Abstract
Assume that Vh is a space of piecewise polynomials of a degree less than r ≥ 1 on a family of quasi-uniform triangulation of size h. There exists the well-known upper bound of the approximation error by Vh for a sufficiently smooth function. In this paper, we prove that, roughly speaking, if the function does not belong to Vh, 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.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 1-13 |
| Number of pages | 13 |
| Journal | Mathematics of Computation |
| Volume | 83 |
| Issue number | 285 |
| DOIs | |
| State | Published - Jan 1 2014 |
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All Science Journal Classification (ASJC) codes
- Algebra and Number Theory
- Computational Mathematics
- Applied Mathematics
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Lower bounds of the discretization error for piecewise polynomials. / Lin, Qun; Xie, Hehu; Xu, Jinchao.
In: Mathematics of Computation, Vol. 83, No. 285, 01.01.2014, p. 1-13.Research output: Contribution to journal › Article
TY - JOUR
T1 - Lower bounds of the discretization error for piecewise polynomials
AU - Lin, Qun
AU - Xie, Hehu
AU - Xu, Jinchao
PY - 2014/1/1
Y1 - 2014/1/1
N2 - Assume that Vh is a space of piecewise polynomials of a degree less than r ≥ 1 on a family of quasi-uniform triangulation of size h. There exists the well-known upper bound of the approximation error by Vh for a sufficiently smooth function. In this paper, we prove that, roughly speaking, if the function does not belong to Vh, 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.
AB - Assume that Vh is a space of piecewise polynomials of a degree less than r ≥ 1 on a family of quasi-uniform triangulation of size h. There exists the well-known upper bound of the approximation error by Vh for a sufficiently smooth function. In this paper, we prove that, roughly speaking, if the function does not belong to Vh, 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.
UR - http://www.scopus.com/inward/record.url?scp=84888021085&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84888021085&partnerID=8YFLogxK
U2 - 10.1090/S0025-5718-2013-02724-X
DO - 10.1090/S0025-5718-2013-02724-X
M3 - Article
AN - SCOPUS:84888021085
VL - 83
SP - 1
EP - 13
JO - Mathematics of Computation
JF - Mathematics of Computation
SN - 0025-5718
IS - 285
ER -