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Brezzi–Douglas–Marini

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De Rham complex families\(\left[S_{1,k}^\unicode{0x25FA}\right]_{d-1}\) / \(\mathcal{P}_{k}\Lambda^{d-1}(\Delta_d)\)
Abbreviated namesBDM
VariantsLegendre: Integral moments are taken against orthonormal polynomials
Lagrange: Integral moments are taken against (Lagrange)[element:lagrange] basis functions
Orders\(1\leqslant k\)
Reference elementstriangle, tetrahedron
Polynomial set\(\mathcal{P}_{k}^d\)
↓ Show polynomial set definitions ↓
DOFsOn each facet: normal integral moments with an order \(k\) Lagrange space
On the interior of the reference element: integral moments with an order \(k-1\) Nédélec (first kind) space
Number of DOFstriangle: \((k+1)(k+2)\) (A002378)
tetrahedron: \((k+1)(k+2)(k+3)/2\) (A027480)
Mappingcontravariant Piola
continuityComponents normal to facets are continuous
CategoriesVector-valued elements, H(div) conforming elements

Implementations

Basixbasix.ElementFamily.BDM
↓ Show Basix examples ↓ This implementation is correct for all the examples below.
Basix.UFLbasix.ElementFamily.BDM
↓ Show Basix.UFL examples ↓ This implementation is correct for all the examples below.
FIATFIAT.BrezziDouglasMarini(..., variant="integral")
↓ Show FIAT examples ↓ This implementation is correct for all the examples below that it supports.
Symfem"N2div", variant="legendre" (Legendre)
"N2div" (Lagrange)
↓ Show Symfem examples ↓ This implementation is used to compute the examples below and verify other implementations.
(legacy) UFL"BDM"
↓ Show (legacy) UFL examples ↓

Examples

triangle
order 1
Lagrange variant

(click to view basis functions)
triangle
order 2
Lagrange variant

(click to view basis functions)
tetrahedron
order 1
Lagrange variant

(click to view basis functions)
tetrahedron
order 2
Lagrange variant

(click to view basis functions)
triangle
order 1
Legendre variant

(click to view basis functions)
triangle
order 2
Legendre variant

(click to view basis functions)
tetrahedron
order 1
Legendre variant

(click to view basis functions)
tetrahedron
order 2
Legendre variant

(click to view basis functions)

References

DefElement stats

Element added30 December 2020
Element last updated15 December 2023