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The previous proved-bound is $C(1+\frac{H^2}{\delta^2})$ for the conditionnumber of the overlapping domain decomposition$\mathrm{H}(\mathrm{curl};\Omega)$ and $\mathrm{H}(\mathrm{div};\Omega)$methods, where $H$ and $\delta$ are the sizes of subdomains and overlapsrespectively. But all numerical results indicate that the best bound is$C(1+\frac{H}{\delta})$. In this work, we solve this long-standing open problemby proving that $C(1+\frac{H}{\delta})$ is indeed the best bound.

On a Sharp Estimate of Overlapping Schwarz Methods in $\mathrm{H}(\mathrm{curl};\Omega)$ and $\mathrm{H}(\mathrm{div};\Omega)$