$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\X}{\boldsymbol{X}}
\renewcommand{\u}{\boldsymbol{u}}
\renewcommand{\v}{\boldsymbol{v}}
\newcommand{\e}{\boldsymbol{e}}
\newcommand{\f}{\boldsymbol{f}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\Iz}{\mathcal{I}_z}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\Ifb}{{I_b}} % for FEM
\newcommand{\sequencei}[1]{\left\{ {#1}_i \right\}_{i\in\If}}
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\psib}{\boldsymbol{\psi}}
\newcommand{\sinL}[1]{\sin\left((#1+1)\pi\frac{x}{L}\right)}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\Xno}[1]{X_{(#1)}}
\newcommand{\xdno}[1]{\boldsymbol{x}_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Modification of the linear system; the raw system
- Drop boundary function
- Compute as if there are not Dirichlet conditions
- Modify the linear system to incorporate Dirichlet conditions
- \( \If \) holds the indices of all nodes \( \{0,1,\ldots,N=N_n\} \)
$$
\begin{align*}
\sum_{j\in\If}
\biggl(\underbrace{\int_\Omega \basphi_i\basphi_j\dx}_{M_{i,j}}\biggr)
c_j &= \sum_{j\in\If}
\biggl(\underbrace{\int_\Omega \basphi_i\basphi_j \dx}_{M_{i,j}} -
\Delta t\underbrace{\int_\Omega
\dfc\nabla \basphi_i\cdot\nabla\basphi_j\dx}_{K_{i,j}}\biggr) c_{1,j}
\\
&\quad \underbrace{-\Delta t\int_\Omega f\basphi_i\dx -
\Delta t\int_{\partial\Omega_N} g\basphi_i\ds}_{f_i},\quad i\in\If
\end{align*}
$$
