Jump and average conventions for symmetric interior-penalty DG

Let $d\ge 2$ and let $\mathcal{T}$ be a finite matching affine simplicial mesh of a bounded polyhedral open set $\mathrm{\Omega }\subset {\mathbb{R}}^d$, with shape-regularity bound $\sigma$, as in Matching affine simplicial meshes and shape regularity. Write ${\mathcal{F}}_h^{\mathrm{i}\mathrm{n}\mathrm{t}}$ for its interior facets and ${\mathcal{F}}_h^{\mathrm{b}\mathrm{d}\mathrm{y}}$ for its boundary facets, and put ${\mathcal{F}}_h={\mathcal{F}}_h^{\mathrm{i}\mathrm{n}\mathrm{t}}\cup {\mathcal{F}}_h^{\mathrm{b}\mathrm{d}\mathrm{y}}$. Fix a degree $k\in {\mathbb{N}}_0$ for this broken space. The trial functions below belong to the broken scalar polynomial space $V_h^{\mathrm{b}\mathrm{r}}={\prod }_{K\in \mathcal{T}}{\mathbb{P}}_k(K;\mathbb{R})$ from Broken and conforming polynomial spaces on a matching mesh.

For each interior facet $F=K^{-}\cap K^{+}$, choose a unit normal $n_F$ pointing from $K^{-}$ to $K^{+}$. For a scalar or vector field $g$ with traces from both cells, define $[g{]}_F:=g^{-}-g^{+},\{g{\}}_F:=\frac{1}{2}(g^{-}+g^{+})\mathrm{.}$ For a boundary facet $F\subset \mathrm{\partial }K\cap \mathrm{\partial }\mathrm{\Omega }$, let $n_F=n_K$ be the outward unit normal and define $[g{]}_F:=g{\mathrm{\mid }}_K$ and $\{g{\}}_F:=g{\mathrm{\mid }}_K$. Reversing the chosen interior normal also reverses $[g{]}_F$, so the scalar jump/average contractions used in the form are independent of that choice.

For one-dimensional interval examples, use the following separate extension of the same formulas. At an interior point shared by a left cell $K^{-}$ and right cell $K^{+}$, orient from left to right and set $[g{]}_F=g^{-}-g^{+}$ and $\{g{\}}_F=(g^{-}+g^{+})\mathrm{/}2$. At either endpoint use the one-sided trace for both jump and average, and use counting measure on the zero-dimensional face. This extension is not an instance of the published $d\ge 2$ mesh definition above.

The following diagram records the interior orientation; the formulas above are the definitions, and the diagram is only a convention aid.

This is the scalar-jump convention: on an interior facet the corresponding vector jump is $[v{]}_F{n}_F$. On a boundary facet the jump is the trace itself; this is the convention that makes the penalty and consistency terms impose homogeneous Dirichlet data weakly. A normal-flux contraction with the scalar jump is unchanged when the interior orientation is reversed.