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A Multi-Level Mixed Element Method for the Eigenvalue Problem of Biharmonic Equation

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Abstract

In this paper, we discuss approximating the eigenvalue problem of biharmonic equation. We first present an equivalent mixed formulation which admits natural nested discretization. Then, we present multi-level finite element schemes by implementing the algorithm as in Lin and Xie (Math Comput 84:71–88, 2015) to the nested discretizations on a series of nested grids. The multi-level mixed scheme for the biharmonic eigenvalue problem possesses optimal convergence rate and optimal computational cost. Both theoretical analysis and numerical verifications are presented.

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Notes

  1. In this paper, \(\lesssim \), \(\gtrsim \), and denote \(\leqslant \), \(\geqslant \), and \(=\) up to a constant respectively. The hidden constants depend on the domain, and, when triangulation is involved, they also depend on the shape-regularity of the triangulation, but they do not depend on h or any other mesh parameter.

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Acknowledgements

S. Zhang is partially supported by the National Natural Science Foundation of China with Grant No. 11471026 and National Centre for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences. X. Ji is partially supported by the National Natural Science Foundation of China with Grant Nos. 11271018 and 91630313, and National Centre for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences. The authors would like to thank Prof. Hehu Xie for his valuable discussion.

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Authors and Affiliations

  1. LSEC, NCMIS, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Shuo Zhang, Yingxia Xi & Xia Ji

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  1. Shuo Zhang
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  2. Yingxia Xi
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  3. Xia Ji
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Correspondence to Xia Ji.

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Zhang, S., Xi, Y. & Ji, X. A Multi-Level Mixed Element Method for the Eigenvalue Problem of Biharmonic Equation. J Sci Comput 75, 1415–1444 (2018). https://doi.org/10.1007/s10915-017-0592-7

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  • DOI: https://doi.org/10.1007/s10915-017-0592-7

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