Finite Element Exterior Calculus (FEEC) provides a cohomology framework for structure-preserving discretisation of a large class of PDEs. There has been a relatively mature FEEC theory with de Rham complexes for problems involving differential forms (skew-symmetric tensors) and vector fields. A canonical discretisation exists, which has a discrete topological interpretation and can be generalized to other discrete structures, e.g., graph cohomology.
In recent years, there has been significant interest in extending FEEC to tensor-valued problems with applications in continuum mechanics, differential geometry and general relativity etc. In this talk, we first review the de Rham sequences and their canonical discretisation with Whitney forms. Then we provide an overview of some efforts towards Finite Element Tensor Calculus (FETC). On the continuous level, we characterise tensors and differential structures using the Bernstein-Gelfand-Gelfand (BGG) machinery and incorporate analysis. On the discrete level in 2D and 3D, we discuss analogies of the Whitney forms and establish their cohomology. A special case is Christiansen’s finite element interpretation of Regge calculus, a discrete geometric scheme for metric and curvature. Moreover, we present a correspondence between BGG sequences, continuum mechanics with microstructure and Riemann-Cartan geometry. These efforts are in the direction of establishing a tensor calculus on triangulation and potentially on other discrete structures.