Oriented interval meshes and broken polynomial spaces

Let $a<b$ and choose an integer $N\ge 1$ and points $a=x_0<x_1<\cdots <x_N=b\mathrm{.}$ The oriented interval mesh is the ordered family of cells $I_j=(x_{j-1},x_j)$, $1\le j\le N$, each oriented from left to right. Write $h_j=x_j-x_{j-1}>0$. For an integer $p\ge 0$, the broken polynomial space is $V_h^{\mathrm{b}\mathrm{r}}={\prod }_{j=1}^N{\mathbb{P}}_p(I_j),$ with functions interpreted cell by cell. At an interior node $x_j$, write $w_j^{-}=w{\mathrm{\mid }}_{I_j}(x_j)$ for the trace from the left cell and $w_j^{+}=w{\mathrm{\mid }}_{I_{j+1}}(x_j)$ for the trace from the right cell; the chosen face normal points from left to right, and the jump is $[w{]}_j=w_j^{+}-w_j^{-}$. At $a$ and $b$, the outward normals are $-1$ and $+1$, respectively.

The published matching-affine-simplicial-mesh convention used elsewhere on the A page has dimension $d\ge 2$. This definition supplies the distinct one-dimensional interval and trace convention; it makes no assertion about multidimensional cells.