The highest order superconvergence for bi-$k$ degree rectangular elements at nodes: A proof of $2k$-conjecture
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- by Chuanmiao Chen and Shufang Hu
- Math. Comp. 82 (2013), 1337-1355
- DOI: https://doi.org/10.1090/S0025-5718-2012-02653-6
- Published electronically: December 5, 2012
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Abstract:
We proved the highest order superconvergence $(u-u_h)(z)=O(h^{2k})|\ln h|$ at nodes $z$, based on Element Orthogonality Analysis (EOA), correction techniques and tensor product, where $u\in W^{2k,\infty }(\Omega )$ is the solution for the Poisson equation $-\Delta u=f$ in a rectangle $\Omega$, $u=0$ on $\Gamma$, and $u_h\in S^h_0$ is its bi-$k$ degree rectangular finite element approximation. This conclusion is also verified by numerical experiments for $k=4,5$.References
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Bibliographic Information
- Chuanmiao Chen
- Affiliation: College of Mathematics and Computer Science, Hunan Normal University, Changsha, 410081 Hunan, People’s Republic of China
- Email: cmchen@hunnu.edu.cn
- Shufang Hu
- Affiliation: College of Mathematics and Computer Science, Hunan Normal University, Changsha, 410081 Hunan, People’s Republic of China
- Email: shufanghu@163.com
- Received by editor(s): November 23, 2009
- Received by editor(s) in revised form: November 1, 2010, June 21, 2011, September 26, 2011, October 3, 2011, and November 22, 2011
- Published electronically: December 5, 2012
- Additional Notes: The first author was supported by The National Natural Science Foundation of China (No. 10771063), Key Laboratory of High Performance Computation and Stochastic Information Processing, Hunan Province and Ministry of Education, Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province, and The Graduate Student Research Innovation Foundation of Hunan (No. CX2011B184)
- © Copyright 2012 American Mathematical Society
- Journal: Math. Comp. 82 (2013), 1337-1355
- MSC (2010): Primary 65N30, 65N15
- DOI: https://doi.org/10.1090/S0025-5718-2012-02653-6
- MathSciNet review: 3042566