= Galerkin coarse-grid operator
{title2=$A_H=RA_hP$}
A Galerkin coarse-grid operator is obtained by composing interpolation, the fine operator and residual restriction. With $R=cP^T$, $c>0$, and positive-definite $A_h$, the exact <coarse-grid correction> $I-P(RA_hP)^{-1}RA_h$ is the energy-orthogonal projection onto the complement of the coarse space. Indeed it annihilates every vector $Pz$, and its corrected error $e'$ satisfies $P^TA_he'=0$. These identities explain why compatible transfers remove representable smooth error rather than merely averaging an approximate solution.
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