$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
We choose \( \baspsi_i \) as finite element basis functions
$$ \baspsi_i = \basphi_{\nu(i)},\quad i\in\If$$
Degree of freedom number \( \nu(i) \) in the mesh corresponds to
unknown number \( i \) (\( c_i \)).
Model problem: \( \nu(i)=i \), \( \If=\{0,\ldots,N_n-2\} \) (last node excluded)
$$ u = D + \sum_{j\in\If} c_j\basphi_{\nu(j)}$$
or with \( \basphi_i \) in the boundary function:
$$ u = D\basphi_{N_n-1} + \sum_{j\in\If} c_j\basphi_{j}$$
