar, H.: Inverse Scattering Theory and Transmission Eigenvalues. SIAM, Philadelphia (2016)

    Book  MATH  Google Scholar 

  2. Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory, 4th edn. Springer, Cham (2019)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Sun, J., Zhou, A.: Finite Element Methods for Eigenvalue Problems. CRC Press, Taylor Francis Group, Boca Raton, London, New York (2016)

    Book  Google Scholar 

  5. An, J., Shen, J.: Spectral approximation to a transmission eigenvalue problem and its applications to an inverse problem. Comput. Math. Appl. 69, 1132–1143 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Zeng, F., Sun, J., Xu, L.: A spectral projection method for transmission eigenvalues. Sci. China Math. 59, 1613–1622 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. 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 

  8. Geng, H., Ji, X., Sun, J., Xu, L.: \(C^0\)IP methods for the transmission eigenvalue problem. J. Sci. Comput. 68, 326–338 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, H., Guo, H., Zhang, Z., Zou, Q.: A \(C^0\) linear finite element method for two fourth-order eigenvalue problems. IMA J. Numer. Anal. 37, 2120–2138 (2017)

    MathSciNet  MATH  Google Scholar 

  10. Li, T., Huang, T.M., Lin, W.W., Wang, J.N.: An efficient numerical algorithm for computing densely distributed positive interior transmission eigenvalues. Inverse Problems 33(3), 035009(2017)

  11. Han, J., Yang, Y.: An \(H^m\)-conforming spectral element method on multi-dimensional domain and its application to transmission eigenvalues. Sci. China Math. 60(8), 1529–1542 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kleefeld, A., Pieronek, L.: The method of fundamental solutions for computing acoustic interior transmission eigenvalues. Inverse Problems 34(3), 035007(2018)

  13. Camaño, J., Rodríguez, R., Venegas, P.: Convergence of a lowest-order finite element method for the transmission eigenvalue problem. Calcolo 55(3), Article 33(2018)

  14. Mora D., Vel\(\acute{a}\)squez, I.: A virtual element method for the transmission eigenvalue problem. Math. Models Methods Appl. Sci. 28(14), 2803-2831(2018)

  15. Yang, Y., Zhang, Y., Bi, H.: A type of adaptive \(C^0\) non-conforming finite element method for the Helmholtz transmission eigenvalue problme. Comput. Methods Appl. Mech. Engrg. 360, 112697 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ji, X., Sun, J.: A multi-level method for transmission eigenvalues of anisotropic media. J. Comput. Phys. 255, 422–435 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Xie, H., Wu, X.: A multilevel correction method for interior transimission eigenvalue problem. J. Sci. Comput. 72, 586–604 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kleefeld, A., Pieronek, L.: Computing interior transmission eigenvalues for homogeneous and anisotropic media. Inverse Problems 34(10), 105007(2018)

  19. Gong, B., Sun, J., Turner, T., Zheng, C.: Finite element/holomorphic operator function method for the transmission eigenvalue problem. Math. Comp. 91, 2517–2537 (2022)

    MathSciNet  MATH  Google Scholar 

  20. Meng, J., Mei, L.: Virtual element method for the Helmholtz transmission eigenvalue problem of anisotropic media. Mathematical Models and Methods in Applied Sciences 32(08), 1493–1529 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  21. Cockburn, B., Karniadakis, G.E., Shu, C.-W.: Discontinuous Galerkin Methods, Thoery. Computation and Applications. Springer-Verlag, Berlin (1999)

    Google Scholar 

  22. Wihler, T.P.: Discontinuous Galerkin FEM for Elliptic Problems in Polygonal Domains. PhD thesis, Swiss Federal Institute of Technology Zurich. Diss. ETH No.14973(2002)

  23. Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods, Algorithms, Analysis, and Applications. Springer-Verlag, New York (2008)

    MATH  Google Scholar 

  24. Rivi\(\grave{e}\)re, B.: Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations. SIAM, Theory and Implementation (2008)

  25. Cangiani, A., Dong, Z., Georgoulis, E.H., Houston, P.: hp-Version Discontinuous Galerkin Method on Polygonal and Polyhedral Meshes. Springer, New York (2010)

    MATH  Google Scholar 

  26. Di Pietro, D.A., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods. Springer, New York (2012)

    Book  MATH  Google Scholar 

  27. Antonietti, P., Buffa, A., Perugia, I.: Discontinuous Galerkin approximation of the Laplace eigenproblem. Comput. Methods Appl. Mech. Eng. 195, 3483–3503 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  28. Wang, L., Xiong, C., Wu, H., Luo, F.: A priori and a posteriori error analysis for discontinuous Galerkin finite element approximations of biharmonic eigenvalue problems. Adv. Comput. Math. 45, 2623–2646 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  29. Buffa, A., Perugia, I.: Discontinuous Galerkin approximation of the Maxwell eigenproblem. SIAM J. Numer. Anal. 44(5), 2198–2226 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  30. Lepe, F., Mora, D.: Symmetric and Nonsymmetric Discontinuous Galerkin Methods for a Pseudostress Formulation of the Stokes Spectral Problem. SIAM Journal on Scientific Computing. 42(2), A698–A722 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  31. Bonnet-Ben Dhia A.S., Ciarlet P.J., Zw\(\ddot{o}\)lf C.M.: Time harmonic wave diffraction problems in materials with sign-shifting coefficients. J. Comput. Appl. Math. 234, 1912-1919(2010)

