An hp-adaptive Newton-Galerkin finite element procedure for semilinear boundary value problems

Amrein, Mario; Melenk, Jens Markus; Wihler, Thomas (2017). An hp-adaptive Newton-Galerkin finite element procedure for semilinear boundary value problems. Mathematical methods in the applied sciences, 40(6), pp. 1973-1985. Wiley 10.1002/mma.4113

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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.

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