$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\X}{\boldsymbol{X}}
\renewcommand{\u}{\boldsymbol{u}}
\renewcommand{\v}{\boldsymbol{v}}
\newcommand{\e}{\boldsymbol{e}}
\newcommand{\f}{\boldsymbol{f}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\Iz}{\mathcal{I}_z}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\Ifb}{{I_b}} % for FEM
\newcommand{\sequencei}[1]{\left\{ {#1}_i \right\}_{i\in\If}}
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\psib}{\boldsymbol{\psi}}
\newcommand{\sinL}[1]{\sin\left((#1+1)\pi\frac{x}{L}\right)}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\Xno}[1]{X_{(#1)}}
\newcommand{\xdno}[1]{\boldsymbol{x}_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Variable coefficient; problem
$$
\begin{equation*}
-\frac{d}{dx}\left( \dfc(x)\frac{du}{dx}\right) = f(x),\quad x\in\Omega =[0,L],\
u(0)=C,\ u(L)=D
\end{equation*}
$$
- Variable coefficient \( \dfc(x) \)
- \( V = \hbox{span}\{\baspsi_0,\ldots,\baspsi_N\} \)
- Nonzero Dirichlet conditions at \( x=0 \) and \( x=L \)
- Must have \( \baspsi_i(0)=\baspsi_i(L)=0 \)
- Any \( v\in V \) has then \( v(0)=v(L)=0 \)
- \( B(x) = C + \frac{1}{L}(D-C)x \)
$$
u(x) = B(x) + \sum_{j\in\If} c_j\baspsi_i(x),\quad
$$
