The Bernstein-B\'ezier framework for polynomial vector splines and linear differential operators on spaces of multivariate (vector) splines serves as a new tool for studying finite element complexes in the numerical solutions of partial differential equations. The results on dimensions and minimal determining sets that follow from new Bernstein-B\'ezier techniques lead to direct constructions of various finite elements. The stable computational algorithms based on Bernstein-B\'ezier tools result in faster and more stable computations for existing finite elements, as well as in computer-aided geometric design related applications.
We will also adress a significant challenge in this work: the phenomenon of intrinsic supersmoothness -- unexpected additional smoothness in the images of differential operators on splines.