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

Amrein, Mario ; Wihler, Thomas P.

In: IMA Journal of Numerical Analysis, 2017, vol. 37, no. 4, p. 2004-2019

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    Title
    • An adaptive space-time Newton-Galerkin approach for semilinear singularly perturbed parabolic evolution equations
    Author
    • Amrein, Mario. Mathematics Institute, University of Bern, CH-3012 Bern, Switzerland
    • Wihler, Thomas P.. Mathematics Institute, University of Bern, CH-3012 Bern, Switzerland
    Document Type
    • Postprint
    Language
    • English
    Published in
    • IMA Journal of Numerical Analysis, 2017, vol. 37, no. 4, p. 2004-2019. Oxford University Press
    Other Version
    OAI-PMH Identifier
    • oai:doc.rero.ch:332853
    Summary
    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.