The author proves the stability of the \(L_{2}\) projection into the finite element trial space of piecewise polynomial, in particular, piecewise linear basis functions in \(H^{2}(\Omega)\) for \(s\in \left( 0,1\right] \). A stability condition needed in numerical analysis of mixed and hybrid boundary finite element methods is proved. Explicit and computable local mesh conditions to be satisfied which depend on the Sobolev index \(s\) are formulated. Finally a numerical example is considered.