Advection-diffusion equation

Strong form

\[\begin{split} \begin{align*} &\text{Find}~u(\textbf{x}, t): \Omega\times[0,\infty) \to \mathbb{R}~\text{such that } \\ &\mathbb{IBVP}\begin{cases} \frac{\partial u}{\partial t} + \textbf{a}\cdot\nabla u= \nabla\cdot(\mathsf{D}\cdot\nabla u) & \forall(\textbf{x}, t)\in\Omega\times[0,\infty) \\ u=u_0 & \forall(\textbf{x},t)\in\Omega\times\{0\}\\ u=u_{\text{D}} & \forall(\textbf{x},t)\in\partial\Omega_{\text{D}}\times[0,\infty) \\ \textbf{n}\cdot(\mathsf{D}\cdot\nabla{u}) = u_{\text{N}} & \forall(\textbf{x},t)\in\partial\Omega_{\text{N}}\times[0,\infty)~,~\partial\Omega_{\text{N}}=\partial\Omega/\partial\Omega_{\text{D}} \end{cases}~. \end{align*} \end{split}\]

Time discretization

\[\frac{u^{n+1} - u^n}{\Delta t^n} + \mathcal{D}_{{\textbf{a}},u}(\textbf{a}\cdot\nabla u) = \nabla\cdot(\mathcal{D}_{{\mathsf{D}},u}(\mathsf{D}\cdot\nabla u))\]

Weak form

\[\begin{split} \begin{aligned} &\text{Find}~u^{n+1}\in V~\text{such that } \\ &\begin{align*} F(u^{n+1},v)&=\int_\Omega\text{d}\Omega~v\frac{u^{n+1} - u^n}{\Delta t^n} + v\mathcal{D}_{{\textbf{a}},u}(\textbf{a}\cdot\nabla u) \\ &\qquad\quad + \nabla v\cdot\mathcal{D}_{{\mathsf{D}},u}(\mathsf{D}\cdot\nabla u) \\ &\quad - \int_{\partial\Omega_{\text{N}}}\text{d}\Gamma~vu_{\text{N}}=0 \quad\forall v\in V~. \end{align*} \end{aligned} \end{split}\]

Specification

\[\begin{split}\mathbb{S}\begin{cases} \Omega \\ u_0(\textbf{x}) \\ u_{\text{D}}(\textbf{x}, t)~,~\partial\Omega_{\text{D}} \\ u_{\text{N}}(\textbf{x}, t)~,~\partial\Omega_{\text{N}} \\ \textbf{a}(\textbf{x}, t) \\ \mathsf{D}(\textbf{x}, t) \\ \end{cases} \end{split}\]