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$ 2k$ superconvergence of $ Q_k$ finite elements by anisotropic mesh approximation in weighted Sobolev spaces


Authors: Wenming He and Zhimin Zhang
Journal: Math. Comp. 86 (2017), 1693-1718
MSC (2010): Primary 65N30, 65N25, 65N15
DOI: https://doi.org/10.1090/mcom/3159
Published electronically: October 7, 2016
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Abstract: We first establish anisotropic finite element estimates for the discrete Green's function of the Poisson equation. Then we prove that the bi-$ k$ rectangular finite element method under a specially designed partition has the highest possible convergent rate at all element vertices for the Poisson equation on the unit square. Furthermore, we provide numerical comparison with the standard uniform rectangular partition and demonstrate that our a priori superconvergent error estimate is sharp up to a logarithm factor.


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Additional Information

Wenming He
Affiliation: Department of Mathematics, Wenzhou University, Wenzhou, Zhejiang 325035, People’s Republic of China
Email: hewenming@wzu.edu.cn, he_wenming@aliyun.com

Zhimin Zhang
Affiliation: Beijing Computational Science Research Center, Beijing 100093, People’s Republic of China – and — Department of Mathematics, Wayne State University, Detroit, Michigan 48202
Email: zmzhang@csrc.ac.cn, ag7761@wayne.edu

DOI: https://doi.org/10.1090/mcom/3159
Keywords: Poisson equation, superconvergence, extrapolation, Green's function
Received by editor(s): January 8, 2015
Received by editor(s) in revised form: August 24, 2015, November 13, 2015, and January 24, 2016
Published electronically: October 7, 2016
Additional Notes: The first author was supported in part by the National Natural Science Foundation of China (11671304, 11171257, 11301396), and the Zhejiang Provincial Natural Science Foundation, China (No. LY15A010015). The second author is the corresponding author and was supported in part by the National Natural Science Foundation of China, grants 11471031, 91430216, U1530401, and the US National Science Foundation, grant DMS-1419040
Article copyright: © Copyright 2016 American Mathematical Society