$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
PDE problem
$$
\begin{align*}
\frac{\partial u}{\partial t} &= \nabla\cdot (\dfc(u)\nabla u) + f(u),\quad
&\x\in\Omega,\ t\in (0,T]
\\
-\dfc(u)\frac{\partial u}{\partial n} &= g,\quad &\x\in\partial\Omega_N,\
t\in (0,T]
\\
u &= u_0,\quad &\x\in\partial\Omega_D,\ t\in (0,T]
\end{align*}
$$
