first time step of length $\Delta t_0>0$ there holds $$\label{eq:O}
u(\Delta t_0)-u_1=\frac{\Delta t_0^2}{2h}\eta_h+\mathcal{O}(\Delta t_0^3)+\mathcal{O}(\Delta t_0^2h\|{\mathsf{N}_{{\mathsf{F}}}}(u_0)\|_X^2),$$ where $u_1$ is the forward Euler solution from . Therefore, upon setting $$\Delta t_0=\sqrt{2\tau h\|\eta_h\|_X^{-1}},$$ we arrive at $$\|u(\Delta t_0)-u_1\|_X\le\tau+\mathcal{O}(\Delta t_0^3)+\mathcal{O}(\Delta t_0^2h\|{\mathsf{N}_{{\mathsf{F}}}}(u_0)\|_X^2).$$ In order to balance the $\mathcal{O}$-terms in  it is sensible to make the choice $$h=\mathcal{O}(\Delta t_0\|{\mathsf{N}_{{\mathsf{F}}}}(u_0)\|_X^{-2}),$$ i.e., $$\label{eq:h}
h=\gamma \Delta t_0\|{\mathsf{N}_{{\mathsf{F}}}}(u_0)\|_X^{-2},$$ for some parameter $\gamma>0$. This leads to the following *adaptive Newton algorithm*.\

 \[al:zs\] Fix a tolerance $\tau>0$ as well as a parameter $\gamma>0$, and set $n\gets0$.

Start the Newton iteration with an initial guess $ u_{0}\in\Xi$. let $\kappa_{n}=\Delta t_{n-1}$, and $h_n=\gamma \kappa_n\|{\mathsf{N}_{{\mathsf{F}}}}(u_n)\|_X^{-2}$ based on ; define the Newton step size $$\label{eq:knew}
\Delta t_n=\min\left\{\sqrt{2\tau h_n{\left\|{\mathsf{N}_{{\mathsf{F}}}}(u_n+h_n{\mathsf{N}_{{\mathsf{F}}}}(u_n))-{\mathsf{N}_{{\mathsf{F}}}}(u_n)\right\|}^{-1}_{X}},1\right\}.$$ Compute $u_{n+1}$ based on the Newton iteration , and go to ([3:]{}) with $n\leftarrow n+1 $.

The following comments are noteworthy.

(i) We remark that the purpose of the above derivations is to work under minimal structural assumptions on the dynamics caused by the Newton transform. This is particularly important in the context of nonlinear singularly perturbed problems (with ${\epsilon}\ll 1$): Indeed, for such problems (as ${\epsilon}\to 0$) the inverse of the derivative of the associated nonlinear operator will typically become highly irregular (even on a local level), 