$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Finite difference scheme
$$ \dfc_{i+\half} \approx
\half(\dfc(u_i) + \dfc(u_{i+1})) = [\overline{\dfc(u)}^x]^{i+\half} $$
results in
$$ [-D_x\overline{\dfc}^x D_x u +au = f]_i\tp$$
$$
\begin{align*}
&-\frac{1}{2\Delta x^2}
\left((\dfc(u_i)+\dfc(u_{i+1}))(u_{i+1}-u_i) -
(\dfc(u_{i-1})+\dfc(u_{i}))(u_{i}-u_{i-1})\right)\\
&\qquad\qquad + au_i = f(u_i)
\end{align*}
$$
