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

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