On the construction of deflation-based preconditioners

In this article we introduce new bounds for the condition numbers of deflated symmetric positive definite systems, used with and without classical preconditioning. For the case of a subdomain deflation such as that of Nicolaides (1987), these lemmas can provide direction in choosing a proper decomposition into subdomains and a proper choice of classical preconditioner. If grid refinement is done keeping the subdomain resolutions fixed, the condition number can be shown to be independent of the number of subdomains.