Error estimate calculation of integral functionals using piecewise linear functions

A. A. Klyachin

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 1 (2015), pp. 6-12

Résumé

Consider the functional given by the integral \begin{equation} I(u)=\int\limits_{\Omega}G(x,u,\nabla u)dx, \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. \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 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 $ C ^ 2 $ 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.

Détail

Keywords: piecewise linear functions, approximation of functional, degree of error, fineness of partition.
Mots-clés : triangulation