SUPG stabilization of the steady advection-diffusion-reaction equation

Strong form

\[\begin{split} \begin{align*} &\text{Find}~u(\textbf{x}): \Omega \to \mathbb{R}~\text{such that} \\ &\mathbb{IBVP}_u\begin{cases} \textbf{a}\cdot\nabla u= \nabla\cdot(\mathsf{D}\cdot\nabla u) + Ru + J & \forall\textbf{x}\in\Omega \\ u=u_{\text{D}} & \forall \textbf{x}\in\partial\Omega_{\text{D}} \\ \textbf{n}\cdot(\mathsf{D}\cdot\nabla{u}) = u_{\text{N}} & \forall\textbf{x}\in\partial\Omega_{\text{N}}=\partial\Omega/\partial\Omega_{\text{D}} \end{cases} \\ &\text{given} \\ &\mathbb{S}_u\begin{cases} \Omega\subset\mathbb{R}^d & \text{domain} \\ u_{\text{D}}(\textbf{x})~,~\partial\Omega_{\text{D}} & \text{Dirichlet boundary condition}\\ u_{\text{N}}(\textbf{x})~,~\partial\Omega_{\text{N}} & \text{Neumann boundary condition}\\ \textbf{a}(\textbf{x}) & \text{velocity} \\ \mathsf{D}(\textbf{x}) & \text{dispersion} \\ R(\textbf{x}) & \text{reaction rate} \\ J(\textbf{x}) & \text{reaction source} \\ \end{cases} \end{align*} \end{split}\]

Weak form

\[\begin{split} \begin{align*} &\text{Find}~u\in V ~\text{such that} \\ &F(u,v)+F_{\text{SUPG}}(u,v)=0 \quad\forall v\in V \\ &\text{where}\\ &F_{\text{SUPG}}(u,v)=\tau P(v)\mathcal{R}(u)\\ &\mathcal{R}(u)= ... \end{align*} \end{split}\]