Here, the jump is defined as $[v] = v^+ - v^-$, where the superscripts refer
to the left ('+') and right ('-') values at the face. The upwind value
-$u^{\text{UP}}$ is defined to be $u^+$ if $\beta \cdot n>0$ and $u^-$ otherwise.
+$u^{\text{upwind}}$ is defined to be $u^+$ if $\beta \cdot n>0$ and $u^-$ otherwise.
As a result, the mesh-dependent weak form reads:
@f[
\sum_{T\in \mathbb T_h} -\bigl(\nabla \phi_i,{\mathbf \beta}\cdot \phi_j \bigr)_T +
-\sum_{F\in\mathbb F_h^i} \bigl< [\phi_i], \phi_i^{UP} \beta\cdot \mathbf n\bigr>_{F} +
+\sum_{F\in\mathbb F_h^i} \bigl< [\phi_i], \phi_i^{upwind} \beta\cdot \mathbf n\bigr>_{F} +
\bigl<\phi_i, \phi_j \beta\cdot \mathbf n\bigr>_{\Gamma_+}
= -\bigl<\phi_i, g \beta\cdot\mathbf n\bigr>_{\Gamma_-}.
@f]
template <int dim>
void AdvectionProblem<dim>::assemble_system()
{
- using ActiveCellIterator = typename DoFHandler<dim>::active_cell_iyerator;
- const BoundaryValues<dim> boundary_function;
+ using Iterator = typename DoFHandler<dim>::active_cell_iterator;
+ const BoundaryValues<dim> boundary_function;
// This is the function that will be executed for each cell.
auto cell_worker = [&](const Iterator & cell,