Prolongation operator ($P$)

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 $R{A}_{h}P$. The correction equation acts on the residual so that a coarse representation of the error can improve the fine-grid approximation.