A Superconvergence result for mixed finite element approximations of the eigenvalue problem
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 4, pp. 797-812.

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.

DOI : 10.1051/m2an/2011065
Classification : 65N30, 65N25, 65L15, 65B99
Mots clés : second order elliptic eigenvalue problem, mixed finite element method, superconvergence
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     title = {A {Superconvergence} result for mixed finite element approximations of the eigenvalue problem},
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Lin, Qun; Xie, Hehu. A Superconvergence result for mixed finite element approximations of the eigenvalue problem. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 4, pp. 797-812. doi : 10.1051/m2an/2011065. http://www.numdam.org/articles/10.1051/m2an/2011065/

[1] I. Babuška and J.E. Osborn, Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems. Math. Comp. 52 (1989) 275-297. | MR | Zbl

[2] 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. | MR | Zbl

[3] C. Bacuta and J.H. Bramble, 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. | MR | Zbl

[4] D. Boffi, Finite element approximation of eigenvalue problems. Acta Numer. 19 (2010) 1-120. | MR | Zbl

[5] D. Boffi, F. Brezzi and L. Gastaldi, On the convergence of eigenvalues for mixed fomulations. Ann. Scuola Norm. Sup. Pisa Cl. Sci 25 (1997) 131-154. | EuDML | Numdam | MR | Zbl

[6] D. Boffi, F. Brezzi and L. Gastaldi, On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form. Math. Comp. 69 (2000) 121-140. | MR | Zbl

[7] S. Brenner and L. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York (1994). | MR | Zbl

[8] F. Brezzi and M. Fortin, MixedandHybrid Finite Element Methods. Springer-Verlag, New York (1991). | MR | Zbl

[9] F. Chatelin, Spectral Approximation of Linear Operators. Academic Press Inc., New York (1983). | MR | Zbl

[10] J. Douglas and J.E. Roberts, Global estimates for mixed methods for second order elliptic equations. Math. Comp. 44 (1985) 39-52. | MR | Zbl

[11] R. Durán, L. Gastaldi and C. Padra, A posteriori error estimators for mixed approximations of eigenvalue problems. Math. Models Methods Appl. Sci. 9 (1999) 1165-1178. | MR | Zbl

[12] F. Gardini, A posteriori error estimates for an eigenvalue problem arising from fluid-structure interaction. Instituto Lombardo (Rend. Sc.) (2004) 138. | MR | Zbl

[13] F. Gardini, Mixed approximation of eigenvalue problems : a superconvergence result. ESAIM : M2AN 43 (2009) 853-865. | Numdam | MR | Zbl

[14] V. Girault and P. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Springer-Verlag, Berlin (1986). | MR | Zbl

[15] P. Grisvard, Singularities in Boundary Problems. MASSON and Springer-Verlag (1985). | Zbl

[16] Q. Lin and J. Lin, Finite Element Methods : Accuracy and Inprovement. China Sci. Tech. Press (2005).

[17] Q. Lin and H. Xie, 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. | MR | Zbl

[18] Q. Lin and N. Yan, The Construction and Analysis of High Efficiency Finite Element Methods. HeBei University Publishers (1995) (in Chinese)

[19] Q. Lin, H. Huang and Z. Li, New expansion of numerical eigenvalue for − Δu = λρu by nonconforming elements. Math. Comp. 77 (2008) 2061-2084. | MR | Zbl

[20] B. Mercier, J. Osborn, J. Rappaz and P.A. Raviart, Eigenvalue approximation by mixed and hybrid methods. Math. Comp. 36 (1981) 427-453. | MR | Zbl

[21] J. Osborn, 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. | MR | Zbl

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