Scaling Arguments

This is a pretty important concept in PDEs and its numerical approximations. Specifically, tt shows up in Bramble-Hilbert lemma, and domain decomposition analysis.
Most of this post is pretty much written right after reading Toselli and Widlund’s book, so there are a lot of resemblance.

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ which is ‘nice’ (say Lipschitz boundary) with radius $h$. Now let $u, v \in H^1(\Omega)$ such that
\begin{align*}
|v|_{H^1(\Omega)} \le C||u||_{H^1(\Omega)}
\end{align*}
and we wish to obtain the $h$ dependence from $C$.

What we do is to first consider a scaled domain $\hat \Omega$ which is just $\Omega$ scaled to be of radius 1, with the change of basis $x = h\hat x$.
If we find the corresponding inequality on $\hat \Omega$, then the constant $C$ will not depend on $h$.
Let $\hat v(\hat x) := v(h\hat x)$, then we note that $\hat \nabla \hat v(\hat x) = h\hat \nabla v(h\hat x)$ where $\hat \nabla$ is the gradient with respect to $\hat x$.
Then,
\begin{align*}
|v|^2_{H^1(\Omega)} &= \int_\Omega |\nabla v(x)|^2 \, dx \\
&= \int_{\hat \Omega} |\hat\nabla v(h \hat x)|^2 h^n \, d\hat x \\
&= \int_{\hat \Omega} |\hat\nabla \hat v(\hat x)|^2 h^{-2} h^n \, d\hat x = h^{n-2}|\hat v|_{H^1(\hat \Omega)}^2
\end{align*}

But for $L^2$ norm, there is no $h^2$ scaling, hence
\begin{align*}
||u||_{L^2(\Omega)}^2 &= \int_\Omega |u(x)|^2 \, dx \\
&= \int_{\hat \Omega} |u(h \hat x)|^2 h^n \, d\hat x = h^n ||\hat u||_{L^2(\hat \Omega)}^2.
\end{align*}
This is why derivatives mixing causes scaling issues.

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