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

Mark Ainsworth, Miangaly Gaelle Andriamaro, Oleg Davydov

Research output: Contribution to journalArticle

44 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

### Fingerprint

Polynomials
Finite Element
Bernstein Polynomials
Piecewise Polynomials
Arbitrary
Basis Functions
Nonlinear Problem
Standards

### Keywords

• finite elements
• optimal assembly
• polynomials

### Cite this

Ainsworth, Mark ; Andriamaro, Miangaly Gaelle ; Davydov, Oleg. / Bernstein–Bézier finite elements of arbitrary order and optimal assembly procedures. In: SIAM Journal on Scientific Computing. 2011 ; Vol. 33, No. 6. pp. 3087-3109.
@article{7e24f101a32c493a933c1fe4bddde938,
title = "Bernstein–B{\'e}zier finite elements of arbitrary order and optimal assembly procedures",
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.",
keywords = "finite elements , optimal assembly , polynomials",
author = "Mark Ainsworth and Andriamaro, {Miangaly Gaelle} and Oleg Davydov",
year = "2011",
doi = "10.1137/11082539X",
language = "English",
volume = "33",
pages = "3087--3109",
journal = "SIAM Journal on Scientific Computing",
issn = "1064-8275",
number = "6",

}

Bernstein–Bézier finite elements of arbitrary order and optimal assembly procedures. / Ainsworth, Mark; Andriamaro, Miangaly Gaelle; Davydov, Oleg.

In: SIAM Journal on Scientific Computing, Vol. 33, No. 6, 2011, p. 3087-3109.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Bernstein–Bézier finite elements of arbitrary order and optimal assembly procedures

AU - Ainsworth, Mark

AU - Andriamaro, Miangaly Gaelle

AU - Davydov, Oleg

PY - 2011

Y1 - 2011

N2 - 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.

AB - 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.

KW - finite elements

KW - optimal assembly

KW - polynomials

U2 - 10.1137/11082539X

DO - 10.1137/11082539X

M3 - Article

VL - 33

SP - 3087

EP - 3109

JO - SIAM Journal on Scientific Computing

JF - SIAM Journal on Scientific Computing

SN - 1064-8275

IS - 6

ER -