Auxiliary space theory for the analysis of iterative methods for semidefinite linear systems

Abstract: We present an auxiliary space theory that provides a unified framework for analyzing various iterative methods for solving linear systems that may be semidefinite. By interpreting a given iterative method for the original system as an equivalent, yet more elementary, iterative method for an auxiliary system defined on a larger space, we derive sharp convergence estimates using elementary linear algebra. In particular, we establish identities for the error propagation operator and the condition number associated with iterative methods, which generalize and refine existing results. The proposed auxiliary space theory is applicable to the analysis of numerous advanced numerical methods in scientific computing. To illustrate its utility, we present three examples -- subspace correction methods, Hiptmair--Xu preconditioners, and auxiliary grid methods -- and demonstrate how the proposed theory yields refined analyses for these cases.