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

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