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

Open Mathematics (2013)

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

## Access Full Article

top## Abstract

top## How to cite

topJosef 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

top- [1] Ainsworth M., Oden J.T., A Posteriori Error Estimation in Finite Element Analysis, Pure Appl. Math. (N.Y.), John Wiley & Sons, New York, 2000 Zbl1008.65076
- [2] Babuška I., Rheinboldt W.C., A-posteriori error estimates for the finite element method, Internat. J. Numer. Methods Engrg., 1978, 12(10), 1597–1615 http://dx.doi.org/10.1002/nme.1620121010 Zbl0396.65068
- [3] Babuška I., Strouboulis T., The Finite Element Method and its Reliability, Numer. Math. Sci. Comput., Clarendon Press, Oxford University Press, New York, 2001 Zbl0995.65501
- [4] Chen C., Huang Y., High Accuracy Theory of Finite Element Methods, Hunan Science and Technology Press, Changsha, 1995 (in Chinese)
- [5] Dalík J., Averaging of directional derivatives in vertices of nonobtuse regular triangulations, Numer. Math., 2010, 116(4), 619–644 http://dx.doi.org/10.1007/s00211-010-0316-5 Zbl1215.65044
- [6] Dalík J., Approximations of the partial derivatives by averaging, Cent. Eur. J. Math., 10(1), 2012, 44–54 http://dx.doi.org/10.2478/s11533-011-0107-y
- [7] Haug E.J., Choi K.K., Komkov V., Design sensitivity analysis of structural systems, Math. Sci. Eng., 177, Academic Press, Orlando, 1986 Zbl0618.73106
- [8] Hlaváček I., Křížek M., Pištora V., How to recover the gradient of linear elements on nonuniform triangulations, Appl. Math., 1996, 41(4), 241–267 Zbl0870.65093
- [9] Křížek M., Neittaanmäki P., Superconvergence phenomenon in the finite element method arising from averaging gradients, Numer. Math., 1984, 45(1), 105–116 http://dx.doi.org/10.1007/BF01379664 Zbl0575.65104
- [10] Lin Q., Yan N., The Construction and Analysis of High Efficiency Finite Elements, Hebei University Press, Hunan, 1996 (in Chinese)
- [11] Verfürth R., A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Teubner Skr. Numer., Wiley-Teubner, Stuttgart, 1996 Zbl0853.65108
- [12] Wahlbin L.B., Superconvergence in Galerkin Finite Element Methods, Lecture Notes in Math., 1605, Springer, Berlin, 1995 Zbl0826.65092
- [13] Zienkiewicz O.C., Cheung Y.K., The Finite Element Method in Structural and Continuum Mechanics, European civil engineering series, McGraw-Hill, London-New York, 1967 Zbl0189.24902
- [14] 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., 1992, 33(7), 1331–1364 http://dx.doi.org/10.1002/nme.1620330702 Zbl0769.73084
- [15] Zlámal M., Superconvergence and reduced integration in the finite element method, Math. Comput., 1978, 32(143), 663–685 Zbl0448.65068

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.