$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\X}{\boldsymbol{X}}
\renewcommand{\u}{\boldsymbol{u}}
\renewcommand{\v}{\boldsymbol{v}}
\newcommand{\e}{\boldsymbol{e}}
\newcommand{\f}{\boldsymbol{f}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\Iz}{\mathcal{I}_z}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\Ifb}{{I_b}} % for FEM
\newcommand{\sequencei}[1]{\left\{ {#1}_i \right\}_{i\in\If}}
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\psib}{\boldsymbol{\psi}}
\newcommand{\sinL}[1]{\sin\left((#1+1)\pi\frac{x}{L}\right)}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\Xno}[1]{X_{(#1)}}
\newcommand{\xdno}[1]{\boldsymbol{x}_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
The collocation or interpolation method; ideas and math
Here is another idea for approximating \( f(x) \) by \( u(x)=\sum_jc_j\baspsi_j \):
- Force \( u(\xno{i}) = f(\xno{i}) \) at some selected collocation points
\( \sequencei{x} \)
- Then \( u \) is said to interpolate \( f \)
- The method is known as interpolation or collocation
$$ u(\xno{i}) = \sum_{j\in\If} c_j \baspsi_j(\xno{i}) = f(\xno{i})
\quad i\in\If,N
$$
This is a linear system with no need for integration:
$$
\begin{align}
\sum_{j\in\If} A_{i,j}c_j &= b_i,\quad i\in\If\\
A_{i,j} &= \baspsi_j(\xno{i})\\
b_i &= f(\xno{i})
\end{align}
$$
No symmetric matrix: \( \baspsi_j(\xno{i})\neq \baspsi_i(\xno{j}) \) in general
