How to show the stability of $L^2$ projection?

Question

If $\mathcal{T}_h$ is a regular and quasi-uniform triangulation of $\Omega$, and $V_h$ is the $H^1$-conforming linear finite element space. Moreover, let $P_h$ be the $L^2$ projection to $V_h\subset H^1(\Omega)$, i.e., \begin{equation} (P_hu,v_h)=(u,v_h),\quad\quad \forall v_h\,\in V_h\quad\quad (1). \end{equation} How to show the stability of the following result \begin{equation} \|P_hu\|_{H^1(\Omega)}\leq C\|u\|_{H^1(\Omega)}\quad\quad (2). \end{equation}

By choosing $v_h=P_hu$, we have \begin{equation} \|P_hu\|_{L^2(\Omega)}^2=(u,p_hu)\leq \|u\|_{L^2(\Omega)}\|P_hu\|_{L^2(\Omega)}, \end{equation} which shows that \begin{equation} \|P_hu\|_{L^2(\Omega)}\leq \|u\|_{L^2(\Omega)}. \end{equation} But, how to show the result $(2)$? Could anyone give some suggestions? Thank you!

Answer 1

I'd assume you want to prove $(2)$, and $u\in V\subset H^1(\Omega)$.

You already proved that $\|P_hu\|_{L^2(\Omega)}\leq \|u\|_{L^2(\Omega)}$, which is just one more step to arrive $$ \|P_hu\|_{H^1(\Omega)}\leq C_0\left(\|u\|_{H^1(\Omega)}+\|u\|_{L^2(\Omega)}\right). $$ Then you prove $$ \|u\|_{L^2(\Omega)} \leq C_1\|u\|_{H^1(\Omega)}. $$

Combine the above two inequalities, you get $$ \|P_hu\|_{H^1(\Omega)}\leq C\|u\|_{H^1(\Omega)}. $$

What listed above is just a process for the proof other than its rigorous detail, which could be found in the literature, such as Brenner & Scott.