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2

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-...


2

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.


4

Post-smoothing reduces the high frequency error that is introduced by the coarse grid correction. If you visualize a correction computed on a mesh of size $2h$, then it has no kinks on half of the nodes of the fine mesh of size $h$, and strong kinks on the other half. Post-smoothing distributes this a bit and leads to a coarse grid correction that doesn't ...


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