# Approximation properties of FEM projections operators on a boundary

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.

• I'm pretty sure you're going to get $h^{k+1/2}$, but I don't have a paper reference at hand. Have you done a literature search? Feb 4 '16 at 15:25

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.