$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Calculation details of Newton's method at the PDE level
Insert \( u^{n,k} +\delta u \) for \( u^n \) in PDE:
$$
\frac{u^{n,k} +\delta u - u^{n-1}}{\Delta t} =
\nabla\cdot (\dfc(u^{n,k} + \delta u)\nabla (u^{n,k}+\delta u))
+ f(u^{n,k}+\delta u)
$$
Taylor expand \( \dfc(u^{n,k} + \delta u) \) and
\( f(u^{n,k}+\delta u) \):
$$
\begin{align*}
\dfc(u^{n,k} + \delta u) & = \dfc(u^{n,k}) + \frac{d\dfc}{du}(u^{n,k})
\delta u + \Oof{\delta u^2}\approx \dfc(u^{n,k}) + \dfc^{\prime}(u^{n,k})\delta u\\
f(u^{n,k}+\delta u) &= f(u^{n,k}) + \frac{df}{du}(u^{n,k})\delta u
+ \Oof{\delta u^2}\approx f(u^{n,k}) + f^{\prime}(u^{n,k})\delta u
\end{align*}
$$
