$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Summary of finite element vs finite difference nonlinear algebraic equations
$$ -(\dfc(u)u^{\prime})^{\prime} +au = f(u)$$
Uniform P1 finite elements:
- Group finite element or Trapezoidal integration at nodes:
\( -(\dfc(u)u^{\prime})^{\prime} \) becomes \( -h[D_x\overline{\dfc(u)}^xD_x u]_i \)
- \( f(u) \) becomes \( hf(u_i) \) with Trapezoidal integration
or the "mass matrix" representation \( h[f(u) - \frac{h}{6}D_xD_x f(u)]_i \)
if group finite elements
- \( au \) leads to the "mass matrix" form \( ah[u - \frac{h}{6}D_xD_x u]_i \)
