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I am looking into full multigrid, FMG, and several sources, including these slides, that a lot of people are referring to, state that the prolongation operator used in FMG the first time you visit a finer grid should be of higher order than the prolongation operator used otherwise.

What is a concrete example of such a higher order operator?

For the prolongation in the other steps of multigrid, I use the operator suggested in the slides page 56.

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I know the methods which are used for prolongation/restriction affects speedup too much. The point is your selected scheme should not introduce any positive or negative source term. In the other word it is better to use a conservative operator particularly if you are using it for conservative equations. So I recommend to plot your coarse and fine points together and make cells (even in 1D) then for interpolation a data in a cell just integrate upon the grid. for some regular simple 1d and 2d cases you result in some trivial popular operators.

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