Study guide: Solving nonlinear ODE and PDE problems

$$ \newcommand{\uex}{{u_{\small\mbox{e}}}} \newcommand{\half}{\frac{1}{2}} \newcommand{\tp}{\thinspace .} \newcommand{\Oof}[1]{\mathcal{O}(#1)} \newcommand{\x}{\boldsymbol{x}} \newcommand{\dfc}{\alpha} % diffusion coefficient \newcommand{\Ix}{\mathcal{I}_x} \newcommand{\Iy}{\mathcal{I}_y} \newcommand{\If}{\mathcal{I}_s} % for FEM \newcommand{\Ifd}{{I_d}} % for FEM \newcommand{\basphi}{\varphi} \newcommand{\baspsi}{\psi} \newcommand{\refphi}{\tilde\basphi} \newcommand{\xno}[1]{x_{#1}} \newcommand{\dX}{\, \mathrm{d}X} \newcommand{\dx}{\, \mathrm{d}x} \newcommand{\ds}{\, \mathrm{d}s} $$

Numerical integration at the nodes

Instead of the group finite element method and exact integration, use Trapezoidal rule in the nodes for $ \int_0^L \dfc(\sum_k u_k\basphi_k)\basphi_i^{\prime}\basphi_j^{\prime}\dx $.

Work at the cell level (most convenient with discontinuous $ \basphi_i' $): $$ \begin{align*} & \int_{-1}^1 \alpha(\sum_t\tilde u_t\refphi_t)\refphi_r'\refphi_s'\frac{h}{2}dX = \int_{-1}^1 \dfc(\sum_{t=0}^1 \tilde u_t\refphi_t)\frac{2}{h}\frac{d\refphi_r}{dX} \frac{2}{h}\frac{d\refphi_s}{dX}\frac{h}{2}dX\\ &\quad = \frac{1}{2h}(-1)^r(-1)^s \int_{-1}^1 \dfc(\sum_{t=0}^1 u_t\refphi_t(X))dX \\ & \qquad \approx \frac{1}{2h}(-1)^r(-1)^s\dfc ( \sum_{t=0}^1\refphi_t(-1)\tilde u_t) + \dfc(\sum_{t=0}^1\refphi_t(1)\tilde u_t)\\ &\quad = \frac{1}{2h}(-1)^r(-1)^s(\dfc(\tilde u_0) + \dfc(\tilde u^{(1)})) \end{align*} $$