energy $E$.

We follow the discussion in (\[sub:Discrete-approximation-of\]) and approximate $$\lambda_{E}\left(\xi\right)\approx\sum_{j=1}^{m}p_{j}\delta\left(\xi-\xi_{j}\right).\label{eq:discrete approx of Young measure, Elsasser}$$ For this example, $p_{j}$ and $\xi$ are independent of $E$. In Figure \[fig:young-measure\], we compare the Young measure (\[eq:analytic Young measure\]) associated with (\[eq:Elsasser band model, intro\]) against its discrete approximation (\[eq:discrete approx of Young measure, Elsasser\]); in this example, we take $\beta=1$ and, in (\[eq:discrete approx of Young measure, Elsasser\]), $m=30$ equispaced values $\sigma_{j}$ between $1$ and $c_{\beta}$, $\beta=1$. Note that the error in the discrete approximation (\[eq:discrete approx of Young measure, Elsasser\]) necessarily rises at the end points, since the density $d\lambda_{E}\left(\xi\right)$ is infinite at $\sigma=1$ and $\sigma=c_{\beta}$.

Using (\[eq:discrete approx of Young measure, Elsasser\]) in (\[eq:mu &gt; 0, homogenized\])-(\[eq:mu &lt; 0, homogenized\]), we then obtain the (approximate) homogenized solution $\psi_{0}\left(x,\mu,E\right)$. In Figure \[fig:transport comparison, Elsasser\], we compare the “exact” scalar flux (\[eq:discrete scalar flux, exact\]) against its homogenized version (\[eq:discrete scalar flux, approx\]), where $\psi_{0}$ is computed from (\[eq:discrete approx of Young measure, Elsasser\]) and (\[eq:mu &gt; 0, homogenized\])-(\[eq:mu &lt; 0, homogenized\]). We use $m=30$ equispaced values $\sigma_{j}$ between $1$ and $c_{\beta}$, $\beta=1$, and a single group $\left[E_{0},E_{1}\right]=\left[0,1\right]$.

![Comparison of the Young measure (\[eq:analytic Young measure\]) and its discrete approximation (\[eq:discrete 