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Hessian recovery for finite element methods


Authors: Hailong Guo, Zhimin Zhang and Ren Zhao
Journal: Math. Comp. 86 (2017), 1671-1692
MSC (2010): Primary 65N50, 65N30; Secondary 65N15
DOI: https://doi.org/10.1090/mcom/3186
Published electronically: September 27, 2016
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Abstract: In this article, we propose and analyze an effective Hessian recovery strategy for the Lagrangian finite element method of arbitrary order. We prove that the proposed Hessian recovery method preserves polynomials of degree $ k+1$ on general unstructured meshes and superconverges at a rate of $ O(h^k)$ on mildly structured meshes. In addition, the method is proved to be ultraconvergent (two orders higher) for the translation invariant finite element space of any order. Numerical examples are presented to support our theoretical results.


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Additional Information

Hailong Guo
Affiliation: Department of Mathematics, Wayne State University, Detroit, Michigan 48202
Address at time of publication: Department of Mathematics, University of California, Santa Barbara, California 93106
Email: hlguo@math.ucsb.edu

Zhimin Zhang
Affiliation: Beijing Computational Science Research Center, Beijing 100193, China – and – Department of Mathematics, Wayne State University, Detroit, Michigan 48202
Email: zmzhang@csrc.ac.cn, zzhang@math.wayne.edu

Ren Zhao
Affiliation: Department of Mathematics, Wayne State University, Detroit, Michigan 48202
Email: rzhao@math.wayne.edu

DOI: https://doi.org/10.1090/mcom/3186
Keywords: Hessian recovery, gradient recovery, ultraconvergence, superconvergence, finite element method, polynomial preserving
Received by editor(s): July 17, 2014
Received by editor(s) in revised form: December 14, 2015
Published electronically: September 27, 2016
Additional Notes: The second author is the corresponding author. The research of the second author was supported in part by the National Natural Science Foundation of China under grants 11471031, 91430216 U1530401, and the U.S. National Science Foundation through grant DMS-1419040.
Article copyright: © Copyright 2016 American Mathematical Society