$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Stopping criteria
Let \( ||\cdot|| \) be the standard Eucledian vector norm. Several termination
criteria are much in use:
- Absolute change in solution: \( ||u - u^{-}||\leq \epsilon_u \)
- Relative change in solution: \( ||u - u^{-}||\leq \epsilon_u ||u_0|| \),
where \( u_0 \) denotes the start value of \( u^{-} \) in the iteration
- Absolute residual: \( ||F(u)|| \leq \epsilon_r \)
- Relative residual: \( ||F(u)|| \leq \epsilon_r ||F(u_0)|| \)
- Max no of iterations: stop when \( k > k_{\max} \)