On the stability of the $L_2$ projection in fractional Sobolev spaces

Summary.

In this paper we prove the stability of the $L_2$ projection onto the finite element trial space of piecewise polynomial, in particular, piecewise linear basis functions in $H^s(\Omega)$ for $s \in (0,1]$. We formulate explicit and computable local mesh conditions to be satisfied which depend on the Sobolev index s. In conclusion we prove a stability condition needed in the numerical analysis of mixed and hybrid boundary element methods as well as in the construction of efficient preconditioners in adaptive boundary and finite element methods.