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# Brezzi–Douglas–Marini

 De Rham complex families $$\left[S_{1,k}^\unicode{0x25FA}\right]_{d-1}$$ / $$\mathcal{P}_{k}\Lambda^{d-1}(\Delta_d)$$ Abbreviated names BDM Variants Legendre: Integral moments are taken against orthonormal polynomialsLagrange: Integral moments are taken against (Lagrange)[element:lagrange] basis functions Orders $$1\leqslant k$$ Reference elements triangle, tetrahedron Polynomial set $$\mathcal{P}_{k}^d$$↓ Show polynomial set definitions ↓ DOFs On 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 DOFs triangle: $$(k+1)(k+2)$$ (A002378)tetrahedron: $$(k+1)(k+2)(k+3)/2$$ (A027480) Mapping contravariant Piola continuity Components normal to facets are continuous Categories Vector-valued elements, H(div) conforming elements

## Implementations

 Basix basix.ElementFamily.BDM↓ Show Basix examples ↓ This implementation is correct for all the examples below. Basix.UFL basix.ElementFamily.BDM↓ Show Basix.UFL examples ↓ This implementation is correct for all the examples below. FIAT FIAT.BrezziDouglasMarini(..., variant="integral")↓ Show FIAT examples ↓ This implementation is correct for all the examples below that it supports.↓ Show more ↓ 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

 triangleorder 1Lagrange variant (click to view basis functions) triangleorder 2Lagrange variant (click to view basis functions) tetrahedronorder 1Lagrange variant (click to view basis functions) tetrahedronorder 2Lagrange variant (click to view basis functions) triangleorder 1Legendre variant (click to view basis functions) triangleorder 2Legendre variant (click to view basis functions) tetrahedronorder 1Legendre variant (click to view basis functions) tetrahedronorder 2Legendre variant (click to view basis functions)

## DefElement stats

 Element added 30 December 2020 Element last updated 15 December 2023