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.

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Modelling, Analysis and Simulation [MAS]
Computational Dynamics

Frank, J., & Vuik, C. (2000). On the construction of deflation-based preconditioners. Modelling, Analysis and Simulation [MAS]. CWI.