In Multigrid, using Poisson's equation, does the equality below always hold regardless of what type of boundary conditions you use? $$ R= c\cdot I^T, \text{ for some constant }c $$ where $R$ and $I$ are the restriction and interpolation operator. Why or why not?
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asked Mar 17, 2013 at 22:18
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It's not a question whether the condition holds or not. You choose $R$ so that it's a multiple of $I^T$ because you want the condition to hold. You want the condition to hold because that's the only way you can ensure that the multigrid operator is symmetric and amenable to, say, being a preconditioner for CG.
answered Mar 19, 2013 at 2:22
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$\begingroup$ Ugh. My restriction operator wasn't handling inhomogeneous Dirichlet boundary conditions properly. So I wrote it separately...Meaning that as of now I have no $c$ s.t. the equality holds. Does this mean I'll have to reconfigure everything? $\endgroup$ Mar 19, 2013 at 2:27
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1$\begingroup$ Probably. You will also note that if you were to naively build restriction and prolongation operators as if you wanted them to represent an interpolation between spaces (which I imagine is what you did) that they will not respect the equality. That's why we call them restriction/prolongation and not up-/down-interpolation or similar. $\endgroup$ Mar 19, 2013 at 2:37
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$\begingroup$ Well i used bilinear interpolation and full weighting. I worked out a few cases by hand and made note of how the matrix changed depending on the size of the discretization. I did the same thing for homogeneous Dirichlet bc and it worked out fine, but as soon as I started messing with inhomogeneous, everything went awry. $\endgroup$ Mar 19, 2013 at 4:42