Fast inversion of the simplicial Bernstein mass matrix

Abstract

We consider the mass matrix arising from Bernstein polynomials as finite element shape functions. In particular, we give an explicit formula for its eigenvalues and exact characterization of the eigenspaces in terms of the Bernstein representation of orthogonal polynomials. We then derive a fast algorithm for solving linear systems involving the element mass matrix. After establishing these results, we describe the application of Bernstein techniques to the discontinuous Galerkin finite element method for hyperbolic conservation laws, obtaining optimal complexity algorithms. Finally, we present numerical results investigating the accuracy of the mass inversion algorithms and the scaling of total run-time for the function evaluation needed in DG time-stepping.