# Approximations of the partial derivatives by averaging

Open Mathematics (2012)

- Volume: 10, Issue: 1, page 44-54
- ISSN: 2391-5455

## Access Full Article

top## Abstract

top## How to cite

topJosef Dalík. "Approximations of the partial derivatives by averaging." Open Mathematics 10.1 (2012): 44-54. <http://eudml.org/doc/269486>.

@article{JosefDalík2012,

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.},

author = {Josef Dalík},

journal = {Open Mathematics},

keywords = {Regular simplicial triangulation; Lagrange finite element; Averaging the partial derivatives; High-order approximation; regular simplicial triangulation; high-order approximation of derivatives; numerical examples},

language = {eng},

number = {1},

pages = {44-54},

title = {Approximations of the partial derivatives by averaging},

url = {http://eudml.org/doc/269486},

volume = {10},

year = {2012},

}

TY - JOUR

AU - Josef Dalík

TI - Approximations of the partial derivatives by averaging

JO - Open Mathematics

PY - 2012

VL - 10

IS - 1

SP - 44

EP - 54

AB - 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.

LA - eng

KW - Regular simplicial triangulation; Lagrange finite element; Averaging the partial derivatives; High-order approximation; regular simplicial triangulation; high-order approximation of derivatives; numerical examples

UR - http://eudml.org/doc/269486

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 http://dx.doi.org/10.1002/9781118032824 Zbl1008.65076
- [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)70178-4
- [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/s00211-010-0316-5 Zbl1215.65044
- [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 Zbl0870.65093
- [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/BF01379664 Zbl0575.65104
- [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.2011.66.70
- [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.2005.5.769 Zbl1078.65108
- [8] Quarteroni A., Valli A., Numerical Approximation of Partial Differential Equations, Springer Ser. Comput. Math., 23, Springer, Berlin, 1994 Zbl0803.65088
- [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/S1064827503402837 Zbl1078.65110
- [10] Zienkiewicz O.C., Cheung Y.K., The Finite Element Method in Structural and Continuum Mechanics, McGraw-Hill, London, 1967 Zbl0189.24902
- [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.1620330702 Zbl0769.73084