Approximations of the partial derivatives by averaging

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 boun...

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Bibliographic Details
Published in:Open mathematics (Warsaw, Poland) Poland), 2012-02, Vol.10 (1), p.44-54
Main Author: Dalík, Josef
Format: Article
Language:English
Subjects:
65D25
Apexes
Approximation
Averaging the partial derivatives
High-order approximation
Lagrange finite element
Mathematical analysis
Polynomials
Regular simplicial triangulation
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Summary: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.
ISSN:2391-5455
DOI:10.2478/s11533-011-0107-y