The approximation of the fourth-order partial differential equations in the class of the piecewise polynomial functions on the triangular grid

The present work determines the deviation of the piecewise cubic almost-solution of the biharmonic equation and derives the general formula (3) for its calculation. Based on this concept, we obtained an approximation of the equation. A number of numerical calculations were carried out in order to confirm the obtained formula (see pictures 1 and 2) experimentally. In general, for all selected biharmonic functions, the deviation value "ΔΔ(𝑢) turned out to be, as expected, rather small even with a relatively small number of triangulation nodes. On average, with 25 ≤ ≤ 35, the absolute error was about 0, 0001, which gives an approximately asymptotic estimate of 𝑂(ℎ2) when the partitioning step is ℎ → 0.