Low-Complexity Finite Element Algorithms for the de Rham Complex on Simplices

Abstract

We combine recently developed finite element algorithms based on Bernstein polynomials [M. Ainsworth, G. Andriamaro, and O. Davydov, SIAM J. Sci. Comput., 33 (2011), pp. 3087--3109; R. C. Kirby, Numer. Math., 117 (2011), pp. 631--652] with the explicit basis construction of the finite element exterior calculus [D. N. Arnold, R. S. Falk, and R. Winther, Comput. Methods Appl. Mech. Engrg., 198 (2009), pp. 1660--1672] to give a family of algorithms for the de Rham complex on simplices that achieves stiffness matrix construction and matrix-free action in optimal complexity. These algorithms are based on realizing the exterior calculus bases as short combinations of Bernstein polynomials. Numerical results confirm optimal scaling of the algorithms and favorable comparison with FEniCS at high polynomial order as well. Additional empirical studies show that very high accuracy is achieved in the mixed discretization of the Poisson equation and the Maxwell eigenvalue problem as the polynomial degree increases.