# Approximations of the partial derivatives by averaging

Open Mathematics (2012)

• Volume: 10, Issue: 1, page 44-54
• ISSN: 2391-5455

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## Abstract

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A straightforward generalization of a classical method of averaging is presented and its essential characteristics are discussed. The method constructs high-order approximations of the l-th partial derivatives of smooth functions u in inner vertices a of conformal simplicial triangulations T of bounded polytopic domains in ℝd for arbitrary d ≥ 2. For any k ≥ l ≥ 1, it uses the interpolants of u in the polynomial Lagrange finite element spaces of degree k on the simplices with vertex a only. The high-order accuracy of the resulting approximations is proved to be a consequence of a certain hypothesis and it is illustrated numerically. The method of averaging studied in [Dalík J., Averaging of directional derivatives in vertices of nonobtuse regular triangulations, Numer. Math., 2010, 116(4), 619–644] provides a solution of this problem in the case d = 2, k = l = 1.

## How to cite

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Josef Dalík. "Approximations of the partial derivatives by averaging." Open Mathematics 10.1 (2012): 44-54. <http://eudml.org/doc/269486>.

@article{JosefDalík2012,
abstract = {A straightforward generalization of a classical method of averaging is presented and its essential characteristics are discussed. The method constructs high-order approximations of the l-th partial derivatives of smooth functions u in inner vertices a of conformal simplicial triangulations T of bounded polytopic domains in ℝd for arbitrary d ≥ 2. For any k ≥ l ≥ 1, it uses the interpolants of u in the polynomial Lagrange finite element spaces of degree k on the simplices with vertex a only. The high-order accuracy of the resulting approximations is proved to be a consequence of a certain hypothesis and it is illustrated numerically. The method of averaging studied in [Dalík J., Averaging of directional derivatives in vertices of nonobtuse regular triangulations, Numer. Math., 2010, 116(4), 619–644] provides a solution of this problem in the case d = 2, k = l = 1.},
author = {Josef Dalík},
journal = {Open Mathematics},
keywords = {Regular simplicial triangulation; Lagrange finite element; Averaging the partial derivatives; High-order approximation; regular simplicial triangulation; high-order approximation of derivatives; numerical examples},
language = {eng},
number = {1},
pages = {44-54},
title = {Approximations of the partial derivatives by averaging},
url = {http://eudml.org/doc/269486},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Josef Dalík
TI - Approximations of the partial derivatives by averaging
JO - Open Mathematics
PY - 2012
VL - 10
IS - 1
SP - 44
EP - 54
AB - A straightforward generalization of a classical method of averaging is presented and its essential characteristics are discussed. The method constructs high-order approximations of the l-th partial derivatives of smooth functions u in inner vertices a of conformal simplicial triangulations T of bounded polytopic domains in ℝd for arbitrary d ≥ 2. For any k ≥ l ≥ 1, it uses the interpolants of u in the polynomial Lagrange finite element spaces of degree k on the simplices with vertex a only. The high-order accuracy of the resulting approximations is proved to be a consequence of a certain hypothesis and it is illustrated numerically. The method of averaging studied in [Dalík J., Averaging of directional derivatives in vertices of nonobtuse regular triangulations, Numer. Math., 2010, 116(4), 619–644] provides a solution of this problem in the case d = 2, k = l = 1.
LA - eng
KW - Regular simplicial triangulation; Lagrange finite element; Averaging the partial derivatives; High-order approximation; regular simplicial triangulation; high-order approximation of derivatives; numerical examples
UR - http://eudml.org/doc/269486
ER -

## References

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