$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Robin condition
Heat conduction problems often apply a kind of Newton's cooling law,
also known as a Robin condition, at the boundary:
$$
-\dfc(u)\frac{\partial u}{\partial u} = h(u)(u-T_s(t)),\quad\x\in\partial\Omega_R
$$
Here:
- \( h(u) \): heat transfer coefficient between the body (\( \Omega \))
and its surroundings
- \( T_s \): temperature of the surroundings
Inserting the condition in the boundary integral
\( \int_{\partial\Omega_N}\dfc(u)\frac{\partial u}{\partial u}v\ds \):
$$ \int_{\partial\Omega_R}h(u)(u-T_s(T))v\ds$$
Use \( h(u^{-})(u-T_s) \) for Picard, differentiate for Newton
