In this article we introduce new bounds on the effective condition number of deflated and preconditioned-deflated symmetric positive definite linear systems. For the case of a subdomain deflation such as that of Nicolaides [SIAM J. Numer. Anal., 24 (1987), pp. 355--365], these theorems can provide direction in choosing a proper decomposition into subdomains. If grid refinement is performed, keeping the subdomain grid resolution fixed, the condition number is insensitive to the grid size. Subdomain deflation is very easy to implement and has been parallelized on a distributed memory system with only a small amount of additional communication. Numerical experiments for a steady-state convection-diffusion problem are included.
deflation, preconditioners, optimal methods, parallel computing, conjugate gradients
Iterative methods for linear systems (msc 65F10), Sparse matrices (msc 65F50), Solution of discretized equations (msc 65N22)
S.I.A.M.
dx.doi.org/10.1137/S1064827500373231
SIAM Journal on Scientific Computing
Computational Dynamics

Frank, J.E, & Vuik, C. (2001). On the construction of deflation-based preconditioners. SIAM Journal on Scientific Computing, 23, 442–462. doi:10.1137/S1064827500373231