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Spectral Indicator Method for a Non-selfadjoint Steklov Eigenvalue Problem

  • J. Liu
  • J. SunEmail author
  • T. Turner
Article
  • 16 Downloads

Abstract

We propose an efficient numerical method for a non-selfadjoint Steklov eigenvalue problem. The Lagrange finite element is used for discretization and the convergence is proved using the spectral perturbation theory for compact operators. The non-selfadjointness of the problem leads to non-Hermitian matrix eigenvalue problem. Due to the existence of complex eigenvalues and lack of a priori spectral information, we employ the recently developed spectral indicator method to compute eigenvalues in a given region on the complex plane. Numerical examples are presented to validate the effectiveness of the proposed method.

Keywords

Non-selfadjoint Steklov eigenvalues Spectral indicator methods Helmholtz equation Finite elements 

Notes

References

  1. 1.
    Armentano, M.G., Padra, C.: A posteriori error estimates for the Steklov eigenvalue problem. Appl. Numer. Math. 58(5), 593–601 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    An, J., Bi, H., Luo, Z.: A highly efficient spectral-Galerkin method based on tensor product for fourth-order Steklov equation with boundary eigenvalue. J. Inequal. Appl. 12, 211 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Andreev, A.B., Todorov, T.D.: Isoparametric finite-element approximation of a Steklov eigenvalue problem. IMA J. Numer. Anal. 24(2), 309–322 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Babuška, I., Osborn, J. In: Ciarlet, P.G., Lions, J.L. (eds.) Eigenvalue Problems, Handbook of Numerical Analysis, vol. II, Finite Element Methods (Part 1). Elsevier Science Publishers B.V., North-Holland (1991)Google Scholar
  5. 5.
    Bai, Z., Demmel, J., Dongarra, J., Ruhe, A., van der Vorst, H. (eds.): Templates for the Solution of Algebraic Eigenvalue Problems: a Practical Guide. Society for Industrial and Applied Mathematics, Philadelphia (2000)zbMATHGoogle Scholar
  6. 6.
    Beyn, W.J.: An integral method for solving nonlinear eigenvalue problems. Linear Algebr. Appl. 436(10), 3839–3863 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bi, H., Li, H., Yang, Y.: An adaptive algorithm based on the shifted inverse iteration for the Steklov eigenvalue problem. Appl. Numer. Math. 105, 64–81 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bi, H., Li, Z., Yang, Y.: Local and parallel finite element algorithms for the Steklov eigenvalue problem. Numer. Methods Partial Differ. Equ. 32(2), 399–417 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Boffi, D.: Finite element approximation of eigenvalue problems. Acta Numer. 19, 1–120 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bramble, J.H., Osborn, J.E.: Approximation of Steklov eigenvalues of non-selfadjoint second order elliptic operators. In: The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations Proceedings Symposyium, University of Maryland, Baltimore, MD, pp. 387–408. Academic Press, New York (1972)Google Scholar
  11. 11.
    Brenner, S.C., Scott, L.R.: The mathematical theory of finite element methods. In: Texts in Applied Mathematics, 3rd edn. vol. 15. Springer, New York (2008)Google Scholar
  12. 12.
    Cakoni, F., Colton, D., Meng, S., Monk, P.: Stekloff eigenvalues in inverse scattering. SIAM J. Appl. Math. 76(4), 1737–1763 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Canavati, J.A., Minzoni, A.A.: A discontinuous Steklov problem with an application to water waves. J. Math. Anal. Appl. 69(2), 540–558 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Cao, L., Zhang, L., Allegretto, W., Lin, Y.: Multiscale asymptotic method for Steklov eigenvalue equations in composite media. SIAM J. Numer. Anal. 51(1), 273–296 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Colton, D., Monk, P., Sun, J.: Analytical and computational methods for transmission eigenvalues. Inverse Probl. 26(4), 045011 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Dello Russo, A., Alonso, A.E.: A posteriori error estimates for nonconforming approximations of Steklov eigenvalue problems. Comput. Math. Appl. 62(11), 4100–4117 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Grisvard, P.: Elliptic Problems in Non Smooth Domains. Pitman, Boston (1985)zbMATHGoogle Scholar
