$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Simplified problem for symbolic calculations
Simple nonlinear problem: \( -u^{\prime\prime}=u^2 \),
\( u'(0)=1 \), \( u'(L)=0 \).
$$ \int_0^L u^{\prime}v^{\prime}\dx = \int_0^L u^2v\dx
- v(0),\quad\forall v\in V$$
Now,
- Focus on \( \int u^2v\dx \)
- Set \( c_j = u(\xno{j}) = u_j \)
(to mimic finite difference interpretation of \( u_j \))
- That is, \( u = \sum_ju_j\basphi_j \)
