A New Family of Preconditioners for Domain Decomposition

This paper presents a new approach to construct preconditioners for the domain decomposition-based preconditioned conjugate gradient methods. The idea is to approximate the spectrum of the Schur complement by simple functions, such as rational approximations, which leads to solving an easy, lower dimension interface problem. Examples are given for nonuniform grids on interfaces; this is a common situation for which no effective preconditioners now exist. Theory and experimental data are given to illustrate the effectiveness of the author’s approach in this application.