Hostname: page-component-7b7567c8c-mnklg Total loading time: 0 Render date: 2026-11-13T03:06:47.366Z Has data issue: false hasContentIssue false
  • English
  • Français

A Superconvergence result for mixed finite element approximations of the eigenvalue problem∗

Published online by Cambridge University Press:  03 February 2012

Qun Lin and
Hehu Xie
Qun Lin
Affiliation:
LSEC, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, P.R. China. linq@lsec.cc.ac.cn ; hhxie@lsec.cc.ac.cn
Hehu Xie
Affiliation:
LSEC, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, P.R. China. linq@lsec.cc.ac.cn ; hhxie@lsec.cc.ac.cn

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

In this paper, we present a superconvergence result for the mixed finite element approximations of general second order elliptic eigenvalue problems. It is known that a superconvergence result has been given by Durán et al. [Math. Models Methods Appl. Sci. 9 (1999) 1165–1178] and Gardini [ESAIM: M2AN 43 (2009) 853–865] for the lowest order Raviart-Thomas approximation of Laplace eigenvalue problems. In this work, we introduce a new way to derive the superconvergence of general second order elliptic eigenvalue problems by general mixed finite element methods which have the commuting diagram property. Some numerical experiments are given to confirm the theoretical analysis.

Keywords

Second order elliptic eigenvalue problemmixed finite element methodsuperconvergence

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 4 , July 2012, pp. 797 - 812
Copyright
© EDP Sciences, SMAI, 2012

References

Références

Babuška, I. and Osborn, J.E., Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems. Math. Comp. 52 (1989) 275–297. Google Scholar
I. Babuška and J. Osborn, Eigenvalue Problems, in Handbook of Numerical Analysis II, Finite Element Methods (Part 1), edited by P.G. Lions and P.G. Ciarlet. North-Holland, Amsterdam (1991) 641–787.
Bacuta, C. and Bramble, J.H., Regularity estimates for the solutions of the equations of linear elasticity in convex plane polygonal domain, Special issue dedicated to Lawrence E. Payne. Z. Angew. Math. Phys. 54 (2003) 874–878. Google Scholar
Boffi, D., Finite element approximation of eigenvalue problems. Acta Numer. 19 (2010) 1–120. Google Scholar
Boffi, D., Brezzi, F. and Gastaldi, L., On the convergence of eigenvalues for mixed fomulations. Ann. Scuola Norm. Sup. Pisa Cl. Sci 25 (1997) 131–154. Google Scholar
Boffi, D., Brezzi, F. and Gastaldi, L., On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form. Math. Comp. 69 (2000) 121–140. Google Scholar
S. Brenner and L. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York (1994).
F. Brezzi and M. Fortin, MixedandHybrid Finite Element Methods. Springer-Verlag, New York (1991).
F. Chatelin, Spectral Approximation of Linear Operators. Academic Press Inc., New York (1983).
Douglas, J. and Roberts, J.E., Global estimates for mixed methods for second order elliptic equations. Math. Comp. 44 (1985) 39–52. Google Scholar
Durán, R., Gastaldi, L. and Padra, C., A posteriori error estimators for mixed approximations of eigenvalue problems. Math. Models Methods Appl. Sci. 9 (1999) 1165–1178. Google Scholar
F. Gardini, A posteriori error estimates for an eigenvalue problem arising from fluid-structure interaction. Instituto Lombardo (Rend. Sc.) (2004) 138.
Gardini, F., Mixed approximation of eigenvalue problems : a superconvergence result. ESAIM : M2AN 43 (2009) 853–865. Google Scholar
V. Girault and P. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Springer-Verlag, Berlin (1986).
P. Grisvard, Singularities in Boundary Problems. MASSON and Springer-Verlag (1985).
Q. Lin and J. Lin, Finite Element Methods : Accuracy and Inprovement. China Sci. Tech. Press (2005).
Lin, Q. and Xie, H., Asymptotic error expansion and Richardson extrapolation of eigenvalue approximations for second order elliptic problems by the mixed finite element method. Appl. Numer. Math. 59 (2009) 1884–1893. Google Scholar
Q. Lin and N. Yan, The Construction and Analysis of High Efficiency Finite Element Methods. HeBei University Publishers (1995) (in Chinese)
Lin, Q., Huang, H. and Li, Z., New expansion of numerical eigenvalue for − Δu = λρu by nonconforming elements. Math. Comp. 77 (2008) 2061–2084. Google Scholar
Mercier, B., Osborn, J., Rappaz, J. and Raviart, P.A., Eigenvalue approximation by mixed and hybrid methods. Math. Comp. 36 (1981) 427–453. Google Scholar
Osborn, J., Approximation of the eigenvalue of a nonselfadjoint operator arising in the study of the stability of stationary solutions of the Navier-Stokes equations. SIAM J. Numer. Anal. 13 (1976) 185–197.Google Scholar