$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\X}{\boldsymbol{X}}
\renewcommand{\u}{\boldsymbol{u}}
\renewcommand{\v}{\boldsymbol{v}}
\newcommand{\e}{\boldsymbol{e}}
\newcommand{\f}{\boldsymbol{f}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\Iz}{\mathcal{I}_z}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\Ifb}{{I_b}} % for FEM
\newcommand{\sequencei}[1]{\left\{ {#1}_i \right\}_{i\in\If}}
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\psib}{\boldsymbol{\psi}}
\newcommand{\sinL}[1]{\sin\left((#1+1)\pi\frac{x}{L}\right)}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\Xno}[1]{X_{(#1)}}
\newcommand{\xdno}[1]{\boldsymbol{x}_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
The element matrix and local vs global node numbers
$$
\tilde A^{(e)} = \{ \tilde A^{(e)}_{r,s}\},\quad
\tilde A^{(e)}_{r,s} =
\int_{\Omega^{(e)}}\basphi_{q(e,r)}\basphi_{q(e,s)}dx,
\quad r,s\in\Ifd=\{0,\ldots,d\}
$$
Now,
- \( r,s \) run over local node numbers in an element: \( 0, 1,\ldots, d \)
- \( i,j \) run over global node numbers \( i,j\in\If = \{0,1,\ldots,N\} \)
- \( i=q(e,r) \): mapping of local node number \( r \) in element
\( e \) to the global node number \( i \) (math equivalent to
i=elements[e][r])
- Add \( \tilde A^{(e)}_{r,s} \) into the global \( A_{i,j} \)
(assembly)
$$
A_{q(e,r),q(e,s)} := A_{q(e,r),q(e,s)} + \tilde A^{(e)}_{r,s},\quad
r,s\in\Ifd
$$
