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An $ hp$-adaptive Newton-discontinuous-Galerkin finite element approach for semilinear elliptic boundary value problems

Authors: Paul Houston and Thomas P. Wihler
Journal: Math. Comp. 87 (2018), 2641-2674
MSC (2010): Primary 65N30
Published electronically: January 24, 2018
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Abstract: In this paper we develop an $ hp$-adaptive procedure for the numerical solution of general second-order semilinear elliptic boundary value problems, with possible singular perturbation. Our approach combines both adaptive Newton schemes and an $ hp$-version adaptive discontinuous Galerkin finite element discretisation, which, in turn, is based on a robust $ hp$-version a posteriori residual analysis. Numerical experiments underline the robustness and reliability of the proposed approach for various examples.

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Additional Information

Paul Houston
Affiliation: School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, United Kingdom

Thomas P. Wihler
Affiliation: Mathematics Institute, University of Bern, CH-3012 Bern, Switzerland

Keywords: Newton method, semilinear elliptic problems, adaptive finite element methods, discontinuous Galerkin methods, $hp$-adaptivity.
Received by editor(s): July 22, 2016
Received by editor(s) in revised form: April 7, 2017, and May 31, 2017
Published electronically: January 24, 2018
Additional Notes: The second author acknowledges the support of the Swiss National Science Foundation (SNF), Grant No. 200021-162990
Article copyright: © Copyright 2018 American Mathematical Society

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