On the optimality of some FAC and AFAC methods for elliptic finite element problems (Q1282131)

The author investigates fast adaptive composite (FAC) and asynchronous FAC (AFAC) methods for solving large-scale systems of linear equations arising from finite element discretizations of second-order elliptic boundary value problems on composite grids. It is proved that the condition number of the operator corresponding to a certain AFAC method is independent of the number of degrees of freedom and the number of refinement levels. For getting this result a strengthened Cauchy inequality is proved using an interpolation theorem for Hilbert scales. Furthermore, a FAC algorithm with inexact solvers is considered. The optimality of this algorithm is proved by using theoretical results developed in this paper for the investigation of the AFAC method and by using some ideas from multigrid methods.