Lower bounds of the discretization error for piecewise polynomials

Qun Lin, Hehu Xie, Jinchao Xu

Research output: Contribution to journalArticle

24 Citations (Scopus)

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

Fingerprint

Discretization Error
Piecewise Polynomials
Polynomials
Lower bound
Upper bound
Sharpness
Approximation Error
Triangulation
Finite Element Approximation
Smooth function
Elliptic Problems
Eigenvalue Problem
Error Estimates
Finite Element
Grid
Family

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory
  • Computational Mathematics
  • Applied Mathematics

Cite this

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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 journalArticle

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