Module polynomials 

Orthonormal polynomials.

The orthonormal polynomials in ndelement span the degree $k$ natural polynomial (or rationomial for a pyramid) space on a cell. As given at https://defelement.org/ciarlet.html#The+degree+of+a+finite+element, these natural spaces are defined for an interval, triangle, quadrilateral, tetrahedron, hexahedron, triangular prism and square-based pyramid by

• $\mathbb{P}^{\text{interval}}_k=\operatorname{span}\left{x^{p_0},\middle|,p_0\in\mathbb{N},,p_0\leqslant k\right},$
• $\mathbb{P}^{\text{triangle}}_k=\operatorname{span}\left{x^{p_0}y^{p_1},\middle|,p_0,p_1\in\mathbb{N}_0,,p_0+p_1\leqslant k\right},$
• $\mathbb{P}^{\text{quadrilateral}}_k=\operatorname{span}\left{x^{p_0}y^{p_1},\middle|,p_0,p_1\in\mathbb{N}_0,,p_0\leqslant k,,p_1\leqslant k\right},$
• $\mathbb{P}^{\text{tetrahedron}}_k=\operatorname{span}\left{x^{p_0}y^{p_1}z^{p_2},\middle|,p_0,p_1,p_2\in\mathbb{N}_0,,p_0+p_1+p_2\leqslant k\right},$
• $\mathbb{P}^{\text{hexahedron}}_k=\operatorname{span}\left{x^{p_0}y^{p_1}z^{p_2},\middle|,p_0,p_1,p_2\in\mathbb{N}_0,,p_0\leqslant k,,p_1\leqslant k,,p_2\leqslant k\right},$
• $\mathbb{P}^{\text{prism}}_k=\operatorname{span}\left{x^{p_0}y^{p_1}z^{p_2},\middle|,p_0,p_1,p_2\in\mathbb{N}_0,,p_0+p_1\leqslant k,,p_2\leqslant k\right},$
• $\mathbb{P}^{\text{pyramid}}_k=\operatorname{span}\left{\frac{x^{p_0}y^{p_1}z^{p_2}}{(1-z)^{p_0+p_1}},\middle|,p_0,p_1,p_2\in\mathbb{N}_0,,p_0\leqslant k,,p_1\leqslant k,,p_2\leqslant k\right}.$

Note that for non-pyramid cells, these coincide with the polynomial space for the degree $k$ Lagrange element on the cell.

Functions

derivative_count

Return the total number of partial derivatives up to a given degree.

legendre_shape

The shape of a table containing the values of Legendre polynomials

polynomial_count

Return the number of polynomials in the natural polynomial set for a given cell type and degree.

tabulate_legendre_polynomials

Tabulate orthonormal polynomials