Science China Mathematics

, Volume 61, Issue 8, pp 1519–1542 | Cite as

An adaptive C0IPG method for the Helmholtz transmission eigenvalue problem

  • Hao Li
  • Yidu YangEmail author


The interior penalty methods using C0 Lagrange elements (C0IPG) developed in the recent decade for the fourth order problems are an interesting topic in academia at present. In this paper, we discuss the adaptive fashion of C0IPG method for the Helmholtz transmission eigenvalue problem. We give the a posteriori error indicators for primal and dual eigenfunctions, and prove their reliability and efficiency. We also give the a posteriori error indicator for eigenvalues and design a C0IPG adaptive algorithm. Numerical experiments show that this algorithm is efficient and can get the optimal convergence rate.


transmission eigenvalues interior penalty Galerkin method Lagrange elements a posteriori error estimates adaptive algorithm 


65N25 65N30 


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This work was supported by National Natural Science Foundation of China (Grant No. 11561014). The authors thank the referees for their valuable comments and suggestions that led to the large improvement of this paper.


  1. 1.
    Ainsworth M, Oden J T. A Posterior Error Estimation in Finite Element Analysis. New York: Wiley-Inter Science, 2011Google Scholar
  2. 2.
    An J, Shen J. A spectral-element method for transmission eigenvalue problems. J Sci Comput, 2013, 57: 670–688MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Babuska I, Osborn J E. Eigenvalue problems. In: Finite Element Methods (Part 1). Handbook of Numerical Analysis, vol. 2. North-Holand: Elsevier, 1991, 640–787Google Scholar
  4. 4.
    Babuska I, Rheinboldt W C. Error estimates for adaptive finite element computations. SIAM J Numer Anal, 1978, 15: 736–754MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brenner S C. C 0 interior penalty methods. In: Frontiers in Numerical Analysis-Durham 2010. Lecture Notes in Computational Science and Engineering, vol. 85. New York: Springer-Verlag, 2012, 79–147Google Scholar
  6. 6.
    Brenner S C, Gedicke J, Sung L-Y. Adaptive C 0 interior penalty method for biharmonic eigenvalue problems. Oberwolfach Rep, 2013, 10: 3265–3267Google Scholar
  7. 7.
    Brenner S C, Monk P, Sun J. C 0IPG Method for Biharmonic Eigenvalue Problems. Spectral and High Order Methods for Partial Differential Equations, ICOSAHOM 2014. Lecture Notes in Computational Science and Engineering, vol. 106. Switzerland: Springer, 2015Google Scholar
  8. 8.
    Brenner S C, Scott L R. The Mathematical Theory of Finite Element Methods, 2nd ed. New york: Springer-Verlag, 2002CrossRefzbMATHGoogle Scholar
  9. 9.
    Brenner S C, Sung L. C 0 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains. J Sci Comput, 2005, 22/23: 83–118CrossRefzbMATHGoogle Scholar
  10. 10.
    Brenner S C, Wang K, Zhao J. Poincaré-Friedrichs inequalities for piecewise H 2 functions. Numer Funct Anal Optim, 2004, 25: 463–478MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cakoni F, Cayoren M, Colton D. Transmission eigenvalues and the nondestructive testing of dielectrics. Inverse Problems, 2009, 24: 065016MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cakoni F, Gintides D, Haddar H. The existence of an infinite discrete set of transmission eigenvalues. SIAM J Math Anal, 2010, 42: 237–255MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Cakoni F, Haddar H. On the existence of transmission eigenvalues in an inhomogeneous medium. Appl Anal, 2009, 88: 475–493MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Cakoni F, Monk P, Sun J. Error analysis for the finite element approximation of transmission eigenvalues. Comput Methods Appl Math, 2014, 14: 419–427MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Chatelin F. Spectral Approximations of Linear Operators. New York: Academic Press, 1983zbMATHGoogle Scholar
  16. 16.
    Chen L. iFEM: An Integrated Finite Element Method Package in MATLAB. Technical Report. Irvine: University of California at Irvine, 2009Google Scholar
  17. 17.
    Ciarlet P G. Basic error estimates for elliptic proplems. In: Finite Element Methods (Part1). Handbook of Numerical Analysis, vol. 2. North-Holand: Elsevier, 1991, 17–351Google Scholar
  18. 18.
    Colton D, Kress R. Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed. Applied Mathematical Sciences, vol. 93. New York: Springer, 1998CrossRefzbMATHGoogle Scholar
  19. 19.
