$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Galerkin-type discretizations
- \( V \): function space with basis functions \( \baspsi_i(x) \), \( i\in\If \)
- Dirichlet conditionat \( x=L \): \( \baspsi_i(L)=0 \), \( i\in\If \) (\( v(L)=0\ \forall v\in V \))
- \( u = D + \sum_{j\in\If}c_j\baspsi_j \)
Galerkin's method for \( -(\dfc(u)u')' + au = f(u) \):
$$
\int_0^L \dfc(u)u^{\prime}v^{\prime}\dx + \int_0^L auv\dx =
\int_0^L f(u)v\dx + [\dfc(u)u^{\prime}v]_0^L,\quad \forall v\in V
$$
Insert Neumann condition:
$$ [\dfc(u)u^{\prime}v]_0^L = \dfc(u(L))u^{\prime}(L)v(L) - \dfc(u(0))u^{\prime}(0)v(0)
= -Cv(0)
$$
