$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
A note on our notation and the different meanings of \( u \) (1)
PDE problem: \( u(\x,t) \) is the exact solution of
$$
u_t = \nabla\cdot(\dfc(u)\nabla u) + f(u)
$$
Time discretization: \( u(\x) \) is the exact solution of the time-discrete
spatial equation
$$ u - \Delta t\nabla\cdot(\dfc(u^n)\nabla u) - \Delta t f(u) = u^{(1)}$$
The same \( u(\x) \) is the exact solution of the
(continuous) variational form:
$$
\int_\Omega (uv + \Delta t\,\dfc(u)\nabla u\cdot\nabla v
- \Delta t f(u)v - u^{(1)} v)\dx,\quad\forall v\in V
$$
