### Abstract

Assume that V_{h} 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 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.

Original language | English (US) |
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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

### Cite this

*Mathematics of Computation*,

*83*(285), 1-13. https://doi.org/10.1090/S0025-5718-2013-02724-X

}

*Mathematics of Computation*, vol. 83, no. 285, pp. 1-13. https://doi.org/10.1090/S0025-5718-2013-02724-X

**Lower bounds of the discretization error for piecewise polynomials.** / Lin, Qun; Xie, Hehu; Xu, Jinchao.

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 -