Bilinear interpolation

On a rectangle, bilinear interpolation is linear in each coordinate separately and matches the four corner values. With normalized coordinates $s,t\in [0,1]$, it is $(1-s)(1-t)u_{00}+s(1-t)u_{10}+(1-s)t{u}_{01}+st{u}_{11}$. A multigrid method can use it to prolong a coarse correction to finer-grid nodes. It is not generally an affine function of the pair $(s,t)$, since it can contain a mixed $st$ term.