Maxwell’s equations have a close connection to the de Rham complex, and the perspective of continuous and discrete differential forms has inspired key progress in computational electromagnetism. The complex point of view also plays an important role in, e.g., continuum theory of defects, intrinsic elasticity and relativity.

In this talk, we briefly review the Finite Element Exterior Calculus for the de Rham complex developed by Arnold, Falk and Winther among others. Then we generate new complexes from the de Rham complexes and study their algebraic and analytic properties. As an example, we construct Sobolev and finite element elasticity complexes (referred to as the Kröner complex in mechanics and the linearized Calabi complex in geometry) and generalize various results in classical elasticity from this cohomological approach, e.g., the Korn inequality and the Cesàro-Volterra path integral.