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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...)?

Paul
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Disclaimer: I don't actually work in this field (just find it interesting), so I could be misunderstanding some of the ideas. Apologies if this happens, and please correct me if you see a mistake.

One side note - finite elements aren't quite equivalent to finite element methods. These are elements defined by Ciarlet - finite dimensional approximation spaces with degrees of freedom defined as linear functionals for space. Finite element methods can be a much broader set of discretizations (for example, stabilizations of the weak form, discrete tricks, etc).

Doug Arnold has a good sampling of current work in the vein of the SIAM table. One neat bit was the extension of this idea to the serendipity group of finite elements, which allowed him to generate a new family of 3D serendipity finite elements. Annalisa Buffa has also fit B-spline discretizations into this framework of differential forms.

Many of the above ideas involve a finite-dimensional reproduction of a De Rham complex to form "compatible discretizations" (the stability of mixed finite elements is tied to the general idea of compatibility in discretizations). Compatibility is also present in Maxwells and curl-curl problems, where this gives stability of the method and an accurate reproduction of the operator spectrum. Outside of FEM, mimetic finite difference methods seem to be related to these concepts as well (though they're closely related to mixed FEM methods, so I'm not sure how special this is).

More recently, Arnold came up with finite elements for elasticity based on a separate "elasticity" complex, and John Evans reproduced this idea for Stokes, defining a basis for incompressible flow problems based on a "Stokes complex". If the full problem including the divergence free condition is discretized, the resulting discretization can be shown to be pointwise (not weakly) divergence free. Gerritsma and Hiemstra argue you can use the same geometric ideas to construct high order discretizations satisfying exact conservation properties for a variety of conservation laws.

TL;DR - for the periodic table of FEM: exotic and nontraditional elements? For the idea of grouping FEM into families: compatible discretizations and geometric modeling of physics?

Jesse Chan
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