I'm looking at this famous paper which is available in the link below:
- Franco Brezzi, LD Marini, Endre Süli, Discontinuous Galerkin methods for first-order hyperbolic problems, Mathematical Models and Methods in Applied Sciences Vol. 14, No. 12 (2004) pp. 1893-1903, doi:10.1142/S0218202504003866, ResearchGate.
I am wonrdering if I've found a typo or (more probably) I am missing something.
I will use the enumeration of the paper. Let me quickly write the setting: they introduce a norm in eq. $(31)$ $$|||\delta|||^2 = \|\delta\|_{0,\Omega}^2 + \sum_{e \in E_j} \|c_e^{\frac{1}{2}} [\delta]\|_{0,e}^2$$ What is relevant is not what $c_e$, but just the definition of the norm. In order to derive an a-priori bound for $\delta$ in this norm, they start from $$C_s |||\delta||| \leq a_h(n ,\delta)+ b_h^s(n,\delta)$$
and they bound both $a(n,\delta)$ and $b_h^s(n,\delta)$ in the following way: $$a_h(n,\delta) \leq C h^{k+1}\|\delta\|_{0,\Omega}\|u\|_{k+1,\Omega} \quad (45)$$
$$b_h^s(n,\delta) \leq C h^{k+\frac{1}{2}} \|u\|_{k+1,\Omega} \Bigl( \sum_{e \in E_h} \|c_e^{\frac{1}{2}}[\delta]\|_{0,e}^2\Bigr)^{\frac{1}{2}} \quad (49)$$
So they bound the two terms in the sum $(43)$ and obtain $$C_s |||\delta||| \leq C h^{k+\frac{1}{2}} \|u\|_{k+1,\Omega} |||\delta||| \quad (50)$$ My problem is exactly in this last bound: I don't see how it's possible to have only $h^{k+\frac{1}{2}}$ in the latter bound, as summing $(45)$ and $(49)$ I have a term with $h^k$ and a term with $h^{k+\frac{1}{2}}$. Also, the constant $C$ should not depend on $h$, since they wrote:
$C$ will denote a generic positive constant which depends only on the degree $k$ of the polynomials, on the minimum angle of the mesh, and on the maximum value of the stabilizing functions $c_e$.
Any help is highly appreciated.
EDIT:
Maybe I spot the tricky part (to me). In computing $(45)$ they're using the inverse inequality $$|\delta|_{1,T} \leq h_T^{-1} \|\delta\|_{0,\Omega}$$ which I know it holds under the assumption $h = \max_T h_T \leq 1$
But if $h \leq 1$, then for sure I can bound $h^{k + 1}$ with $h^{k+\frac{1}{2}}$ and I would be able to prove this. Does it sound reasonable to you? I've seen that inverse inequalities hold under less severe conditions, but I don't know which one they're using.
