  Many classical finite elements such as the Argyris and Bell elements have
long been absent from high-level PDE software. Building on recent theoretical
work, we describe how to implement very general finite element transformations
in FInAT and hence into the Firedrake finite element system. Numerical results
evaluate the new elements, comparing them to existing methods for classical
problems. For a second order model problem, we find that new elements give
smooth solutions at a mild increase in cost over standard Lagrange elements.
For fourth-order problems, however, the newly-enabled methods significantly
outperform interior penalty formulations. We also give some advanced use cases,
solving the nonlinear Cahn-Hilliard equation and some biharmonic eigenvalue
problems (including Chladni plates) using $C^1$ discretizations.
