$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\X}{\boldsymbol{X}}
\renewcommand{\u}{\boldsymbol{u}}
\renewcommand{\v}{\boldsymbol{v}}
\newcommand{\e}{\boldsymbol{e}}
\newcommand{\f}{\boldsymbol{f}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\Iz}{\mathcal{I}_z}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\Ifb}{{I_b}} % for FEM
\newcommand{\sequencei}[1]{\left\{ {#1}_i \right\}_{i\in\If}}
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\psib}{\boldsymbol{\psi}}
\newcommand{\sinL}[1]{\sin\left((#1+1)\pi\frac{x}{L}\right)}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\Xno}[1]{X_{(#1)}}
\newcommand{\xdno}[1]{\boldsymbol{x}_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Assembly of the right-hand side
$$
b_i = \int_\Omega f(x)\basphi_i(x)dx =
\sum_{e} \int_{\Omega^{(e)}} f(x)\basphi_i(x)dx,\quad
b^{(e)}_{i}=\int_{\Omega^{(e)}} f(x)\basphi_i(x)dx
$$
Important observations:
- \( b_i^{(e)}\neq 0 \) if and only if global node \( i \) is a node in element \( e \)
(otherwise \( \basphi_i=0 \))
- The \( d+1 \) nonzero \( b_i^{(e)} \) can be collected in an element vector
\( \tilde b_r^{(e)}=\{ \tilde b_r^{(e)}\} \), \( r\in\Ifd \)
Assembly:
$$
b_{q(e,r)} := b_{q(e,r)} + \tilde b^{(e)}_{r},\quad
r,s\in\Ifd
$$
