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If $I_h$ is the bilinear interpolation operator, i.e. $I_h: H^2(\Omega)\rightarrow V_h$, where $V_h=\{v_h\in L^2(\Omega)|v_h|_K\in \mathrm{span}\{1,x,y,xy\},~~K \in\mathcal{T}_h\}$

i know that $\|I_hu\|_{0,\infty}\leq C(u)$ for $u\in H^2(\Omega)$. However, i dont know how to prove it. Can any one give me some suggestions?

In addition, if $I_h$ is a nonconforming interpolation operator(such as Crouzeix-Raviart element, the degree of freedom is $\int_l uds$). More precisely, $I_h: H^1(\Omega)\rightarrow V_h$, where $V_h=\{v_h\in L^2(\Omega)|v_h|_K\in \mathrm{span}\{1,x,y\},~~K \in\mathcal{T}_h\}$. Does the above relationship still hold? If so, how to prove it?

If the $\|\nabla I_hu\|_{0,\infty}\leq C(u)$ holds, $u$ should belong to $H^{?}(\Omega)$?

I'm very glad to receive some suggestions.

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  • $\begingroup$ It is worth noting that in all of your examples, $C=C(u)$. $\endgroup$ Mar 29, 2017 at 19:29
  • $\begingroup$ @WolfgangBangerth Yes, could you give me more detail explanations about how to prove the above relationship. Thank you! $\endgroup$
    – Feng Young
    Mar 30, 2017 at 2:03
  • $\begingroup$ Can you look somewhere the precise definition of $I_h$? Which spaces is it defined for? Maybe you can just use the boundedness of the interpolation operator in the appropriate Sobolev spaces with well-known embedding theorems for $L_{\infty}$? $\endgroup$
    – VorKir
    Mar 30, 2017 at 17:52
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    $\begingroup$ The first place to look is in the book by Brenner and Scott; chapter 8 is devoted to infinity-norm estimates such as the ones you're interested in. $\endgroup$ Mar 31, 2017 at 8:15
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    $\begingroup$ I don't know where specifically to look, but the search terms you ought to use for a literature search are "stability of the interpolation operator". $\endgroup$ Mar 31, 2017 at 17:04

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