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# Lagrange

 Alternative names Polynomial, Galerkin, DGT (facets), Hdiv trace (facets), Q (quadrilateral and hexahedron), Gauss–Lobatto–Legendre (GLL variant), Lobatto (Lobatto variant) De Rham complex families $$\left[S_{2,k}^\unicode{0x25FA}\right]_{0}$$ / $$\mathcal{P}^-_{k}\Lambda^{0}(\Delta_d)$$, $$\left[S_{1,k}^\unicode{0x25FA}\right]_{0}$$ / $$\mathcal{P}_{k}\Lambda^{0}(\Delta_d)$$, $$\left[S_{4,k}^\square\right]_{0}$$ / $$\mathcal{Q}^-_{k}\Lambda^{0}(\square_d)$$, $$\left[S_{2,k}^\unicode{0x25FA}\right]_{d}$$ / $$\mathcal{P}^-_{k}\Lambda^{d}(\Delta_d)$$, $$\left[S_{1,k}^\unicode{0x25FA}\right]_{d}$$ / $$\mathcal{P}_{k}\Lambda^{d}(\Delta_d)$$, $$\left[S_{4,k}^\square\right]_{d}$$ / $$\mathcal{Q}^-_{k}\Lambda^{d}(\square_d)$$, $$\left[S_{3,k}^\square\right]_{d}$$ Abbreviated names P, CG, DG, GLL (GLL variant) Variants equispaced: The variant has its point evaluations at equally spaced points.GLL: This variant has its point evaluations at GLL points.Lobatto: This variant uses integrals against L2 duals of Lobatto polynomials in the place of point evaluations Orders $$1\leqslant k$$ Reference elements interval, triangle, tetrahedron, quadrilateral, hexahedron, prism, pyramid Polynomial set $$\mathcal{P}_{k}$$ (interval, triangle, tetrahedron) $$\mathcal{Q}_{k}$$ (quadrilateral, hexahedron) $$\mathcal{Z}^{(15)}_{k}$$ (prism) $$\mathcal{P}_{k} \oplus \mathcal{Z}^{(16)}_{k}$$ (pyramid) ↓ Show polynomial set definitions ↓ DOFs On each vertex: point evaluations On each edge: point evaluations On each face: point evaluations On each volume: point evaluations Number of DOFs interval: $$k+1$$ (A000027)triangle: $$(k+1)(k+2)/2$$ (A000217)tetrahedron: $$(k+1)(k+2)(k+3)/6$$ (A000292)quadrilateral: $$(k+1)^2$$ (A000290)hexahedron: $$(k+1)^3$$ (A000578)prism: $$(k+1)^2(k+2)/2$$ (A002411)pyramid: $$(k+1)(k+2)(2k+3)/6$$ (A000330) Number of DOFs on subentities vertices: $$1$$ (A000012)edges: $$k-1$$ (A000027)faces: $$(k-1)(k-2)/2$$ (A000217) (triangle), $$(k-1)^2$$ (A000290) (quadrilateral)volumes: $$(k-1)(k-2)(k-3)/6$$ (A000292) (tetrahedron), $$(k-1)^3$$ (A000578) (hexahedron), $$(k-1)^2(k-2)/2$$ (A002411) (prism), $$(k-1)(k-2)(2k-3)/6$$ (A000330) (pyramid) Mapping identity continuity Function values are continuous. Notes DGT and Hdiv trace are names given to this element when it is defined on the facets of a mesh. For the Lobatto variant, the derivatives of most of the basis functions are orthogonal. Categories Scalar-valued elements

## Implementations

 Basix basix.ElementFamily.P↓ Show Basix examples ↓ This implementation is correct for all the examples below that it supports.↓ Show more ↓ Basix.UFL basix.ElementFamily.P↓ Show Basix.UFL examples ↓ This implementation is correct for all the examples below that it supports.↓ Show more ↓ Bempp "P"↓ Show Bempp examples ↓ FIAT FIAT.Lagrange↓ Show FIAT examples ↓ This implementation is correct for all the examples below that it supports.↓ Show more ↓ Symfem "Lagrange" (interval, equispaced; triangle, equispaced; tetrahedron, equispaced; prism, equispaced; pyramid, equispaced)"Q" (quadrilateral, equispaced; hexahedron, equispaced)"Lagrange", variant="gll" (interval, GLL)"Q", variant="gll" (quadrilateral, GLL; hexahedron, GLL)"Lagrange", variant="lobatto" (interval, Lobatto)"Q", variant="lobatto" (quadrilateral, Lobatto; hexahedron, Lobatto)↓ Show Symfem examples ↓ This implementation is used to compute the examples below and verify other implementations. (legacy) UFL "Lagrange" (interval, equispaced; triangle, equispaced; tetrahedron, equispaced)"Q" (quadrilateral, equispaced; hexahedron, equispaced)"Lobatto" (interval, Lobatto)↓ Show (legacy) UFL examples ↓

## Examples

 intervalorder 1equispaced variant (click to view basis functions) intervalorder 2equispaced variant (click to view basis functions) intervalorder 3equispaced variant (click to view basis functions) triangleorder 1equispaced variant (click to view basis functions) triangleorder 2equispaced variant (click to view basis functions) triangleorder 3equispaced variant (click to view basis functions) quadrilateralorder 1equispaced variant (click to view basis functions) quadrilateralorder 2equispaced variant (click to view basis functions) quadrilateralorder 3equispaced variant (click to view basis functions) tetrahedronorder 1equispaced variant (click to view basis functions) tetrahedronorder 2equispaced variant (click to view basis functions) hexahedronorder 1equispaced variant (click to view basis functions) hexahedronorder 2equispaced variant (click to view basis functions) prismorder 1equispaced variant (click to view basis functions) prismorder 2equispaced variant (click to view basis functions) pyramidorder 1equispaced variant (click to view basis functions) pyramidorder 2equispaced variant (click to view basis functions) intervalorder 1GLL variant (click to view basis functions) intervalorder 2GLL variant (click to view basis functions) intervalorder 3GLL variant (click to view basis functions) intervalorder 4GLL variant (click to view basis functions) quadrilateralorder 1GLL variant (click to view basis functions) quadrilateralorder 2GLL variant (click to view basis functions) intervalorder 1Lobatto variant (click to view basis functions) intervalorder 2Lobatto variant (click to view basis functions) intervalorder 3Lobatto variant (click to view basis functions) quadrilateralorder 1Lobatto variant (click to view basis functions) quadrilateralorder 2Lobatto variant (click to view basis functions) quadrilateralorder 3Lobatto variant (click to view basis functions) hexahedronorder 1Lobatto variant (click to view basis functions) hexahedronorder 2Lobatto variant (click to view basis functions)

## DefElement stats

 Element added 16 January 2021 Element last updated 14 June 2024