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2013 | 11 | 4 | 597-608
Tytuł artykułu

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

Treść / Zawartość
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Języki publikacji
EN
Abstrakty
EN
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.
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
11
Numer
4
Strony
597-608
Opis fizyczny
Daty
wydano
2013-04-01
online
2013-01-29
Twórcy
autor
Bibliografia
  • [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
  • [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
  • [3] Babuška I., Strouboulis T., The Finite Element Method and its Reliability, Numer. Math. Sci. Comput., Clarendon Press, Oxford University Press, New York, 2001
  • [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
  • [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
  • [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
  • [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
  • [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
  • [12] Wahlbin L.B., Superconvergence in Galerkin Finite Element Methods, Lecture Notes in Math., 1605, Springer, Berlin, 1995
  • [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
  • [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
  • [15] Zlámal M., Superconvergence and reduced integration in the finite element method, Math. Comput., 1978, 32(143), 663–685
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-012-0159-7
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