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In this paper we develop an adaptive procedure for the numerical solution ofsemilinear parabolic problems, with possible singular perturbations. Ourapproach combines a linearization technique using Newton's method with anadaptive discretization-which is based on a spatial finite element method andthe backward Euler time stepping scheme-of the resulting sequence of linearproblems. Upon deriving a robust a posteriori error analysis, we design a fullyadaptive Newton-Galerkin time stepping algorithm. Numerical experimentsunderline the robustness and reliability of the proposed approach for variousexamples.

Fully Adaptive Newton-Galerkin Time Stepping Methods for Singularly Perturbed Parabolic Evolution Equations