an encyclopedia of finite element definitions

Serendipity

 De Rham complex families $$\left[S_{1,k}^\square\right]_{0}$$ / $$\mathcal{S}_{k}\Lambda^{0}(\square_d)$$, $$\left[S_{2,k-d}^\square\right]_{0}$$ / $$\mathcal{S}^-_{k-d}\Lambda^{0}(\square_d)$$ Abbreviated names S Orders $$1\leqslant k$$ Reference elements interval, quadrilateral, hexahedron Polynomial set $$\mathcal{P}_{k} \oplus \mathcal{X}_{k}$$↓ Show polynomial set definitions ↓ DOFs On each vertex: point evaluations On each edge: integral moments with an order $$k-2$$ dPc space On each face: integral moments with an order $$k-4$$ dPc space On each volume: integral moments with an order $$k-6$$ dPc space Number of DOFs interval: $$k+1$$ (A000027)quadrilateral: $$\begin{cases}4&k=1\\k(k+3)/2+3&k>1\end{cases}$$ (A340266)hexahedron: $$\begin{cases}12k-4&k=1,2,3\\3k^2-3k+14&k=4,5\\k(k-1)(k+1)/6+k^2+5k+4&k>6\end{cases}$$ Mapping identity continuity Function values are continuous. Categories Scalar-valued elements

Implementations

 Basix basix.ElementFamily.serendipity, lagrange_variant=basix.LagrangeVariant.equispaced, dpc_variant=basix.DPCVariant.simplex_equispaced↓ Show Basix examples ↓ This implementation is correct for all the examples below. Basix.UFL basix.ElementFamily.serendipity, lagrange_variant=basix.LagrangeVariant.equispaced, dpc_variant=basix.DPCVariant.simplex_equispaced↓ Show Basix.UFL examples ↓ This implementation is correct for all the examples below. FIAT FIAT.Serendipity↓ Show FIAT examples ↓ This implementation is correct for all the examples below. Symfem "serendipity"↓ Show Symfem examples ↓ This implementation is used to compute the examples below and verify other implementations. (legacy) UFL "S"↓ 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) quadrilateralorder 1 (click to view basis functions) quadrilateralorder 2 (click to view basis functions) quadrilateralorder 3 (click to view basis functions)

DefElement stats

 Element added 02 January 2021 Element last updated 16 September 2023