On $C^1$-convergence of piecewise polynomial solutions to a fourth order variational equation

In the present work we consider a boundary value problem in a polygonal domain for a fourth order variational equation. We assume that this domain is partitioned into finitely many triangles forming its triangulation. We introduce a class of piecewise polynomial functions of a given degree and for a considered equation we define the notion of a piecewise polynomial solution on a triangle net. We prove a theorem on existence and uniqueness of such solution. Moreover, we establish that under certain conditions for the triangulation of the domain, the second derivatives of the piecewise polynomial solutions are estimated by a constant independent of the fineness of the partition. This fact allows us to prove $C^1$-convergence of piecewise polynomial solutions to the equations as the fineness of grid tends to zero.