Study guide: Solving nonlinear ODE and PDE problems

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

Newton's method (2)

The Jacobian becomes $$ \begin{align*} J_{i,j} = \frac{\partial F_i}{\partial c_j} = \int_\Omega & (\baspsi_j\baspsi_i + \Delta t\,\dfc^{\prime}(u)\baspsi_j \nabla u\cdot\nabla \baspsi_i + \Delta t\,\dfc(u)\nabla\baspsi_j\cdot\nabla\baspsi_i - \\ &\ \Delta t f^{\prime}(u)\baspsi_j\baspsi_i)\dx \end{align*} $$

Evaluation of $ J_{i,j} $ as the coefficient matrix in the Newton system $ J\delta u = -F $ means $ J(u^{-}) $: $$ \begin{align*} J_{i,j} = \int_\Omega & (\baspsi_j\baspsi_i + \Delta t\,\dfc^{\prime}(u^{-})\baspsi_j \nabla u^{-}\cdot\nabla \baspsi_i + \Delta t\,\dfc(u^{-})\nabla\baspsi_j\cdot\nabla\baspsi_i - \\ &\ \Delta t f^{\prime}(u^{-})\baspsi_j\baspsi_i)\dx \end{align*} $$