Open Access
Issue
ESAIM: M2AN
Volume 55, Number 5, September-October 2021
Page(s) 1779 - 1802
DOI https://doi.org/10.1051/m2an/2021039
Published online 17 September 2021
  1. M. Armentano and C. Padra, A posteriori error estimates for the Steklov eigenvalue problem. Appl. Numer. Math. 58 (2008) 593–601. [CrossRef] [MathSciNet] [Google Scholar]
  2. I. Babuška and J. Osborn, Eigenvalue Problems, edited by J. Lions and P. Ciarlet, Vol. II. In: Handbook of numerical analysis, Finite element methods (Part 1), North-Holland, Amsterdam (1991) 641–787. [Google Scholar]
  3. R.E. Bank and T. Dupont, An optimal order process for solving finite element equations. Math. Comput. 36 (1981) 35–51. [Google Scholar]
  4. H. Bi and Y. Yang, A two-grid method of the non-conforming Crouzeix-Raviart element for the Steklov eigenvalue problem. Appl. Math. Comput. 217 (2011) 9669–9678. [Google Scholar]
  5. H. Bi, Y. Zhang and Y. Yang, Two-grid discretizations and a local finite element scheme for a non-selfadjoint Stekloff eigenvalue problem. Comput. Math. Appl. 79 (2020) 1895–1913. [Google Scholar]
  6. J. Bramble and J. Osborn, Approximation of Steklov Eigenvalues of Non-selfadjoint Second Order Elliptic Operators, edited by A. Aziz, In: Mathematical foundations of the finite element method with applications to PDEs. Academic Press, New York (1972) 387–408. [Google Scholar]
  7. J. Bramble and J. Pasciak, New convergence estimates for multigrid algorithms. Math. Comput. 49 (1987) 311–329. [Google Scholar]
  8. S. Brenner and L. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York (1994). [Google Scholar]
  9. F. Cakoni, D. Colton, S. Meng and P. Monk, Steklov eigenvalues in inverse scattering. SIAM J. Appl. Math. 76 (2016) 1737–1763. [Google Scholar]
  10. J.A. Canavati and A.A. Minzoni, A discontinuous Steklov problem with an application to water waves. J. Math. Anal. Appl. 69 (1979) 540–558. [Google Scholar]
  11. L. Cao, L. Zhang, W. Allegretto and Y. Lin, Multiscale asymptotic method for Steklov eigenvalue equations in composite media. SIAM J. Numer. Anal. 51 (2013) 273–296. [Google Scholar]
  12. C. Carstensen and J. Gedicke, An adaptive finite element eigenvalue solver of asymptotic quasi-Optimal computational complexity. SIAM J. Numer. Anal. 50 (2012) 1029–1057. [Google Scholar]
  13. J.M. Cascon, C. Kreuzer, R.H. Nochetto and K.G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method. SIAM J. Numer. Anal. 46 (2008) 2524–2550. [Google Scholar]
  14. Z. Chen and S. Dai, On the efficiency of adaptive finite element methods for elliptic problems with discontinous coefficients. SIAM J. Sci. Comput. 24 (2002) 443–462. [Google Scholar]
  15. H. Chen, H. Xie and F. Xu, A full multigrid method for eigenvalue problems. J. Comput. Phys. 322 (2016) 747–759. [Google Scholar]
  16. P. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978). [Google Scholar]
  17. K. Cliffe, E. Hall and P. Houston, Adaptive discontinuous Galerkin methods for eigenvalue problems arising in incompressible fluid flows. SIAM J. Sci. Comput. 31 (2008) 4607–4632. [Google Scholar]
  18. X. Dai, J. Xu and A. Zhou, Convergence and optimal complexity of adaptive finite element eigenvalue computations. Numer. Math. 110 (2008) 313–355. [Google Scholar]
  19. S. Giani and I.G. Graham, A convergent adaptive method for elliptic eigenvalue problems. SIAM J. Numer. Anal. 47 (2009) 1067–1091. [Google Scholar]
  20. W. Hackbusch, Multi-Grid Methods and Applications. Vol.4 of: Computational Mathematics. Springer-Verlag, Berlin-Heidelberg (1985). [Google Scholar]
  21. X. Han, H. Xie and F. Xu, A cascadic multigrid method for eigenvalue problem. J. Comput. Math. 35 (2017) 56–72. [Google Scholar]
  22. V. Heuveline and R. Rannacher, A posteriori error control for finite element approximations of elliptic eigenvalue problems. Adv. Comput. Math. 15 (2001) 107–138. [CrossRef] [MathSciNet] [Google Scholar]
  23. Q. Hong, H. Xie and F. Xu, A Multilevel correction type of adaptive finite element method for eigenvalue problems. SIAM J. Sci. Comput. 40 (2018) A4208–A4235. [Google Scholar]
  24. G. Hu, H. Xie and F. Xu, A multilevel correction adaptive finite element method for Kohn-Sham equation. J. Comput. Phys. 355 (2018) 436–449. [Google Scholar]
  25. X. Ji, J. Sun and H. Xie, A multigrid method for Helmholtz transmission eigenvalue problems. J. Sci. Comput. 60 (2014) 276–294. [Google Scholar]
  26. S. Jia, H. Xie, M. Xie and F. Xu, A full multigrid method for nonlinear eigenvalue problems. Sci. China Math. 59 (2016) 2037–2048. [Google Scholar]
