Influence of smoothness on the error of approximation of derivatives under local interpolation on triangulations

The paper is concerned with one problem of function interpolation on a triangle. We consider a large class of interpolation conditions guaranteeing the smoothness of order $m$ of the resulting piecewise polynomial function on the triangulated domain. It is known that, for smoothness $m\ge1$, the known upper estimates for the error of approximation of derivatives of order $2$ and above by derivatives of interpolation polynomials defined on a triangulation element contain the sine of the smallest angle in the denominator. As a result, the “smallest angle condition” must be imposed on the triangulation. It was shown earlier that the influence of the smallest angle could be weakened (which does not mean that it can be eliminated in all cases). The principal aim of this paper is to show that, for a large number of methods of choosing interpolation conditions, including traditional conditions, the influence of the smallest angle of the triangle on the error of approximation of derivatives of a function by derivatives of the interpolation polynomial is essential for a number of derivatives of order $2$ and above for $m\ge1$. In the case $m=0$, the influence of the middle (largest) angle is important. As a consequence, the results on the unimprovability of the upper estimates obtained earlier are strengthened.