$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\X}{\boldsymbol{X}}
\renewcommand{\u}{\boldsymbol{u}}
\renewcommand{\v}{\boldsymbol{v}}
\newcommand{\e}{\boldsymbol{e}}
\newcommand{\f}{\boldsymbol{f}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\Iz}{\mathcal{I}_z}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\Ifb}{{I_b}} % for FEM
\newcommand{\sequencei}[1]{\left\{ {#1}_i \right\}_{i\in\If}}
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\psib}{\boldsymbol{\psi}}
\newcommand{\sinL}[1]{\sin\left((#1+1)\pi\frac{x}{L}\right)}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\Xno}[1]{X_{(#1)}}
\newcommand{\xdno}[1]{\boldsymbol{x}_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Dirichlet and Neumann conditions; problem
$$
\begin{equation*}
-u''(x)=f(x),\quad x\in \Omega=[0,1],\quad u'(0)=C,\ u(1)=D
\end{equation*}
$$
- Use a global polynomial basis \( \baspsi_i\sim x^i \) on \( [0,1] \)
- Because of \( u(1)=D \): \( \baspsi_i(1)=0 \)
- Basis: \( \baspsi_i(x)=(1-x)^{i+1},\quad i\in\If \)
- Boundary function: \( B(x)=Dx \)
- \( u(x) = B(x) + \sum_{j\in\If}c_j\basphi_j = Dx + \sum_{j\in\If} c_j(1-x)^{i+1} \)
Variational formulation: find \( (u-B)\in V \) such that
$$
(u,\baspsi_i') = (f,\baspsi_i)-(B',\baspsi_i) - C\baspsi_i(0),\ i\in\If
$$
