Abstract

Powerful software projects such as FEniCS enable the efficient numerical solution of systems governed by partial differential equations via finite element methods. Historically, one critical aspect of enabling these simulations has been the specification of the local basis functions, which can be rather difficult when unstructured simplicial geometry is required. Thanks to the FIAT project, FEniCS supports a very general suite of possible basis functions. However, recent research on Bernstein polynomials, already widely used in approximation theory and computational geometry, presents a new approach to finite element basis functions. They have a remarkably simple and concise definition, generalize to any degree and spatial dimension, and possess remarkable structure that enables new optimal-complexity algorithms for a wide suite of finite element calculations