Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1. Mathematica. Physica
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Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1. Mathematica. Physica, 2016, Issue 3(34), Pages 73–83
DOI: https://doi.org/10.15688/jvolsu1.2016.3.7
(Mi vvgum112)
 

This article is cited in 2 scientific papers (total in 2 papers)

Computer modelling

Modeling minimum triangulated surfaces: error estimation calculating the area of the design of facilities

A. A. Klyachin, A. G. Panchånko

Volgograd State University
Full-text PDF (452 kB) Citations (2)
References:
Abstract: Consider the functional given by the integral
\begin{equation} I(u)=\int\limits_{\Omega}G(x,u,\nabla u)dx, \tag{1} \end{equation}
defined for functions $u\in C^1(\Omega)\cap C(\overline{\Omega})$. Note that the Euler–Lagrange equation of the variational problem for this functional has the form
\begin{equation} Q[u]\equiv \sum_{i=1}^n\left(G'_{\xi_i}(x,u,\nabla u)\right)'_{x_i}-G'_u(x,u,\nabla u)=0. \tag{2} \end{equation}
Where $G(x,u,\nabla u)=\sqrt{1+|\nabla u|^2}$. Equation (2) is the equation of a minimal surface. Another example is the Poisson equation $\Delta u=f(x)$, which corresponds to the function $G(x,u,\nabla u) = |\nabla u|^2+2f(x)u(x)$.
Next, we examine the question of the degree of approximation of the functional (1) by piecewise quadratic functions. For such problems lead the convergence of variational methods for some boundary value problems. Note that the derivatives of a continuously differentiable function approach derived piecewise quadratic function with an error of the second order with respect to the diameter of the triangles of the triangulation. We obtain that the value of the integral (1) for functions in $ C ^ 2 $ is possible to bring a greater degree of accuracy. Note also that in [3; 8] estimates the error calculation of the surface triangulation, built on a rectangular grid.
Keywords: piecewise quadratic function, area of a surface, the approximation of functional, triangulation, minimal surface.
Funding agency Grant number
Russian Foundation for Basic Research 15-41-02517-ð_ïîâîëæüå_à
Document Type: Article
UDC: 517.951, 519.632
BBC: 22.161, 22.19
Language: Russian
Citation: A. A. Klyachin, A. G. Panchånko, “Modeling minimum triangulated surfaces: error estimation calculating the area of the design of facilities”, Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1. Mathematica. Physica, 2016, no. 3(34), 73–83
Citation in format AMSBIB
\Bibitem{KlyPan16}
\by A.~A.~Klyachin, A.~G.~Panchånko
\paper Modeling minimum triangulated surfaces: error estimation calculating the area of the design of facilities
\jour Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1. Mathematica. Physica
\yr 2016
\issue 3(34)
\pages 73--83
\mathnet{http://mi.mathnet.ru/vvgum112}
\crossref{https://doi.org/10.15688/jvolsu1.2016.3.7}
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  • This publication is cited in the following 2 articles:
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