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

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.