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)


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 languageEnglish
Pages (from-to)3087-3109
Number of pages23
JournalSIAM Journal on Scientific Computing
Issue number6
Early online date1 Nov 2011
Publication statusPublished - 2011


  • finite elements
  • optimal assembly
  • polynomials

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