Algorithms for polynomials in Bernstein form
Computer Aided Geometric Design
Spectral methods on triangles and other domains
Journal of Scientific Computing
Fundamentals of computer aided geometric design
Fundamentals of computer aided geometric design
ACM Transactions on Mathematical Software (TOMS)
Computer graphics (2nd ed. in C): principles and practice
Computer graphics (2nd ed. in C): principles and practice
Basis Functions for Triangular and Quadrilateral High-Order Elements
SIAM Journal on Scientific Computing
Curves and surfaces for CAGD: a practical guide
Curves and surfaces for CAGD: a practical guide
Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics (Scientific Computation)
Fast simplicial finite element algorithms using Bernstein polynomials
Numerische Mathematik
The Bernstein polynomial basis: A centennial retrospective
Computer Aided Geometric Design
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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.