×

Asymptotic expansion and extrapolation for the eigenvalue approximation of the biharmonic eigenvalue problem by Ciarlet-Raviart scheme. (English) Zbl 1122.65106

The paper deals with a new Ciarlet-Raviart mixed scheme for solving the biharmonic eigenvalue problem with bilinear finite elements. This method provides higher order convergence rate for eigenvalues and eigenfunctions approximations. Furthermore, an asymptotic expansion of the eigenvalue error is given which is then used to derive an efficient extrapolation and a posteriori error estimate for eigenvalues. The results are illustrated by some numerical experiments.

MSC:

65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
35P15 Estimates of eigenvalues in context of PDEs
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Babuška, I., Osborn, J.: Eigenvalue problems. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. II, Finite element method (Part I) (1991) · Zbl 0875.65087
[2] Bjørstad, P. E.; Tjøstheim, B. P., High precision solutions of two fourth order eigenvalue problems, Computing, 63, 97-107 (1999) · Zbl 0940.65119
[3] Boffi, D.; Brezzi, F.; Gastaldi, L., On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form, Math. Comput., 69, 121-140 (2000) · Zbl 0938.65126
[4] Brezzi, F.; Fortin, M., Mixed and hybrid finite element method, Springer Series in Computational Mathematics, vol. 15 (1991), Berlin Heidelberg New York: Springer, Berlin Heidelberg New York · Zbl 0788.73002
[5] Brown, B. M.; Davies, E. B.; Jimack, P. K.; Mihajlović, M. D., A numerical investigation of the solution of a class of fourth order eigenvalue problem, Proc. Roy. Soc. London Ser. A, 456, 1505-1521 (2000) · Zbl 0977.65101
[6] Ciarlet, P. G., Finite Element Method for Elliptic Problems (1978), Amsterdam: North-Holland, Amsterdam · Zbl 0383.65058
[7] Ciarlet, P. G.; Raviart, P. A.; de Boor, C., A mixed finite element method for the biharmonic equation, Symposium on Mathematical Aspects of Finite Elements in Partial DIfferential Equations, 125-143 (1974), New York: Academic, New York
[8] Heuveline, V.; Rannacher, R., A posteriori error control for finite element approximations of elliptic eigenvalue problems, Adv. Comput. Math., 15, 107-138 (2001) · Zbl 0995.65111
[9] Hu, Q.; Zou, J., Two new variants of nonlinear inexact Uzawa algorithms for saddle-point problems, Numer. Math., 93, 333-359 (2002) · Zbl 1019.65024
[10] Ishihara, K., A mixed finite element method for the biharmonic eigenvalue problem of plate bending, Publ. Res. Inst. Math. Sci., 14, 399-414 (1978) · Zbl 0389.73075
[11] Lin, Q., Lin, J.: High Performance FEMs. China Sci. Technol. Press (2006)
[12] Lin, Q.; Li, J.; Zhou, A., A rectangle test for Ciarlet-Raviart scheme, Proc. Sys. Sci. & Sys. Engrg., 230-231 (1991), China: Great Wall Culture Publishing Co., China
[13] Lin, Q.; Lu, T., Asymptotic expansions for finite element eigenvalues and finite element, Bonner Math. Schriften, 158, 1-10 (1984) · Zbl 0549.65072
[14] Lin, Q.; Yan, N., High Efficiency FEM Construction and Analysis (1996), China: Hebei University Press, China
[15] Mercier, B.; Osborn, J.; Rappaz, J.; Raviart, P. A., Eigenvalue approximation by mixed and hybird methods, Math. Comput., 36, 427-453 (1981) · Zbl 0472.65080
[16] Osborn, J.: Spectral approximation for compact operators. Math. Comput. 29, 712-725 · Zbl 0315.35068
[17] Wieners, C., Bounds for the N lowest eigenvalues of fourth-order boundary value problems, Computing, 59, 29-41 (1997) · Zbl 0883.65082
[18] Xu, J.; Zhou, A., A two-grid discretization scheme for eigenvalue problems, Math. Comput., 70, 17-25 (2001) · Zbl 0959.65119
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.