On a Sharp Estimate of Overlapping Schwarz Methods in $\mathrm{H}(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l};\mathrm{\Omega })$ and $\mathrm{H}(\mathrm{d}\mathrm{i}\mathrm{v};\mathrm{\Omega })$

The previous proved-bound is $C(1+\frac{H^2}{{\delta }^2})$ for the condition number of the overlapping domain decomposition $\mathrm{H}(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l};\mathrm{\Omega })$ and $\mathrm{H}(\mathrm{d}\mathrm{i}\mathrm{v};\mathrm{\Omega })$ methods, where $H$ and $\delta$ are the sizes of subdomains and overlaps respectively. But all numerical results indicate that the best bound is $C(1+\frac{H}{\delta })$. In this work, we solve this long-standing open problem by proving that $C(1+\frac{H}{\delta })$ is indeed the best bound.