Let $\Omega \in \mathbb{R}^d$ be an open bounded set, and consider a function $u$ and a sequence $(u_h)_h$ in the Sobolev space $u \in H^2(\Omega)$. I wish to use a theorem in literature that proves the existence of a constant $C$, independent on $h$, such that $$ \Vert u-u_h \Vert_{L^\infty(\Omega)} \leq C h^q \Vert u \Vert_{H^2(\Omega)} $$
I am in a situation where I can, and in fact I must, assume that $u$ and $(u_h)$ are sequences in the space $C^2(D)$, where $D$ is closed and bounded.
All the hypotheses of the theorem in the literature are verified, and I am hoping to conclude that $\Vert u-u_h \Vert_{\infty,D}$, the supremum norm over $D$ is an $O(h^q)$, hence using norms that include the boundary of the compact $D$. It seems to me that I can only conclude $$ \sup_{x \in D^\circ} |u(x)-u_h(x)| \leq C h^2 \Vert u \Vert_{C^2(D)} $$ where $D^\circ$ is the interior of $D$. Do you think I can in any way include the boundary in the statement, given that $u$ and $u_h$ are of class $C^2$ on the closed domain?