Finite Element Exterior Calculus (FEEC) is a powerful framework for developing stable discretizations of PDEs. In problems related to the de Rham complex, FEEC has played a critical role in the development of stable finite elements. However, in contrast to the plethora of structure-preserving discretizations available for the de Rham complex, there are two areas of theoretical and practical interest which remain relatively under-explored: (1) adaptive FEEC-based discretizations, and (2) applications to higher-order differential complexes related to problems in continuum mechanics, relativity, etc.
We will present recent results on both topics. We discuss the development of adaptive, structure-preserving discretizations of the de Rham complex. Then, leveraging the Bernstein-Gelfand-Gelfand construction, we will discuss extensions which apply to general high-order complexes. In particular, we will discuss the elasticity and the div-div complexes as examples.