Hello everyone,
I am working with a transport equation of the form
\frac{\partial u(t,\boldsymbol{x})}{\partial t} + \theta(\boldsymbol{x}) \cdot \nabla_{x} u(t,\boldsymbol{x}) = 0
where \theta is a given vector field, and u is the unknown function.
To solve this equation, I would like to apply the Discontinuous Galerkin (DG) approach. In my variational formulation, the following term appears:
\begin{cases}
\sum_{K\in\mathcal{T}_{h}}\int_{\partial K^{in}}\theta\cdot n_{K}[u]v_{+}, & K\cap\partial\Omega^{in}=\emptyset,\\
\sum_{K\in\mathcal{T}_{h}}\int_{\partial K^{in}}\theta\cdot n_{K}u_{+}v_{+}, & K\cap\partial\Omega^{in}\neq\emptyset.
\end{cases}
Here, \mathcal{T}_{h} is the set of finite elements, K is a finite element (triangle or tetrahedron), and n_{K} is the outward unit normal vector on \partial K. The set \partial K^{in} is defined as
\partial K^{in}=\left\{ \boldsymbol{x}\in\partial K\ \middle|\ \theta(\boldsymbol{x})\cdot n_{K}(\boldsymbol{x})<0\right\},
and the jump and trace operators are given by
v_{+}(\boldsymbol{x})=\lim_{s\rightarrow0^{+}} v(\boldsymbol{x}+s\theta), \quad
u_{\pm}(\boldsymbol{x})=\lim_{s\rightarrow0^{\pm}} u(\boldsymbol{x}+s\theta),
[u]=u_{+}(\boldsymbol{x})-u_{-}(\boldsymbol{x}).
The inflow boundary subset \partial\Omega^{in} is defined as
\partial\Omega^{in}=\left\{ \boldsymbol{x}\in\partial\Omega\ \middle|\ \theta(\boldsymbol{x})\cdot n(\boldsymbol{x})<0\right\},
where n is the unit normal to \partial\Omega.
Of course, I am using a time discretization to solve this problem. However, implementing the above term in FEniCSx (0.9) has been quite challenging for me. Could you provide some guidance?