$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
The effect of relaxation can potentially be great!
- \( \Delta t=0.9 \): Picard required 32 iterations on average
- \( \omega =0.8 \): 7 iterations
- \( \omega =0.5 \): 2 iterations (!) - optimal choice
Other \( \omega=1 \) experiments:
| \( \Delta t \) | \( \epsilon_r \) | Picard | Newton |
|---|
| \( 0.2 \) | \( 10^{-7} \) | 5 | 2 |
| \( 0.2 \) | \( 10^{-3} \) | 2 | 1 |
| \( 0.4 \) | \( 10^{-7} \) | 12 | 3 |
| \( 0.4 \) | \( 10^{-3} \) | 4 | 2 |
| \( 0.8 \) | \( 10^{-7} \) | 58 | 3 |
| \( 0.8 \) | \( 10^{-3} \) | 4 | 2 |
