Error estimation of area calculation for piecewise polynomial approximation

A. A. Klyachin ; A. G. Panchеnko

Matematičeskaâ fizika i kompʹûternoe modelirovanie, Tome 23 (2020) no. 2, pp. 22-30.

Résumé

The paper considers the functional given by the integral $$ I(f)=\int\limits_{\Omega}G(x,f,\nabla f)dx, {(1)} $$ defined for functions $f\in C^{m+1}(\Omega)\cap C(\overline{\Omega})$. The authors note that the Euler — Lagrange equation of the variational problem for this functional has the form $$ Q[f]\equiv \sum_{i=1}^2\left(G'_{\xi_i}(x,f,\nabla f)\right)'_{x_i}-G'_f(x,f,\nabla f)=0, {(2)} $$ where $G(x,f,\nabla f)=\sqrt{1+|\nabla f|^2}$. Equation (2) is the equation of a minimal surface. Another example is the Poisson equation $\Delta f=g(x)$, which corresponds to the function $G(x,f,\nabla f)=|\nabla f|^2+2g(x)f(x)$. Next, the article examines the issue of the degree of approximation of the functional (1) by piecewise polynomial functions. This leads to the convergence of variational methods for some boundary value problems. The authors note that the derivatives of a continuously differentiable function approach derived piecewise polynomial function with an error of the $m$-order with respect to the diameter of the triangles of the triangulation. The reasechers have found that for functions from $ C^{m+1}(\Omega) $ functional (1) is calculated with accuracy $O(h^{m+1})$ in the class of piecewise polynomial functions of degree $m$.

Détail

Keywords: piecewise polynomial function, area of a surface, approximation of functional, minimal surface.
Mots-clés : triangulation