A new cubic element in the FEM

In the paper, a new two-dimensional cubic element in the finite element method is suggested. It is proved that, in contrast to the classical element with interpolation at the center of gravity, the new element under the approximation of any admissible derivatives is free of the known condition of “sine of the smallest angle” of triangulation. It proved well to replace this condition by a weaker condition of “sine of the greatest angle” of triangulation. It is established, up to absolute constants, that the obtained estimates of approximation errors of derivatives are unimprovable. For the new element, the estimates of approximation error become worse only for triangles with two small angles. In terms of barycentric coordinates, fundamental interpolating polynomials are explicitly written out for the suggested element.