$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Combination of absolute and relative stopping criteria
Problem with relative criterion: a small
\( ||F(u_0)|| \) (because \( u_0\approx u \), perhaps because of small \( \Delta t \))
must be significantly reduced. Better with absolute criterion.
- Can make combined absolute-relative criterion
- \( \epsilon_{rr} \): tolerance for relative part
- \( \epsilon_{ra} \): tolerance for absolute part
$$
||F(u)|| \leq \epsilon_{rr} ||F(u_0)|| + \epsilon_{ra}
$$
$$
||F(u)|| \leq \epsilon_{rr} ||F(u_0)|| + \epsilon_{ra}
\quad\hbox{or}\quad
||\delta u|| \leq \epsilon_{ur} ||u_0|| + \epsilon_{ua}
\quad\hbox{or}\quad
k>k_{\max}
$$
