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I am reading the following article:

Characteristic-Galerkin and Galerkin/least-squares space-time formulations for the advectiondiffusion equation with time-dependent domains

by O. Pironneau, et al.

Available at: (if you have an account)

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This article is very relevant to my thesis, so I have spent a lot of time reading it, but there is an inequality I just don't understand. I'm hoping that you can help me out :)

Proposition 4 gives a convergence rate for the scheme to be $\mathcal{O}(h^2/k + k + h)$. I understand the proof up until the very last line (eq, 29).

The norm $\| \cdot \|_\nu$ is defined as

$$ \| \phi^{n+1} \|_\nu^2 = |\phi^{n+1}|^2 + \nu k |\nabla \phi^{n+1}|$$

where $|\cdot|$ (or $| \cdot |_0$) represents the $L^2$-norm. Then, if $\Pi_h$ is linear interpolation and we consider the inequality just before equation (29), then I think that we should get

$$\|\phi^{n+1} - \Pi_h \phi^{n+1} \|_\nu \leq \sqrt{C h^4 + C \nu k h^2} \leq C(h^2 + h\sqrt{\nu k}).$$

And for the other term

$$(1+Ck)|\phi^n - \Pi_h \phi^n|_0 \leq (1+Ck)Ch^2 $$

Putting these two together we get $C(h^2 + \sqrt{\nu k}h + k h^2)$ but according to equation (29) they seem to have got $C(h^2 + \nu k h)$. How is this possible?

Thanks in advance!

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