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

Amrein, Mario; Wihler, Thomas (2017). An adaptive space-time Newton-Galerkin approach for semilinear singularly perturbed parabolic evolution equations. IMA journal of numerical analysis, 37(4), pp. 2004-2019. Oxford University Press 10.1093/imanum/drw049

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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.

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