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 R^d, to be computed in optimal complexity O(n^2d). The algorithms (i) take account of numerical quadrature; (ii) are applicable to non-linear problems; and, (iii) do not rely on pre-computed arrays containing values of one-dimensional basis functions at quadrature points (although these can be used if desired). The elements are based on Bernstein-Bézier polynomials and are the first to achieve optimal complexity for the standard finite element spaces on simplicial elements.

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