  We develop a method to compute the $H^2$-conforming finite element
approximation to planar fourth order elliptic problems without having to
implement $C^1$ elements. The algorithm consists of replacing the original
$H^2$-conforming scheme with pre-processing and post-processing steps that
require only an $H^1$-conforming Poisson type solve and an inner Stokes-like
problem that again only requires at most $H^1$-conformity. We then demonstrate
the method applied to the Morgan-Scott elements with three numerical examples.
