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Lower bounds of the discretization error for piecewise polynomials


Authors: Qun Lin, Hehu Xie and Jinchao Xu
Journal: Math. Comp. 83 (2014), 1-13
MSC (2010): Primary 65N30, 41A10, 65N15; Secondary 65N25, 35J57, 35J58
DOI: https://doi.org/10.1090/S0025-5718-2013-02724-X
Published electronically: June 5, 2013
MathSciNet review: 3120579
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Abstract | References | Similar Articles | Additional Information

Abstract: Assume that $ V_h$ is a space of piecewise polynomials of a degree less than $ r\geq 1$ on a family of quasi-uniform triangulation of size $ h$. There exists the well-known upper bound of the approximation error by $ V_h$ for a sufficiently smooth function. In this paper, we prove that, roughly speaking, if the function does not belong to $ V_h$, the upper-bound error estimate is also sharp.

This result is further extended to various situations including general shape regular grids and many different types of finite element spaces. As an application, the sharpness of finite element approximation of elliptic problems and the corresponding eigenvalue problems is established.


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Additional Information

Qun Lin
Affiliation: LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
Email: linq@lsec.cc.ac.cn

Hehu Xie
Affiliation: LSEC, ICMSEC, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
Email: hhxie@lsec.cc.ac.cn

Jinchao Xu
Affiliation: Center for Computational Mathematics and Applications and Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA
Email: xu@math.psu.edu

DOI: https://doi.org/10.1090/S0025-5718-2013-02724-X
Keywords: Lower bound, error estimate, finite element method, elliptic problem, eigenpair problem
Received by editor(s): July 26, 2011
Received by editor(s) in revised form: May 2, 2012
Published electronically: June 5, 2013
Additional Notes: The first author was supported in part by the National Natural Science Foundation of China through 11031006, 2011CB309703 and 2010DFR00700
The work of the second author was supported in part by the National Science Foundation of China through NSFC 11001259, the National Center for Mathematics and Interdisciplinary Science and the President Foundation of AMSS-CAS.
The work of the third author was partially supported by the US National Science Foundation through DMS 0915153 and DMS 0749202
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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