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So if I've understood it correctly, the Algebraic Multigrid Method (AMG) basically takes a fine mesh, coarsens it, solves the coarse mesh and projects the solution back on the fine mesh.

Wouldn't it make sense to skip all the extra steps, just start directly with the coarse mesh, solve it normally and be done with it? Instead of starting with too much information, then trimming it down only to artificially blow it up again? Intuitively speaking, what benefit do all the extra steps bring?

(I'm new to the subject so I hope it's not too stupid of a question, but I couldn't find any explanation I was able to make sense of.)

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  • $\begingroup$ You may want to look into Full Multigrid method and Cascadic Conjugate Gradients or Cascadic Multigrid. $\endgroup$
    – cos_theta
    Feb 28 at 8:27

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It does seem like that should help at first but it doesn't. The thing to understand is that that would entail a great deal of wasted computation as in multigrid you don't solve the coarse grid problem, you smooth the low frequency errors on the fine grid that show up as high frequency errors on the coarse grid. In reality you solve a series of coarse grid error problems, couple these with the fine grid discretization, and this allows for an accurate fine grid solution to the physical problem of interest at optimal scaling. You'll find that if you solve sequentially on the coarse grids (grid sequencing) you tend to see more robustness but not too much of a computational decrease as far as I know.

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  • $\begingroup$ Thanks! I guess the smoothing part is what I'm struggling with. If I've understood it correctly, smoothing basically means evening out small local variations of a curve while keeping the overall shape intact? But how can we smooth the curve before we have actually calculated it? (At least I think that's what the method is supposed to do?) Maybe you can point me in the right direction here? $\endgroup$
    – MaxD
    Feb 28 at 17:09
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    $\begingroup$ You're not smoothing the curve itself, you're smoothing the error in your calculation of the curve on the coarse grids. So you begin on a fine mesh, perform some smoothing passes, then restrict the error on to a coarser mesh, do some more smoothing, and repeat till you reach the coarsest mesh. Then you solve the much cheaper error equation and prolongate the solution correction from the coarsest mesh to the next coarsest one and smooth and repeat. That forms a V-cycle and begins on the finest mesh. I do not know of any cycles that begin from the coarsest mesh. $\endgroup$
    – EMP
    Feb 28 at 19:56

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