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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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

**1.**R. A. Adams.*Sobolev Spaces*. Academic Press, New York, 1975. MR**0450957 (56:9247)****2.**T. Arbogast and Z. Chen. On the implementation of mixed methods and nonconforming methods for second-order elliptic problems.*Math. Comp.*, 64: 943-972, 1995. MR**1303084 (95k:65102)****3.**I. Babuška, R. B. Kellogg, and J. K. Pitäranta. Direct and inverse error estimates for finite elements with mesh refinements.*Numer. Math.*, 33: 447-471, 1979. MR**553353 (81c:65054)****4.**I. Babuška and A. Miller. A feedback element method with a posteriori error estimation: Part I. The finite element method and some basic properties of the a posteriori error estimator.*Comput. Methods Appl. Mech. Engrg.*, 61(1): 1-40, 1987. MR**880421 (88d:73036)****5.**I. Babuška and T. Strouboulis.*The Finite Element Method and its Reliability*. Clarendon Press, 2001. MR**1857191 (2002k:65001)****6.**S. Brenner and L. Scott.*The Mathematical Theory of Finite Element Methods*. Springer-Verlag, 2008. MR**2373954 (2008m:65001)****7.**H. S. Chen and B. Li. Superconvergence analysis and error expansion for the Wilson nonconforming finite element.*Numer. Math.*, 69: 125-140, 1990. MR**1310313 (95k:65105)****8.**P. G. Ciarlet.*The finite Element Method for Elliptic Problem*. North-Holland, Amsterdam, 1978. MR**0520174 (58:25001)****9.**M. Crouzeix and P.-A. Raviart. Conforming and nonconforming finite element methods for solving the stationary Stokes equations.*I. RAIRO Numer. Anal.*, 3: 33-75, 1973. MR**0343661 (49:8401)****10.**J. Hu, Y. Huang, and Q. Lin.*The lower bounds for eigenvalues of elliptic operators-by nonconforming finite element methods*, arxiv: 1112.1145, 2011.**11.**D. Gilbarg and N. Trudinger.*Elliptic Partial Differential Equations of Second Order*. Springer-Verlag, Berlin and Heidelberg, 2001. MR**1814364 (2001k:35004)****12.**P. Grisvard.*Elliptic Problems in Nonsmooth Domains, Pitman Advanced Publishing Program*. Boston, 1985. MR**775683 (86m:35044)****13.**M. Křížek, H. Roos, and W. Chen. Two-sided bounds of the discretization error for finite elements.*ESAIM: M2AN*, 45: 915-924, 2011. MR**2817550****14.**Q. Lin, L. Tobiska, and A. Zhou. Superconvergence and extrapolation of non-conforming low order finite elements applied to the Poisson equation.*Numer. Anal.*, 25: 160-181, 2005. MR**2110239 (2005k:65256)****15.**Q. Lin, H. Xie, F. Luo, Y. Li, and Y. Yang. Stokes eigenvalue approximation from below with nonconforming mixed finite element methods.*Math. in Practice and Theory*, 19:157-168, 2010 (in Chinese). MR**2768711****16.**S. Liu and Y. Xu. Graded Galerkin methods for the high-order convection-diffusion problem.*Numer. Methods Partial Differential Equations*, 25(6): 1261-1282, 2009. MR**2561549 (2010i:65276)****17.**R. Rannacher and S. Turek. Simple nonconforming quadrilateral Stokes element.*Numer. Methods PDEs.*, 8: 97-111, 1992. MR**1148797 (92i:65170)****18.**M. Wang and J. Xu. Minimal finite-element spaces for -th order partial differential equations in .*Research Report*, 29 (2006), School of Mathematical Sciences and Institute of Mathematics, Peking University.**19.**M. Wang and J. Xu. The Morley element for fourth order elliptic equations in any dimensions.*Numer. Math.*, 103: 155-169, 2006. MR**2207619 (2006i:65205)****20.**O. Widlund. Some results on best possible error bounds for finite element methods and approximation with piecewise polynomial functions.*Proc. Roy. Soc. Lond. A.*, 323: 167-177, 1971. MR**0353699 (50:6182)****21.**O. Widlund. On best error bounds for approximation by piecewise polynomial functions.*Numer. Math.*, 27: 327-338, 1977. MR**0458031 (56:16234)**

Retrieve articles in *Mathematics of Computation*
with MSC (2010):
65N30,
41A10,
65N15,
65N25,
35J57,
35J58

Retrieve articles in all journals with MSC (2010): 65N30, 41A10, 65N15, 65N25, 35J57, 35J58

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.