Estimation of the error of calculating the functional containing higher-order derivatives on a triangular grid

In the present paper, it is proved that a functional containing second derivatives can be calculated with an error of order $ O (h ^ {4m+1}) $ with a triangular mesh of $h \to 0$ if the piecewise polynomial functions of degree $ 4m+1 $ for $m\geq 1$. For $ n = 2 $ we give an example of the fact that the piecewise quadratic approximation gives the second order of accuracy for calculating the functional for a special kind of triangulation.