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«THE BULLETIN OF IRKUTSK STATE UNIVERSITY». SERIES «MATHEMATICS»
«IZVESTIYA IRKUTSKOGO GOSUDARSTVENNOGO UNIVERSITETA». SERIYA «MATEMATIKA»
ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2024. Vol 49

Estimates for Piecewise Linear Approximation of Derivative Functions of Sobolev Classes

Author(s)
Vladimir A. Klyachin1,2

1Volgograd State University, Volgograd, Russian Federation

2Novosibirsk State University, Novosibirsk, Russian Federation

Abstract
The article considers the problem of estimating the error in calculating the gradient of functions of Sobolev classes for piecewise linear approximation on triangulations. Traditionally, problems of this kind are considered for continuously differentiable functions. In this case, the corresponding estimates reflect both the smoothness class of the functions and the quality of the triangulation simplexes. However, in problems of substantiating the existence of a solution to a variational problem in the nonlinear theory of elasticity, conditions arise for permissible deformations in terms of generalized derivatives. Therefore, for the numerical solution of these problems, conditions are required that provide the necessary approximation of the derivatives of continuous functions of the Sobolev classes. In this article, an integral estimate of the indicated error for continuously differentiable functions is obtained in terms of the norms of the corresponding spaces for functions that reflect, on the one hand, the quality of triangulation of the polygonal region, and on the other hand, the class of functions with generalized derivatives. The latter is expressed in terms of the majorant of the modulus of continuity of the gradient. To obtain the final estimate, the possibility of passing to the limit using the norm of the Sobolev space is proven.
About the Authors
Vladimir A. Klyachin, Dr. Sci. (Phys.–Math.), Prof., Volgograd State University, Volgograd, 400062, Russian Federation, klchnv@mail.ru
For citation

Klyachin V. A. Estimates for Piecewise Linear Approximation of Derivative Functions of Sobolev Classes. The Bulletin of Irkutsk State University. Series Mathematics, 2024, vol. 49, pp. 78–89. (in Russian)

https://doi.org/10.26516/1997-7670.2024.49.78

Keywords
triangulation, Delaunay triangulation, piecewise linear approximation, gradient approximation, numerical methods
UDC
514.174.3 + 519.652
MSC
65D25, 65D05, 41A05
DOI
https://doi.org/10.26516/1997-7670.2024.49.78
References
  1. Vodop’yanov S.K., Molchanova A.O. Variational problems of nonlinear elasticity in certain classes of mappings with finite distortion. Doklady Mathematics, 2015, vol. 92, no. 3, pp. 739–742. https://doi.org/10.7868/S086956521535008X (in Russian)
  2. Bernard R. Gelbaum, John M.H. Olmsted Counterexamples in analysis. HoldenDay, San Francisco, London, Amsterdam, 1964.
  3. Gilbarg D., Trudinger N.S. Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983.
  4. Klyachin V.A. On a multidimensional analogue of the Schwarz example. Izv. Math., 2012, vol. 76, no. 4, pp. 681–687. https://doi.org/10.1070/IM2012v076n04ABEH002601
  5. Klyachin V.A., Kuzmin V.V., Khizhnyakova E.V. Triangulation Method for Approximate Solving of Variational Problems in Nonlinear Elasticity. The Bulletin of Irkutsk State University. Series Mathematics, 2023, vol. 45, pp. 54–72. https://doi.org/10.26516/1997-7670.2023.45.54 (in Russian)
  6. Klyachin V.A., Shirokii A.A. The Delaunay triangulation for multidimensional surfaces and its approximative properties. Russian Mathematics, 2012. vol. 56, pp. 27–34. https://doi.org/10.3103/S1066369X12010045
  7. Klyachin V.A., Shiroky A.A. Delaunay triangulation of multidimensional surfaces.Vestnik Samarskogo Universiteta. Estestvenno-Nauchnaya Seriya, 2010, no. 4, pp. 51–55. (in Russian)
  8. Miklyukov V.M. Some conditions for the existence of the total differential. Siberian mathematical journal, 2010, vol. 51, pp. 639–647. https://doi.org/10.1007/s11202- 010-0065-9
  9. Miklyukov V. M. Some conditions for the existence of a total differential at a point. Sb. Math., 2010, vol. 201, no. 8, pp. 1135–1152. https://doi.org/10.4213/sm7566
  10. Molchanova A. A variational approximation scheme for the elastodynamic problems in the new class of admissible mappings. Siberian Journal of Pure and Applied Mathematics, 2016. vol. 16, no. 3. pp. 55–60. https://doi.org/10.17377/PAM.2016.16.305 (in Russian)
  11. Skvortsov A.V., Mirza N.S. Algoritmy postroeniya i analiza triangulacii [Constructing and Analisys of Triangulation Algorithms.] Tomsk, Tomsk Univ. Publ., 2006.
  12. Subbotin Y.N. The error in multidimensional piecewise polynomial approximation. Theory of functions and related questions of analysis, Proceedings of a conference on function theory honoring the 80th birthday of Academician Sergej Mikhajlovich Nikolskij. Collection of papers, Trudy Mat. Inst. Steklov., vol. 180, Moscow, Nauka Publ., 1987, pp. 208–209.
  13. Subbotin Y.N. Dependence of estimates of a multidimensional piecewise polynomial approximation on the geometric characteristics of the triangulation. Proc. Steklov Inst. Math., 1990, vol. 189, no. 4, pp. 135–159.
  14. Subbotin Y.N. Error of the approximation by interpolation polynomials of small degrees on n-simplices. Mathematical notes of the Academy of Sciences of the USSR, 1990, vol. 48, pp. 1030–1037. https://doi.org/10.1007/BF01139604
  15. Shewchuk J.R. What is a Good Linear Element? Interpolation, Conditioning, and Quality Measures. Department of Electrical Engineering and Computer Sciences University of California at Berkeley Berkeley, CA 94720, 2002. Preprint.
  16. Vodopyanov S.K., Molchanova A. Injectivity almost everywhere and mappings with finite distortion in nonlinear elasticity. Calculus of Variations and PDE, 2020, vol. 59, no. 17, pp. 1–25. https://doi.org/10.1007/s00526-019-1671-4

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