$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Nonlinear algebraic equations arising from the variational form
$$
\int_\Omega (uv + \Delta t\,\dfc(u)\nabla u\cdot\nabla v
- \Delta t f(u)v - u^{(1)} v)\dx = 0
$$
$$
F_i =
\int_\Omega (u\baspsi_i + \Delta t\,\dfc(u)\nabla u\cdot\nabla \baspsi_i
- \Delta t f(u)\baspsi_i - u^{(1)}\baspsi_i)\dx = 0
$$
Picard iteration:
$$
F_i \approx \hat F_i =
\int_\Omega (u\baspsi_i + \Delta t\,\dfc(u^{-})\nabla u\cdot\nabla \baspsi_i
- \Delta t f(u^{-})\baspsi_i - u^{(1)}\baspsi_i)\dx = 0
$$
This is a variable coefficient problem like \( au - \nabla\cdot\dfc(\x)\nabla u
= f(\x,t) \) and results in a linear system
