$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\X}{\boldsymbol{X}}
\renewcommand{\u}{\boldsymbol{u}}
\renewcommand{\v}{\boldsymbol{v}}
\newcommand{\e}{\boldsymbol{e}}
\newcommand{\f}{\boldsymbol{f}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\Iz}{\mathcal{I}_z}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\Ifb}{{I_b}} % for FEM
\newcommand{\sequencei}[1]{\left\{ {#1}_i \right\}_{i\in\If}}
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\psib}{\boldsymbol{\psi}}
\newcommand{\sinL}[1]{\sin\left((#1+1)\pi\frac{x}{L}\right)}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\Xno}[1]{X_{(#1)}}
\newcommand{\xdno}[1]{\boldsymbol{x}_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Derivation of the linear system
- \( \forall v\in V \) is replaced by for all \( \baspsi_i \), \( i\in\If \)
- Insert \( u = B + \sum_{j\in\If} c_j\baspsi_j \), \( B = u_0 \), in the
variational form
- Identify \( i,j \) terms (matrix) and \( i \) terms (right-hand side)
- Write on form \( \sum_{i\in\If}A_{i,j}c_j = b_i \), \( i\in\If \)
$$
A_{i,j} = (\v\cdot\nabla \baspsi_j, \baspsi_i) +
(\alpha \baspsi_j ,\baspsi_i) + (a\nabla \baspsi_j,\nabla \baspsi_i)
$$
$$
b_i = (g,\baspsi_i)_{N} + (f,\baspsi_i) -
(\v\cdot\nabla u_0, \baspsi_i) + (\alpha u_0 ,\baspsi_i) +
(a\nabla u_0,\nabla \baspsi_i)
$$
