On the optimality of some FAC and AFAC methods for elliptic finite elementproblems

We consider some solution methods for large sparse linear systems of equati ons which arise from second-order elliptic finite element problems defined on composite meshes. Historically these methods were called FAC and AFAC me thods. Optimal bounds of the condition number for certain AFAC iterative op erator are established by proving a strengthened Cauchy-Schwarz inequality using an interpolation theorem for Hilbert scales. This work completes earl ier work by Dryja and Widlund. We also apply an extension theorem for finit e element functions to get a weaker bound under some more general assumptio ns. The optimality of the FAC methods, with exact solvers or spectrally equ ivalent inexact solvers being used, is also proved by using similar techniq ues and some ideas from multigrid theory.