The convergence speed of multigrid methods is difficult to determine precisely, because it depends on different basic iterations: the coarse-grid correction, pre-smoothing and post-smoothing. The coarse-grid correction is a nonconvergent itaration (it bahaves as the identity in a subspace determined by the rows of the restriction operator, but is the zero matrix - so infinite convergence speed - in the orthogonal), the pre and post smoothings are in general simple and slowly convergent stationary methods. The high convergence speed of the multigrid method depends on the complementarity, that is the pre - smoothing is fast where the coarse-grid corrrection is nonconvergent and the post-smoothing adds substantial convergence speed if it is fast where the combination of pre-smoothing and coarse-grid correction is less effective.....

In conclusion often the bounds depend on the number of smoothing steps but the effective convergece speed depend on pre and post smoothing separately and, in addition, it is often more effective (for the previous reasoning on the complementarity) that the post-smoothing is a stationary method different from the pre-smoothing

Stefano Serra-Capizzano Thank you for the answer!But in case we use same stationary method for both pre and post smoothing what is the difference for example if we use (1,1) ,(0,2) or (2,0) smoothing steps?

Dear Spyros, if you look to the iteration matrix of the two-grid method (TGM), then, using the same stationary method, you discover that the TGM iteration matrices in case of (s,t), (u,v) smoothing steps are similar as long as s+t=u+v. Therefore the asymptotic convergence speed is identical, but not the practical behavior, since the fastest convergence even at the initial iterations is observed when the global iteration matrix is the most close to a normal matrix..... In many sitiations, coming back at your example, (1,1) is better than (0,2) or (2,0).... But to prove it is somehow tricky.