# Article

 Title: Operators approximating partial derivatives at vertices of triangulations by averaging (English) Author: Dalík, Josef Language: English Journal: Mathematica Bohemica ISSN: 0862-7959 (print) ISSN: 2464-7136 (online) Volume: 135 Issue: 4 Year: 2010 Pages: 363-372 Summary lang: English . Category: math . Summary: 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 $a$. (English) Keyword: partial derivative Keyword: high-order approximation Keyword: recovery operator MSC: 65D25 idZBL: Zbl 1224.65057 idMR: MR2681010 DOI: 10.21136/MB.2010.140827 . Date available: 2010-11-24T08:24:05Z Last updated: 2020-07-29 Stable URL: http://hdl.handle.net/10338.dmlcz/140827 . Reference: [1] Ainsworth, M., Craig, A.: A posteriori error estimators in the finite element method.Numer. Math. 60 429-463 (1992). Zbl 0757.65109, MR 1142306, 10.1007/BF01385730 Reference: [2] Ainsworth, M., Oden, J.: A Posteriori Error Estimation in Finite Element Analysis.Wiley, New York (2000). Zbl 1008.65076, MR 1885308 Reference: [3] Dalík, J.: Local quadratic interpolation in vertices of regular triangulations.East J. Approx. 14 81-102 (2008). Zbl 1217.41005, MR 2391625 Reference: [4] Dalík, J.: Averaging of directional derivatives in vertices of nonobtuse regular triangulations.Submitted for publication to Numer. Math. Reference: [5] Hlaváček, I., Křížek, M., Pištora, V.: How to recover the gradient of linear elements on nonuniform triangulations.Appl. Math. 41 241-267 (1996). MR 1395685 Reference: [6] Vacek, J.: Dual variational principles for an elliptic partial differential equation.Appl. Math. 21 5-27 (1976). Zbl 0345.35035, MR 0412594 Reference: [7] Zhang, Z., Naga, A.: A new finite element gradient recovery method: Superconvergence property.SIAM J. Sci. Comp. 26 1192-1213 (2005). Zbl 1078.65110, MR 2143481, 10.1137/S1064827503402837 Reference: [8] Zienkiewicz, O. C., Cheung, Y. K.: The Finite Element Method in Structural and Continuum Mechanics.McGraw Hill, London (1967). Zbl 0189.24902 Reference: [9] Zienkiewicz, O. C., Zhu, J. Z.: The superconvergent patch recovery and a posteriori error estimates.Part 1: The recovery technique. Internat. J. Numer. Methods Engrg. 33 1331-1364 (1992). Zbl 0769.73085, MR 1161557, 10.1002/nme.1620330702 .

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