## 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 |

## All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Computational Mathematics
- Applied Mathematics