$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\X}{\boldsymbol{X}}
\renewcommand{\u}{\boldsymbol{u}}
\renewcommand{\v}{\boldsymbol{v}}
\newcommand{\e}{\boldsymbol{e}}
\newcommand{\f}{\boldsymbol{f}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\Iz}{\mathcal{I}_z}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\Ifb}{{I_b}} % for FEM
\newcommand{\sequencei}[1]{\left\{ {#1}_i \right\}_{i\in\If}}
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\psib}{\boldsymbol{\psi}}
\newcommand{\sinL}[1]{\sin\left((#1+1)\pi\frac{x}{L}\right)}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\Xno}[1]{X_{(#1)}}
\newcommand{\xdno}[1]{\boldsymbol{x}_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Example with two Dirichlet values; cellwise computations
- All element matrices are as in the previous example
- New element vector in the first and last cell
From the last cell:
$$
\tilde b_0^{(N_e)} = \int_{-1}^1 \left(f\refphi_0 - D\frac{2}{h}
\frac{d\refphi_1}{dX}\frac{2}{h}\frac{d\refphi_0}{dX}\right)
\frac{h}{2} \dX
= h + \frac{D}{h}
$$
From the first cell:
$$
\tilde b_0^{(0)} = \int_{-1}^1 \left(f\refphi_1 - C\frac{2}{h}
\frac{d\refphi_0}{dX}\frac{2}{h}\frac{\refphi_1}{dX}\right)
\frac{h}{2} \dX
= h + \frac{C}{h}
$$
(hpl 1: These calculations had some errors - redo them in detail!)
