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Journal of Scientific Computing

, Volume 57, Issue 3, pp 670–688 | Cite as

A Spectral-Element Method for Transmission Eigenvalue Problems

  • Jing An
  • Jie ShenEmail author
Article

Abstract

We develop an efficient spectral-element method for computing the transmission eigenvalues in two-dimensional radially stratified media. Our method is based on a dimension reduction approach which reduces the problem to a sequence of one-dimensional eigenvalue problems that can be efficiently solved by a spectral-element method. We provide an error analysis which shows that the convergence rate of the eigenvalues is twice that of the eigenfunctions in energy norm. We present ample numerical results to show that the method convergences exponentially fast for piecewise stratified media, and is very effective, particularly for computing the few smallest eigenvalues.

Keywords

Spectral method Transmission eigenvalue Helmholtz equation 

Mathematics Subject Classification (1991)

78M22 78A46 65N35 35J05 41A58 

References

  1. 1.
    Bruno, O.P., Reitich, F.: High-order boundary perturbation methods. In: Mathematical Modeling in Optical Science, volume 22 of Frontiers Appl. Math., pp. 71–109. SIAM, Philadelphia, PA (2001)Google Scholar
  2. 2.
    Cakoni, F., Cayoren, M., Colton, D.: Transmission eigenvalues and the nondestructive testing of dielectrics. Inverse Probl. 24, 065016 (2008)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Cakoni, F., Colton, D., Haddar, H.: On the determination of dirichlet or transmission eigenvalues from far field data. Comptes Rendus Math. 348(7–8), 379–383 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cakoni, F., Colton, D., Monk, P.: On the use of transmission eigenvalues to estimate the index of refraction from far field data. Inverse Probl. 23, 507 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cakoni, F., Colton, D., Monk, P., Sun, J.: The inverse electromagnetic scattering problem for anisotropic media. Inverse Probl. 26, 074 (2010)MathSciNetGoogle Scholar
  6. 6.
    Cakoni, F., Gintides, D., Haddar, H.: The existence of an infinite discrete set of transmission eigenvalues. SIAM J. Math. Anal. 42(1), 237–255 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cakoni, F., Haddar, H.: On the existence of transmission eigenvalues in an inhomogeneous medium. Appl. Anal. 88(4), 475–493 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Colton, D., Monk, P., Sun, J.: Analytical and computational methods for transmission eigenvalues. Inverse Probl. 26, 045011 (2010)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Colton, D., Paivarinta, L., Sylvester, J.: The interior transmission problem. Inverse Probl. Imaging 1(1), 13 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fang, Q., Nicholls, D., Shen, J.: A stable, high-order method for two-dimensional bounded-obstacle scattering. J. Comput. Phys. 224, 1145–1169 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Golub, G.H., Van Loan, C.F.: Matrix Computations. The John Hopkins University Press, Baltimore (1989)zbMATHGoogle Scholar
  12. 12.
    He, Y., Nicholls, D.P., Shen, J.: An efficient and stable spectral method for electromagnetic scattering from a layered periodic structure. J. Comput. Phys. 231(8), 3007–3022 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ji, X., Sun, J., Turner, T.: Al- gorithm 922: a mixed finite element method for helmholtz trans- mission eigenvalues. ACM Transaction on Math. Soft. 38 (2012)Google Scholar
  14. 14.
    Kirsch, A.: On the existence of transmission eigenvalues. Inverse Probl. Imaging 3(2), 155–172 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kwan, Y., Shen, J.: An efficient direct parallel elliptic solver by the spectral element method. J. Comput. Phys. 225, 1721–1735 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Nicholls, D., Shen, J.: A stable, high-order method for two-dimensional bounded-obstacle scattering. SIAM J. Sci. Comput. 28, 1398–1419 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Nicholls, D.P., Reitich, F.: A new approach to analyticity of Dirichlet-Neumann operators. Proc. R. Soc. Edinburgh Sect. A 131(6), 1411–1433 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Päivärinta, L., Sylvester, J.: Transmission eigenvalues. SIAM J. Math. Anal 40(2), 738–753 (2008)Google Scholar
  19. 19.
    Rynne, B.P., Sleeman, B.D.: The interior transmission problem and inverse scattering from inhomogeneous media. SIAM J. Math. Anal. 22, 1755 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Shen, J.: Efficient spectral-Galerkin method I. direct solvers for second- and fourth-order equations by using Legendre polynomials. SIAM J. Sci. Comput. 15, 1489–1505 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Shen, J.: Efficient spectral-galerkin methods iii: polar and cylindrical geometries. SIAM J. Sci. Comput. 18(6), 1583–1604 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Shen, J., Tang, T., Wang, L.: Spectral Methods: Algorithms, Analysis and Applications, volume 41 of Springer Series in Computational Mathematics. Springer, Berlin (2011)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.School of Mathematics and ScienceXiamen UniversityXiamenChina
  2. 2.Department of MathematicsPurdue UniversityWest LafayetteUSA

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