# Bernstein–Bézier finite elements of arbitrary order and optimal assembly procedures

Mark Ainsworth, Miangaly Gaelle Andriamaro, Oleg Davydov

Research output: Contribution to journalArticle

50 Citations (Scopus)

## Abstract

Algorithms are presented that enable the element matrices for the standard finite element space, consisting of continuous piecewise polynomials of degree $n$ on simplicial elements in $\mathbb{R}^d$, to be computed in optimal complexity $\mathcal{O}(n^{2d})$. The algorithms (i) take into account numerical quadrature; (ii) are applicable to nonlinear problems; and (iii) do not rely on precomputed arrays containing values of one-dimensional basis functions at quadrature points (although these can be used if desired). The elements are based on Bernstein polynomials and are the first to achieve optimal complexity for the standard finite element spaces on simplicial elements.
Original language English 3087-3109 23 SIAM Journal on Scientific Computing 33 6 1 Nov 2011 https://doi.org/10.1137/11082539X Published - 2011

## Keywords

• finite elements
• optimal assembly
• polynomials