  32. Ciarlet, P.J.: \(T\)-coercivity: application to the discretization of Helmholtz-like problems. Comput. Math. Appl. 64, 22–34 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  33. Bonnet-Ben Dhia A.S., Chesnel L., Haddar H.: On the use of T-coercivity to study the interior transmission eigenvalue problem. C. R. Acad. Sci. Paris Ser.I. 349 647-651(2011)

  34. Yang, Y., Wang, S., Bi, H.: The finite element method for the elastic transmission eigenvalue problme with different elastic tensors. J. Sci. Comput. 93, 65 (2022)

    Article  MATH  Google Scholar 

  35. Buffa, A., Ortner, C.: Compact embeddings of broken Sobolev spaces and applications. IMA Journal of Numerical Analysis. 29, 827–855 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  36. Osborn, J.E.: Spectral approximation for compact operators. Math Comput. 26, 712–725 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  37. Cogar, S., Colton, D., Meng, S., Monk, P.: Modified transmission eigenvalues in inverse scattering theory. Inverse Prob. 33, 125002 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  38. Audibert, L., Cakoni, F., Haddar, H.: New sets of eigenvalues in inverse scattering for inhomogeneous media and their determination from scattering data. Inverse Prob. 33, 125011 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  39. Gintides, D., Pallikarakis, N., Stratouras, K.: On the modified transmission eigenvalue problem with an artificial metamaterial background. Res. Math. Sci. 8, 40 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  40. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  41. P. Houston, I. Perugia, D. Sch\(\ddot{o}\)tzau.: An a posteriori error indicator for discontinuous Galerkin discretizations of H(curl)-elliptic partial differential equations. IMA J. Numer. Anal. 27, 122-150(2007)

  42. Riviere, B., Wheeler, M.F., Girault, V.: A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems. SIAM J. Numer. Anal. 39, 902–931 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  43. Gudi, T., Nataraj, N., Pani, A.K.: Mixed discontinuous Galerkin finite element method for the biharmonic equation. J Sci Comput. 37, 139–161 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  44. Di Pietro, D.A., Ern, A.: Discrete functional analysis tools for Discontinuous Galerkin methods with application to the incompressible Navier-Stokes equations. Mathematics of Computation. 79(271), 1303–1330 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  45. Babu\(\check{s}\)ka, I., Osborn, J.E.: Eigenvalue problems. In: Ciarlet, P.G., Lions, J.L. (eds.) Finite Element Methods(Part 1), Handbook of Numerical Analysis, vol. 2, pp. 640–787. Elsevier Science Publishers, North-Holand (1991)

  46. Ern, A., Guermond, J.-L.: Finite Elements II, Galerkin Approximation. Elliptic and Mixed PDEs. Springer, Cham (2021)

    Book  MATH  Google Scholar 

  47. Chatelin, F.: Spectral Approximations of Linear Operators. Academic Press, New York (1983)

    MATH  Google Scholar 

  48. Yang, Y., Zhang, Z., Lin, F.: Eigenvalue approximation from below using non-conforming finite elements. Sci. China Math. 53(1), 137–150 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  49. Gustafsson, T., McBain, G.D.: scikit-fem: A Python package for finite element assembly. J. Open Source Softw. 5(52), 2369 (2020)

    Article  Google Scholar 

  50. Shen, J., Tang, T., Wang, L.-L.: Spectral Methods: Algorithms. Analysis and Applications. Springer Ser. Comput. Math, Springer, Heidelberg, Germany (2011)

    Book  MATH  Google Scholar 

  51. Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985)

    MATH  Google Scholar 

Download references

Acknowledgements

The authors cordially thank the editors and the referees for their valuable comments and suggestions that lead to the improvement of this paper.

Funding

Projects supported by the National Natural Science Foundation of China (Grant Nos. 12261024, 11561014, 11761022).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Guizhou Normal University, Guiyang, 550001, China

    Shixi Wang, Hai Bi & Yidu Yang

Authors
  1. Shixi Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hai Bi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yidu Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yidu Yang.

Ethics declarations

Competing interests

The authors have no relevant financial or non-financial interests to disclose, and have no competing interests to declare that are relevant to the content of this article.

Additional information

Publisher's Note

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

Projects supported by the National Natural Science Foundation of China (Grant Nos. 12261024, 11561014, 11761022).

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, S., Bi, H. & Yang, Y. The Mixed Discontinuous Galerkin Method for Transmission Eigenvalues for Anisotropic Medium. J Sci Comput 96, 22 (2023). https://doi.org/10.1007/s10915-023-02244-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-023-02244-z

Keywords

Mathematics Subject Classification

Search

Navigation