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The correction operator for the canonical interpolation operator of the Adini element and the lower bounds of eigenvalues. (English) Zbl 1248.65117

The lower bound property of the eigenvalue by the Adini element method in the two dimensions was first analyzed by Y. Yang [J. Comput. Math. 18, No. 4, 413–418 (2000; Zbl 0957.65092)]. In this paper, the authors develop a correction operator and then use it to analyze the lower bound property of eigenvalues of the Adini element for the fourth order elliptic eigenvalue problem. Comparing these results with earlier results, following two fold advantages are highlighted. Firstly, it can be extended to the Wilson element in any dimension, see [Z. C. Shi and M. Wang, The finite element method. Beijing: Science Press, (2010)] and other nonconforming elements as well; secondly, it provides a new motivation for the analysis of super-convergence for the finite element method, especially for the high order methods.

MSC:

65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35J40 Boundary value problems for higher-order elliptic equations
35P15 Estimates of eigenvalues in context of PDEs

Citations:

Zbl 0957.65092
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Full Text: DOI

References:

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