  18. 18.
    Han, X., Li, Y., Xie, H.: A multilevel correction method for Steklov eigenvalue problem by nonconforming finite element methods. Numer. Math. Theory Methods Appl. 8(3), 383–405 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hsiao, G.C., Wendland, W.L.: Boundary integral equations. In: Applied Mathematical Sciences, vol. 164. Springer, Berlin (2008)Google Scholar
  20. 20.
    Huang, R., Struthers, A., Sun, J., Zhang, R.: Recursive integral method for transmission eigenvalues. J. Comput. Phys. 327, 830–840 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Huang, R., Sun, J., Yang, C.: Recursive integral method with Cayley transformation. Numer. Linear Algebr. Appl. 25(6), (2018).  https://doi.org/10.1002/nla.2199
  22. 22.
    Kato, T.: Perturbation Theory of Linear Operators, Classics in Mathematics. Springer, Berlin (1995)Google Scholar
  23. 23.
    Kumar, P., Kumar, M.: Simulation of a nonlinear Steklov eigenvalue problem using finite-element approximation. Comput. Math. Model. 21(1), 109–116 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kuznetsov, N., Kulczycki, T., Kwaśnicki, M., Nazarov, A., Poborchi, S., Polterovich, I., Siudeja, B.: The legacy of Vladimir Andreevich Steklov. Not. Am. Math. Soc. 61(1), 9–22 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Li, Q., Lin, Q., Xie, H.: Nonconforming finite element approximations of the Steklov eigenvalue problem and its lower bound approximations. Appl. Math. 58(2), 129–151 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Mora, D., Rivera, G., Rodríguez, R.: A virtual element method for the Steklov eigenvalue problem. Math. Models Methods Appl. Sci. 25(8), 1421–1445 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Osborn, J.: Spectral approximation for compact operators. Math. Comp. 29(131), 712–725 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Peng, Z., Bi, H., Li, H., Yang, Y.: A multilevel correction method for convection–diffusion eigenvalue problems. Math. Probl. Eng. 904347 (2015)Google Scholar
  29. 29.
    Polizzi, E.: Density-matrix-based algorithms for solving eigenvalue problems. Phys. Rev. B 79, 115112 (2009)CrossRefGoogle Scholar
  30. 30.
    Saad, Y.: Numerical Methods for Large Eigenvalue Problems, Classics in Applied Mathematics, vol. 66. SIAM, Philadelphia (2011)CrossRefGoogle Scholar
  31. 31.
    Sakurai, T., Sugiura, H.: A projection method for generalized eigenvalue problems using numerical integration. J. Comput. Appl. Math. 159(1), 119–128 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Sun, J.: Iterative methods for transmission eigenvalues. SIAM J. Numer. Anal. 49(5), 1860–1874 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Sun, J.: Estimation of transmission eigenvalues and the index of refraction from Cauchy data. Inverse Probl. 27, 015009 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Sun, J., Zhou, A.: Finite Element Methods for Eigenvalue Problems. CRC Press, Taylor & Francis Group, Boca Raton (2016)CrossRefGoogle Scholar
  35. 35.
    Xie, H.: A type of multilevel method for the Steklov eigenvalue problem. IMA J. Numer. Anal. 34, 592–608 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Xie, H., Wu, X.: A multilevel correction method for interior transmission eigenvalue problem. J. Sci. Comput. 72, 586–604 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Xie, H., Zhang, Z.: A Multilevel Correction Scheme for Nonsymmetric Eigenvalue Problems by Finite Element Methods. arXiv:1505.06288
  38. 38.
    Yang, Y., Li, Q., Li, S.: Nonconforming finite element approximation of the Steklov eigenvalue problem. Appl. Numer. Math. 59, 2388–2401 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mathematical SciencesJinan UniversityGuangzhouChina
  2. 2.Department of Mathematical SciencesMichigan Technological UniversityHoughtonUSA
  3. 3.Department of Mathematics and Computer ScienceUniversity of Maryland Eastern ShorePrincess AnneUSA

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