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Classic H^2 conforming elements would be nice, but there are now alternatives already available in FEniCS.
For example, take a look at the thesis of Lizao Li (https://conservancy.umn.edu/handle/11299/199048) particularly Section 4.2.3.
Note that this approach is easily extendible to the biharmonic in three dimensions.
The elements exist in FIAT, but the transformation between physical and reference space is far more complicated (different basis functions are transformed differently), which requires special support from the form compiler.
Many applications would benefit from the support of H2 conforming elements such as Hermite, Morley, Argyris and so on, already available in FIAT.
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