# How to prove the stability of the interpolation?

From the book (Vidar Thomee, Galerkin finite element methods for parabolic problems), there holds (see Lemma 13.3) \begin{align}\label{eq} \|\nabla I_h u\|_{L_{\infty}}\leq C\|\nabla u\|_{L_{\infty}},\quad\quad (1). \end{align} where $$I_h$$ is the linear interpolation operator,i.e., \begin{align} I_h: H^2(\Omega)\to S_h, ~~and ~~S_h=\{v_h\in C(\bar{\Omega}),~v_h|_K\in span{1,x,y}\} \end{align} with \begin{align} I_hv(a_i)=v(a_i), \end{align} $$a_i$$ $$(i=1,2,3)$$ are three vertices of an element $$K$$.

How to prove this result (1) and this result (1) can be extended to the bilinear Lagrange interpolation operator?

• Does the book have a proof of the lemma? Does it have a reference to a publication that has a proof? Jun 29 at 15:53