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Approximation of the Zero-Index Transmission Eigenvalues with a Conductive Boundary and Parameter Estimation

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Abstract

In this paper, we present a spectral-Galerkin method to approximate the zero-index transmission eigenvalues with a conductive boundary condition. This is a new eigenvalue problem derived from the scalar inverse scattering problem for an isotropic media with a conductive boundary condition. In our analysis, we will consider the equivalent fourth-order eigenvalue problem where we establish the convergence when the approximation space is the span of finitely many Dirichlet eigenfunctions for the Laplacian. We establish the convergence rate of the spectral approximation by appealing to Weyl’s law. Numerical examples for computing the eigenvalues and eigenfunctions for the unit disk and unit square are presented. Lastly, we provide a method for estimating the refractive index assuming the conductivity parameter is either sufficiently large or small but otherwise unknown.

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References

  1. Agranovich, M., Katsenelenbaum, B., Sivov, A., Voitovich, N.: Generalized Method of Eigenoscillations in Diffraction Theory. Wiley-VCH, Weinheim (1999)

    MATH  Google Scholar 

  2. An, J.: A legendre-Galerkin spectral approximation and estimation of the index of refraction for transmission eigenvalues. Appl. Numer. Math. 108, 1132–1143 (2016)

    Article  MathSciNet  Google Scholar 

  3. An, J., Shen, J.: Spectral approximation to a transmission eigenvalue problem and its applications to an inverse problem. Comput. Math. Appl. 69(10), 1132–1143 (2015)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. Arendt, W., Nittka, R., Peter, W., Steiner, F.: Weyl’s law: spectral properties of the Laplacian in mathematics and physics, mathematical analysis of evolution, information, and complexity (2009) 1–71

  6. Atkinson, K., Han, W.: Theoretical Numerical Analysis: A Functional Analysis Framework, 3rd edn. Springer, New York (2009)

    MATH  Google Scholar 

  7. Audibert, L., Chesnel, L., Haddar, H.: Transmission eigenvalues with artificial background for explicit material index identification. C. R. Acad. Sci. Paris Ser. I. 356(6), 626–631 (2018)

    Article  MathSciNet  Google Scholar 

  8. Babuska, I., Osborn, J.E.: Eigenvalue Problems, Handbook of Numerical Analysis, vol. 2, pp. 641–787. Elseveier Science Publishers, Holland (1991)

  9. Bondarenko, O., Harris, I., Kleefeld, A.: The interior transmission eigenvalue problem for an inhomogeneous media with a conductive boundary. Appl. Anal. 96(1), 2–22 (2017)

    Article  MathSciNet  Google Scholar 

  10. Brenner, S.C., Monk, P., Sun, J.: \(C^0\) Interior Penalty Galerkin Method for Biharmonic Eigenvalue Problems, Spectral and High Order Methods for Partial Differential Equations (2014) Lecture Notes in Computational Science and Engineering, vol. 106, pp. 3–15. Springer, New York (2015)

  11. Cakoni, F., Colton, D., Monk, P.: The Linear Sampling Method in Inverse Electromagnetic Scattering, CBMS Series. SIAM 80, Philadelphia (2011)

  12. Cakoni, F., Colton, D., Monk, P., Sun, J.: The inverse electromagnetic scattering problem for anisotropic media. Inverse Probl. 26, 074004 (2010)

    Article  MathSciNet  Google Scholar 

  13. Cakoni, F., Colton, D., Haddar, H.: Inverse Scattering Theory and Transmission Eigenvalues, CBMS Series. SIAM 88, Philadelphia (2016)

  14. Cakoni, F., Colton, D., Haddar, H.: On the determination of Dirichlet or transmission eigenvalues from far field data. C. R. Acad. Sci. Paris 348, 379–383 (2010)

    Article  MathSciNet  Google Scholar 

  15. Cakoni, F., Haddar, H., Harris, I.: Homogenization of the transmission eigenvalue problem for periodic media and application to the inverse problem. Inverse Probl. Imaging 9(4), 1025–1049 (2015)

    Article  MathSciNet  Google Scholar 

  16. Cakoni, F., Gintides, D.: New results on transmission eigenvalues. Inverse Probl. Imaging 4, 39–48 (2010)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  18. Evans, L.: Partial Differential Equations, 2nd edn. AMS, Providence (2010)

    MATH  Google Scholar 

  19. 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  Google Scholar 

  20. Gintides, D., Pallikarakis, N.: A computational method for the inverse transmission eigenvalue problem. Inverse Probl. 29, 104010 (2013)

    Article  MathSciNet  Google Scholar 

  21. Harris, I.: Non-destructive testing of anisotropic materials, Ph.D. Thesis, University of Delaware. (2015)

  22. Harris, I.: Analysis of two transmission eigenvalue problems with a coated boundary condition. Appl. Anal. (2019). https://doi.org/10.1080/00036811.2019.1672869

    Article  Google Scholar 

  23. Harris, I., Kleefeld, A.: The inverse scattering problem for a conductive boundary condition and transmission eigenvalues. Appl. Anal. 96(3), 508–529 (2020)

    Article  MathSciNet  Google Scholar 

  24. Kirsch, A., Lechleiter, A.: The inside-outside duality for scattering problems by inhomogeneous media. Inverse Probl. 29, 104011 (2013)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  26. Liu, Y.: Strong maximum principle for multi-term time-fractional diffusion equations and its application to an inverse source problem. Comput Math. Appl. 71(1), 96–108 (2017)

    Article  MathSciNet  Google Scholar 

  27. Osborn, J.: Spectral approximation for compact operators. Math. Comput. 29, 712–725 (1975)

    Article  MathSciNet  Google Scholar 

  28. Steinbach, O., Unger, G.: Convergence analysis of a Galerkin boundary element method for the Dirichlet Laplacian eigenvalue problem. SIAM J. Numer. Anal. 50(2), 710–728 (2012)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  30. Sun, J., Zhou, A.: Finite Element Methods for Eigenvalue Problems, 1st edn. Chapman and Hall/CRC Publications, Boca Raton (2016)

    Book  Google Scholar 

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  1. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Isaac Harris

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  1. Isaac Harris
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Harris, I. Approximation of the Zero-Index Transmission Eigenvalues with a Conductive Boundary and Parameter Estimation. J Sci Comput 82, 80 (2020). https://doi.org/10.1007/s10915-020-01183-3

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  • DOI: https://doi.org/10.1007/s10915-020-01183-3

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