We have an elliptic projection
$$P: V \rightarrow V_{h}$$
which satisfies
$$\Vert u - Pu \Vert_{L^{2}(\Omega_{e})} \leq Ch^{k+1} \enspace .$$
Can we say anything about $\Vert u - Pu \Vert_{L^{2}( \partial \Omega_{e})}$?
I know if we have use the Trace theorem we can bound it by $h^{k}$, but I would not like to lose the order of convergence.