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If we use an aggresive coarsening how does this affect the convergence of a MG cycle?Is the convergence slower?Do we need more cycles in this case?

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Yes, quicker coarsening means that your coarse grid correction is not as good and that you need more MG cycles. For a well-tuned multigrid, you reduce the error by about a factor of 5-10 for each CG iteration with MG V-cycle preconditioner. If you were to coarsen too aggressively, the reduction factor would be smaller.

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  • $\begingroup$ And the reason for this is that with a more aggressive coarsening we lose information about some nodes?(assume canonical restriction) $\endgroup$ – spyros Feb 4 '20 at 3:28
  • $\begingroup$ Correct. The solution on the more-coarser level is just not as good an approximation as the one on the less-coarser level. $\endgroup$ – Wolfgang Bangerth Feb 4 '20 at 17:23

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