  In this paper we develop an adaptive procedure for the numerical solution of
semilinear parabolic problems, with possible singular perturbations. Our
approach combines a linearization technique using Newton's method with an
adaptive discretization-which is based on a spatial finite element method and
the backward Euler time stepping scheme-of the resulting sequence of linear
problems. Upon deriving a robust a posteriori error analysis, we design a fully
adaptive Newton-Galerkin time stepping algorithm. Numerical experiments
underline the robustness and reliability of the proposed approach for various
examples.
