What's the definition of $L^{\infty}$-norm for nonconforming finite element?

Question

We know that \begin{align*} \|u\|_{0,\infty,\Omega}={\text{ess} \sup}_{x\in \Omega}|u|. \end{align*} Moreover, the nonconforming Crouzeix-Raviart finite element space is \begin{align*} V_h=\{ v|_K\in P_1(K),\ v ~\text{is continuous at the midpoints of edges and } v=0 \text{ at the midpoints on boundary edges}\}. \end{align*} For a function $w\in V_h$, we know that $w\notin H_0^1(\Omega)$.

My question is what is the definition of $\|w\|_{0,\infty,\Omega}$ and $\|\nabla w\|_{0,\infty,\Omega}$?

In addition, does the inversive inequality \begin{align*} \|\nabla w\|_{0,\infty,\Omega}\leq Ch^{-1}\|w\|_{0,2,\Omega}, w\in V_h \end{align*} hold?

Answer 1

As @VorKir mentions, for nonconforming spaces one has to use broken Sobolev norms that are defined elementwise (in your notation) as $$\begin{aligned} \|v\|_{k,p,\Omega} &= \left(\sum_{K} \|v\|_{k,p,K}^p\right)^{1/p},\quad 1\leq p<\infty,\\ \|v\|_{k,\infty,\Omega} &= \max_K \|v\|_{k,\infty,K} \end{aligned}$$ for $v\in V_h$, where $\|v\|_{k,p,K}$ is the usual $W^{k,p}(K)$ norm of $v|_K$ (which by definition is a polynomial and hence arbitrarily often differentiable).

There are inverse estimates for broken norms, but you lose additional powers of $h$ by having a larger value of $p$ on the left-hand side. If $\Omega\subset \mathbb{R}^n$, there exists a $C$ independent of $h$ such that $$ \|v\|_{1,\infty,\Omega} \leq C h^{-1-n/2} \|v\|_{0,2,\Omega} \quad\text{for all }v\in V_h.$$ (This is a special case of Theorem 4.5.11 in Brenner, Susanne C.; Scott, L.Ridgway, The mathematical theory of finite element methods, Texts in Applied Mathematics 15. New York, NY: Springer (ISBN 978-0-387-75933-3/hbk). xvii, 397~p. (2008). ZBL1135.65042.)