ON THE OPTIMALITY OF SOME FAC AND AFAC METHODS FOR ELLIPTIC FINITE ELEMENT PROBLEMS

Abstract

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