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.
@article{bwmeta1.element.doi-10_2478_s11533-011-0107-y,
author = {Josef Dal\'\i k},
title = {Approximations of the partial derivatives by averaging},
journal = {Open Mathematics},
volume = {10},
year = {2012},
pages = {44-54},
zbl = {1259.65049},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0107-y}
}
Josef Dalík. Approximations of the partial derivatives by averaging. Open Mathematics, Tome 10 (2012) pp. 44-54. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0107-y/
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