On the Convergence of Cascadic Iterations for Elliptic Problems

A b s t r a c t. We consider nested iterations, in which the multigrid method is replaced by some simple basic iteration procedure, and call them cascadic iterations. They were introduced by Deuflhard, who used the conjugate gradient method as basic iteration (CCG method). He demonstrated by numerical experiments tha t the CCG method works within a few iterations if the linear systems on coarser triangulations are solved accurately enough. Shaidurov subsequently proved multigrid complexity for the CCG method in the case of i?2-regular two-dimensional problems with quasi-uniform triangulations. We show tha t his result still holds true for a large class of smoothing iterations as basic iteration procedure in the case of two- and three-dimensional _ff1+a-regular problems. Moreover we show how to use cascadic iterations in adaptive codes and give in particular a new termination criterion for the CCG method. K e y W o r d s. Finite element approximation, cascadic iteration, nested iteration, smoothing iteration, conjugate gradient method, adaptive triangulations A M S ( M O S) subjec t c lass i f icat ion. 65F10, 65N30, 65N55 Introduction. Let 0 C Rd b a polygonal Lipschitz doman. W co