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?
Prolongation/Restriction Operator in Multigrid
interpolation;multigrid;poisson
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.