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As far as I can tell, nodal interpolation estimates proven with Bramble / Hilbert require higher regularity of the functions being interpolated. E.g. with linear elements in 2 dimensions, one needs $v \in W^{m,p}$ with $m p > 2, p \geq 1$ in order to have

$$ |v - I(v)|_{W^{r,p}} \leq C h^{m-r} |v|_{W^{m,p}}, $$

where $I$ is the nodal interpolant, $| \cdot |$ are seminorms in the indicated spaces and $0 \leq r \leq m$. In particular note that the bound is using the $m$-th derivative of $v$.

For my application I only have $m = 1$ and $p \leq 2$. More specifically this is what I want:

Let $V_h$ be a continuous $P_1$ space over a polygonal domain $U \subset \mathbb{R}^2$ with a quasi-uniform mesh, and $(v_h) \subset V_h$ a sequence bounded in $W^{1,2}(U)$. For each $h$ let $I_h:C^0 \rightarrow V_h$ be some interpolant (I'd use the nodal interpolant but I'm not working with $W^{2,2}$) and $q$ some quadratic function. To simplify maybe take $q(x)=x^2$. I want to show

$$ \|q(v_h) - I_h(q(v_h))\|_{L^2(U)} \leq C h^\alpha \|D q(v_h) \|_{L^2(U)}$$

for any $\alpha > 0$ which makes this work. Reading some papers, I see people change $V_h$ to be discontinuous and claim to have bounds using just the first derivative of $q \circ v_h$, but I don't know where to find precise statements and proofs of the properties of the interpolant in this case. I'm happy to make $V_h$ discontinuous, I just want the damn thing to converge :)

Any help, anyone?

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  • $\begingroup$ It's not exactly clear to me what you are looking for. If $v$ is really only in $W^{1,2+\varepsilon}$, then the estimate $\|v-I_h v\|_{L^{2+\alpha}} = {\cal O}(h^{1+\alpha})$ is likely sharp. You will of course get something better if you knew that $v$ is quadratic, but the point is that quadratic functions are far smoother than just in $W^{1,2+\alpha}$. $\endgroup$ – Wolfgang Bangerth Jan 26 '18 at 0:13
  • $\begingroup$ I don't know if I understand you well, but I am not looking for faster rates. I am looking for a bound on $\|q(v_h) - I_h(q(v_h)) \|$ which uses the first derivative $D q(v_h)$ instead of $D^2 q(v_h)$, because I don't have it. Since $v_h \in V_h$ I do have it element-wise and I thought maybe one can reason in an element, then use some sort of reverse-Poincaré / Caccioppoli inequality, then patch everything together but I don't have any equation that $v_h$ solves locally and there were more issues... well, it didn't get me far. $\endgroup$ – Miguel Jan 26 '18 at 7:55
  • $\begingroup$ I think I'm confused because on the right hand side of your first equation, you have $|v|_{W^{m,p}}$, which only has first derivatives if you choose $m=1$. $\endgroup$ – Wolfgang Bangerth Jan 26 '18 at 13:30
  • $\begingroup$ Indeed, but the problem is that I can't just plug in $m=1$ because I need $m p \gt 2$ and have $p=2$. Oh! I see now that I forgot to specify that and only wrote it in the final paragraph, sorry. $\endgroup$ – Miguel Jan 26 '18 at 14:10
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    $\begingroup$ I know you stated $mp>2$, but you can choose $m=1$, $p=2+\alpha$. A small $alpha$ isn't going to change much in your approach, and you still only need first derivatives. $\endgroup$ – Wolfgang Bangerth Jan 26 '18 at 16:59
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What I was looking for is Clément interpolation, which is defined by averaging a target function $v \in L^1$ over cells sharing a node and assigning this value to the interpolant at that node (or equivalently by local $L^2$ projections). It is explained in detail in §6.3 of these lecture notes of Volker John at the WIAS.

Specifically, the estimate obtained for $v \in W^{1,q}$ is

$$ \| D^k(v - I_h^{\operatorname{cle}}(v)) \|_{L^q} \leq C h^{l-k} \| D^l v\|_{L^q}$$

for all $0 \leq k \leq l \leq 2 $ and $1 \leq q \leq \infty$, assuming a nice enough mesh.

Alternatively, I guess I could just use global $L^2$ projection instead of interpolation, which guarantees the convergence I want, but I need to use the same operator in other places and I don't want my code to be performing costly projections.

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