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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?

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

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