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Computable Error Estimates for a Nonsymmetric Eigenvalue Problem

Part of: Partial differential equations, boundary value problems Numerical analysis: Ordinary differential equations Acceleration of convergence

Published online by Cambridge University Press:  07 September 2017

Hehu Xie ,
Manting Xie ,
Xiaobo Yin  and
Meiling Yue
Hehu Xie*
Affiliation:
LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China; and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China
Manting Xie*
Affiliation:
LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China; and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China
Xiaobo Yin*
Affiliation:
School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, China
Meiling Yue*
Affiliation:
LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China; and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China
*
*Corresponding author. Email addresses:hhxie@lsec.cc.ac.cn (H. Xie), xiemanting@lsec.cc.ac.cn (M. Xie), yinxb@mail.ccnu.edu.cn (X. Yin), yuemeiling@lsec.cc.ac.cn (M. Yue)
*Corresponding author. Email addresses:hhxie@lsec.cc.ac.cn (H. Xie), xiemanting@lsec.cc.ac.cn (M. Xie), yinxb@mail.ccnu.edu.cn (X. Yin), yuemeiling@lsec.cc.ac.cn (M. Yue)
*Corresponding author. Email addresses:hhxie@lsec.cc.ac.cn (H. Xie), xiemanting@lsec.cc.ac.cn (M. Xie), yinxb@mail.ccnu.edu.cn (X. Yin), yuemeiling@lsec.cc.ac.cn (M. Yue)
*Corresponding author. Email addresses:hhxie@lsec.cc.ac.cn (H. Xie), xiemanting@lsec.cc.ac.cn (M. Xie), yinxb@mail.ccnu.edu.cn (X. Yin), yuemeiling@lsec.cc.ac.cn (M. Yue)
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Abstract

We provide some computable error estimates in solving a nonsymmetric eigenvalue problem by general conforming finite element methods on general meshes. Based on the complementary method, we first give computable error estimates for both the original eigenfunctions and the corresponding adjoint eigenfunctions, and then we introduce a generalised Rayleigh quotient to deduce a computable error estimate for the eigenvalue approximations. Some numerical examples are presented to illustrate our theoretical results.

Keywords

Nonsymmetric eigenvalue problemcomputable error estimatesasymptotical exactnessfinite element methodcomplementary method

MSC classification

Secondary: 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65N25: Eigenvalue problems 65L15: Eigenvalue problems 65B99: None of the above, but in this section

Information

Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 3 , August 2017 , pp. 583 - 602
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Adams, R. A., Sobolev Spaces, Academic Press, New York (1975).Google Scholar
[2] Ainsworth, M. and Oden, J., A Posteriori Error Estimation in Finite Element Analysis, Wiley & Sons, New York (2000).Google Scholar
[3] Brezzi, F. and Fortin, M., Mixed and Hybrid Finite Element Methods, Springer, New York (1991).Google Scholar
[4] Babuška, I. and Osborn, J., Eigenvalue Problems, in Handbook of Numerical Analysis Vol. II, Finite Element Methods (Part 1), Lions, P.G. and Ciarlet, P.G. (Eds.), pp. 641–787, North-Holland, Amsterdam (1991).Google Scholar
[5] Babuška, I. and Rheinboldt, W., Error estimates for adaptive finite element computations, SIAM J. Numer. Anal. 15, 736–754 (1978).Google Scholar
[6] Babuška, I. and Rheinboldt, W., A-posteriori error estimates for the finite element method, Int. J. Numer. Methods Eng. 12, 1597–1615 (1978).Google Scholar
[7] Brenner, S. and Scott, L., The Mathematical Theory of Finite Element Methods, Springer, New York (1994).Google Scholar
[8] Ciarlet, P. G., The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam (1978).Google Scholar
[9] Cliffe, K. A., Hall, E. J. C. and Houston, P., Adapative discontinuous Galerkin methods for eigen-value problems arising in incompressible fluid flows, SIAM J. Sci. Comput. 31, 4607–4632 (2010).Google Scholar
[10] Gedicke, J. and Carstensen, C., A posteriori error estimators for non-symmetric eigenvalue problems, Preprint 659, DFG Research Center Matheon, Berlin (2009).Google Scholar
[11] Haslinger, J. and Hlaváček, I., Convergence of a finite element method based on the dual variational formulation, Appl. Math. 21, 43–65 (1976).Google Scholar
[12] Heuveline, V. and Rannacher, R., A posteriori error control for finite element approximations of elliptic eigenvalue problems, Adv. Comput. Math. 15, 107–138 (2001).Google Scholar
[13] Heuveline, V. and Rannacher, R., Adaptive FEM for eigenvalue problems with application in hydrodynamic stability analysis, in Advances in Numerical Mathematics, Proc. Int. Conf., Sept. 16-17, 2005, Moscow, Moscow: Institute of Numerical Mathematics RAS, 2006.Google Scholar
[14] Lin, Q. and Lin, J., Finite Element Methods: Accuracy and Improvement, Science Press, Beijing (2006).Google Scholar
[15] Neittaanmäki, P. and Repin, S., Reliable Methods for Computer Simulation, Error Control and a Posteriori Estimates, vol. 33 of Studies in Mathematics and Its Applications, Elsevier Science, Amsterdam (2004).Google Scholar
[16] Repin, S., A Posteriori Estimates for Partial Differential Equations, vol. 4 of Radon Series on Computational and Applied Mathematics, Walter de Gruyter GmbH & Co. KG, Berlin (2008).Google Scholar
[17] Vejchodský, T., Complementarity based a posteriori error estimates and their properties, Math. Comput. Simulation 82, 2033–2046 (2012).Google Scholar
[18] Vejchodský, T., Computing upper bounds on Friedrichs’ constant, in Applications of Mathematics, Brandts, J., Chleboun, J., Korotov, S., Segeth, K., Šístek, J. and Vejchodský, T. (Eds.), pp. 278–289, Institute of Mathematics, ASCR, Prague (2012).Google Scholar
[19] Wu, H. and Zhang, Z., Enhancing eigenbalue approximation by gradient recovery on adaptive meshes, IMA J. Numer. Anal. 29, 1008–1022 (2009).Google Scholar
[20] Xie, H. and Xie, M., Computable error estimates for ground state solution of Bose-Einstein condensates, arXiv:1604.05228, http://arxiv.org/abs/1604.05228 (2016).Google Scholar
[21] Xie, H., Yue, M. and Zhang, N., Fully computable error bounds for eigenvalue problem, arXiv:1601.01561, http://arxiv.org/abs/1601.01561 (2016).Google Scholar
[22] Xie, H. and Zhang, Z., A multilevel correction scheme for nonsymmetric eigenvalue problems by finite element methods, arXiv:1505.06288, http://arxiv.org/abs/1505.06288 (2015).Google Scholar
[23] Zienkiewicz, O. C. and Zhu, J., The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique, Int. J. Numer. Methods Eng. 33, 1331–1364 (1992).Google Scholar