$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\X}{\boldsymbol{X}}
\renewcommand{\u}{\boldsymbol{u}}
\renewcommand{\v}{\boldsymbol{v}}
\newcommand{\e}{\boldsymbol{e}}
\newcommand{\f}{\boldsymbol{f}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\Iz}{\mathcal{I}_z}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\Ifb}{{I_b}} % for FEM
\newcommand{\sequencei}[1]{\left\{ {#1}_i \right\}_{i\in\If}}
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\psib}{\boldsymbol{\psi}}
\newcommand{\sinL}[1]{\sin\left((#1+1)\pi\frac{x}{L}\right)}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\Xno}[1]{X_{(#1)}}
\newcommand{\xdno}[1]{\boldsymbol{x}_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Computing integrals
Integrals must be transformed from \( \Omega^{(e)} \) (physical cell)
to \( \tilde\Omega^r \) (reference cell):
$$
\begin{align}
\int_{\Omega^{(e)}}\basphi_i (\x) \basphi_j (\x) \dx &=
\int_{\tilde\Omega^r} \refphi_i (\X) \refphi_j (\X)
\det J\, \dX\\
\int_{\Omega^{(e)}}\basphi_i (\x) f(\x) \dx &=
\int_{\tilde\Omega^r} \refphi_i (\X) f(\x(\X)) \det J\, \dX
\end{align}
$$
where \( \dx = dx dy \) or \( \dx = dxdydz \) and \( \det J \) is the determinant of the
Jacobian of the mapping \( \x(\X) \).
$$
J = \left[\begin{array}{cc}
\frac{\partial x}{\partial X} & \frac{\partial x}{\partial Y}\\
\frac{\partial y}{\partial X} & \frac{\partial y}{\partial Y}
\end{array}\right], \quad
\det J = \frac{\partial x}{\partial X}\frac{\partial y}{\partial Y}
- \frac{\partial x}{\partial Y}\frac{\partial y}{\partial X}
$$
Affine mapping
(12): \( \det J=2\Delta \), \( \Delta = \hbox{cell volume} \)
