and, for instance, local or global Lipschitz type or boundedness assumptions (involving, e.g., second derivative information) will in general not be available with practically feasible constants. We remark, however, that more sophisticated step size prediction schemes can be derived if the structure of the nonlinearities can be exploited further; see, in particular, the monograph [@5].

(ii) The minimum in  ensures that the step size $\Delta t_n$ is chosen to be 1 whenever possible. Indeed, this is required in order to guarantee quadratic convergence of the Newton iteration close to a root (provided that the root is simple).

(iii) The preset tolerance $ \tau $ in the above adaptive strategy will typically be fixed [*a priori*]{}. Here, for highly nonlinear problems featuring numerous or even infinitely many solutions, it is typically mandatory to select $\tau\ll 1$ small in order to remain within the attractor of the given initial guess. This is particularly important if the starting value is relatively far away from a solution.

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Newton-Galerkin $hp$-FEM for Semilinear Elliptic Problems {#sc:FEM}
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The purpose of this section is to derive an $hp$-version Newton-Galerkin finite element formulation for the solution of . To this end, we will apply the abstract adaptive 