$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Finite elements for a variable coefficient Laplace term
Consider the term \( (\dfc u^{\prime})^{\prime} \), with
the group finite element method: \( \dfc(u)\approx \sum_k\alpha(u_k)\basphi_k \),
and the variational counterpart
$$
\int_0^L \dfc(\sum_k c_k\basphi_k)\basphi_i^{\prime}\basphi_j^{\prime}\dx
\approx
\sum_k (\int_0^L \basphi_k\basphi_i^{\prime}\basphi_j^{\prime}\dx)
\dfc(u_k) = \ldots
$$
Further calculations (see text) lead to
$$
-\frac{1}{h}(\half(\dfc(u_i) + \dfc(u_{i+1}))(u_{i+1}-u_i)
- \half(\dfc(u_{i-1}) + \dfc(u_{i}))(u_{i}-u_{i-1}))
$$
= standard finite difference discretization
of \( -(\dfc(u)u^{\prime})^{\prime} \) with an arithmetic mean of \( \dfc(u) \)
