$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Integrating very simple nonlinear functions results in complicated expressions in the finite element method
Consider \( \int u^2v\dx \) with \( u = \sum_ku_k\basphi_k \) and \( v=\basphi_i \):
$$ \int_0^L (\sum_ku_k\basphi_k)^2\basphi_i\dx$$
Tedious exact evaluation on uniform P1 elements:
$$ \frac{h}{12}(u_{i-1}^2 + 2u_i(u_{i-1} + u_{i+1}) + 6u_i^2
+ u_{i+1}^2)$$
Finite difference counterpart: \( u_i^2 \) (!)
