= Full-weighting restriction
{title2=$R=2^{-d}P^T$}
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 $\left(\begin{smallmatrix}1&2&1\\2&4&2\\1&2&1\end{smallmatrix}\right)/16$. 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.
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