Legendre-Gauss-Lobatto nodes, weights, and tensor quadrature

Fix an integer $p\ge 1$. Let ${\xi }_0=-1$, ${\xi }_p=1$, and let ${\xi }_1<\cdots <{\xi }_{p-1}$ be the roots of $P_p^{\mathrm{\prime }}$ in $(-1,1)$; for $p=1$ this interior list is empty. By Legendre-Gauss-Lobatto nodes are distinct and interior critical points are simple these $p+1$ nodes are distinct. Define ${\mathrm{\ell }}_j(x)={\prod }_{0\le i\le p,i\mathrm{\ne }j}(x-{\xi }_i)\mathrm{/}({\xi }_j-{\xi }_i)$, set $w_j={\int }_{-1}^1{\mathrm{\ell }}_j(x)dx$, and define the Legendre–Gauss–Lobatto rule by $Q_p(f)={\sum }_{j=0}^p{w}_{j}f({\xi }_j)$.

On $[-1,1{]}^d$ its tensor rule is $Q_p^{\otimes d}(f)={\sum }_{j_1,\dots ,j_d=0}^p(w_{j_1}\cdots w_{j_d})f({\xi }_{j_1},\dots ,{\xi }_{j_d})$. For an affine box map $F_K(\xi )=b_K+J_K\xi$, define the transported rule by $Q_{K,p}(f)=\mathrm{\mid }\det J_K\mathrm{\mid }{Q}_p^{\otimes d}(f\circ F_K)$. The nodal basis on $K$ is the pullback of ${\prod }_{r=1}^d{\mathrm{\ell }}_{j_r}({\xi }_r)$. The local mass and stiffness forms are respectively $m_{K,p}(v,w)=Q_{K,p}(vw)$ and $a_{K,p}(v,w)=Q_{K,p}(\mathrm{\nabla }v\cdot \mathrm{\nabla }w)$; any sampled coefficient is evaluated at the same quadrature nodes. Global forms are assembled by summing the cell forms.