  27. N. Kuznetsov, T. Kulczycki, M. Kwaśnicki, A. Nazarov, S. Poborchi, I. Polterovich and B. Siudeja, The legacy of Vladimir Andreevich Steklov. Not. Am. Math. Soc. 61 (2014) 9–22. [Google Scholar]
  28. Q. Lin and J. Lin, Finite Element Methods: Accuracy and Inprovement. Science Press, Beijing (2006). [Google Scholar]
  29. Q. Lin and H. Xie, A multi-level correction scheme for eigenvalue problems. Math. Comput. 84 (2015) 71–88. [Google Scholar]
  30. J. Liu, Y. Liu and J. Sun, An inverse medium problem using Stekloff eigenvalues and a Bayesian approach. Inverse Prob. 35 (2019). [Google Scholar]
  31. J. Liu, J. Sun and T. Turner, Spectral indicator method for a non-selfadjoint Steklov eigenvalue problem. J. Sci. Comput. 79 (2019) 1814–1831. [Google Scholar]
  32. J. Meng and L. Mei, Discontinuous Galerkin methods of the non-selfadjoint Steklov eigenvalue problem in inverse scattering. Appl. Math. Comput. 381 (2020) 125–307. [Google Scholar]
  33. P. Morin, R.H. Nochetto and K. Siebert, Convergence of adaptive finite element methods. SIAM Rev. 44 (2002) 631–658. [CrossRef] [MathSciNet] [Google Scholar]
  34. A.D. Russo and A.E. Alonso, A posteriori error estimates for nonconforming approximations of Steklov eigenvalue problem. Comput. Appl. Math. 62 (2011) 4100–4117. [Google Scholar]
  35. R. Verfürth, A Review of a Posteriori Error Estimation Andadaptive Mesh-refinement Techniques. Wiley-Teubner, New York (1996). [Google Scholar]
  36. H. Xie, A multigrid method for eigenvalue problem. J. Comput. Phys. 274 (2014) 550–561. [Google Scholar]
  37. H. Xie, A type of multilevel method for the Steklov eigenvalue problem. IMA J. Numer. Anal. 34 (2014) 592–608. [Google Scholar]
  38. H. Xie and X. Wu, A multilevel correction method for interior transmission eigenvalue problem. J. Sci. Comput. 72 (2017) 586–604. [Google Scholar]
  39. H. Xie and M. Xie, A multigrid method for the ground state solution of Bose-Einstein condensates. Commun. Comput. Phys. 19 (2016) 648–662. [Google Scholar]
  40. H. Xie and M. Xie, Computable error estimates for ground state solution of Bose-Einstein condensates. J. Sci. Comput. 81 (2019) 1072–1087. [Google Scholar]
  41. H. Xie, M. Xie, X. Yin and M. Yue, Computable error estimates for a nonsymmetric eigenvalue problem. East Asian J. Appl. Math. 7 (2017) 583–602. [Google Scholar]
  42. H. Xie and Z. Zhang, A multilevel correction scheme for nonsymmetric eigenvalue problems by finite element methods. arXiv:1505.06288 (2015). http://arxiv.org/abs/1505.06288. [Google Scholar]
  43. H. Xie and T. Zhou, A multilevel finite element method for Fredholm integral eigenvalue problems. J. Comput. Phys. 303 (2015) 173–184. [Google Scholar]
  44. F. Xu, H. Xie and N. Zhang, A parallel augmented subspace method for eigenvalue problems. SIAM J. Sci. Comput. 42 (2020) A2655–A2677. [Google Scholar]
  45. J. Xu and Z. Zhang, Analysis of recovery type a posteriori error estimators for mildly structured grids. Math. Comput. 73 (2003) 1139–1152. [Google Scholar]
  46. F. Xu, M. Yue, Q. Huang and H. Ma, An asymptotically exact a posteriori error estimator for non-selfadjoint Steklov eigenvalue problem. Appl. Numer. Math. 156 (2020) 210–227. [Google Scholar]
  47. Y. Yang, J. Han and H. Bi, Error estimates and a two grid scheme for approximating transmission eigenvalues. arXiv:1506.06486v2 (2016). http://arxiv.org/abs/1506.06486v2. [Google Scholar]
  48. M. Yue, H. Xie and M. Xie, A cascadic multigrid method for nonsymmetric eigenvalue problem. Appl. Numer. Math. 146 (2019) 55–72. [Google Scholar]
  49. Y. Zeng and F. Wang, A posteriori error estimates for a discontinuous Galerkin approximation of Steklov eigenvalue problems. Appl. Math. 62 (2017) 243–267. [Google Scholar]
  50. S. Zhang, Y. Xi and X. Ji, A multi-level mixed element method for the eigenvalue problem of biharmonic equation. J. Sci. Comput. 75 (2018) 1415–1444. [Google Scholar]
  51. Y. Zhang, H. Bi and Y. Yang, A multigrid correction scheme for a new Steklov eigenvalue problem in inverse scattering. Int. J. Comput. Math. (2019). DOI:10.1080/00207160.2019.1622686. [PubMed] [Google Scholar]
  52. Z. Zhang and A. Naga, A new finite element gradient recovery method: superconvergence property. SIAM J. Sci. Comput. 26 (2005) 1192–1213. [Google Scholar]
  53. O.C. Zienkiewicz and J. Zhu, The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique. Int. J. Numer. Methods Eng. 33 (1992) 1331–1364. [Google Scholar]

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