Lower estimates for the error of approximation of derivatives for composite finite elements with smoothness properties

We consider a natural class of composite finite elements that provides the $m$th-order smoothness of the resulting piecewise polynomial function on the triangulated domain and does not require information on neighboring elements. It is known that, to provide the required convergence rate, the “smallest angle condition” must be often imposed on the triangulation in the finite element method; i.e., the smallest possible values of the smallest angles of the triangles must be lower bounded. On the other hand, the negative role of the smallest angle can be weakened (but not excluded completely) by choosing appropriate interpolation conditions. As shown earlier, for a large number of methods of choosing interpolation conditions in the construction of simple (noncomposite) finite elements, 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 present paper, a similar result is proved for some class of composite finite elements.