Korean J. Math.  Vol 28, No 3 (2020)  pp.587-601
DOI: https://doi.org/10.11568/kjm.2020.28.3.587

Enhancing eigenvalue approximation with Bank--Weiser error estimators

Kwang-Yeon Kim

Abstract


In this paper we propose a way of enhancing eigenvalue approximations with the Bank--Weiser error estimators for the $P1$ and $P2$ conforming finite element methods of the Laplace eigenvalue problem. It is shown that we can achieve two extra orders of convergence than those of the original eigenvalue approximations when the corresponding eigenfunctions are smooth and the underlying triangulations are strongly regular. Some numerical results are presented to demonstrate the accuracy of the enhanced eigenvalue approximations.


Keywords


eigenvalue problem; finite element method; error estimator; superconvergence

Subject classification

65N15; 65N25; 65N30

Sponsor(s)



Full Text:

PDF

References


P. Amore, J. P. Boyd, F. M. Ferna ́ndez, and B. Ro ̈sler, High or- der eigenvalues for the Helmholtz equation in complicated non-tensor domains through Richardson extrapolation of second order finite differences, J. Comput. Phys. 312 (2016), 252–271. (Google Scholar)

I. Babuˇska and J. E. Osborn, Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems, Math. Comp. 52 (1989), 275–297. (Google Scholar)

I. Babuˇska and J. E. Osborn, Eigenvalue Problems, in Handbook of Numeri- cal Analysis II, Finite Element Methods (Part 1), edited by P.G. Lions and P.G. Ciarlet, North-Holland, Amsterdam, 1991, 641–787. (Google Scholar)

R. E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp. 44 (1985), 283–301. (Google Scholar)

R. E. Bank and J. Xu, Asymptotically exact a posteriori error estimators, Part I: Grids with superconvergence, SIAM J. Numer. Anal. 41 (2003), 2294– 2312. (Google Scholar)

D. Boffi, Finite element approximation of eigenvalue problems, Acta Numer. 19 (2010), 1–120. (Google Scholar)

R. G. Dura ́n, C. Padra, and R. Rodr ́ıguez, A posteriori error estimates for the finite element approximation of eigenvalue problems, Math. Models Methods Appl. Sci. 13 (2003), 1219–1229. (Google Scholar)

R. G. Dura ́n and R. Rodr ́ıguez, On the asymptotic exactness of Bank– Weiser’s estimator, Numer. Math. 62 (1992), 297–303. (Google Scholar)

H. Guo, Z. Zhang, and R. Zhao, Superconvergent two-grid methods for el- liptic eigenvalue problems, J. Sci. Comput. 70 (2017), 125–148. (Google Scholar)

J. Hu, Y. Huang, and Q. Shen, A high accuracy post-processing algorithm for the eigenvalues of elliptic operators, Numer. Math. 52 (2012), 426–445. (Google Scholar)

Y. Huang and J. Xu, Superconvergence of quadratic finite elements on mildly structured grids, Math. Comp. 77 (2008), 1253–1268. (Google Scholar)

O. Karakashian and F. Pascal, A posteriori error estimates for a discontin- uous Galerkin approximation of second-order elliptic problems, SIAM J. Numer. Anal. 41 (2003), 2374–2399. (Google Scholar)

K.-Y. Kim and J.-S. Park, Asymptotic exactness of some Bank–Weiser error estimator for quadratic triangular finite element, Bull. Korean Math. Soc. 57 (2020), 393–406. (Google Scholar)

M. G. Larson, A posteriori and a priori error analysis for finite element ap- proximations of self-adjoint elliptic eigenvalue problems, SIAM J. Numer. Anal. 38 (2000), 608–625. (Google Scholar)

Q. Lin, H. Xie, and J. Xu, Lower bounds of the discretization error for piece- wise polynomials, Math. Comp. 83 (2014), 1–13. (Google Scholar)

A. Maxim, Asymptotic exactness of an a posteriori error estimator based on the equilibrated residual method, Numer. Math. 106 (2007), 225–253. (Google Scholar)

A. Naga and Z. Zhang, Function value recovery and its application in eigen- value problems, SIAM J. Numer. Anal. 50 (2012), 272–286. (Google Scholar)

A. Naga, Z. Zhang, and A. Zhou, Enhancing eigenvalue approximation by gradient recovery, SIAM J. Sci. Comput. 28 (2006), 1289–1300. (Google Scholar)

M. R. Racheva and A. B. Andreev, Superconvergence postprocessing for eigenvalues, Comp. Methods Appl. Math. 2 (2002), 171–185. (Google Scholar)

H. Wu and Z. Zhang, Enhancing eigenvalue approximation by gradient recov- ery on adaptive meshes, IMA J. Numer. Anal. 29 (2009), 1008–1022. (Google Scholar)

J. Xu and Z. Zhang, Analysis of recovery type a posteriori error estimators for mildly structured grids, Math. Comp. 73 (2004), 1139–1152. (Google Scholar)

J. Xu and A. Zhou, A two-grid discretization scheme for eigenvalue problems, Math. Comp. 70 (2001), 17–25. (Google Scholar)


Refbacks

  • There are currently no refbacks.


ISSN: 1976-8605 (Print), 2288-1433 (Online)

Copyright(c) 2013 By The Kangwon-Kyungki Mathematical Society, Department of Mathematics, Kangwon National University Chuncheon 21341, Korea Fax: +82-33-259-5662 E-mail: kkms@kangwon.ac.kr