On some interpolation third-degree polynomials on a three-dimensional simplex

The interpolation problem under consideration is connected with the finite element method in $\mathbb R^3$. In most cases, when finite elements are constructed by means of the partition of a given domain in $\mathbb R^2$ into triangles and interpolation of the Hermite or Birkhoff type, the sine of the smallest angle of the triangle appears in the denominators of the error estimates for the derivatives. In the case of $\mathbb R^m$ ($m\ge3$), the ratio of the radius of the inscribed sphere to the diameter of the simplex is used as an analog of this characteristic. This makes it necessary to impose constraints on the triangulation of the domain. The recent investigations by a number of authors reveal that, in the case of triangles, the smallest angle in the error estimates for some interpolation processes can be replaced by the middle or the greatest one, which makes it possible to weaken the triangulation requirements. There are fewer works of this kind for $m\ge3$, and the error estimates are given there in terms of other characteristics of the simplex. In this paper, methods are suggested for constructing an interpolation third-degree polynomial on a simplex in $\mathbb R^3$. These methods allow one to obtain estimates in terms of a new characteristic of a rather simple form and weaken the triangulation requirements.