Abstract
A straightforward generalization of a classical method of averaging is presented and its essential characteristics are discussed. The method constructs high-order approximations of the l-th partial derivatives of smooth functions u in inner vertices a of conformal simplicial triangulations T of bounded polytopic domains in ℝd for arbitrary d ≥ 2. For any k ≥ l ≥ 1, it uses the interpolants of u in the polynomial Lagrange finite element spaces of degree k on the simplices with vertex a only. The high-order accuracy of the resulting approximations is proved to be a consequence of a certain hypothesis and it is illustrated numerically. The method of averaging studied in [Dalík J., Averaging of directional derivatives in vertices of nonobtuse regular triangulations, Numer. Math., 2010, 116(4), 619–644] provides a solution of this problem in the case d = 2, k = l = 1.
[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 http://dx.doi.org/10.1002/978111803282410.1002/9781118032824Search in Google Scholar
[2] Ciarlet P.G., The Finite Element Method for Elliptic Problems, Stud. Math. Appl., 4, North Holland, Amsterdam-New York-Oxford, 1978 http://dx.doi.org/10.1016/S0168-2024(08)70178-410.1016/S0168-2024(08)70178-4Search in Google Scholar
[3] 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/s00211-010-0316-510.1007/s00211-010-0316-5Search in Google Scholar
[4] Hlaváček I., Křížek M., Pištora V., How to recover the gradient of linear elements on nonuniform triangulations, Appl. Math., 1996, 41(4), 241–267 10.21136/AM.1996.134325Search in Google Scholar
[5] 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/BF0137966410.1007/BF01379664Search in Google Scholar
[6] Li S., Concise formulas for the area and volume of a hyperspherical cap, Asian J. Math. Stat., 2011, 4(1), 66–70 http://dx.doi.org/10.3923/ajms.2011.66.7010.3923/ajms.2011.66.70Search in Google Scholar
[7] Naga A., Zhang Z., The polynomial-preserving recovery for higher order finite element methods in 2D and 3D, Discrete Contin. Dyn. Syst. Ser. B, 2005, 5(3), 769–798 http://dx.doi.org/10.3934/dcdsb.2005.5.76910.3934/dcdsb.2005.5.769Search in Google Scholar
[8] Quarteroni A., Valli A., Numerical Approximation of Partial Differential Equations, Springer Ser. Comput. Math., 23, Springer, Berlin, 1994 10.1007/978-3-540-85268-1Search in Google Scholar
[9] Zhang Z., Naga A., A new finite element gradient recovery method: superconvergence property, SIAM J. Sci. Comput. 2005, 26(4), 1192–1213 http://dx.doi.org/10.1137/S106482750340283710.1137/S1064827503402837Search in Google Scholar
[10] Zienkiewicz O.C., Cheung Y.K., The Finite Element Method in Structural and Continuum Mechanics, McGraw-Hill, London, 1967 Search in Google Scholar
[11] Zienkiewicz O.C., Zhu J.Z., The superconvergent patch recovery and a posteriori error estimates. Part I: The recovery technique, Internat. J. Numer. Methods Engrg., 1992, 33(7), 1331–1364 http://dx.doi.org/10.1002/nme.162033070210.1002/nme.1620330702Search in Google Scholar
© 2012 Versita Warsaw
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.
