Previous |  Up |  Next


partial derivative; high-order approximation; recovery operator
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 $a$.
[1] Ainsworth, M., Craig, A.: A posteriori error estimators in the finite element method. Numer. Math. 60 429-463 (1992). DOI 10.1007/BF01385730 | MR 1142306 | Zbl 0757.65109
[2] Ainsworth, M., Oden, J.: A Posteriori Error Estimation in Finite Element Analysis. Wiley, New York (2000). MR 1885308 | Zbl 1008.65076
[3] Dalík, J.: Local quadratic interpolation in vertices of regular triangulations. East J. Approx. 14 81-102 (2008). MR 2391625 | Zbl 1217.41005
[4] Dalík, J.: Averaging of directional derivatives in vertices of nonobtuse regular triangulations. Submitted for publication to Numer. Math.
[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
[6] Vacek, J.: Dual variational principles for an elliptic partial differential equation. Appl. Math. 21 5-27 (1976). MR 0412594 | Zbl 0345.35035
[7] Zhang, Z., Naga, A.: A new finite element gradient recovery method: Superconvergence property. SIAM J. Sci. Comp. 26 1192-1213 (2005). DOI 10.1137/S1064827503402837 | MR 2143481 | Zbl 1078.65110
[8] Zienkiewicz, O. C., Cheung, Y. K.: The Finite Element Method in Structural and Continuum Mechanics. McGraw Hill, London (1967). Zbl 0189.24902
[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). DOI 10.1002/nme.1620330702 | MR 1161557 | Zbl 0769.73085
Partner of
EuDML logo