## Asymptotic expansions of eigenvalues by both the Crouzeix–Raviart and enriched Crouzeix–Raviart elements

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Jun Hu and Limin Ma
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## Abstract:

Asymptotic expansions are derived for eigenvalues produced by both the Crouzeix-Raviart element and the enriched Crouzeix–Raviart element. The expansions are optimal in the sense that extrapolation eigenvalues based on them admit a fourth order convergence provided that exact eigenfunctions are smooth enough. The major challenge in establishing the expansions comes from the fact that the canonical interpolation of both nonconforming elements lacks a crucial superclose property, and the nonconformity of both elements. The main idea is to employ the relation between the lowest-order mixed Raviart–Thomas element and the two nonconforming elements, and consequently make use of the superclose property of the canonical interpolation of the lowest-order mixed Raviart–Thomas element. To overcome the difficulty caused by the nonconformity, the commuting property of the canonical interpolation operators of both nonconforming elements is further used, which turns the consistency error problem into an interpolation error problem. Then, a series of new results are obtained to show the final expansions.## References

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## Additional Information

**Jun Hu**- Affiliation: LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China
- MR Author ID: 714525
- Email: hujun@math.pku.edu.cn
**Limin Ma**- Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
- MR Author ID: 1261964
- Email: maliminpku@gmail.com
- Received by editor(s): May 6, 2020
- Received by editor(s) in revised form: November 5, 2020, December 28, 2020, and January 24, 2021
- Published electronically: September 21, 2021
- Additional Notes: The first author was supported in part by NSFC #11625101. The second author was partially supported by Center for Computational Mathematics and Applications, The Pennsylvania State University.
- © Copyright 2021 American Mathematical Society
- Journal: Math. Comp.
**91**(2022), 75-109 - MSC (2020): Primary 65N30
- DOI: https://doi.org/10.1090/mcom/3635
- MathSciNet review: 4350533