An adaptive space-time Newton–Galerkin approach for semilinear singularly perturbed parabolic evolution equations

Erworben im Rahmen der Schweizer Nationallizenzen (http://www.nationallizenzen.ch)In this article, 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