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The highest order superconvergence for bi-$ k$ degree rectangular elements at nodes: A proof of $ 2k$-conjecture


Authors: Chuanmiao Chen and Shufang Hu
Journal: Math. Comp. 82 (2013), 1337-1355
MSC (2010): Primary 65N30, 65N15
DOI: https://doi.org/10.1090/S0025-5718-2012-02653-6
Published electronically: December 5, 2012
MathSciNet review: 3042566
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Abstract: We proved the highest order superconvergence $ (u-u_h)(z)=O(h^{2k})\vert\ln h\vert$ 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$.


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Additional 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

DOI: https://doi.org/10.1090/S0025-5718-2012-02653-6
Keywords: bi-$k$ degree rectangular element, highest order superconvergence, element orthogonality analysis, correction function, tensor product
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)
Article copyright: © Copyright 2012 American Mathematical Society

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