Lower bounds of the discretization error for piecewise polynomials

Qun Lin, Hehu Xie, Jinchao Xu

Research output: Contribution to journalArticlepeer-review

29 Scopus citations

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 languageEnglish (US)
Pages (from-to)1-13
Number of pages13
JournalMathematics of Computation
Volume83
Issue number285
DOIs
StatePublished - Jan 1 2014

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory
  • Computational Mathematics
  • Applied Mathematics

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