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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.
If you use a sharper trace inequality you get $$\begin{aligned}\|u-Pu\|_{0,\partial \Omega}^2 &\lesssim \|u-Pu\|_0 \|u-Pu\|_1 \\ &\lesssim h^{k+1} h^{k}|u|^2_{k+1} \end{aligned} $$ implying $$\|u-Pu\|_{0,\partial \Omega} \lesssim h^{k+\frac12},$$ just as Wolfgang suggested in the comments.
