$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\X}{\boldsymbol{X}}
\renewcommand{\u}{\boldsymbol{u}}
\renewcommand{\v}{\boldsymbol{v}}
\newcommand{\e}{\boldsymbol{e}}
\newcommand{\f}{\boldsymbol{f}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\Iz}{\mathcal{I}_z}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\Ifb}{{I_b}} % for FEM
\newcommand{\sequencei}[1]{\left\{ {#1}_i \right\}_{i\in\If}}
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\psib}{\boldsymbol{\psi}}
\newcommand{\sinL}[1]{\sin\left((#1+1)\pi\frac{x}{L}\right)}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\Xno}[1]{X_{(#1)}}
\newcommand{\xdno}[1]{\boldsymbol{x}_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Deriving the linear systems
- \( u = \sum_{j=0}^{N}c_j\baspsi_j(\x) \)
- \( u_1 = \sum_{j=0}^{N} c_{1,j}\baspsi_j(\x) \)
- \( \forall v\in V \): for \( v=\baspsi_i \), \( i=0,\ldots,N \)
Insert these in
$$
(u, \baspsi_i) = (u_1,\baspsi_i) -
\Delta t (\dfc\nabla u_1,\nabla\baspsi_i) +
(f^n,\baspsi_i)
$$
and order terms as matrix-vector products (\( \quad i=0,\ldots,N \)):
$$
\begin{equation*}
\sum_{j=0}^{N} \underbrace{(\baspsi_i,\baspsi_j)}_{M_{i,j}} c_j =
\sum_{j=0}^{N} \underbrace{(\baspsi_i,\baspsi_j)}_{M_{i,j}} c_{1,j}
-\Delta t \sum_{j=0}^{N}
\underbrace{(\nabla\baspsi_i,\dfc\nabla\baspsi_j)}_{K_{i,j}} c_{1,j}
+ (f^n,\baspsi_i)
\end{equation*}
$$
