Averaging of gradient in the space of linear triangular and bilinear rectangular finite elements

Josef Dalík; Václav Valenta

Open Mathematics (2013)

  • Volume: 11, Issue: 4, page 597-608
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Josef Dalík, and Václav Valenta. "Averaging of gradient in the space of linear triangular and bilinear rectangular finite elements." Open Mathematics 11.4 (2013): 597-608. <http://eudml.org/doc/269494>.

@article{JosefDalík2013,
abstract = {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.},
author = {Josef Dalík, Václav Valenta},
journal = {Open Mathematics},
keywords = {Linear triangular and bilinear rectangular finite element; Nonobtuse regular triangulation; Averaging partial derivatives; A posteriori error estimator; Adaptive solution of elliptic differential problems in 2D; linear triangular and bilinear rectangular finite element; nonobtuse regular triangulation; averaging partial derivatives; a posteriori error estimator; adaptive solution of elliptic differential problems in 2D; numerical examples; elliptic boundary-value problems},
language = {eng},
number = {4},
pages = {597-608},
title = {Averaging of gradient in the space of linear triangular and bilinear rectangular finite elements},
url = {http://eudml.org/doc/269494},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Josef Dalík
AU - Václav Valenta
TI - Averaging of gradient in the space of linear triangular and bilinear rectangular finite elements
JO - Open Mathematics
PY - 2013
VL - 11
IS - 4
SP - 597
EP - 608
AB - 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.
LA - eng
KW - Linear triangular and bilinear rectangular finite element; Nonobtuse regular triangulation; Averaging partial derivatives; A posteriori error estimator; Adaptive solution of elliptic differential problems in 2D; linear triangular and bilinear rectangular finite element; nonobtuse regular triangulation; averaging partial derivatives; a posteriori error estimator; adaptive solution of elliptic differential problems in 2D; numerical examples; elliptic boundary-value problems
UR - http://eudml.org/doc/269494
ER -

References

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