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M. Amrein, J. Melenk, T. Wihler:
"An hp-adaptive Newton-Galerkin finite element procedure for semilinear boundary value problems";
in: "ASC Report 6/2016", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2016, ISBN: 978-3-902627-09-4, 1 - 15.

In this paper we develop an hp-adaptive procedure for the numerical solution of general, semilinear elliptic boundary value problems in 1d, with possible singular perturbations. Our approach combines both a prediction-type adaptive Newton method and an hp-version adaptive finite element discretization (based on a robust a posteriori residual analysis), thereby leading to a fully hp-adaptive Newton-Galerkin scheme. Numerical experiments underline the robustness and reliability of the proposed approach for various examples.

Adaptive Newton methods, semilinear elliptic problems, singularly perturbed problems, adaptive finite element methods, hp-FEM, hp-adaptivity problems

http://www.asc.tuwien.ac.at/preprint/2016/asc6x2016.pdf