Discrete tensor product BGG sequences: Splines and finite elements

Abstract

In this paper, we provide a systematic discretization of the Bernstein-Gelfand-Gelfand diagrams and complexes over cubical meshes in arbitrary dimension via the use of tensor product structures of one-dimensional piecewise-polynomial spaces, such as spline and finite element spaces. We demonstrate the construction of the Hessian, the elasticity, and div ⁡ div \operatorname {div}\operatorname {div} complexes as examples for our construction.