An averaging method for the second-order approximation of the values of the gradient of an arbitrary smooth function u = u(x 1, x 2) at the vertices of a regular triangulation T h composed both of rectangles and triangles is presented. The method assumes that only the interpolant Πh[u] of u in the finite element space of the linear triangular and bilinear rectangular finite elements from T h is known. A complete analysis of this method is an extension of the complete analysis concerning the finite element spaces of linear triangular elements from [Dalík J., Averaging of directional derivatives in vertices of nonobtuse regular triangulations, Numer. Math., 2010, 116(4), 619–644]. The second-order approximation of the gradient is extended from the vertices to the whole domain and applied to the a posteriori error estimates of the finite element solutions of the planar elliptic boundary-value problems of second order. Numerical illustrations of the accuracy of the averaging method and of the quality of the a posteriori error estimates are also presented.
@article{bwmeta1.element.doi-10_2478_s11533-012-0159-7,
author = {Josef Dal\'\i k and V\'aclav Valenta},
title = {Averaging of gradient in the space of linear triangular and bilinear rectangular finite elements},
journal = {Open Mathematics},
volume = {11},
year = {2013},
pages = {597-608},
zbl = {1263.65111},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0159-7}
}
Josef Dalík; Václav Valenta. Averaging of gradient in the space of linear triangular and bilinear rectangular finite elements. Open Mathematics, Tome 11 (2013) pp. 597-608. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0159-7/
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