Geodesic
The approximation of the fourth-order partial differential equations in the class of the piecewise polynomial functions on the triangular grid
Matematičeskaâ fizika i kompʹûternoe modelirovanie, Tome 22 (2019) no. 2, pp. 65-72.

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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 $\varepsilon_{\Delta \Delta}(u)$ turned out to be, as expected, rather small even with a relatively small number of triangulation nodes. On average, with $ 25 \leq N \leq 35 $, the absolute error was about $0,0001 $, which gives an approximately asymptotic estimate of $O(h^2)$ when the partitioning step is $h \to 0$.
Keywords: piecewise cubic function, almost solution, biharmonic equation, approximation of the equation, deviation of the piecewise cubic almost solution. 
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     author = {A. A. Klyachin and V. A. Klyachin},
     title = {The approximation of the fourth-order partial differential equations in the class of the piecewise polynomial functions on the triangular grid},
     journal = {Matemati\v{c}eska\^a fizika i kompʹ\^uternoe modelirovanie},
     pages = {65--72},
     publisher = {mathdoc},
     volume = {22},
     number = {2},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VVGUM_2019_22_2_a4/}
}
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A. A. Klyachin; V. A. Klyachin. The approximation of the fourth-order partial differential equations in the class of the piecewise polynomial functions on the triangular grid. Matematičeskaâ fizika i kompʹûternoe modelirovanie, Tome 22 (2019) no. 2, pp. 65-72. http://geodesic.mathdoc.fr/item/VVGUM_2019_22_2_a4/