Upper estimates for the error of approximation of derivatives in a finite element of Hsieh–Clough–Tocher type

For a triangle $T$, we obtain upper estimates for the error of approximation of derivatives of a function $f\in W^4M$ by derivatives of a piecewise polynomial function $P_3$ that defines a composite Hsieh–Clough–Tocher element. In the obtained error estimates, the negative influence of the smallest angle $\alpha$ of the triangle $T$ on the error of approximation of derivatives is decreased as compared to most often used classical estimates for noncomposite elements. Contrary to expectations, the behavior of the obtained upper estimates with respect to the angle $\alpha$ turned out to be similar to the estimates for the fifth-order polynomial $\widetilde P_5$ defining a “purely polynomial” (noncomposite) finite element that were found by Yu. N. Subbotin. However, the Hsieh–Clough–Tocher element may have an advantage over the polynomial $\widetilde P_5$, which provides the same smoothness, because the implementation of the finite element method for finding $P_3$ requires 12 free parameters, whereas the implementation of this method for finding $\widetilde P_5$ requires 21 parameters.