an encyclopedia of finite element definitions

Bernstein

 Alternative names Bernstein–Bézier Orders $$0\leqslant k$$ Reference elements interval, triangle, tetrahedron Polynomial set $$\mathcal{P}_{k}$$↓ Show polynomial set definitions ↓ DOFs On each vertex: point evaluations On each edge: evaluation of Bernstein coefficients On each face: evaluation of Bernstein coefficients On each volume: evaluation of Bernstein coefficients Number of DOFs interval: $$k+1$$ (A000027)triangle: $$(k+1)(k+2)/2$$ (A000217)tetrahedron: $$(k+1)(k+2)(k+3)/6$$ (A000292) Mapping identity continuity Function values are continuous. Categories Scalar-valued elements

Implementations

 FIAT FIAT.Bernstein↓ Show FIAT examples ↓ This implementation is correct for all the examples below. Symfem "Bernstein"↓ Show Symfem examples ↓ This implementation is used to compute the examples below and verify other implementations. (legacy) UFL "Bernstein"↓ Show (legacy) UFL examples ↓

Examples

 intervalorder 1 (click to view basis functions) intervalorder 2 (click to view basis functions) intervalorder 3 (click to view basis functions) triangleorder 1 (click to view basis functions) triangleorder 2 (click to view basis functions) triangleorder 3 (click to view basis functions)

References

• Ainsworth, Mark, Andriamaro, Gaelle, and Davydov, Oleg. Bernstein–Bézier finite elements of arbitrary order and optimal assembly procedures, SIAM Journal on Scientific Computing 33(6), 3087–3109, 2011. [DOI: 10.1137/11082539X] [BibTeX]
• Kirby, Robert C. Fast simplicial finite element algorithms using Bernstein polynomials, Numerische Mathematik 117(4), 631–652, 2011. [DOI: 10.1007/s00211-010-0327-2] [BibTeX]

DefElement stats

 Element added 20 February 2021 Element last updated 16 September 2023