On the multigrid F-cycle

In a recent paper, a bound was derived on the convergence of the multigrid V-cycle for the case when the solution is in the Sobolev space H^1^+^@a but not in H^1^+^@a^', @a' #62;; @a, showing that the convergence factor approaches one only as 1 - O(k^(^@a^-^1^)^/^@a) for a large number of levels k. We now extend the technique to obtain the asymptotically better bound 1 - O(k^-^(^1^-^@a^)^^^2^ ^/^ ^@a) on the multigrid F-cycle. We also show that in many cases, for practical values of k, one gets the same bound for the F-cycle as for the V-cycle with @a = 1.