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

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