Galerkin coarse-grid operator ($A_H=R{A}_{h}P$)

A Galerkin coarse-grid operator is obtained by composing interpolation, the fine operator and residual restriction. With $R=c{P}^T$, $c>0$, and positive-definite $A_h$, the exact coarse-grid correction $I-P(R{A}_{h}P{)}^{-1}R{A}_h$ is the energy-orthogonal projection onto the complement of the coarse space. Indeed it annihilates every vector $Pz$, and its corrected error $e^{\prime }$ satisfies $P^T{A}_h{e}^{\prime }=0$. These identities explain why compatible transfers remove representable smooth error rather than merely averaging an approximate solution.