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New expansions of numerical eigenvalues for $ -\Delta u=\lambda \rho u$ by nonconforming elements


Authors: Qun Lin, Hung-Tsai Huang and Zi-Cai Li
Journal: Math. Comp. 77 (2008), 2061-2084
MSC (2000): Primary 65N30
DOI: https://doi.org/10.1090/S0025-5718-08-02098-X
Published electronically: May 29, 2008
MathSciNet review: 2429874
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Abstract: The paper explores new expansions of the eigenvalues for $ -\Delta u=\lambda \rho u$ in $ S$ with Dirichlet boundary conditions by the bilinear element (denoted $ Q_1$) and three nonconforming elements, the rotated bilinear element (denoted $ Q_1^{rot}$), the extension of $ Q_1^{rot}$ (denoted $ EQ_1^{rot}$) and Wilson's elements. The expansions indicate that $ Q_1$ and $ Q_1^{rot}$ provide upper bounds of the eigenvalues, and that $ EQ_1^{rot}$ and Wilson's elements provide lower bounds of the eigenvalues. By extrapolation, the $ O(h^4)$ convergence rate can be obtained, where $ h$ is the maximal boundary length of uniform rectangles. Numerical experiments are carried out to verify the theoretical analysis made.


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Additional Information

Qun Lin
Affiliation: Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, P.O. Box 2719, Beijing 1000080, China
Email: qlin@lsec.cc.ac.cn

Hung-Tsai Huang
Affiliation: Department of Applied Mathematics, I-Shou University, Taiwan 840
Email: huanght@isu.edu.tw

Zi-Cai Li
Affiliation: Department of Applied Mathematics, and Department of Computer Science and Engineering, National Sun Yat-sen University, Kaohsiung, Taiwan 80424
Email: zcli@math.nsysu.edu.tw

DOI: https://doi.org/10.1090/S0025-5718-08-02098-X
Keywords: Bilinear elements, rotated bilinear element, the extension of rotated bilinear element, Wilson's element, eigenvalue problem, extrapolation, global superconvergence.
Received by editor(s): February 24, 2006
Received by editor(s) in revised form: February 14, 2007
Published electronically: May 29, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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