Computing $H^2$-conforming finite element approximations without having to implement $C^1$-elements

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.