= Prolongation operator
{title2=$P$}
A prolongation operator transfers a coarse-grid correction to a finer <mesh>. For nested one-dimensional grids of spacings $2h$ and $h$, linear interpolation sets $(Pe)_{2j}=e_j$ and $(Pe)_{2j+1}=(e_j+e_{j+1})/2$, with zero endpoint values for homogeneous correction boundary conditions. The associated <full-weighting restriction> is $R=P^T/2$, and the <Galerkin coarse-grid operator> is $RA_hP$. The correction equation acts on the residual so that a coarse representation of the error can improve the fine-grid approximation.
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