    Colton D, Monk P, Sun J. Analytical and computational methods for transmission eigenvalues. Inverse Problems, 2010, 26: 045011MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Dai X, Xu J, Zhou A. Convergence and optimal complexity of adaptive finite element eigenvalue computations. Numer Math, 2008, 110: 313–355MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Dörfler W. A convergent adaptive algorithm for Poisson’s equation, SIAM J Numer Anal, 1996, 33: 1106–1124MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Engel G, Garikipati K, Hughes T, et al. Continuous/discontinuous finite element approximations of fourth order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity. Comput Methods Appl Mech Engrg, 2001, 191: 3669–3750MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Geng H, Ji X, Sun J, et al. C 0IP methods for the transmission eigenvalue problem. J Sci Comput, 2016, 68: 326–338MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Gudi T. A new error analysis for discontinuous finite element methods for the linear elliptic problems. Math Comp, 2010, 79: 2169–2189MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Han J, Yang Y. An adaptive finite element method for the transmission eigenvalue problem. J Sci Comput, 2016, 69: 326–338MathSciNetCrossRefGoogle Scholar
  26. 26.
    Han J, Yang Y. An H m-conforming spectral element method on multi-dimensional domain and its application to transmission eigenvalues. Sci China Math, 2017, 60: 1529–1542MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Ji X, Sun J, Turner T. Algorithm 922: A mixed finite element method for Helmholtz transmission eigenvalues. ACM Trans Math Software, 2012, 38: 1–8MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Ji X, Sun J, Xie H. A multigrid method for Helmholtz transmission eigenvalue problems. J Sci Comput, 2014, 60: 276–294MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Ji X, Sun J, Yang Y. Optimal penalty parameter for C 0IPDG. Appl Math Lett, 2014, 37: 112–117MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Kleefeld A. A numerical method to compute interior transmission eigenvalues. Inverse Problems, 2013, 29: 104012MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Li H, Yang Y. C 0 IPG adaptive algorithms for biharmonic eigenvalue problem. Numer Algorithms, 2018, 78: 553–567MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Monk P, Sun J. Finite element methods of Maxwell transmission eigenvalues. SIAM J Sci Comput, 2012, 34: 247–264MathSciNetCrossRefGoogle Scholar
  33. 33.
    Morin P, Nochetto R H, Siebert K. Convergence of adaptive finite element methods. SIAM Rev, 2002, 44: 631–658MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Oden J T, Reddy J N. An Introduction to the Mathematical Theory of Finite Elements. New York: Courier Dover Publications, 2012zbMATHGoogle Scholar
  35. 35.
    Rynne B P, Sleeman B D. The interior transmission problem and inverse scattering from inhomogeneous media. SIAM J Math Anal, 1991, 22: 1755–1762MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Shi Z, Wang M. Finite Element Methods. Beijing: Scientific Publishers, 2013Google Scholar
  37. 37.
    Sun J. Estimation of transmission eigenvalues and the index of refraction from Cauchy data. Inverse Problems, 2011, 27: 015009MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Sun J. Iterative methods for transmission eigenvalues. SIAM J Numer Anal, 2014, 49: 1860–1874MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Sun J, Xu L. Computation of Maxwell’s transmission eigenvalues and its applications in inverse medium problems. Inverse Problems, 2013, 29: 104013MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Verfürth R. A Posteriori Error Estimation Techniques. Oxford: Oxford University Press, 2013CrossRefzbMATHGoogle Scholar
  41. 41.
    Wells G N, Dung N T. A C 0 discontinuous Galerkin formulation for Kirhhoff plates. Comput Methods Appl. Mech Engrg, 2007, 196: 3370–3380MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Yang Y, Bi H, Li H, et al. Mixed method for the Helmholtz transmission eigenvalues. SIAM J Sci Comput, 2016, 38: 1383–1403MathSciNetCrossRefGoogle Scholar
  43. 43.
    Yang Y, Bi H, Li H, et al. A C 0IPG method and its error estimates for the Helmholtz transmission eigenvalue problem. J Comput Appl Math, 2017, 326: 71–86MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Yang Y, Han J, Bi H. Error estimates and a two grid scheme for approximating transmission eigenvalues. ArXiv: 1506.06486, 2016Google Scholar
  45. 45.
    Yang Y, Han J, Bi H. Non-conforming finite element methods for transmission eigenvalue problem. Comput Methods Appl Mech Engrg, 2016, 307: 144–163MathSciNetCrossRefGoogle Scholar
  46. 46.
    Zeng F, Sun J, Xu L. A spectral projection method for transmission eigenvalues. Sci China Math, 2016, 59: 1613–1622MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesGuizhou Normal UniversityGuiyangChina

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