Diagonal-norm summation-by-parts element data

On an affine element in dimension $d\ge 1$, diagonal-norm SBP data consist of a nodal state vector with $N_K$ entries, a positive diagonal matrix $H_K\in {\mathbb{R}}^{N_K\times N_K}$, and derivative matrices $D_{K,\mathrm{\ell }}$, $1\le \mathrm{\ell }\le d$. Put $Q_{K,\mathrm{\ell }}=H_K{D}_{K,\mathrm{\ell }}$. For each face $F$ let $R_{K,F}$ select its face nodes from the volume nodes, let $M_{K,F}$ be a positive diagonal face quadrature matrix, and let $n_{K,F}$ be its constant outward unit normal. The data are SBP compatible when $D_{K,\mathrm{\ell }}\mathbf{1}=0,R_{K,F}\mathbf{1}={\mathbf{1}}_F,Q_{K,\mathrm{\ell }}+Q_{K,\mathrm{\ell }}^T={\sum }_{F\subset \mathrm{\partial }K}{n}_{K,F,\mathrm{\ell }}{R}_{K,F}^T{M}_{K,F}{R}_{K,F}$ for each coordinate $\mathrm{\ell }$. The rows of $R_{K,F}$ select the nodes used by the face quadrature; on an interior face, the two cells have the same face nodes and weights under the face identification and opposite normals. In dimension one, $R_L$ and $R_R$ select the left and right endpoint nodes with normals $-1$ and $+1$.

This is conditional algebraic data: the definition does not assert that every mesh has such a rule. The identities include exact affine metric factors and no curved-element metric approximation.