$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Backward Euler scheme and Newton's method
Normally, Newton's method is defined for systems of algebraic equations,
but the idea of the method can be applied at the PDE level too!
Let \( u^{n,k} \) be an approximation to the unknown \( u^n \).
We seek a better approximation
$$
u^{n} = u^{n,k} + \delta u
$$
- Insert \( u^{n} = u^{n,k} + \delta u \) in the PDE
- Taylor expand the nonlinearities
and keep only terms that are linear in \( \delta u \)
Result: linear PDE for the approximate correction \( \delta u \)
