What is the smallest possible constant in Céa's lemma?

Wei Chen; Michal Křížek

Applications of Mathematics (2006)

  • Volume: 51, Issue: 2, page 129-144
  • ISSN: 0862-7940

Abstract

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We consider finite element approximations of a second order elliptic problem on a bounded polytopic domain in ℝ d with d ∈ { 1 , 2 , 3 , ... } . The constant C ≥ 1 appearing in Céa’s lemma and coming from its standard proof can be very large when the coefficients of an elliptic operator attain considerably different values. We restrict ourselves to regular families of uniform partitions and linear simplicial elements. Using a lower bound of the interpolation error and the supercloseness between the finite element solution and the Lagrange interpolant of the exact solution, we show that the ratio between discretization and interpolation errors is equal to 1 + 𝒪 ( h ) as the discretization parameter h tends to zero. Numerical results in one and two-dimensional case illustrating this phenomenon are presented.

How to cite

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Chen, Wei, and Křížek, Michal. "What is the smallest possible constant in Céa's lemma?." Applications of Mathematics 51.2 (2006): 129-144. <http://eudml.org/doc/33248>.

@article{Chen2006,
abstract = {We consider finite element approximations of a second order elliptic problem on a bounded polytopic domain in $\mathbb \{R\}^d$ with $d\in \lbrace 1,2,3,\ldots \rbrace $. The constant $C\ge 1$ appearing in Céa’s lemma and coming from its standard proof can be very large when the coefficients of an elliptic operator attain considerably different values. We restrict ourselves to regular families of uniform partitions and linear simplicial elements. Using a lower bound of the interpolation error and the supercloseness between the finite element solution and the Lagrange interpolant of the exact solution, we show that the ratio between discretization and interpolation errors is equal to $1+\mathcal \{O\}(h)$ as the discretization parameter $h$ tends to zero. Numerical results in one and two-dimensional case illustrating this phenomenon are presented.},
author = {Chen, Wei, Křížek, Michal},
journal = {Applications of Mathematics},
keywords = {supercloseness; Lagrange finite elements; Lagrange remainder; lower estimates; elliptic problems; $d$-simplex; uniform partitions; supercloseness; Lagrange finite elements; Lagrange remainder; lower estimates; second order elliptic problems; -simplex; uniform partitions; linear simplicial elements; numerical results; interpolation error},
language = {eng},
number = {2},
pages = {129-144},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {What is the smallest possible constant in Céa's lemma?},
url = {http://eudml.org/doc/33248},
volume = {51},
year = {2006},
}

TY - JOUR
AU - Chen, Wei
AU - Křížek, Michal
TI - What is the smallest possible constant in Céa's lemma?
JO - Applications of Mathematics
PY - 2006
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 51
IS - 2
SP - 129
EP - 144
AB - We consider finite element approximations of a second order elliptic problem on a bounded polytopic domain in $\mathbb {R}^d$ with $d\in \lbrace 1,2,3,\ldots \rbrace $. The constant $C\ge 1$ appearing in Céa’s lemma and coming from its standard proof can be very large when the coefficients of an elliptic operator attain considerably different values. We restrict ourselves to regular families of uniform partitions and linear simplicial elements. Using a lower bound of the interpolation error and the supercloseness between the finite element solution and the Lagrange interpolant of the exact solution, we show that the ratio between discretization and interpolation errors is equal to $1+\mathcal {O}(h)$ as the discretization parameter $h$ tends to zero. Numerical results in one and two-dimensional case illustrating this phenomenon are presented.
LA - eng
KW - supercloseness; Lagrange finite elements; Lagrange remainder; lower estimates; elliptic problems; $d$-simplex; uniform partitions; supercloseness; Lagrange finite elements; Lagrange remainder; lower estimates; second order elliptic problems; -simplex; uniform partitions; linear simplicial elements; numerical results; interpolation error
UR - http://eudml.org/doc/33248
ER -

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