Klyachin A.A. Error Estimate Calculation of Integral Functionals Using Piecewise Linear Functions

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http://dx.doi.org/10.15688/jvolsu1.2015.1.1

Klyachin Alеksеy Alеksandrovich

Doctor of Physical and Mathematical Sciences, Head of Department of Mathematical
Analysis and Function Theory, Volgograd State University
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Prosp. Universitetsky, 100, 400062 Volgograd, Russian Federation

Abstract. Consider the functional given by the integral: I(u) =Ω∫︁G(x,u,∇u)dx, (1)

defined for functions u ∈ C¹(Ω)∩C(Ω). Note that the Euler — Lagrange equation of the variational problem for this functional has the form:

Q[u] ≡ ∑︁ (︀G′_ξi(x,u,∇u))︀′_xi− G′_u(x,u,∇u) = 0. (2)

Where G(x,u,∇u) =√︀(1 + |∇u|²⁾. Equation (2) is the equation of a minimal surface. Another example is the Poisson equation Δu = f(x), which corresponds to the function G(x,u,∇u) = |∇u|²+ 2f(x)u(x).

Next, we examine the question of the degree of approximation of the functional (1) by piecewise linear 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 linear function with an error of the first order with respect to the diameter of the triangles of the triangulation. We obtain that the value of the integral (1) for functions in C2 is possible to bring a greater degree of accuracy. Note also that in [1;6] estimates the error calculation of the surface triangulation, built on a rectangular grid. 

Key words: piecewise linear functions, approximation of functional, triangulation, degree of error, fineness of partition.

Error Estimate Calculation of Integral Functionals Using Piecewise Linear Functions by Klyachin A.A. is licensed under a Creative Commons Attribution 4.0 International License.

Citation in English: Science Journal of Volgograd State University. Mathematics. Physics. №1 (26) 2015 pp. 6-12

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