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.

  • $\begingroup$ 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
  • 1
    $\begingroup$ 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
  • 1
    $\begingroup$ 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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.