In a recent SIAM News article, there is a long article describing a systematic organization of the finite elements, aptly dubbed the Periodic Table of Finite Elements. Its really quite fascinating to see how classification can be accomplished via finite element exterior calculus. As the authors indicate:
Just as the arrangement of the chemical elements in a periodic table led to the discovery of new elements, the periodic table of finite elements has not only clarified existing elements but also highlighted holes in our knowledge and led to new families of finite elements suited for certain purposes.
The analogy fascinates me, and makes me wonder if it is possible to fill all the possible "holes" in the same way that missing material elements have been found. Perhaps this may be stretching the analogy too far, but I'm curious if all the possible "gaps" in finite elements have been fully explored and developed according to this finite element exterior calculus classification approach. If not, what are the more important "missing methods" that research is currently focused on developing and why? Furthermore, are there any finite element methods that cannot be classified by this approach (aside from the obvious omission of arbitrary shaped simplices...)?