Operators approximating partial derivatives at vertices of triangulations by averaging

Josef Dalík

Mathematica Bohemica (2010)

  • Volume: 135, Issue: 4, page 363-372
  • ISSN: 0862-7959

Abstract

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Let 𝒯 h be a triangulation of a bounded polygonal domain Ω ⊂ ℜ 2 , ℒ h the space of the functions from C ( Ω ¯ ) linear on the triangles from 𝒯 h and Π h the interpolation operator from C ( Ω ¯ ) to ℒ h . For a unit vector z and an inner vertex a of 𝒯 h , we describe the set of vectors of coefficients such that the related linear combinations of the constant derivatives ∂ Π h ( u ) / ∂ z on the triangles surrounding a are equal to ∂ u / ∂ z ( a ) for all polynomials u of the total degree less than or equal to two. Then we prove that, generally, the values of the so-called recovery operators approximating the gradient ∇ u ( a ) cannot be expressed as linear combinations of the constant gradients ∇ Π h ( u ) on the triangles surrounding a .

How to cite

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Dalík, Josef. "Operators approximating partial derivatives at vertices of triangulations by averaging." Mathematica Bohemica 135.4 (2010): 363-372. <http://eudml.org/doc/197018>.

@article{Dalík2010,
abstract = {Let $\mathcal \{T\}_h$ be a triangulation of a bounded polygonal domain $\Omega \subset \Re ^2$, $\mathcal \{L\}_h$ the space of the functions from $C(\overline\{\Omega \})$ linear on the triangles from $\mathcal \{T\}_h$ and $\Pi _h$ the interpolation operator from $C(\overline\{\Omega \})$ to $\mathcal \{L\}_h$. For a unit vector $z$ and an inner vertex $a$ of $\mathcal \{T\}_h$, we describe the set of vectors of coefficients such that the related linear combinations of the constant derivatives $\partial \Pi _h(u)/\partial z$ on the triangles surrounding $a$ are equal to $\partial u/\partial z(a)$ for all polynomials $u$ of the total degree less than or equal to two. Then we prove that, generally, the values of the so-called recovery operators approximating the gradient $\nabla u(a)$ cannot be expressed as linear combinations of the constant gradients $\nabla \Pi _h(u)$ on the triangles surrounding $a$.},
author = {Dalík, Josef},
journal = {Mathematica Bohemica},
keywords = {partial derivative; high-order approximation; recovery operator; partial derivative; high-order approximation; recovery operator; gradient},
language = {eng},
number = {4},
pages = {363-372},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Operators approximating partial derivatives at vertices of triangulations by averaging},
url = {http://eudml.org/doc/197018},
volume = {135},
year = {2010},
}

TY - JOUR
AU - Dalík, Josef
TI - Operators approximating partial derivatives at vertices of triangulations by averaging
JO - Mathematica Bohemica
PY - 2010
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 135
IS - 4
SP - 363
EP - 372
AB - Let $\mathcal {T}_h$ be a triangulation of a bounded polygonal domain $\Omega \subset \Re ^2$, $\mathcal {L}_h$ the space of the functions from $C(\overline{\Omega })$ linear on the triangles from $\mathcal {T}_h$ and $\Pi _h$ the interpolation operator from $C(\overline{\Omega })$ to $\mathcal {L}_h$. For a unit vector $z$ and an inner vertex $a$ of $\mathcal {T}_h$, we describe the set of vectors of coefficients such that the related linear combinations of the constant derivatives $\partial \Pi _h(u)/\partial z$ on the triangles surrounding $a$ are equal to $\partial u/\partial z(a)$ for all polynomials $u$ of the total degree less than or equal to two. Then we prove that, generally, the values of the so-called recovery operators approximating the gradient $\nabla u(a)$ cannot be expressed as linear combinations of the constant gradients $\nabla \Pi _h(u)$ on the triangles surrounding $a$.
LA - eng
KW - partial derivative; high-order approximation; recovery operator; partial derivative; high-order approximation; recovery operator; gradient
UR - http://eudml.org/doc/197018
ER -

References

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  8. Zienkiewicz, O. C., Cheung, Y. K., The Finite Element Method in Structural and Continuum Mechanics, McGraw Hill, London (1967). (1967) Zbl0189.24902
  9. Zienkiewicz, O. C., Zhu, J. Z., 10.1002/nme.1620330702, Part 1: The recovery technique. Internat. J. Numer. Methods Engrg. 33 1331-1364 (1992). (1992) Zbl0769.73085MR1161557DOI10.1002/nme.1620330702

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