Does the convergence rate of multigrid depend on the total number of smoothing steps or on the number of pre and post smoothing steps seperately?


1 Answer 1


Separately, but it does depend. Not very strongly, however: A very large number of pre- and post-smoothing steps only improves the convergence rate a little bit over a large number of steps. The difference is most between using one, two, or three pre- and post-smoothing steps.

And you need both pre- and post-smoothing steps. You cannot compensate for no pre-smoothing steps by using more post-smoothing steps.

  • $\begingroup$ In this case what happened?If for example use (1,1) or (0,2) or (2,0) smoothing steps for example.Thank you for your help! $\endgroup$
    – spyros
    Commented Feb 12, 2020 at 13:31
  • $\begingroup$ I'm not an expert in MG behavior. I suspect that for the (0,2) and (2,0) case, you can't prove convergence, but it might converge (slowly) anyway in practice. $\endgroup$ Commented Feb 12, 2020 at 14:27
  • $\begingroup$ I did find an excellent reference: The book of Elman, Silvester, and Wathen on "Finite elements and fast solvers" has exactly the kind of table you are look for on p. 105. You really do need both pre- and post-smoothing to get mesh-independent convergence rates. $\endgroup$ Commented Feb 23, 2020 at 19:36
  • $\begingroup$ Is this book online?I couldn't find any reference.Can you send me a link or something?Thank you! $\endgroup$
    – spyros
    Commented Feb 24, 2020 at 16:14
  • $\begingroup$ I don't know. I happen to have the book. $\endgroup$ Commented Feb 24, 2020 at 20:53

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