Lower Bounds in Domain Decomposition

Abstract

An important indicator of the efficiency of a domain decomposition preconditioner is the condition number of the preconditioned system. Upper bounds for the condition numbers of the preconditioned systems have been the focus of most analyses in domain decomposition [21, 20, 23]. However, in order to have a fair comparison of two preconditioners, the sharpness of the respective upper bounds must first be established, which means that we need to derive lower bounds for the condition numbers of the preconditioned systems.