$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Continuation method: solve difficult problem as a sequence of simpler problems
- Introduce a continuation parameter \( \Lambda \)
- \( \Lambda =0 \): simple version of the PDE problem
- \( \Lambda =1 \): desired PDE problem
- Increase \( \Lambda \) in steps: \( \Lambda_0=0 ,\Lambda_1 < \cdots < \Lambda_n=1 \)
- Use the solution from \( \Lambda_{i-1} \) as
initial guess for the iterations for \( \Lambda_i \)
