Full-weighting restriction ($R=2^{-d}{P}^T$)

Full-weighting restriction transfers a fine-grid residual to a coarser grid by local averaging. In one dimension its weights are $(1,2,1)/4$; in two dimensions they are the tensor-product stencil $\left(\begin{array}{ccc}1& 2& 1\\ 2& 4& 2\\ 1& 2& 1\end{array}\right)/16$. For linear or bilinear interpolation $P$ and coarsening by two in every direction, it equals $2^{-d}{P}^T$ in unweighted coordinates. This scaling makes restriction the adjoint of interpolation for mesh-weighted inner products.