On a sharp estimate of overlapping Schwarz methods in H(curl; Ω) and H(div; Ω)

Abstract

Abstract The previously proved upper bound is $C\big(1+\frac{H^{2}}{\delta ^{2}}\big)$ for the condition number of the discrete system arising from the classical two-level overlapping Schwarz method for symmetric positive definite problems in $\boldsymbol{H}(\textbf{curl}; \varOmega )$ and $\boldsymbol{H}(\textrm{div}; \varOmega )$, where $H$ and $\delta $ are the sizes of subdomains and overlap, respectively. But all numerical results indicate that a sharper bound is $C\big (1+ \frac{H}{\delta } \big )$. In this work, we prove that the condition number is indeed bounded above by $C\big(1+\frac{H}{\delta }\big)$.